Control Theory and Related Topics: In Memory of Prof Xunjing Li, Fudan University, China, 3-5 June 2005

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Control Theory and Related Topics

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edited by

Shanjian Tang Fudan University, China

Jiongmin Yong University of Central Florida, USA

Control Theory and Related Topics

In Memory of Xunjing Li Fudan University, China

3 – 5 June 2005

World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

CONTROL THEORY AND RELATED TOPICS IN MEMORY OF PROFESSOR XUNJING LI (1935–2003) Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-270-582-2 ISBN-10 981-270-582-1

Printed in Singapore.

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Xunjing Li (1935–2003)

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PREFACE

In the past several decades extensive research has been devoted to the mathematical control theory, in particular for distributed parameter and stochastic systems, and its application in mathematical finance. Professor Xunjing Li, a distinguished educator and researcher, played a special leading role in the development of these research efforts, particularly related to the maximum principle for optimal controls in infinite dimensional state space from the late seventies of the last century until his untimely death in February, 2003. This commemorative volume collects research articles devoted to reviewing the state of the art of this and other related rapidly developing research and to exploring new directions of research in these fields. It is a tribute to the Life and Work of Professor Xunjing Li by his students, friends, and colleagues whose personal and professional lives he has deeply touched through his generous insights and dedication to his profession. During June 3–5, 2005, Fudan University organized in Shanghai the “Workshop on Control of Distributed Parameters and Stochastic Systems in Memory of Professor Xunjing Li’s Seventieth Birthday”. We would like to thank all the participants for their presence. We also thank our colleague Yuan Zhou, our post doctors Juan Li and Mingyu Xu, and our graduate students Yashan Xu, Xiaobo Bao, Lei Wang, and Liang Zhu for their efficient services for the workshop. On behalf of all the participants, we also thank Jianxiong Huang for organizing a very nice excursion to Nanxun, Huzhou. Many of the contributors in this volume are speakers at the workshop. There are a few others as well. Due to various reasons, we were not able to invite all the students, colleagues and co-authors of Professor Xunjing Li, which we feel very regret. The topics covered include several aspects of linear quadratic optimal control of deterministic and stochastic systems, controllability of stochastic and/or parabolic systems, nonlinear observers and stabilization, dynamical systems for PDEs, the maximum principle for optimally controlled quasi-linear elliptic obstacle problems, optimization of Navier-Stokes equation, assets and insurance pricing, and applied BSDEs and reflected BSDEs in the nonlinear probability. It took concerted and collective efforts from many people to produce this volume. We would like to take this opportunity to express our gratitude to them. Our sincere thanks first go to all the contributors for their contributions and for their cooperation, and, in particular, to Jin Ma and Yuncheng You for their thoughtful article: “A Tribute in Memory of Professor Xunjing Li on His Seventieth Birthday.” Our special thanks also go to the World Scientific staff and, in particular, to Editor vii

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Ms E. H. Chionh and Kim Wei Lee for their cooperation and kind patience in bringing out this volume in a timely fashion. Finally, we would like to take this opportunity to recognize the special role played by Liwan Liu, Professor Xunjing Li’s wife, whose long term support and influence to Professor Xunjing Li was extremely significant. We express to her our special gratitude for her presence at the workshop. The publication of this volume is supported by the National Basic Research Program of China (973 Program) with Grant No. 2007CB814904, which is greatly appreciated.

June 30, 2007 Shanjian Tang Jiongmin Yong

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CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Conference Photo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix Part One: Xunjing Li’s Academic Life A Tribute in Memory of Professor Xunjing Li on His Seventieth Birthday . . . . . . .3 Jin Ma and Yuncheng You Publications of Xunjing Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Students and Post-doctors Advised by Xunjing Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Part Two: Stochastic Control, Mathematical Finance, and Backward Stochastic Differential Equations Axiomatic Characteristics for Solutions of Reflected Backward Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Xiaobo Bao and Shanjian Tang A Linear Quadratic Optimal Control Problem for Stochastic Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Shuping Chen and Jiongmin Yong An Additivity of Maximum Expectations and Its Applications . . . . . . . . . . . . . . . . .67 Zengjing Chen, Matt Davison, Mark Reesor, and Ying Zhang Stochastic Control and BSDEs with Quadratic Growth . . . . . . . . . . . . . . . . . . . . . . . . 80 Marco Fuhrman, Ying Hu, and Gianmario Tessitore A Fundamental Theorem of Asset Pricing in Continuous Time with Square Integrable Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Hanqing Jin and Xun Yu Zhou

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Indifference Pricing of Universal Variable Life Insurance . . . . . . . . . . . . . . . . . . . . . . 107 Jin Ma and Yuhua Yu gΓ -Expectations and the Related Nonlinear Doob–Meyer Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Shige Peng and Mingyu Xu Nonconvexity Phenomenon on Itˆ o’s Integrals and on Stochastic Attainable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Shanjian Tang Unique Continuation and Observability for Stochastic Parabolic Equations and Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Xu Zhang Part Three: Deterministic Control Systems Design of Dynamic High-Gain Observers for a Class of MIMO Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Hao Lei, Jianfeng Wei, and Wei Lin Some Counterexamples in Existence Theory of Optimal Control . . . . . . . . . . . . . . 183 Hongwei Lou A Generalized Framework for Global Output Feedback Stabilization of Inherently Nonlinear Systems with Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Jason Polendo and Chunjiang Qian A Linear Quadratic Constrained Optimal Feedback Control Problem . . . . . . . . . 215 Yashan Xu On Finite-Time Stabilization of a Class of Nonsmoothly Stabilizable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Bo Yang and Wei Lin The Algebraic Criterion for Delay-Independent Stability of Linear Systems . . . 241 Xin Yu, Chao Xu and Kangsheng Liu Transformation Method of Solving a Time-Optimal Control Problem with Pointwise State Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Shangwei Zhu and Xunjing Li

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Part Four: Dynamics and Optimal Control of Partial Differential Equations Optimal Control of Quasilinear Elliptic Obstacle Problems . . . . . . . . . . . . . . . . . . . 263 Qihong Chen and Yuquan Ye Controllability of a Nonlinear Degenerate Parabolic System with Bilinear Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Ping Lin, Hang Gao, and Xu Liu Near-Optimal Controls to Infinite Dimensional Linear-Quadratic Optimal Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Liping Pan and Qihong Chen Shape Optimization of Stationary Navier–Stokes Equation . . . . . . . . . . . . . . . . . . . 323 Gengsheng Wang, Lijuan Wang, and Donghui Yang Solution Map of Strongly Nonlinear Impulsive Evolution Inclusions . . . . . . . . . . .338 Xiaoling Xiang, Yunfei Peng, and Wei Wei Study on Repairable Series System with Two Components — A Semigroup Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 Hong Xu and Dinghua Shi Finite Dimensional Reduction of Global Dynamics and Lattice Dynamics of a Damped Nonlinear Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 Yuncheng You Recent Progress on Nonlinear Wave Equations via KAM Theory . . . . . . . . . . . . . . 387 Xiaoping Yuan

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From left: back row — Zengjing Chen, Jianxiong Huang, Liping Pan, Qihong Chen, Gengsheng Wang, Hongwei Lou, Hanzhong Wu and Xiaobo Bao. Middle row — Hang Gao, Yuan Zhou, Kangsheng Liu, Chunjiang Qian, Xu Zhang, Xun Li, Yashan Xu and Lei Wang. Front row — Xiaoling Xiang, Shanjian Tang, Shige Peng, Shuping Chen, Yuncheng You, Jin Ma, Xunyu Zhou, Wei Lin and Ying Hu.

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LIST OF PARTICIPANTS

Xiaobo Bao School of Mathematical Sciences, Fudan University, Shanghai, China [email protected] Qihong Chen Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai, China [email protected] Shuping Chen Department of Mathematics, Guizhou University, Guizhou, and Department of Mathematics, Zhejiang University, Hangzhou, China [email protected] Zengjing Chen Department of Mathematics, Shandong University, Jinan, China [email protected] Hang Gao School of Mathematics & Statistics, Northeast Normal University, Changchun, China [email protected] Ying Hu IRMAR, Universit´e Rennes 1, Rennes, France [email protected] Jianxiong Huang Shanghai College of Electric Power, Shanghai, China [email protected] Xun Li National University of Singapore, Singapore [email protected]

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Wei Lin Department of Electrical Engineering and Computer Science, Case Western Reserve University, Ohio, USA [email protected] Daobai Liu Fudan Univeristy, Shanghai, China [email protected] Kangsheng Liu Zhejiang University, Hangzhou, China [email protected] Hongwei Lou School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Jin Ma Department of Mathematics, Purdue University, Indiana, USA [email protected] Liping Pan School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Shige Peng School of Mathematical Sciences, Fudan Univeristy, Shanghai, and School of Mathematics and System Science, Shandong University, Jinan, China [email protected] Chunjiang Qian Department of Electrical and Computer Engineering, University of Texas at San Antonio, San Antonio, Texas, USA [email protected], [email protected] Shanjian Tang School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Gengsheng Wang Department of Mathematics, Wuhan University, Wuhan, Hubei, China [email protected]

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Jianbo Wang School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Lei Wang Fudan Univeristy, Shanghai, China [email protected] Hanzhong Wu School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Xiaoling Xiang Department of Mathematics, Guizhou University, Guiyang 550025, Guizhou, China [email protected] Yashan Xu School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Yuncheng You Department of Mathematics, University of South Florida,Tampa, Florida, USA [email protected] Xiaoping Yuan School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Xu Zhang School of Mathematics, Sichuan University, Chengdu, China [email protected] Hailang Zhou Shanghai Bank, China Xun Yu Zhou Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong, China [email protected]

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Yuan Zhou School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Daxun Zhu School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Huimin Zhu School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Liang Zhu School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Shangwei Zhu Department of Applied Mathematics, Shanxi Finance & Economics University, Taiyuan, China [email protected]

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LIST OF CONTRIBUTORS

Xiaobo Bao School of Mathematical Sciences, Fudan University, Shanghai, China [email protected] Qihong Chen Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai, China [email protected] Shuping Chen Department of Mathematics, Guizhou University, Guizhou, and Department of Mathematics, Zhejiang University, Hangzhou, China [email protected] Zengjing Chen Department of Mathematics, Shandong University, Jinan, China [email protected] Matt Davison Department of Applied Mathematics, University of Western Ontario London, ON, Canada [email protected] Marco Fuhrman Dipartimento di Matematica, Politecnico di Milano, Milano, Italy [email protected] Hang Gao School of Mathematics & Statistics, Northeast Normal University, Changchun, China [email protected] Ying Hu IRMAR, Universit´e Rennes 1, Rennes, France [email protected]

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Hanqing Jin Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong, China [email protected] Hao Lei Department of Electrical Engineering and Computer Science, Case Western Reserve University, Ohio, USA [email protected] Ping Lin Graduate Student, School of Mathematics & Statistics, Northeast Normal University, Changchun, China [email protected] Wei Lin Department of Electrical Engineering and Computer Science, Case Western Reserve University, Ohio, USA [email protected] Kangsheng Liu Zhejiang University, Hangzhou, China [email protected] Xu Liu Graduate Student, Department of Mathematics, College of Science, Zhejiang University, Hangzhou and School of Mathematics & Statistics, Northeast Normal University, Changchun, China Hongwei Lou School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Jin Ma Department of Mathematics, Purdue University, Indiana, USA [email protected] Liping Pan School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected]

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Shige Peng School of Mathematical Sciences, Fudan Univeristy, Shanghai, and School of Mathematics and System Science, Shandong University, Jinan, China [email protected] Yunfei Peng Department of Mathematics, Guizhou University, Guiyang, Guizhou, China [email protected] Jason Polendo Department of Electrical and Computer Engineering, University of Texas at San Antonio, San Antonio, Texas, USA [email protected] Chunjiang Qian Department of Electrical and Computer Engineering, University of Texas at San Antonio, San Antonio, Texas, USA [email protected], [email protected] Mark Reesor Department of Applied Mathematics, University of Western Ontario London, ON, Canada [email protected] Dinghua Shi Department of Mathematics, Shanghai University, Shanghai, China Shanjian Tang School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Gianmario Tessitore Dipartimento di Matematica, Universit` a di Parma, Parma, Italy [email protected] Gengsheng Wang Department of Mathematics, Wuhan University, Wuhan, Hubei, China [email protected] Lijuan Wang Department of Mathematics, Wuhan University, Wuhan, Hubei, China [email protected]

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Jianfeng Wei Department of Electrical Engineering and Computer Science, Case Western Reserve University, Ohio, USA [email protected] Wei Wei Department of Mathematics, Guizhou University, Guiyang, Guizhou, China [email protected] Xiaoling Xiang Department of Mathematics, Guizhou University, Guiyang 550025, Guizhou, China [email protected] Chao Xu Zhejiang University, Hangzhou, China [email protected] Hong Xu Department of Mathematics, Shanghai University, Shanghai, China Mingyu Xu School of Mathematics and System Science, Shandong University, Jinan, China [email protected] Yashan Xu School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Bo Yang Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas, USA [email protected] Donghui Yang Department of Mathematics, Wuhan University, Wuhan, Hubei, China [email protected] Yuquan Ye Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai, China [email protected]

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Jiongmin Yong Department of Mathematics, University of Central Florida, Orlando, Florida, USA, and School of Mathematical Sciences, Fudan University, Shanghai, China [email protected] Yuncheng You Department of Mathematics, University of South Florida,Tampa, Florida, USA [email protected] Xin Yu Zhejiang University, Hangzhou, China [email protected] Yuhua Yu Department of Mathematics, Purdue University, Indiana, USA [email protected] Xiaoping Yuan School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Xu Zhang School of Mathematics, Sichuan University, Chengdu, China [email protected] Ying Zhang Department of Mathematics and Statistics Acadia University, Canada [email protected] Xun Yu Zhou Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong, China [email protected] Shangwei Zhu Department of Applied Mathematics, Shanxi Finance & Economics University, Taiyuan, China [email protected]

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PART ONE XUNJING LI’S ACADEMIC LIFE

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Chap10-Tribute

A TRIBUTE IN MEMORY OF PROFESSOR XUNJING LI ON HIS SEVENTIETH BIRTHDAY

JIN MA Department of Mathematics, Purdue University, West Lafayette, IN 47907-1395, USA E-mail: [email protected] YUNCHENG YOU Department of Mathematics, University of South Florida, Tampa, FL 33620-5700, USA E-mail: [email protected]

Professor Xunjing Li passed away in February, 2003, at age 68, in Shanghai, where he had lived for about half a century, almost equivalent to all his professional life. Professor Li was known to many of his students as “Lao Ban”, which means “Boss” in English. This is not only because of his absolute authority sensed by his students, but most importantly is because of his vision in finding new directions of research, his rigorous attitude towards each and every detail in the research work, and his role as a mentor to many young mathematicians he had fostered. He has been greatly missed by all his students, colleagues, collaborators, especially in the event such as the international conferences on stochastic control, mathematical finance, and/or backward stochastic differential equations, like the one held this year (2005) in Fudan University. Professor Xunjing Li was born in Qingdao, Shandong Province, China, on June 13, 1935. He came to Shanghai in 1956, at age of 21, as a graduate student in the Department of Mathematics, Fudan University, after receiving a Bachelor degree of Mathematics from Shandong University. He spent three years for his graduate study, under the supervision of the renowned mathematician Professor Jiangong Chen, in the area of approximation theory of functions. He started his teaching and research career in 1959, when he became an assistant professor in the Department of Mathematics, Fudan University. He was promoted to Lecturer in 1962, to Associated Professor in 1980a , and to Full Professor in 1984. In 1985, he became a nationally appointed Doctoral Supervisor, and was named as a Distinguished a From

1966 to 1976, China was in the period of “Cultural Revolution”. All academic titles in universities were abolished during that period 3

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Professor of Fudan University in 1997. Professor Li retired in 2001. During the forty-five year span of his professional life, Professor Xunjing Li grew from a student to a researcher and an educator, and made ineffaceable contributions to the advances in control theory and related fields in China. He was best known for his works on Maximum Principle for optimal controls of infinite dimensional control systems, and is one of the most prominent pioneers in the area of the optimal control theory of distributed parameter systems in China. Professor Li’s another major contribution to the Chinese and the international control community was, arguably, the fact that he almost single-handedly found and fostered a stochastic control group in Fudan University, which has produced a flock of scholars who later become influential players in the area across China, Europe and North America. Professor Li’s research achievements can be chronologically summarized according to the following main periods.

1. 1959—1976. Almost as soon as Professor Li finished his graduate study and started work as an assistant lecturer in the Department of Mathematics of Fudan University, he switched his research direction from Function Theory to Ordinary Differential Equations. This was partly due to a call from the department to expand the academic areas, and as a young faculty member Professor Li enthusiastically answered. Following the leadership of Professor Fulin Jin, Professor Li spent tremendous amount of time in the teaching and research in this new area. In 1962, when he was only 27 years old, he co-authored the text book “Ordinary Differential Equations” with Professor Fulin Jin. The book has been widely used as a main text/reference book by researchers and students in China for many years. In the meantime, as a natural extension of the theory of ordinary differential equations, Professor Li began to explore the area of control theory in early 1960’s. While working on the subject of absolute stability of (finite dimensional) dynamic systems, he participated in another important service activity to the Chinese control theory community: this time was the translation of the celebrated monograph “Mathematical Theory of Optimal Processes” by L. S. Pontryagin et al. In the middle of 1960’s, when Professor Li’s research activities just started to take off, the whole direction of China took an unfortunate turn. During the ten-year period of 19661976, the normal education and scientific research were strongly discouraged and even interrupted due to the “Cultural Revolution”. Taking the only opportunity in late 1960’s to apply his expertise in optimal control theory, Professor Li turned to industrial and applied mathematics. He participated in several projects associated with Shanghai Petroleum Refinery Factory, as well as Jin-Shan Petroleum Chemical Cooperation. These experiences later became an important factor for his perspectives towards control theory throughout his research.

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2. 1977—1987. The year 1977 marked a resurrection and a new era of Chinese education system, as well as a turning point in Professor Li’s research career. Starting from that year, many traditional teaching and research activities began to be restored, including the national college entrance examination, a long tradition in China for universities to enroll freshman students, at both undergraduate and graduate levelb . However, the scar in the scholastic activities and scientific research caused by the ten-year turmoil period was clearly visible, the research results achieved by many scholars like Professor Li but shelved for a decade became a little out-of-date. Inspired by the new spring in the scientific field, he started to ponder over the new germing points of his research. After a careful survey of articles and evaluating his strength, he decided to attack the infinite dimensional optimal control problems. The first break through came out in 1978, in a joint work with Professor Yunlong Yao, then an assistant professor. For a time optimal control problem of infinite dimensional linear systems, they realized, in general the attainable sets is not necessarily convex (a substantial difference from the finite dimensional case), but they discovered that its closure must be. Such an observation, together with the separation theorem for convex sets in infinite dimensional spaces, lead to a proof of maximum principle of time optimal control for infinite dimensional linear systems. Their work was published in the top journal in China, Scientia Sinica (“Science in China”), and was later presented in the 8th International Federation of Automation Conference (IFAC), Kyoto, Japan. While this might be considered usual by today’s standard, but back then when China was just opened up, it was indeed a highly recognizable event. The subsequent several years then witnessed a series of research accomplishments by Professor Li and his group, including the second author. The vector-valued measures in the infinite dimensional optimal control theory was investigated in depth, and the Pontryagin’s maximum principle was extended to various cases of general semi-linear evolutionary distributed-parameter systems. Among many other results, the one that involves terminal constraints is particularly worth mentioning. It was known that in the finite-dimensional case, the maximum principle requires only the differentiability of the coefficients, provided the the terminal constraint set is closed and convex. But there exist counter-examples showing that this is no longer the case in general for the infinite dimensional systems. As a consequence, seeking the proper conditions under which the maximum principle remains valid became a long-standing challenging problem. In 1985, Li and Yao successfully resolved the problem with rigor and elegance. They proved that, for the general semi-linear evolutionary distributed-parameter systems, if the terminal constraints satisfy the finite co-dimension condition, then the maximum principle holds. This result was highly recognized by the international control community, and was later regarded as the foundation of the “Fudan School” research on the infinite-dimensional optimal control theory.

b Both

authors of this article were the beneficiaries of this new policy, though at different levels.

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3. 1987—2001. Since 1985, a group of new researchers, mostly the new generation of graduate students after the class of 1977, started to join Professor Li’s research group. By 1988, some new Ph.D’s from Europe and US returned to China and became the important new addition. Among others, most notable were the Post-doctor Shige Peng (France) and Associate Professor Jiongmin Yong (USA). In 1989, Professor Li, in collaboration with Jiongmin Yong, further extended the maximum principle to the general semilinear evolutionary distributed systems with mixed initial-terminal constraints, by using Ekeland’s Variational Principle and an improved “spike” variational method. This ignited another wave of activities, and a variety of infinite dimensional versions of the maximum principle were knocked down. It is commonly recognized that finite-dimensional optimal control theory has three milestones: the maximum principle by L. S. Pontryagin, the method of dynamic programming by R. Bellman, and the linear quadratic optimal control theory by R. E. Kalman. Professor Li’s research covered essentially all the areas, although the main focus in his earlier years was on the representation of the Pontryagin maximum principle in the infinite dimensional spaces. Many of Professor Li’s works and thoughts, along with many results obtained by the control theory research group of Fudan University (a.k.a. “Fudan School”), can be found in the book “Optimal Control Theory for Infinite Dimensional Systems”, co-authored by Xunjing Li and Jiongmin Yong, and published by Birkh¨ auser in 1995. The book summarized quite exhaustively the latest results in the optimal control theory of nonlinear, deterministic, infinite dimensional systems up to that point, from the perspectives of the aforementioned three milestones. It was very well commented by researchers in the field of control theory. Although for the most part of his professional life, Professor Li considered himself a “deterministic person”, he was nevertheless in essence the main reason of the existence of several research groups, including the stochastic control group and later the mathematical finance group, in Fudan University. Since 1985, especially after he visited several universities in USA, he had a vision that Fudan had to develop the research direction on stochastic control. He began by organizing a stochastic control seminar, and directing several of his graduate students, including the first authorc , to study and to explore new problems in that area. With Professor Li’s cultivation, the scope of the research on stochastic control was quickly expanded to most of the subjects in the field. In his late years, Professor Li personally involved in many research projects on stochastic control theory. Collaborating with Shuping Chen, Ying Hu, Shige Peng, Shanjian Tang, Jionming Yong, Xunyu Zhou, and others, he worked on various problems in linear quadratic control problems, Maximum Principle for stochastic control systems with partial observations, and with jumps. Apart from these works, Professor Li also made important contributions in many other areas such as multi-player differential games, infinite dimensional c The

first author later went on to complete his Ph.D dissertation, on singular stochastic control problems, at University of Minnesota.

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linear quadratic unbounded optimal control, and optimal control of elliptic partial differential equations. As a closely related subject to his research, Professor Li also had far-reaching perspectives toward infinite-dimensional dynamical systems and applications, an area that has been rapidly and richly expanding since 1980’s. In 1982 and 1983, he invited three leading mathematicians in this area, Professors Jack Hale, George Sell, and Shui-Nee Chow, to visit Fudan University. Inspired by these successful visits and encouraged by Professor Li himself, Xiaobiao Lin (now a professor at the NCSU, Raleigh) and the second author later finished their doctoral dissertations at Brown university and at the University of Minnesota, respectively, on topics of infinitedimensional dynamical systems and then become active and prolific researchers in this area. In addition to his mathematical research, Professor Li’s life-long pedagogical achievements constitute another highlight of his life. Professor Li supervised four postdoctors, eleven Ph.D students, thirteen Master students, as well as many junior faculty members. Professor Li was famously known of being strict to his students (this in effect earned him the name “Lao Ban”, as we mentioned before). In fact, looking back, almost all his former advisees could tell some anecdotes where he or she learned lessons, sometimes embarrassing, from Professor Li. However, this might exactly be one of the main reasons that many of them became successful later on, when they became professors, researchers, supervisors, and principal investigators themselves. As one of the main figures in dynamic system and control theory in Fudan University and in China, Professor Li showed tremendous leadership by not only encouraging young faculty in his research group to boldly explore new areas, but also guiding his graduate students in their studies and investigations in areas unfamiliar to himself. This philosophy of Professor Li was the key for success in many cases with his graduate students. It was because of these efforts that the research directions of the Fudan (control theory) group expanded and developed progressively, from distributed parameter control systems in 1970’s and 1980’s, to stochastic control theory in 1980’s and 1990’s, and to mathematical finance throughout the 1990’s and continuing through the 2000’s. It would not be exaggerating to say that without Professor Li, the Fudan University would not have a control group of a history like what it is seen today. This year when we cherish the memory of Professor Xunjing Li on his seventieth birthday, we all feel that his adamant scholastic spirit and his rigorous scientific approach have more or less become a part of our own professional life in conducting research and in educating younger generations of graduate students. We are proud to be a part of “Fudan School”, and glad to see that the named of the group is being carried on by many more talented and dedicated mathematicians year after year, and hopefully for years to come. We believe that this is what Professor Li would be pleased to see as well.

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Acknowledgments Another important obituary that describes Professor Xunjing Li’s role in the development of control theory and related fields in China is the preface (in Chinese) of his collection of articles1 . Our tribute can be considered an adapted version of that article, and we owe our sincere gratitude to the authors of that article—Shuping Chen, Shige Peng, and Jiongmin Yong, for their effort of collecting all the historical information regarding Professor Li’s professional life, which is indeed the basis of this tribute. References 1. Xunjing Li, Selection of Mathematical Papers by Xunjing Li, Fudan University Press, 2003.

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PUBLICATIONS OF XUNJING LI

I. Books 1. Ordinary Differential Equations, Shanghai Science and Technology Press, 1962, 2nd ed., 1984, (with Fulin Jin et al., in Chinese). 2. Control Theory Applied to Computers, Fudan University press, 1988 (with Laixiang Sun and Yougen Chen, in Chinese). 3. Theory of Ordinary Differential Equations for Optimal Control Systems, Higher Education Press, 1989 (with Xueming Zhang and Zuhao Chen, in Chinese). 4. Optimal Control Theory for Infinite Dimensional Systems, Birkh¨ auser, 1995 (with Jiongmin Yong). 5. Basics of Control Theory, Higher Education Press, 2002 (with Jiongmin Yong and Yuan Zhou, in Chinese). 6. Collection of Mathematical Papers, Fudan University press, 2003 (forwarded in Chinese by Shuping Chen, Shige Peng and Jiongmin Yong). II. Edited Conference Proceedings 1. Control Theory of Distributed Parameter Systems and Applications, Lecture Notes in Control and Information Sciences, vol. 159, Springer-Verlag, 1991 (with Jiongmin Yong). 2. Control of Distributed Parameter and Stochastic Systems, Kluwer Academic Publishers, 1999 (with Shuping Chen, Jiongmin Yong and Xun Yu Zhou). III. Research Papers 1. Stability of second-order linear ordinary differential equations with periodical coefficients, Collection of Students’ Papers, Shandong University, 2 (1956), 1–4 (in Chinese).

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2. Four-point common circles and circles of invariance under bilinear transformation, Collection of Students’ Papers, Shandong University, 2 (1957), 5–9 (in Chinese). 3. An extension of Nikolisky’s theorem, Journal of Fudan University (Natural Science) no. 2 (1957), 264–269 (in Chinese, with an English summary). 4. Ces` aro summability in Banach space, Acta Mathematica Sinica, 10 (1960), 41–54 (in Chinese); English Translation in Chinese Mathematics, 2 (1960), 40–52. 5. Uniform approximation of a function by the Ces` aro means of its Faber series, Journal of Fudan University (Natural Science), 5 (1960), no. 2, 159–166 (in Chinese, with an English summary). 6. On the absolute stability of the indirect control systems Journal of Fudan University (Natural Science), 7 (1962), no. 1, 25–34 (with Huimin Xie and Junben Chen; in Chinese, with an English summary). 7. On the absolute stability of systems with time lags, Acta Mathematica Sinica, 13 (1963), 558–573 (in Chinese); English Translation in Chinese Mathematics, 4 (1963), 609–626. 8. Time optimal control of linear systems, Journal of Fudan University (Natural Science), 9 (1964), no. 4, 501–512 (with Huimin Xie; in Chinese, with a Russian summary). 9. Stability and time optimal control for automatic regulated systems, Collection of Papers, Research Institute of Mathematics, Fudan University, (1964), 95–111 (with Huimin Xie et al.; in Chinese). 10. Mathematical principle for electro-magnetic power-driven compressors (I) — resonance analysis in the case of free loads, Journal of Fudan University (Natural Science), 10 (1965), nos. 3–4, 151–154 (in Chinese). 11. Period-analysis approach to time series, Selection of Long-Term Forecast Methods, (Eastern China 1973 meeting on precipitation forecast in the flood season), Eastern China Meteorological Observatory Center, (1973), 71–73 (in Chinese). 12. Derived dynamic equation and feed-forward control for controlled plants — an application of electronic digital computers in the control of atmospheric

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distillation, Acta Math. Appl. Sinica, 1 (1976), no. 2, 33–45 (in Chinese). 13. Analytical design of control equations, Journal of Fudan University (Natural Science), 17 (1978), no. 2, 38–48 (in Chinese). 14. On the analytic design of the optimal regulator for the systems with time lags, Journal of Fudan University (Natural Science), 19 (1980), no. 2, 189– 195 (in Chinese, with an English summary). 15. An analysis of the mathematical model to the furnace of thermal dissociation for generating ethene, Journal of Fudan University (Natural Science), 19 (1980), no. 1, 115–116 (in Chinese). 16. Time optimal boundary control for systems governed by parabolic equations, Chin. Ann. Math., 1 (1980), nos. 3–4, 453–458 (in Chinese, with an English summary). 17. Time optimal control of distributed parameter systems, Scientia Sinica, 24 (1981), 455–465 (with Yao Yunlong). 18. On the evolution equation in Banach space, Chin. Ann. Math., 2 (1981), no. 4, 479–489 (in Chinese, with an English summary). 19. On applications of vector measure to the optimal control theory for distributed parameter systems, Chin. Ann. Math., 3 (1982), no. 5, 655–662 (in Chinese, with an English summary). 20. Methods and applications of control theory Nature Magazine, 5 (1982), 435– 438 (in Chinese). 21. On the stability of nonlinear control systems with time-lag, Control Theory and Applications, 1 (1984), no. 3, 117–123 (in Chinese, with English summary). 22. Vector-valued measure and the necessary conditions for the optimal control problems of linear systems, Journal of Mathematical Research and Exposition, 4 (1984), no. 4, 51–56. 23. Maximum principle of distributed parameter systems with time-lags, Springer Lecture Notes in Control & Information Science, vol. 75 (1985), 410–427 (with Yunlong Yao).

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24. Margin of stability for the optimal regulator of distributed parameter systems Control Theory and Applications, 3 (1986), no. 1, 76–82 (in Chinese, with an English summary). 25. N-person differential games governed by infinite dimensional systems, J. Optim. Theory & Appl., 50 (1986), 431–450. 26. Maximum principle of optimal periodic control for functional differential systems, J. Optim. Theory & Appl., 50 (1986), 421–429. 27. Maximum principle of optimal controls for functional differential systems, J. Optim. Theory & Appl., 54 (1987), 335–360 (with S. N. Chow). 28. On optimal control of functional differential systems, Springer Lecture Notes in Control & Information Sciences, vol. 102 (1987), 112–119 (with Fulin Jin). 29. Optimal control for infinite dimensional systems, Springer Lecture Notes in Control & Information Sciences, vol. 159 (1991), 96–105. 30. Maximum principle for optimal control of nonlinear generalized systems — finite dimensional case, Acta Automatica Sinica, 17 (1991), 17–23 (with Shige Peng; in Chinese). 31. Necessary conditions for optimal control of distributed parameter systems, SIAM J. Control & Optim., 29 (1991), 895–906 (with Jiongmin Yong). 32. Dynamical model of neural network (I), Acta Biophysica Sinica, vol. 8, no. 2 (1992), 339–345 (with Fanji Gu and Jiong Ruan). 33. Dynamical model of neural network (II), Acta Biophysica Sinica, vol. 8, no. 3 (1992), 412–418 (with Fanji Gu and Jiong Ruan). 34. A realistic model of neural networks, J. of Electronics, vol. 9, no. 4 (1992), 289–295 (with Fanji Gu and Jiong Ruan). 35. Progress in control theory (distributed parameter systems), 1990 Yearbook of Natural Science, 3.24–3.26, Shanghai Translation Publishing Company, Shanghai, 1992 (in Chinese). 36. Optimal control theory for infinite dimensional systems, Progress in Natural Science, 2 (1992), 104–112 (with Jiongmin Yong; in Chinese).

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37. Maximum principle for optimal control problem of nonlinear generalized systems — infinite dimensional case, Acta Math. Appl. Sinica, 15 (1992), no. 1, 99–104 (with Ying Hu and Shige Peng; in Chinese with an English summary). 38. A linear quadratic optimal control problem with disturbances — an algebraic Riccati equation and differential games approach, Appl. Math. Optim., 30 (1994), 267–305 (with Shuping Chen, Shige Peng and Jiongmin Yong). 39. Necessary conditions for optimal control of stochastic systems with random jumps, SIAM J. Control & Optim., 32 (1994), 1447–1475 (with Shanjian Tang). 40. Maximum principle for optimal control of distributed parameter stochastic system with random jumps, Lecture Notes in Pure & Appl. Math., 152 (1994), 867–890 (with Shanjian Tang). 41. The effect of small time delays in the feedback on boundary stabilization, Science in China (Series A), 36 (1993), 1435–1443 (with Kangsheng Liu). 42. General necessary conditions for partially observed optimal stochastic controls, J. Appl. Prob., 32 (1995), 1118–1137 (with Shanjian Tang). 43. Stochastic verification theorems within the framework of viscosity solutions, SIAM J. Control & Optim., 35 (1997), 243–253 (with Jiongmin Yong and Xunyu Zhou). 44. Tracking control for nonlinear affine systems, J. Math. Control & Information, 14 (1997), 307–318 (with K. L. Teo and W. Q. Liu). 45. Stochastic linear quadratic regulator with indefinite control weight costs, SIAM J. Control & Optim., 36 (1998), 1685–1702 (with Shuping Chen and Xunyu Zhou). 46. Minimum period control problem for infinite dimensional system, Chin. Ann. Math. (Ser. B), 19 (1998), no. 1, 113–128 (with Liping Pan). 47. The equivalence between two types of exponential stabilities, Chinese Science Bulletin, 43 (Chinese Series, 1998), no. 16, 1787–1788; 43 (English Series, 1998), no. 18, 1583–1584 (with Hanzhong Wu).

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48. Linear quadratic problem with unbounded control in Hilbert spaces, Chinese Science Bulletin, 43 (1998), no. 20, 1712–1717 (with Hanzhong Wu). 49. Synthesis of upper-triangular non-linear systems with marginally unstable free dynamics using state-dependent saturation, Int. J. Control, 72 (1999), 1078–1086 (with Wei Lin). 50. Research into unexpected events: a key project of mathematical finance, Chinese Science, vol. 51, no. 2 6–9, March, 1999 (with Zhiyuan Shen and Jiongmin Yong; in Chinese). 51. Social responsibility of scientists, Chinese Science, vol. 52, no. 1, 24–25, January, 2000 (with Zhiyuan Shen; in Chinese). 52. A linear quadratic problem with unbounded control in Hilbert spaces, Differential Integral Equations, 13 (2000), no. 4–6, 529–566 (with Hanzhong Wu). 53. Necessary conditions for optimal control of elliptic systems, J. Austral. Math. Soc. (Ser. B), 41 (2000), 542–567 (with Hang Gao). IV. Conference Proceedings Papers 1. On optimal control for distributed parameter systems, Proc. of 8th IFAC World Congress, Kyoto, Japan, 1981, 207–212 (with Yunlong Yao). 2. Time optimal control of distributed Parameter systems (in Chinese), Proc. National Exchange Meeting on Control Theory and Applied Mathematics, Science Press, Beijing, 1981, 210–211 (with Yunlong Yao). 3. Some problems on optimal control theory and computer control, Proc. National Exchange Meeting on Control Theory and Applied Mathematics, Science Press, Beijing, 1981, 230–232 (in Chinese). 4. Vector-valued measure and the necessary conditions for the optimal control problems of linear systems, Proc. IFAC 3rd Symposium on Control of Distributed Parameter Systems, Toulouse, France, 1982, 503–506. 5. On the stability of nonlinear control systems with time-lags, Proc. 1983 Beijing Symposium on Differential Geometry and Differential Equations, Science Press, Beijing, China, 1986, 477–480. 6. Bounded real lemma and stability of the time lag nonlinear control systems, Proc. 9th IFAC World Congress, Budapest, Hungary, 1984, 67–71.

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7. Quadratic optimal control for generalized linear stochastic control systems, Preprint of First China-Japan Symposium on System Control & Its Appl., Hangzhou, China, 1989 (with Ying Hu and Shige Peng). 8. Robust design of regulator for controlled systems via LQ differential games, Preprint of 2nd Japan-China Joint Symposium on System control Theory & its Appl., Osaka, Japan, 1990 (with Shuping Chen). 9. Optimal control for a class of distributed parameter systems, Proc. 29th CDC, Honolulu, USA, 1990, 2319–2320 (with Jiongmin Yong). 10. Optimality conditions for controls of infinite dimensional systems, Proc. 1st World Congress on Nonlinear Analysis, Tampa, Florida, USA, August, 1992. 11. Some Advances in the Theory of Optimal Control, Proc. National Annual Meeting of Control Theory and Applications, 1992 (in Chinese). 12. On stochastic linear controlled systems, SIAM Conference on Control and Appl., Minneapolis, USA, Sept. 1992 (with Shuping Chen, Shige Peng, and Jiongmin Yong). 13. Works on Control Science of Fudan University, Proc. Annual Meeting of Shanghai Society of Automation, 1992 (with Jiongmin Yong; in Chinese). 14. Contributions to the Theory of Optimal Stochastic Controls, Differential Equations and Control Theory (Wuhan, 1994), 169-175, Lecture Notes in Pure & Appl. Math., vol. 176 (1996), Marcel Dekker (with Shanjian Tang). 15. Necessary conditions for optimal control of infinite dimensional systems, Proc. 1994 Hong Kong International Workshop on New Directions of Control and Manufacturing, Hong Kong, 1994, 214–221. 16. A Class of Non-conventional Stochastic Linear Quadratic Regulators, Proc. 2nd Asia Control Conference, 1997, 1-317-320 (with Shuping Chen and Xunyu Zhou). 17. Necessary Conditions for Optimal Controls of Infinite Dimensional Systems, Proc. 2nd Asia Control Conference, 1997, 1429–432.

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18. Global Stabilization of Nonlinear Systems with Marginally Unstable Free Dynamics by Small Controls, Proc. of 1998 American Control Conference, Philadelphia, 1998, 303–307 (with Wei Lin). 19. The Linear quadratic Optimal Control in Hilbert Spaces with Unbounded Controls, Proc. 14th World Congress of IFAC, vol. F, 1999, 121–124. 20. Optimal Control Theory: from Finite Dimensions to Infinite Dimensions, Control of Distributed Parameter and Stochastic Systems (Proc. of the IFIP WG7.2 International Conference on Control of Distributed Parameter and Stochastic Systems, June 19–22, 1998), (eds.: Shuping Chen, Xunjing Li, Jiongmin Yong and Xun Yu Zhou), Kluwer Academic Publishers, Boston, 1999, 85–94.

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STUDENTS AND POST-DOCTORS ADVISED BY XUNJING LI

I. Post-doctors. 1. Shige Peng Final research report: General Stochastic Maximum Principle, Backward Stochastic Differential Equations, and Application of Singular Perturbation in Annealing Simulation and Neutral Networks, (March, 1988—November, 1989). 2. Daode Gao Final research report: Estimation of Potential Taxation, (June, 1996—June, 1998). 3. Aiguo Kong Final research report: Research into Capital Structure of Financial Markets, (December, 1996—July, 1998). 4. Jiang Yu Final research report: Existence of Discrete Breather for a Class of Infinite Dimensional Coupled Oscillators, (June, 1999–June, 2001).

II. Doctors. 1. Shuping Chen Riccati Equation: a New Approach, and Its Applications, Ph.D. thesis, 1985. 2. Xunyu Zhou, Maximum Principle, Dynamic Programming and Their Relationship in Optimal Control Theory, Ph.D. thesis, July, 1989. 3. Ying Hu, Maximum Principle for Optimal Control of Stochastic Systems, Ph.D. thesis, July, 1989. 4. Liping Pan, Optimal Control of Distributed Parameter Systems with Time Lags, and Infinite Dimensional Leader-Follower Differential Games, Ph.D. thesis, March, 1991.

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5. Kangsheng Liu, Stabilization and Local Boundary Control of Distributed Parameter Systems, Ph.D. thesis, December, 1991. 6. Jianxiong Huang, Bifurcation and Chaos of Homoclinic Orbits in Infinite Dimensional Dynamical Systems, Ph.D. thesis, July, 1992. 7. Shanjian Tang, Optimal Control in Hilbert Space of Stochastic Systems with Random Jumps, Ph.D. thesis, January, 1993. 8. Pingjian Zhang, Differential Riccati Equations with Uncertain Coefficients and Their Applications in H ∞ -Optimization and L-Q Games, Ph.D. thesis, 1994. 9. Hang Gao, Domain Optimization and Optimal Control of Systems Governed by Elliptic Equations, Ph.D. thesis, March, 1996. 10. Hanzhong Wu, The Linear Quadratic Optimal Control Problem with Unbounded Control in Hilbert Spaces, Ph.D. thesis, July, 1998. 11. Qihong Chen, Indirect Obstacle Optimal Control Problem for Variational Inequalities, Ph.D. thesis, July, 1999. 12. Shangwei Zhu, Two Problems in Optimal Control Theory and Applications, Ph.D. thesis, July, 2005. III. Masters. 1. Yuncheng You, Optimal Control of Linear Systems in Abstract Spaces with Indefinite Quadratic Criteria, Master thesis, January, 1981. 2. Guozhu Gao, Uniform Asymptotic Stability of Functional Differential Equations of Neutral Type, Master thesis, July, 1981. 3. Jin Ma, On Infinite-Time State Estimation, Master thesis, July, 1984. 4. Yinping Wang, Convergence of Self-Tuned Regulators, Master thesis, July, 1985. 5. Huiheng Zheng, Stability of Nonlinear Control Systems with Time Lags, Master thesis, July, 1986.

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6. Yan Qin, Singular Perturbation Approach to Computation of Optimal Control Problems, Master thesis, July, 1987. 7. Yuan Zhou, Singular Quadratic Optimal Control of Infinite Dimensional Linear Systems, Master thesis, July, 1987. 8. Hong Xu, Necessary Conditions for Optimal Control of Distributed Parameter Systems, Master thesis, July, 1988. 9. Chunjiang Qian, Positive Real Lemma and Linear Quadratic Optimal Control of Time Delayed Systems, Master thesis, July, 1994. 10. Guanghui Li, The Positive Real Lemma and Absolute Stability for Neutral Differential-Difference Equation, Master thesis, July, 1994. 11. Qi Zhou, High-Order Necessary Conditions for Singular Optimal Control of Distributed Parameter Systems, Master thesis, March, 1995. 12. Shihong Wang, Averaging of Hamilton-Jacobi Equation in Infinite Dimensions and Application, Master thesis, July, 1997. 13. Zuoyi Zhou, H∞ Control Problem for Nonlinear Systems, Master thesis, July, 1997.

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PART TWO STOCHASTIC CONTROL, MATHEMATICAL FINANCE, AND BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

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AXIOMATIC CHARACTERISTICS FOR SOLUTIONS OF REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS∗

XIAOBO BAO Institute of Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China & Key Laboratory of Mathematics for Nonlinear Sciences, (Fudan University), Ministry of Education, China SHANJIAN TANG Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433, China, & Key Laboratory of Mathematics for Nonlinear Sciences, (Fudan University), Ministry of Education, China E-mail: [email protected]

In this paper, we introduce the notion of an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator with a floor in terms of four axioms. We show that an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator {Es,t , 0 ≤ s ≤ t ≤ T } with a continuous upper-bounded floor {St , 0 ≤ t ≤ T }, is necessarily represented by the solutions of a backward stochastic differential equation reflected upwards on the floor {St , 0 ≤ t ≤ T }, if it is E µ -super-dominated for some µ > 0 and if it has the non-increasing and floor-above-invariant property of forward translation. Keywords: American contingent claims, reflected BSDE, dynamic nonlinear evaluation, filtration-consistent nonlinear expectation, nonlinear Doob-Meyer decomposition. 2000 Mathematics Subject Classification: Primary 60H10; Secondary 60H30, 60A05, 49N90, 91B30, 91B24.

1. Introduction Let {Bt , 0 ≤ t ≤ T } be a d-dimensional standard Brownian motion defined on a probability space (Ω, F, P ). Let {Ft , 0 ≤ t ≤ T } be the natural filtration of ∗ This

work is partially supported by the NSF of China under Grant No. 10325101 (distinguished youth foundation), the National Basic Research Program of China (973 program) with Grant No. 2007CB814904, and the Chang Jiang Scholars Program. 23

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{Bt , t ∈ [0, T ]}, augmented by all P -null sets of F. Set L2 (FT ) := {ξ : ξ is an FT -measurable random variable s.t. E|ξ|2 < +∞}, L2F (0, T ; Rm) := φ : φ is Rm -valued and {Ft , 0 ≤ t ≤ T }-adapted s.t. Z T E |ϕt |2 dt < ∞ , 0 n o H2 := ϕ ∈ L2F (0, T ; R) : ϕ is predictable , n o S2 := ϕ ∈ H2 : ϕ is a continuous process s.t. E max |ϕt |2 < +∞ . 0≤t≤T

2

2

2

For ξ ∈ L (FT ), denote by L (FT ; ξ) the subset of L (FT ) whose elements are not less than ξ. We shall use L2+ (FT ) to stand for L2 (FT ; 0). Consider the following reflected backward stochastic differential equation (RBSDE):  Z T Z T    Ys = ξ + g(r, Yr , Zr )dr + KT − Ks − hZr , dBr i, 0 ≤ s ≤ T; s sZ (1) T    Ys ≥ Ss , a.s.0 ≤ s ≤ T ; K0 = 0 and (Yr − Sr )dKr = 0. 0

Here, the three objects are previously given: a terminal value ξ, a random field g : Ω × [0, T ] × R × Rd −→ R, and a continuous progressively measurable real-valued random process {St , 0 ≤ t ≤ T }. Assume that (C1) ξ ∈ L2 (FT ). (C2) g(·, y, z) ∈ H2 for (y, z) ∈ R × Rd . (C3) |g(t, y, z) − g(t, y 0 , z 0 )| ≤ α(|y − y 0 | + |z − z 0 |) a.s. with y, y 0 ∈ R and 0 z, z ∈ Rd for some positive constant α. And (C4) S + ∈ S2 and ST ≤ ξ a.s.. The solution to RBSDE (1) is a triple {(Yt , Zt , Kt ), 0 ≤ t ≤ T } of {Ft , 0 ≤ t ≤ T }-progressively measurable processes taking values in R × Rd × R+ such that (i) Z ∈ H2 , and (ii) {Kt , 0 ≤ t ≤ T } is continuous and increasing. In view of El Karoui et al., 5 we know that there exists unique solution {(Yt , Zt , Kt ), 0 ≤ t ≤ T } of RBSDE (1) if the four conditions (C1)–(C4) are satisfied. Define r;g,S Et,T [ξ] := Yt ,

∀ ξ ∈ L2 (FT ; ST )

and r,g r;g,0 Et,T [ξ] := Et,T [ξ],

∀ ξ ∈ L2+ (FT ).

Here, the superscript r indicates that the underlying operators are generated by a reflected BSDE, and the superscripts (g, S) specify the generator and the obstacle of the underlying RBSDE. El Karoui and Quenez 8 argued that the operators r;g,S , 0 ≤ s ≤ t ≤ T } introduce a nonlinear pricing system for square-integrable {Es,t American contingent claims, possessing the following properties:

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(A1) Monotonicity. Es,t [Y ] ≥ Es,t [Y 0 ] if Y, Y 0 ∈ L2 (Ft ; St ) such that Y ≥ Y 0 . (A2) Time consistency. Er,s [Es,t [Y ]] = Er,t [Y ] if r ≤ s ≤ t ≤ T and Y ∈ L2 (Ft ; St ). A financial theoretically interesting problem is the converse one: Is a dynamic operator that possesses similar properties like (A1) and (A2), necessarily represented by a RBSDE? That is, we are concerned with the axiomatic characteristics for a RBSDE. Throughout the paper, we make the following assumption (denoted by (H0)): (H0) The predictable process S is continuous, S + ∈ S2 , and there is a positive constant C such that ess sup0≤t≤T St ≤ C a.s.. The process S is given, either as the (lower) obstacle of the underlying RBSDE or as the floor of the underlying dynamic operator (see Definition (1.1) below). In this paper, we shall formulate and discuss the above converse problem for RBSDE (1) with a given general upper-bounded floor S > −∞. More precisely, we introduce the notion of a dynamic operator with a floor S, and then represent the dynamic operator by a BSDE reflected upwards on the floor S. In this way, we characterize on one hand the solutions of RBSDEs in terms of axioms, and on the other hand, we give a representation for the dynamic operator in terms of RBSDEs (1). We first introduce the following notion of an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator with floor S. Definition 1.1. A time parameterized system of operators Es,t [·] : L2 (Ft ; St ) −→ L2 (Fs ; Ss ),

0≤s≤t≤T

(2)

is called an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator with floor S if it satisfies the following: Es,t [Y ] is continuous in (s, Y ) ∈ [0, t] × L2 (Ft ; St ) for t ∈ [0, T ], and furthermore, it satisfies the following four axioms. (D1) Floor-above strict monotonicity Es,t [Y ] ≥ Es,t [Y 0 ] a.s. for Y and 0 Y ∈ L2 (Ft ; St ) such that Y ≥ Y 0 a.s.. If Es,t [Y ] > Ss a.s. for any s ∈ [r, t] and some r ∈ [0, t], then Y 0 = Y a.s. if Y 0 ≥ Y a.s. and Er,t [Y 0 ] = Er,t [Y ] a.s.. (D2) Es,t [Y ] = Y a.s. for each Y ∈ L2 (Fs ; C). (D3) Time consistency Er,s [Es,t [Y ]] = Er,t [Y ] a.s. for Y ∈ L2 (Ft ; C) if r ≤ s ≤ t ≤ T. ˜ − C˜ = 1A (Es,t [Y + C] ˜ − C), ˜ (D4) Zero-one law For each s ≤ t, Es,t [1A Y + C] a.s., ∀A ∈ Fs for any constant C˜ dominating the floor. The main result of the paper is stated as follows. Theorem 1.1. Consider an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator {Es,t , 0 ≤ s ≤ t ≤ T } with the floor S satisfying assumption (H0). We make the following two assumptions (H1) and (H2): (H1) E µ -super-domination. There is some µ > 0 such that µ Et,T [X + Y ] − Et,T [X] ≤ Et,T [Y ], a.s.

(3)

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26 µ for t ∈ [0, T ], X ∈ L2 (FT ; ST ) and Y ∈ L2+ (FT ). Here, Et,T [Y ] := yt with {yt , 0 ≤ t ≤ T } being the first component of the solution of the following BSDE: Z T Z T ys = Y + µ|zr |dr − hzr , dBr i, 0 ≤ s ≤ T. (4) s

s

(H2) The non-increasing and floor-above-invariant property of forward translation. Et,T [X + Y ] ≤ Et,T [X] + Y , a.s. for X ∈ L2 (FT ; ST ) and Y ∈ L2+ (Ft ). Furthermore, for X ∈ L2 (FT ; ST ) such that Et,T [X] > St a.s. for t ∈ [0, T ], We have Et,T [X + Y ] = Et,T [X] + Y for any Y ∈ L2+ (Ft ). Then, there is a random field g : Ω × [0, T ] × Rd → R such that the following are satisfied: (i) g(t, 0) = 0 for a.e. t ∈ [0, T ], (ii) |g(t, z1 ) − g(t, z2 )| ≤ µ|z1 − z2 |, and r;g,S (iii) Es,t (Y ) = Es,t (Y ) for any Y ∈ L2 (Ft ; St ) and s ∈ [t, T ] with t ∈ [0, T ]. For the particular case of the zero floor (S = 0), the last assertion is still true if the assumption (H1) is replaced by the weaker one (H1)’: (H1)’ E µ -domination. There is some µ > 0 such that µ E0,T [X + Y ] − E0,T [X] ≤ E0,T [Y ], for any X ∈ L2 (FT ; ST ) and Y ∈ L2+ (FT ). (5)

Note that the assumption (H1)’ is much weaker than that of E µ -domination used by Coquet,et al., 3 in that Y is here restricted within L2+ (FT ) instead of taking values in the whole space L2 (FT ) like the latter. This difference will complicate our subsequent arguments. Remark 1.1. For the general case of the upper bounded floor S, the primal dyC namic operator {Es,t , 0 ≤ s ≤ t ≤ T } deduces the following new one {Es,t ,0 ≤ s ≤ t ≤ T} : C Es,t [·] : L2 (Ft ; St − C) −→ L2 (Fs ; Ss − C), 0 ≤ s ≤ t ≤ T

(6)

C Es,t [X] := Es,t [X + C] − C, ∀X ∈ L2 (Ft ; St − C).

(7)

with

C It is easy to prove that Es,t [X] for X ∈ L2 (Ft ; St − C), satisfies all the conditions (H1), (H2), and (D1)–(D4). Then, it follows immediately that if Theorem 1.1 is true for the negative floor S − C, it is also true for the general upper bounded floor S. Hence, it is sufficient to prove Theorem 1.1 for the case of the negative floor.

Even in the simpler case of negative floors, two key points are still worth to be mentioned here for the proof of Theorem 1.1. One is to make full use of the non-increasing and floor-above-invariant assumption (H2) of forward translation and the assumption (H1) of E µ -super-domination for some µ > 0, to extend the underlying {Ft , 0 ≤ t ≤ T }-consistent dynamic operator from the subset L2 (FT ; ST ) of floor-dominating square-integrable random variables to the whole space L 2 (FT )

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of square-integrable random variables. The extended dynamic operators are shown e The generator to be identified to an {Ft , 0 ≤ t ≤ T }-consistent expectation E. g of its BSDE representation given by Theorem 2.3 turns out to be that of the desired RBSDE. The second key point is the following observation: the process e {Et,T [X], 0 ≤ t ≤ T } turns out to be an E-supermartingale for each X ∈ L2 (FT ; ST ). This fact allows us to apply a nonlinear Doob-Meyer’s decomposition theorem in Ref. 3, to give the increasing process of {Et,T [X], 0 ≤ t ≤ T } as a solution to some BSDE reflected upwards on the floor. Eventually, our proof of Theorem 1.1 is both natural and elegant. To diverse the difficulty in the above arguments, the whole proof is divided into Sections 2 and 3. The above two key points are exposed in detail separately in these two sections. The main result of this work has been announced in Bao and Tang 1 . In Bao and Tang 2 , we study a dynamic operator with a very general continuous floor S which may be unbounded from the above. We define a dynamic operator as a stopping-times parameterized system of operators, and also give the representation by RBSDEs of a dynamic operator. The rest of the paper is organized as follows. In Section 2, we consider the case of the zero floor. We concentrate our attention to show how to extend the family of dynamic operators defined on the subset L2+ (FT ) of L2 (FT ) to an {Ft , 0 ≤ t ≤ T }expectation, which is defined on the whole space L2 (FT ). Section 3 is devoted to the case of the negative floor. Restricting the underlying {Et,T , 0 ≤ t ≤ T }-consistent dynamic operator to L2+ (FT ), we get an {Ft , 0 ≤ t ≤ T }-expectation by extending the restriction to L2 (FT ) in the way as shown in the preceding section. The {Ft , 0 ≤ t ≤ T }-expectation gives the generator g of an RBSDE by Theorem 2.3. In addition, we have to give the amount {KtX , 0 ≤ t ≤ T } to push upwards for {Et,T [X], 0 ≤ t ≤ T } with X ∈ L2 (FT ; ST ), that is, the increasing process in relevant to an RBSDE. For this purpose, a nonlinear Doob-Meyer’s decomposition theorems is shown how to be used. The key point is to observe that the process {Et,T [X], 0 ≤ t ≤ T } turns e out to be an E-supermartingale for each X ∈ L2 (FT ; ST ). 2. The Case of the Zero Floor In the case of the zero floor, the properties (D2) and (D4) read (D2)’ Es,t [Y ] = Y a.s. for Y ∈ L2+ (Fs ) and s ∈ [0, t]. (D4)’ Zero-one law Es,t [1A Y ] = 1A Es,t [Y ], a.s. for s ∈ [0, t], A ∈ Fs , and Y ∈ L2+ (Ft ). From (D1), (D2)’, (H1) and (H2), we can give the following: (H2)’ We have for t ∈ [0, T ], X ∈ L2+ (FT ) and Y ∈ L2+ (Ft ), Et,T [X + Y ] = Et,T [X] + Y a.s. . In fact, consider X ∈ L2+ (FT ) and Y ∈ L2+ (Ft ). It follows from (D2)’ that Et,T [] = for t ∈ [0, T ] and very constant > 0. Therefore, from (D1), we deduce that Et,T [ + X] ≥ Et,T [] = > 0 a.s. for t ∈ [0, T ]. Consequently, by (H2), we

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conclude that Et,T [ + X + Y ] = Et,T [ + X] + Y a.s. .

(8)

By (H1), we can take the limit → 0, and we have the desired equality. Remark 2.1. Assume that conditions (C2) and (C3) are satisfied. Moreover, assume that g(·, ·, 0) ≡ 0. Let (y, z) be the adapted solution to the following BSDE: Z T Z T Yt = ξ + g(s, Ys , Zs )ds − hZs , dBs i, 0 ≤ t ≤ T. (9) t

t

L2+ (FT ).

with the terminal condition ξ ∈ Then the triple (y, z, 0) is the adapted solution of RBSDE (1) with the parameter (Y, g, 0). This shows that the adapted solution of RBSDE (1) gives the one of the corresponding BSDE (9) for any terminal value ξ ∈ L2+ (FT ) since the obstacle {St , 0 ≤ t ≤ T } is not active in this case. The following lemma is immediate and will be used later. Lemma 2.1. We have µ (i) E Et,T [X]p ≤ exp(2−1 p(p − 1)−1 µ2 (T − t))E[X p ] for all µ > 0, t ∈ [0, T ], p and X ∈ L (FT ), with p ∈ (1, 2]; −µ µ (ii) Et,T [X + Y ] = Y − Et,T [−X] a.s. for all µ > 0, t ∈ [0, T ], X ∈ L2 (FT ), and 2 Y ∈ L (Ft ); And µ −µ (iii) E0,T [·] and E0,T [·] are strongly continuous in L2 (FT ). Proof. We only prove the first assertion (i). The other two assertions are easy to see. µ For simplicity of notations, set yt := Et,T [X]. Then, by definition, there is unique 2 d z ∈ LF (0, T ; R ) such that (y, z) is the unique adapted solution of BSDE (4). Using Itˆ o’s formula, we have Z T Z T 1 E|yt |p + p(p−1) |ys |p−2 |zs |2 ds = E|X|p +2µp |ys |p/2 |ys |p/2−1 |zs | ds. (10) 2 t t Since µp|ys |p/2 |ys |p/2−1 |zs | ≤

1 p µ2 |ys |p + p(p − 1)|ys |p−2 |zs |2 , 2(p − 1) 2

(11)

we have E|yt |p ≤ E|X|p +

p µ2 2(p − 1)

Z

T

|ys |p ds.

(12)

t

The standard arguments of using Gronwall’s inequality then gives the desired inequality. Remark 2.2. See Coquet et al. 3 for the detailed proof of the first assertion in the case of p = 2, which is easy and standard.

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In view of Remark 2.1, from Proposition 3.6, page 164, Lemma 3.2 and Proposition 3.7, page 165 of Ref. 12 concerning the properties of the solutions to BSDEs, we have the following Theorem 2.1. Assume that conditions (C2) and (C3) are satisfied. Moreover, r,g assume that g is independent of y and g(·, ·, 0) ≡ 0. Then, {Es,t ,0 ≤ s ≤ t ≤ T} satisfies (H1), (D1), (D2)’, (D3), (D4)’ and (H2)’. Therefore, it is an {F t , 0 ≤ t ≤ T }-consistent dynamic operator with the zero floor. Roughly speaking, Theorem 2.1 asserts that an RBSDE with the obstacle being zero introduces an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator. In what follows, we shall consider the converse problem. That is, we shall associate an {Ft , 0 ≤ t ≤ T }consistent dynamic operator with the zero floor to a BSDE reflected on the zero floor. For this purpose, we establish the following six preliminary lemmas. First, we introduce some notations. Definition 2.1. For an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator {Es,t [·], 0 ≤ s ≤ t ≤ T }, define the system of operators E[·|Ft ] : L2+ (FT ) −→ L2+ (Ft ), 0 ≤ t < T by E[Y |Ft ] := Et,T [Y ], a.s. for Y ∈ L2+ (FT )

(13)

2

and the nonlinear functional E[·] : L (FT ) → R by E[Y ] := E0,T [Y ] for Y ∈ L2+ (FT ).

(14)

The two notations E[·] and E[·|Ft ] behave in L2+ (FT ) exactly like an {Ft , 0 ≤ t ≤ T }-consistent expectation and its conditional {Ft , 0 ≤ t ≤ T }-consistent expectation on Ft . The only difference lies in the domains of variables: the former’s are L2+ (FT ), while the latter’s are L2 (FT ). This can be seen from the following obvious lemma. Lemma 2.2. For an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator {Es,t [·], 0 ≤ s ≤ t ≤ T }, E[·] and E[·|Ft ] have the following properties (E1)’-(E3)’. (E1)’ E[Y1 ] ≤ E[Y2 ] if Y1 , Y2 ∈ L2+ (FT ) and Y1 ≤ Y2 . Furthermore, if Y1 ≤ Y2 , then Y1 = Y2 if E[Y1 ] = E[Y2 ]. e > 0. And (E2)’ E[C] = C for any constant C (E3)’ For any Y ∈ L2+ (FT ), there is unique η ∈ L2+ (Ft ) such that E[η1A ] = E[Y 1A ],

∀A ∈ Ft ,

(15)

which is equal to E[Y |Ft ]. Therefore, E[Y |Ft ] can be viewed as the E-expectation of Y ∈ L2+ (FT ) conditioned on Ft for any t ∈ [0, T ], though E is in general not an {Ft , 0 ≤ t ≤ T }-consistent expectation at all. Lemma 2.3. Assume that E[Y |Ft ] satisfies conditions (H1)’ for µ > 0. Then, 1 |E[ξ1 ] − E[ξ2 ]| ≤ exp( µ2 T )||ξ1 − ξ2 ||L2 , 2

∀ξ1 , ξ2 ∈ L2+ (FT ).

(16)

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Therefore, E[·] is strongly continuous in L2 (FT ). Proof. Using (D1) and (H1)’, we have E[ξ1 ] − E[ξ2 ] ≤ E[|ξ1 − ξ2 | + ξ2 ] − E[ξ2 ] ≤ E µ [|ξ1 − ξ2 |]. Using Lemma 2.1, we have E µ [|ξ1 − ξ2 |]2 ≤ exp (µ2 T )E[|ξ1 − ξ2 |2 ].

(17)

Therefore, 1 1 E[ξ1 ]−E[ξ2 ] ≤ exp( µ2 T )(E|ξ1 −ξ2 |2 )1/2 = exp( µ2 T )||ξ1 −ξ2 ||L2 , 2 2

∀ξ1 , ξ2 ∈ L2+ (FT ). (18)

Identically, we can show 1 E[ξ2 ] − E[ξ1 ] ≤ exp( µ2 T )||ξ1 − ξ2 ||L2 , 2 The proof is then complete.

∀ξ1 , ξ2 ∈ L2+ (FT ).

(19)

Lemma 2.4. (E µ -domination) Assume that E[Y |Ft ] satisfies (H1)’ and (H2)’. Then, we have E −µ [Y ] ≤ E[X + Y ] − E[X] ≤ E µ [Y ],

∀X, Y ∈ L2+ (FT ).

(20)

Proof. In view of (H1)’, it is sufficient to prove the following E[X + Y ] − E[Y ] ≥ E −µ [Y ],

∀X, Y ∈ L2+ (FT ).

(21)

If Y ≤ n a.s. for some integer n, then using (H2)’, (H1)’, and Lemma 2.1, we have n − (E[X + Y ] − E[X]) = n + E[X] − E[X + Y ] = E[X + n] − E[X + Y ] ≤ E µ [n − Y ] ≤ n − E −µ [Y ]. Therefore, E[X + Y ] − E[X] ≥ E −µ [Y ].

(22)

In general, consider Y ∈ L2+ (FT ). Define Yn := Y 1{Y ≤n} . Then, Yn converges to Y strongly in L2 (FT ). The above arguments show that E[X + Yn ] − E[X] ≥ E −µ [Yn ]. Then using Lemma 2.3, the desired result (21) follows by passing to the limit n → ∞ in the last inequality. Remark 2.3. The above proof of Lemma 2.4 is more complicated than that of Ref. 3 since both assumptions (H1)’ and (H2)’ are weaker.

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31 ζ For ζ ∈ L2+ (FT ), define the operator Es,t [·] : L2+ (Ft ) → L2 (Fs ) by ζ Es,t [X] := Es,t [X + ζ] − Es,t [ζ],

∀X ∈ L2+ (FT ).

(23)

Identically as in the case of an {Ft , 0 ≤ t ≤ T }-expectation (see Ref. 3 for detailed proof), we can show the following lemma. Lemma 2.5. Let ζ ∈ L2+ (FT ). If {Es,t [·], 0 ≤ s ≤ t ≤ T } is an {Ft , 0 ≤ t ≤ T }consistent dynamic operator defined on L2+ (FT ) and satisfies (H1)’ and (H2)’, then ζ the operator {Es,t [·], 0 ≤ s ≤ t ≤ T } is also an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator defined on L2+ (FT ), and satisfy (H1)’ and (H2)’. The expectation E ζ [X|Ft ] of X ∈ L2+ (FT ) conditioned on Ft is given by the formula: E ζ [X|Ft ] = E[X + ζ|Ft ] − E[ζ|Ft ].

(24)

Lemma 2.6. Assume that the two F-consistent dynamic operators E 1 [·] and E 2 [·] defined on L2+ (FT ) satisfy (H1)’ and (H2)’. If E 1 [X] ≤ E 2 [X],

∀X ∈ L2+ (FT ),

then for all t, E 1 [X|Ft ] ≤ E 2 [X|Ft ],

a.s. for all X ∈ L2+ (FT ).

Proof. The proof is divided into the two steps. Step 1. The case of X ≤ n for some positive integer n. Set η = E 2 [X|Ft ] − E 1 [X|Ft ]. Then −η1{η≤0} = E 1 [X1{η≤0} |Ft ] − E 2 [X1{η≤0} |Ft ] ≥ 0 and n ≤ E 1 [−η1{η≤0} + n] = E 1 [E 1 [X1{η≤0} |Ft ] − E 2 [X1{η≤0} |Ft ] + n] = E 1 [X1{η≤0} − E 2 [X1{η≤0} |Ft ] + n] ≤ E 2 [X1{η≤0} − E 2 [X1{η≤0} |Ft ] + n] = E 2 [E 2 [X1{η≤0} − E 2 [X1{η≤0} |Ft ] + n|Ft ]] = E 2 [E 2 [X1{η≤0} |Ft ] + n − E 2 [X1{η≤0} |Ft ]] = n. Thus, E 1 [−η1{η≤0} + n] = n. From the floor-above strict monotonicity (D1), we have −η1{η≤0} = 0.

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That is, P ({η ≤ 0}) = 0,

a.s.. Thus, E 1 [X|Ft ] ≤ E 2 [X|Ft ],

a.s..

Step 2. The general case of X ∈ L2+ (FT ). For X ∈ L2+ (FT ), define the truncation Xn := X1{X≤n} . Then it follows from Lemma 2.3 that lim E 1 [Xn ] = E 1 [X].

n→∞

From Step 1, we have E 1 [Xn |Ft ] ≤ E 2 [Xn |Ft ],

n = 1, 2, . . . .

Obviously, it is sufficient to prove the following lim E 1 [Xn |Ft ] = E 1 [X|Ft ], ∀X ∈ L2+ (FT )

n→∞

(25)

We now prove it by contradiction. Otherwise, there exists 0 ≤ ≤ 1 and A ∈ Ft such that P (A) > 0 and E 1 [Xn |Ft ]1A ≤ (E 1 [X|Ft ] − )1A . Since E 1 [Xn + 1|Ft ] = E 1 [Xn |Ft ] + 1 and E 1 [X + 1|Ft ] = E 1 [X|Ft ] + 1 (by (H2)’), we have E 1 [Xn + 1|Ft ]1A ≤ (E 1 [X + 1|Ft ] − )1A . Using (D1), we have E 1 [E 1 [Xn + 1|Ft ]1A ] ≤ E 1 [(E 1 [X + 1|Ft ] − )1A ]. Then letting n → ∞, we have lim E 1 [E 1 [Xn + 1|Ft ]1A ] ≤ lim E 1 [(E 1 [X + 1|Ft ] − )1A ]

n→∞

n→∞

< E 1 [(X + 1)1A ].

(26)

While we have by Lemma 2.1 the following lim E 1 [E 1 [Xn + 1|Ft ]1A ] = lim E 1 [(Xn + 1)1A ]

n→∞

n→∞

= E 1 [(X + 1)1A ].

(27)

This contradicts (26). Therefore, (25) is true. Combining Lemmas 2.6 and 2.4, we have Lemma 2.7. Let {Es,t [·], 0 ≤ s ≤ t ≤ T } be an F-consistent dynamic operator defined on L2+ (FT ) and satisfy (H1)’ and (H2)’. Then, for each t ≤ T , we have E −µ [Y |Ft ] ≤ E[X +Y |Ft ]−E[X|Ft ] ≤ E µ [Y |Ft ], a.s. for all X, Y ∈ L2+ (FT ). (28) Remark 2.4. Lemma 2.7 shows that (H1)’ together with (H2)’ and (D1)–(D4) implies (H1), as pointed in the introduction.

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In the following, we extend the underlying {Ft , 0 ≤ t ≤ T }-consistent dynamic operator from the subset L2+ (FT ) of nonnegative square-integrable random variables to the whole space L2 (FT ) of square-integrable random variables. As a preliminary, we have the following fact. Lemma 2.8. Let {Es,t [·], 0 ≤ s ≤ t ≤ T } be an F-consistent dynamic operator defined on L2+ (FT ) and satisfy (H1)’ and (H2)’. Let Xn := X1{X≥−n} and Yn := E[Xn + n|Ft ] − n for X ∈ L2 (FT ) and n = 1, 2, . .p .. Then {Yn }∞ n=1 is a Cauchy 2 sequence in L (Ft ) equipped with the norm || · || = E| · |2 . If X ∈ L2+ (FT ), then Xn = X and Yn = E[X|Ft ] for n = 1, 2, . . .. Proof. For the two positive integers m and n such that m > n, we have Ym − Yn = E[Xm + m|Ft ] − m − (E[Xn + n|Ft ] − n) = E[Xm + m|Ft ] − E[Xn + m|Ft ] = E[Xn + X1{−m≤X≤−n} + m|Ft ] − E[Xn + m|Ft ].

(29)

Thus, from Lemmas 2.7 and 2.1, we have E(Ym − Yn )2 = E(E[Xn + m|Ft ] − E[Xn + X1{−m≤X≤−n} + m|Ft ])2 ≤ E(E µ [−X1{−m≤X≤−n} |Ft ])2 ≤ eµ

2

(T −t)

E[X 2 1{−m≤X≤−n} ].

(30)

Since X ∈ L2 (FT ), we have lim E[X 2 1{−m≤X≤−n} ] = 0.

m,n→∞

Therefore, lim E(Ym − Yn )2 = 0.

n,m→∞

b Lemma 2.8 shows that E[X|F t ] introduced below is well defined for any X ∈ L (FT ). 2

Definition 2.2. For X ∈ L2 (FT ), denote Xn := X1{X≥−n} .

(31)

We introduce a dynamic operator {Ebs,t [X], X ∈ L2 (FT ); 0 ≤ s ≤ t ≤ T } by and

b n |Ft ] := E[Xn + n|Ft ] − n, E[X

b b E[X|F t ] := lim E[Xn |Ft ] n→∞

for any X ∈ L2 (FT ).

n = 1, 2, . . . ;

(32)

(33)

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b Remark 2.5. It is obvious that if X ∈ L2+ (FT ), we have E[X|F t ] = E[X|Ft ] a.s. for t ∈ [0, T ]. b t ]. We have the following continuity of the extended operator E[·|F

Theorem 2.2. Let {Es,t [·], 0 ≤ s ≤ t ≤ T } be an F-consistent dynamic operator defined on L2+ (FT ) and satisfy (H1)’ and (H2)’. For each t ∈ [0, T ], the conditional b t ] is strongly continuous from L2+ (FT ) to L2 (Ft ). That expectation operator E[·|F is, if lim Yn = Y, strongly in L2 (FT ),

n→∞

then b n |Ft ] = E[Y b |Ft ], lim E[Y

strongly in L2 (Ft ).

n→∞

(34)

Proof. For the two positive integers m and n such that m > n, define b n |Ft ] − E[Yn 1{Y ≥−m} + m|Ft ] − m δ1 := E[Y n

and

δ2 := (E[Yn 1{Yn ≥−m} + m|Ft ] − m) − (E[Y 1{Y ≥−m} + m|Ft ] − m). Using Lemma 2.7, we have |δ2 | ≤ |E[Y 1{Y ≥−m} + m + |Yn 1{Yn ≥−m} − Y 1{Y ≥−m} ||Ft ] −E[Y 1{Y ≥−m} + m|Ft ]| ≤ E µ [|Yn 1{Yn ≥−m} − Y 1{Y ≥−m} ||Ft ] ≤ E µ [|Yn 1{Yn ≥−m} − Yn ||Ft ] + E µ [|Yn − Y ||Ft ] + E µ [|Y − Y 1{Y ≥−m} ||Ft ]. Thus, using Lemma 2.1, we have E|δ2 |2 ≤ 3EE µ [|Yn 1{Yn ≥−m} − Yn ||Ft ]2 + 3EE µ [|Yn − Y ||Ft ]2 +3EE µ [|Y − Y 1{Y ≥−m} ||Ft ]2 ≤ 3eµ

2

(T −t)

+3eµ ≤ 3eµ

2

2

T

E|Yn 1{Yn ≥−m} − Yn |2

(T −t)

(E|Yn − Y |2 + E|Y − Y 1{Y ≥−m} |2 )

E|Yn 1{Yn <−m} |2 + 3eµ

2

T

(E|Yn − Y |2 + E|Y 1{Y <−m} |2 ).

Letting m → ∞ in the following b n |Ft ] − E[Y b |Ft ]|2 E|E[Y

b |Ft ]|2 ) ≤ 3(E|δ1 |2 + E|δ2 |2 + E|E[Y 1{Y ≥−m} + m|Ft ] − m − E[Y ≤ C1 Eδ12 + E|Yn 1{Yn <−m} |2 + E|Yn − Y |2 + E|Y 1{Y <−m} |2 2 b +E|E[Y 1{Y ≥−m} + m|Ft ] − m − E[Y |Ft ]| ,

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we have

Therefore,

b n |Ft ] − E[Y b |Ft ]|2 ≤ C1 E|Yn − Y |2 . E|E[Y b n |Ft ] = E[Y b |Ft ]. lim E[Y

n→∞

Now consider BSDE (9), which can be viewed as the extremal case of the negative infinite floor (that is, S = −∞) for RBSDE (1). Under conditions (C1)–(C3), it has g a unique adapted solution (Y, Z). In the following, denote Yt by Et,T [ξ] to emphasize the dependence on the parameter (ξ, g) and the initial and terminal times pair (t, T ). Coquet et al. 3 introduces the following notion of an {Ft , 0 ≤ t ≤ T }-consistent expectation. Definition 2.3. An {Ft , 0 ≤ t ≤ T }-consistent expectation is defined to be a nonlinear functional E[·] on L2 (FT ) which satisfies the following three axioms (E1)– (E3). (E1) E[Y1 ] ≤ E[Y2 ] if Y1 , Y2 ∈ L2 (FT ) and Y1 ≤ Y2 . Furthermore, if Y1 ≤ Y2 , then Y1 = Y2 if E[Y1 ] = E[Y2 ]. e =C e for any constant C. e And (E2) E[C] (E3) The E-expectation (denoted by E[Y |Ft ]) of Y ∈ L2 (FT ) conditioned on Ft exists uniquely for any t ∈ [0, T ]. That is, there is unique η ∈ L2 (Ft ) such that E[η1A ] = E[Y 1A ],

∀A ∈ Ft .

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The third axiom (E3) shows that an {Ft , 0 ≤ t ≤ T }-consistent expectation E[·] induces an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator, which will be denoted by {Es,t [·], 0 ≤ s ≤ t ≤ T }. We shall identify them each other. Coquet et al. 3 proved the following representation result. Theorem 2.3. Assume that an {Ft , 0 ≤ t ≤ T }-consistent expectation {Es,t , 0 ≤ s ≤ t ≤ T } satisfies (H1)’ for some positive constant µ and the following assumption: (H2)’ Translation invariance. Es,t [X + Y ] = Es,t [X] + Y a.s. for s ≤ t, X ∈ L2 (Ft ) and Y ∈ L2 (Fs ). Then, there is a random field g : Ω × [0, T ] × Rd → R such that (i) g(t, 0) = 0 for a.e. t ∈ [0, T ] and g(·, z) ∈ L2F (0, T ) for z ∈ Rd , (ii) |g(t, z1 ) − g(t, z2 )| ≤ µ|z1 − z2 | for z1 , z2 ∈ Rd and a.e. t ∈ [0, T ], and g (iii) E0,T [Y ] = E0,T [Y ] for any Y ∈ L2 (FT ). Note that an {Ft , 0 ≤ t ≤ T }-consistent expectation {Es,t , 0 ≤ s ≤ t ≤ T } has the following four properties. (A1) Strict Monotonicity. Es,t [Y ] ≥ Es,t [Y 0 ] if Y, Y 0 ∈ L2 (Ft ) and Y ≥ Y 0 . Furthermore, if Y 0 ≥ Y and Es,t [Y 0 ] = Es,t [Y ], then Y 0 = Y . (A2) Es,t [Y ] = Y a.s. for s ∈ [0, t] and Y ∈ L2 (Fs ).

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(A3) Time consistency. Er,s [Es,t [Y ]] = Er,t [Y ] if r ≤ s ≤ t ≤ T and Y ∈ L2 (Ft ). (A4) Zero-One law. Es,t [1A Y ] = 1A Es,t [Y ] a.s. for s ≤ t, A ∈ Fs , and Y ∈ L2 (Ft ). Theorem 2.3 means that an {Ft , 0 ≤ t ≤ T }-consistent expectation always has a BSDE representation if it is both E µ -dominated for some µ > 0 and constantpreserving. Note that (H1) is stronger than (H1)’ and however, the latter together with (H2)’ and (A1)-(A4) implies the former (see Remark 2.4 for further details). Further, (H2)’ is stronger than (H2). Therefore, Theorem 2.3 is the special case of S ≡ −∞ for Theorem 1.1. The above extended dynamic operator Eb can be shown to be identified to an b {Ft , 0 ≤ t ≤ T }-consistent expectation E. Theorem 2.4. Let {Es,t [·], 0 ≤ s ≤ t ≤ T } be an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator defined on L2+ (FT ) and satisfy (H1)’ and (H2)’. Then, the dynamic b t ] defined by (33) satisfies (H1)’, (H2)’, (D1), (D2)’, (D3), and (D4)’. operator E[·|F Therefore, it is an {Ft , 0 ≤ t ≤ T }-consistent expectation. Proof. For X ∈ L2 (FT ) and Y ∈ L2 (FT ), define Xn := X1{X≥−n} , Yn = Y 1{Y ≥−n} . It is easy to check all the properties of Monotonicity (D1), Lemma 17, (D2)’, (D3), (D4)’, (H2)’ and (H1)’ for Xn and Yn . From Theorem 2.2, let n → ∞, monotonicity (D1), Lemma 17, (D2)’, (D3), (D4)’, (H2)’, and (H1)’ then follow immediately. b |Ft ] = E[X|F b If Y ≥ X and E[Y t ], then it follows from Lemma 17 that Thus,

−µ b |Ft ] − E[X|F b 0 = E[Y [Y − X|Ft ] ≥ 0. t] ≥ E

E −µ [Y − X|Ft ] = 0. Therefore, Y = X. This shows the strict monotonicity. The generator g of the BSDE representation of the {Ft , 0 ≤ t ≤ T }-consistent b given by Theorem 2.3 turns out to be that of the desired RBSDE. expectation E[·] Theorem 2.5. Theorem 1.1 is true in the case of the zero floor S ≡ 0. Moreover, the assumption (H1) may be weakened to (H1)’.

b Proof. Theorem 2.4 shows that E[X], X ∈ L2 (FT ), defined by (33) is an {Ft , 0 ≤ t ≤ T }-expectation, and satisfy (H1)’ and (H2)’. From Theorem 2.3, there exists a function g : Ω × [0, T ] × Rd → R, satisfying (C2) and (C3) and the folb lowing properties: g(0, z) = 0, |g(t, z)| ≤ µ|z| for a.e. t ∈ [0, T ], and E[X] = E g [X] 2 for any X ∈ L (FT ).

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In version of Remark 2.5 and Lemma 2.1, we have b E[X] = E[X] = E g [X] = E r,g [X] for anyX ∈ L2+ (FT ).

Hence, for any A ∈ Ft and X ∈ L2+ (FT ), we have

E[X1A ] = E r,g [X1A ].

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This shows that E[X|Ft ] = E r,g [X|Ft ] for any X ∈ L2+ (FT ). That is, Et,T [X] = r,g Et,T [X] for any X ∈ L2+ (FT ). 3. The Case of the Negative Floor Consider the dynamic nonlinear operator with a negative floor S ≤ 0. In what follows, we discuss the properties of the solution Y to RBSDE (1) from a view point of operators. Suppose that the generator g satisfies conditions (C2) and (C3). Definition 3.1. Define, for 0 ≤ s ≤ t < ∞ and ξ ∈ L2 (Ft ; St ), r;g,S Es,t [ξ] := ys ,

(37)

where (y, z, K) is the solution of RBSDE (1) with the parameter (ξ, g, S). Theorem 3.1. Let the random field g satisfy (C2) and (C3). Moreover, assume that g(t, y, z) does not depend on the variable y and that g(·, 0) = 0. Then if r;g,S St ≤ 0 a.s. for t ∈ [0, T ], the dynamic operator {Es,t , 0 ≤ s ≤ t ≤ T } satisfies Axioms (H1), (H2), and (D1)–(D4). Proof. Let us prove (H1) first. Consider ξi ∈ L2 (Ft ) for i = 1, 2 such that ξ1 ≥ ξ2 ≥ ST . Let {(Ys1 , Zs1 , Ks1 ), 0 ≤ s ≤ t} and {(Ys2 , Zs2 , Ks2 ), 0 ≤ s ≤ t} be the adapted solutions of RBSDE (1) with the values ξ = ξ1 and ξ = ξ2 at the terminal time t, respectively. Denote by {(ys , zs ), 0 ≤ s ≤ t} the unique adapted solution of BSDE (4) with the value ξ1 − ξ2 at the terminal time t. Note that the terminal time is takes to be t, that is T = t in RBSDE (1) and BSDE (4). Applying Itˆ o’s formula to |(Ys1 − Ys2 − ys )+ |2 , and taking the expectation, we have: Z t 2 + 2 1 E|(Ys − Ys − ys ) | + E 1{Yr1 −Yr2 ≥yr } |Zr1 − Zr2 − zr |2 dr s Z t ≤ 2E (Yr1 − Yr2 − yr )+ [g(r, Zr1 ) − g(r, Zr2 ) − µ|zr |]dr s Z t + 2E (Yr1 − Yr2 − yr )+ (dKr1 − dKr2 ). s

Since

Yr1 Z

−

Yr2

− yr ≤ Yr1 − Sr , we have

t s

(Yr1

−

Yr2

− yr )

+

(dKr1

−

dKr2 )

=−

Z

s

t

(Yr1 − Yr2 − yr )+ dKr2 ≤ 0.

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It follows from the conditions (C2) and (C3) that Z t E|(Ys1 − Ys2 − ys )+ |2 + E 1{Yr1 −Yr2 ≥yr } |Zr1 − Zr2 − zr |2 dr s Z t ≤ 2E (Yr1 − Yr2 − yr )+ µ[Zr1 − Zr2 − zr |]dr s Z t Z t 1 2 2 ≤E 1{Yr1 −Yr2 ≥yr } |Zr − Zr − zr | dr + E µ2 [Zr1 − Zr2 − zr |]2 ds. s

s

Hence E|(Ys1

−

Ys2

+ 2

− ys ) | ≤ E

Z

t s

µ2 [Zr1 − Zr2 − zr |]2 ds,

and from Gronwall’s lemma, (Ys1 − Ys2 − ys )+ = 0, 0 ≤ s ≤ t. Therefore, r;g,S r;g,S µ Es,t [ξ1 ] − Es,t [ξ2 ] ≤ Es,t [ξ1 − ξ2 ],

∀ξ2 ∈ L2 (Ft ; St ) and ξ1 − ξ2 ∈ L2+ (Ft ).

Similarly, we can show the following inequality in (H2): r;g,S r;g,S Es,t [ξ1 ] ≤ Es,t [ξ2 ] + ξ1 − ξ2 ,

∀ξ2 ∈ L2 (Ft ; St ), ξ1 − ξ2 ∈ L2+ (Fs ).

r;g,S If Es,t [ξ2 ] > Ss a.s. for s ∈ [r, t], in version of the following equality Z t r;g,S [Es,t [ξ2 ] − Ss ]dKs2 = 0,

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0

we have Ks2 ≡ K02 = 0, a.s. ∀s ∈ [r, t]. That is K 2 ≡ 0, a.s.. Thus the solution of the RBSDE with the terminal condition ξ2 ≥ ξ1 is actually that of a BSDE. It follows from the strict monotonicity of BSDEs that r;g,S r;g,S ξ2 = ξ1 , a.s. ⇐⇒ ξ2 ≥ ξ1 , a.s. and Er,t [ξ2 ] = Er,t [ξ1 ], a.s..

The proof of (D1)–(D4) can be found in El Karoui and Quenez

8

and Xu

13

.

Let us recall the definitions of E-martingales and E-supermartingales for an {Ft , 0 ≤ t ≤ T }-consistent expectation E, introduced by Coquet et al.. 3 Definition 3.2. Let E[·] be an {Ft , 0 ≤ t ≤ T }-consistent expectation. A squareintegrable process {Mt , 0 ≤ t ≤ T } is called an E-martingale (E-supermartingale, respectively) if for 0 ≤ s ≤ t ≤ T , Ms = E[Mt |Fs ]

(Ms ≥ E[Mt |Fs ], respectively).

The following nonlinear Doob-Meyer’s decomposition theorem, which is Theorem 6.3 in page 20 of Coquet et al., 3 will play a key role in the following arguments. Lemma 3.1. Let E[·] be an {Ft , 0 ≤ t ≤ T }-consistent expectation, and let {Yt , 0 ≤ t ≤ T } be a continuous E-supermartingale such that E[ sup |Yt |2 ] < ∞. t∈[0,T ]

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Then there exists an A ∈ L2F (0, T ) such that A is continuous and increasing with A0 = 0, and such that Y + A is an E-martingale. Let {Es,t [·], 0 ≤ s ≤ t ≤ T } be an F-consistent dynamic operator with floor S ≤ 0 and satisfy (H1) and (H2). Under suitable conditions, we can show that it can be represented as a BSDE reflected upwards on a negative obstacle. For this purpose, consider its restriction on L2+ (FT ) and denote it as + Et,T [X] := Et,T [X],

∀X ∈ L2+ (FT ).

(40)

+ It is straightforward to check that Et,T satisfies conditions (H1)’, (H2)’, (D1), (D2)’, (D3) and (D4). Proceeding identically as in Definition 2.2, we define an {Ft , 0 ≤ t ≤ T }e t ], 0 ≤ t ≤ T } by extending E + from L2+ (FT ) to consistent expectation {E[·|F t,T L2 (FT ).

Lemma 3.2. Let {Es,t [·], 0 ≤ s ≤ t ≤ T } be an F-consistent dynamic operator with the negative floor S ≤ 0 and satisfy (H1) and (H2). For X ∈ L2 (FT ; ST ), (i) we have for t ∈ [0, T ], e E[X|F t ] ≤ Et,T [X].

(41)

Furthermore, if Et,T [X] > St a.s. for any t ∈ [r, T ] with some r ≥ 0, then e E[X|F t ] = Et,T [X]

a.s. for any t ∈ [r, T ].

e (ii) The process {E[X|Ft ], 0 ≤ t ≤ T } is an E-supermartingale.

Proof. First, we prove assertion (i). Set Xn = X1{X≥−n} . From (H2), we have lim Et,T [Xn ] = Et,T [X].

n→∞

e t ] and (H1), we have From the definition of E[·|F

e E[X|F t ] = lim (E[Xn + n|Ft ] − n) n→∞

= lim (Et,T [Xn + n] − n) n→∞

≤ lim Et,T [Xn ] = Et,T [X]. n→∞

(42)

Therefore, we have e E[X|F t ] ≤ Et,T [X].

If Et,T [X] > St a.s. for t ∈ [r, T ], then we have from (D1) that Et,T [Xn ] ≥ Et,T [X] > St a.s. for t ∈ [r, T ]. In view of (H2), we see that the inequality in (42) is actually an equality for t ∈ [r, T ]. Therefore, e E[X|F t ] = Et,T [X] a.s. for t ∈ [r, T ].

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We now prove assertion (ii). Set Xt = Et,T [X] for t ∈ [0, T ]. Then from assertion (i), we have for all 0 ≤ s ≤ t ≤ T e t |Fs ] ≤ Es,t [Xt ] = Xs . E[X

e Therefore, {Et,T [X], 0 ≤ t ≤ T } is an E-supermartingale. Using Lemmas 3.2 and 3.1, we can show the following. Theorem 3.2. Let {Es,t [·], 0 ≤ s ≤ t ≤ T } be an F-consistent dynamic operator + with the negative floor S ≤ 0 and satisfy (H1) and (H2). Let Et,T [·] denote the + 2 e restriction of Et,T [·] on L+ (FT ), and E[·|Ft ] be the extension of E [·] given in t,T

Section 2. Then for X ∈ L2 (FT ; ST ), there exists an {Ft , 0 ≤ t ≤ T }-adapted X continuous increasing process {AX t , 0 ≤ t ≤ T } with A0 = 0 such that e + AX |Ft ] − AX for any t ∈ [0, T ]. Et,T [X] = E[X T t

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Proof. Assertion (ii) of Lemma 3.2 shows that {Et,T [X], 0 ≤ t ≤ T } is an e E-supermartingale. In version of (D2) and (H1), we have St ≤ = ≤ ≤

Et,T [X] Et,T [X − ST + ST ] µ Et,T [ST ] + Et,T [X − ST ] µ Et,T [X − ST ].

Therefore, from the first assertion of Lemma 2.1, we have for some positive constant C1 , E sup |Et,T [X]|2 t∈[0,T ]

µ ≤ E sup Et,T [X − ST ]2 + E sup |St |2 t∈[0,T ]

t∈[0,T ]

µ ≤ 2E sup Et,T [|X|]2 + 2E|ST |2 + E sup |St |2 t∈[0,T ]

t∈[0,T ]

µ ≤ 2E sup Et,T [|X|]2 + 3E sup |St |2 t∈[0,T ]

t∈[0,T ]

≤ C1 E[X 2 ] + E sup |St |2 < ∞. t∈[0,T ]

In view of Lemma 3.1, there exists a continuous increasing {Ft , 0 ≤ t ≤ T }X X adapted process {AX t , 0 ≤ t ≤ T } with A0 = 0 such that {Et,T [X]+At , 0 ≤ t ≤ T } e is an E-martingale. Therefore, e T,T [X] + AX |Ft ] = Et,T [X] + AX , E[E T t

∀t ∈ [0, T ].

While ET,T [X] = X, the proof is then complete. In the following, we introduce three lemmas for subsequent arguments. Identical to the case of an {Ft , 0 ≤ t ≤ T }-consistent expectation, we can show

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Lemma 3.3. Let {Es,t [·], 0 ≤ s ≤ t ≤ T } be an F-consistent dynamic operator with the negative floor S ≤ 0 and satisfy (H1) and (H2). For all 0 ≤ s ≤ t ≤ T , X, Y ∈ L2 (Ft ; St ), and B ∈ Fs , we have Es,t [X1B + Y 1B c ] = Es,t [X]1B + Es,t [Y ]1B c .

(44)

Lemma 3.4. Let {Es,t [·], 0 ≤ s ≤ t ≤ T } be an F-consistent dynamic operator with the negative floor S ≤ 0 and satisfy (H1) and (H2). Let the process {A X t ,0 ≤ t ≤ T} 2 with AX = 0 be given in Theorem 3.2. For a given X ∈ L (F ; S T T ) and some 0 r ∈ [0, t), if Es,T [X](ω) > Ss (ω),

s ∈ [r, t], a.s. ω ∈ B, B ∈ Fr ,

then X AX s 1B = A t 1B ,

a.s. for s ∈ [r, t].

(45)

Proof. For > 0, set Yt = Et,T [X]1B + 1B c . Then Es,t [Yt ] > Ss , s ∈ [r, t]. From Lemma 3.2, Yt satisfies e t |Fs ]. Es,t [Yt ] = E[Y

(46)

Then from Lemma 3.4, we have for ∀s ∈ [r, t],

e t + (AX − AX )1B |Fs ] = E[ e E[X e + AX − AX |Ft ] + (AX − AX )|Fs ]1B + 1B c E[Y t s T t t s X X e e = E[E[X + AT − As |Ft ]|Fs ]1B + 1B c e + AX − AX |Fs ]1B + 1B c = E[X T

s

= Es,T [X]1B + 1B c , = Ys , a.s.

(47)

and e t |Fs ] = Es,t [Yt ] E[Y

= Es,t [Et,T [X]]1B + 1B c = Ys ,

a.s..

(48)

X Since AX t 1B − As 1B ≥ 0 , we have from (47) and (48) that X AX s 1B = A t 1B ,

a.s. for ∀s ∈ [r, t].

Lemma 3.5. Let {Es,t [·], 0 ≤ s ≤ t ≤ T } be an F-consistent dynamic operator with the negative floor S ≤ 0 and satisfy (H1) and (H2). Let the process {A X t ,0 ≤ 2 t ≤ T } with AX = 0 be given in Theorem 3.2. For a given X ∈ L (F T ; ST ), 0 {Et,T [X], 0 ≤ t ≤ T } satisfies the following Z T (Et,T [X] − St )dAX (49) t = 0, a.s. . 0

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Proof. We firstly prove 1{Et,T [X]>St } dAX t = 0 a.s.. Set Bn := {ω : Er,T [X](ω) > Sr (ω), ∀[(t − 1/n) ∨ 0] < r < [(t + 1/n) ∧ T ])}; Cn := ∪{C ∈ F(t−1/n)∨0 : C ⊆ Bn }. Since {Ft , 0 ≤ t ≤ T } is the natural filtration of Bt , augmented by all P -null sets of F, we have ∪∞ n=1 Cn = {Et,T [X] > St }. Lemma 3.4 implies that 1Cn dAX t = 0, a.s. . Therefore, 1{Et,T [X]>St } dAX t =0 For Et,T [X] − St ≥ 0, we have Z T Z (Et,T [X] − St )dAX = t 0

T

a.s. .

(Et,T [X] − St )1{Et,T [X]>St } dAX t = 0.

(50)

0

Theorem 3.3. Theorem 1.1 is true in the case of the negative floor S ≤ 0. Proof. From Theorem 2.3, there exists a function g = g(t, z) : Ω × [0, T ] × Rd satisfying (C2) and (C3) and g(·, ·, 0) ≡ 0, such that the following holds: e |Ft ] = E g [Y |Ft ], ∀Y ∈ L2 (FT ), t ∈ [0, T ]. E[Y

e + AX |Ft ] = E g [X + AX |Ft ] for t ∈ Therefore, for X ∈ L2 (FT ; ST ), we have E[X T T [0, T ]. g X From the definition of E g [X + AX T ] and E [X + AT |Ft ], we know that there is 2 d unique Z ∈ LF (0, T ; R ) such that Z T Z T g X X E [X + AT |Ft ] = X + AT + g(s, Zs )ds − hZs , dBs i, a.s. for any t ∈ [0, T ]. t

Set et := X + X

Z

t

T

g(s, Zs )ds − t

Z

T t

X Zs dBs + AX T − At .

(51)

Then from Lemma 3.2, we have e + AX |Ft ] − AX Et,T [X] = E[X T t

X e (52) = E g [X + AX T |Ft ] − At = Xt . X et = Et,T [X] ≥ St , it follows from Lemma 3.5 that {(X et , Zt , A ), 0 ≤ t ≤ T } Since X t is the solution of RBSDE (X, g, S). That is, et = E r;g,S [X], ∀X ∈ L2 (FT ; ST ). Et,T [X] = X (53) t,T

Remark 3.1. From the proof of Theorem 3.3, {(Et,T [X], Zt , AX t ), 0 ≤ t ≤ T } is the solution of RBSDE (X, g, S). Therefore, the increasing process {AX t ,0 ≤ t ≤ T} is unique for a given X ∈ L2 (FT ; ST ).

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Acknowledgments Both authors thank Professor Shige Peng for his helpful comments. References 1. X. Bao and S. Tang, Representation via Reflected Backward Stochastic Differential Equations of a Filtration-Consistent Nonlinear Dynamic Operator with a Floor, submitted to CRAS. 2. X. Bao and S. Tang, Filtration-consistent dynamic nonlinear evaluation with a general continuous floor and associated reflected backward stochastic differential equations, preprint. 3. F. Coquet, Y. Hu, J. M´emin, and S. Peng, Filtration-consistent nonlinear expectations and related g-expectations, Probability Theory and related Fields, 123, 1–27 (2002). 4. N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7, 1–71 (1997). 5. N. El Karoui, C. Kapoudian, E. Pardoux, S. Peng and M. C. Quenez, Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s, The Annals of Probability, 25, 702–737 (1997). 6. N. El Karoui and M. C. Quenez, Programmation dynamique et ´evalution des actifs contingents en march´es incomplet, C. R. Acad. Sci. Paris, S´er. I, 313, 851–854 (1991). 7. N. El Karoui and M. C. Quenez, Dynamic programming and pricing of contingent claims in incomplete market, SIAM J. Control Optim., 33, 29–66 (1995). 8. N. El Karoui and M. C. Quenez, Nonlinear pricing theory and backward stochastic differential equations, in : Financial Mathematics (ed.: W. J. Runggaldier), Lecture Notes in Mathematics 1656, Springer Verlag, 191-246 (1996). 9. I. Karatzas and S. Shreve, Methods of Mathematical Finance, World Publishing Corporation, Beijing (2004). 10. S. Peng, Backward SDE and related g-expectation, in: N. El Karoui and L. Mazliak (eds.), Backward Stochastic Differential Equations, Pitman Research Notes in Mathematics 364, 141–159 (1997). 11. S. Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer’s type, Probability Theory and related Fields, 113, 473–499 (1999). 12. S. Peng, Nonlinear expectations, nonlinear evaluations and risk measures, Stochastic methods in finance, Lecture Notes in Math., 1856, Springer, Berlin, 165–253 (2004). 13. M. Xu, Contributions to Reflected Backward Stochastic Differential Equations: theory, numerical analysis and simulations, thesis, Shandong University, 57–67 (2005).

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A LINEAR QUADRATIC OPTIMAL CONTROL PROBLEM FOR STOCHASTIC VOLTERRA INTEGRAL EQUATIONS

SHUPING CHEN Department of Mathematics, Zhejiang University, Hangzhou 310027, CHINA E-mail: [email protected] JIONGMIN YONG Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA, and School of Mathematical Sciences, Fudan University, Shanghai 200433, CHINA E-mail: [email protected]

A linear quadratic optimal control problem is considered for a stochastic Volterra integral equation. As a necessary condition for the optimality, a forward-backward stochastic Volterra integral equation (FBSVIE, for short) is derived, via a duality principle for stochastic integral equations. Keywords: Linear quadratic optimal control problem, forward-backward stochastic integral equations, duality principle. AMS Mathematics subject classification: 60H10.

1. Introduction Let (Ω, F, lF, P) be a complete filtered probability space, on which a d-dimensional ∆ standard Brownian motion W (·) is defined with lF ={Ft }t≥0 being its natural filtration augmented by all the P-null sets. We consider the following controlled linear (forward) stochastic Volterra integral equation (FSVIE, for short): X(t) = ϕ(t) +

Z th 0

+

i A0 (t, s)X(s) + B0 (t, s)u(s) ds

d Z th X i=1

i

Ai (t, s)X(s) + Bi (t, s)u(s) dWi (s),

0

(1.1) t ∈ [0, T ],

where X(·) is the state and u(·) is the control, taking values in Euclidean spaces lRn and lRm of dimensions n and m, respectively, ϕ(·), Ai (· , ·) and Bi (· , ·) are given 44

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deterministic functions of suitable dimensions. The cost functional associated with (1.1) is given by J(u(·)) = E

Z

T 0

h

i h Q(t)X(t), X(t) i +2 h S(t)X(t), u(t) i + h R(t)u(t), u(t) i dt,

(1.2) where Q(·), S(·), and R(·) are matrix-valued deterministic functions of suitable dimensions. Our optimal control problem is to minimize cost functional (1.2) subject to the state equation (1.1) with the control u(·) taken from a space U of admissible controls. Such a problem is referred to as linear quadratic (LQ, for short) problem for a stochastic Volterra integral equation. LQ problem for ordinary differential equations (ODEs, for short) was studied by Bellman–Glicksberg–Gross in 1958 the first time 5 . Kalman 14 and Letov 16 solved LQ problem (for ODEs) in terms of linear feedback control and Riccati equation in 1960. Since then, many authors extend the theory to various situations. See Refs. 2, 27, and 29 for some summaries of LQ theory for ODEs; see Refs. 20, 18, and 17 for LQ theory of infinite-dimensional (deterministic) systems; see Refs. 28, 8–11, for LQ theory of stochastic differential equations. Some detailed historic remarks on LQ theory can be found in Ref. 34. For some recently relevant works, see Refs. 30, 1, 32, and 31. LQ problem for (deterministic) Volterra integral equations was firstly studied in 1967 by Vinokurov 26 . See also Ref. 22. Since then some extensions were developed (see, for example, Refs. 7, 3, 4, 24, 35, and references cited therein). On the other hand, stochastic Voterra integral equations were studied by several authors 6,25,23 , and some interesting applications were indicated in Ref. 13. In this paper, we are going to study the LQ problem for stochastic Volterra integral equations with the state equation of form (1.1) and cost functional (1.2). Due to the nature of our state equation, one could not use Itˆ o’s formula; all the derivations have to be carried out under integral(s). In our discussion, we will extend/modify some results of backward stochastic Volterra integral equations (BSVIEs, for short) developed in Ref. 33 (see also Ref. 19). By a duality principle for linear stochastic integral equations, we derive a forward-backward stochastic Volterra integral equations (FBSVIEs, for short) whose solvability leads to the existence of critical point of the cost functional. Then under some kind of nonnegativity condition on the cost functional, an optimal control will exist. Note that, in our discussion, we do not a priori impose nonnegativity condition on matrix-valued functions Q(·) and R(·). The case of random coefficients and/or a term like h GX(T ), X(T ) i appears in the cost functional will have much more things involved, and we hope to present the relevant results in a forthcoming paper. The rest of the paper is organized as follows. Some preliminary results concerning the state equation will be presented in Section 2. Section 3 is devoted to a discussion on a minimization problem of a quadratic functional in a Hilbert space which will give us some abstract idea of our LQ problem. In Section 4, we present

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some results on linear BSVIEs, among other things, we mainly modify some results from Ref. 33. In Section 5, we introduce a two-point boundary value problem as a necessary condition for optimal control of our LQ problem. This leads to a linear coupled FBSVIE. 2. State Equation and Optimal Control Problem In this section, we present some preliminaries. To begin with, let us introduce some spaces. First of all, we let lRn×m be the set of all (n × m) matrices, and S n be the set of all (n × n) symmetric matrices. For H = lRm , lRm×d , S n , etc., we define Z T n o L2F (0, T ; H) = ϕ : (0, T ) × Ω → H ϕ(·) is lF-adapted, E |ϕ(t)|2 dt < ∞ , 0 n 2 2 CF ([0, T ]; L (Ω; H)) = ϕ(·) ∈ LF (0, T ; H) t 7→ ϕ(t, ·) is continuous o from [0, T ] to L2 (Ω; lRn ), sup E|ϕ(t)|2 < ∞ , t∈[0,T ] n L2F (Ω; C([0, T ]; H)) = ϕ(·) ∈ L2F (0, T ; H) ϕ(·) has continuous paths, h i o E sup |ϕ(t)|2 < ∞ . t∈[0,T ]

∞ ∞ Spaces L∞ F (0, T ; H), CF ([0, T ]; L (Ω; H)), and LF (Ω; C([0, T ]; H)) can be defined in an obvious way. We point out that any process ϕ(·) ∈ CF ([0, T ]; L2(Ω; H)) is continuous as a map from [0, T ] to L2 (Ω; H), and does not necessarily have continuous paths. Next, for any RBanach space Y, we let L2 (0, T ; Y) be the set of all maps ϕ : T [0, T ] → Y such that 0 |ϕ(t)|2 dt < ∞. The spaces L∞ (0, T ; Y) and C([0, T ]; Y) can be defined in a similar way. Note that we may take Y = L2F (0, T ; H), L∞ (0, T ; H), ∞ 2 2 L∞ F (0, T ; H), LF (Ω; L (0, T ; H)), etc. For example, process Z : [0, T ] × Ω → H 2 2 2 belongs to L (0, T ; LF (0, T ; H)) if it is B([0, T ] ) ⊗ FT -measurable; for almost all t ∈ [0, T ], Z(t, ·) is lF-adapted; and Z TZ T E |Z(t, s)|2 dsdt < ∞. 0

0

We now make the following standing assumptions. (H1) Suppose the following hold:  n×n 2 2  )),   A0 (· , ·) ∈ L (0, T ; L (0, T ; lR   n×n 2 ∞ Ai (· , ·) ∈ L (0, T ; L (0, T ; lR )), 1 ≤ i ≤ d, n×m 2 ∞  B0 (· , ·) ∈ L (0, T ; L (0, T ; lR  )),    n×m 2 ∞ Bi (· , ·) ∈ L (0, T ; L (0, T ; lR )), 1 ≤ i ≤ d.

(2.1)

(H2) Suppose the following hold: Q(·) ∈ L∞ (0, T ; S n ),

S(·) ∈ L∞ (0, T ; lRm×n ),

R(·) ∈ L∞ (0, T ; S m ).

(2.2)

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From (1.1), we see that the value of Ai (t, s) and Bi (t, s) with 0 ≤ t < s ≤ T are irrelevant. Hence, in what follows, we make the following convention: Ai (t, s) = 0,

Bi (t, s) = 0,

0 ≤ t < s ≤ T,

0 ≤ i ≤ d.

(2.3)

Next, we let X = L2F (0, T ; lRn ),

U = L2F (0, T ; lRm ),

and define A : X → X and B : U → X as follows:  Z t d Z t  X    Ai (t, s)X(s)dWi (s), A0 (t, s)X(s)ds +   (AX)(t) = 0

0

i=1

(2.4)

Z t d Z t  X     (Bu)(t) = B (t, s)u(s)ds + Bi (t, s)u(s)dWi (s). 0  0

i=1

0

The following result is concerned with the well-posedness of the state equation (1.1). Proposition 2.1. Let (H1) hold. Then operators A : X → X and B : U → X are bounded and A is quasi-nilpotent, i.e., 1

lim kAk k k = 0.

(2.5)

k→∞

Consequently, (I −A)−1 : X → X is bounded, hence, for any ϕ(·) ∈ X and u(·) ∈ U, state equation (1.1) admits a unique solution X(·) ∈ X . Proof. By (2.4), we have that for any X(·) ∈ X , and t ∈ [0, T ], (note (2.3)) n 2 E (AX)(t) ≤ (d + 1)E ≤ (d + 1)E

Z

t 0

nh Z

d Z t 2 X o Ai (t, s)X(s)|2 ds A0 (t, s)X(s)ds +

T

|A0 (t, s)|2 ds + 0

i=1 d X

esssup |Ai (t, s)|2

i=1 s∈[0,T ]

Thus, by (H1), for some L > 0, Z T Z 2 E (AX)(t) dt ≤ LE 0

0

iZ

T

0

o |X(s)|2 ds . (2.6)

T

|X(s)|2 ds.

(2.7)

0

This implies the boundedness of the operator A. The boundedness of B can be proved similarly. From (2.7), we see that for any k ≥ 1, Z t Z tZ s k 2 k−1 2 2 2 E (A X)(t) ≤ L E (A X)(s) ds ≤ L E (Ak−2 X)(r) drds 0 0 0 Z t k Z t L 2 = L2 (t − s)E (Ak−2 X)(s) ds ≤ · · · ≤ (t − s)k E|X(s)|2 ds, t ∈ [0, T ]. k! 0 0

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Hence, k Z T Z t k (A X)(t)|2 dt ≤ L (t − s)k E|X(s)|2 dsdt k! 0 0 0 Z T Lk T k+1 Lk k+1 2 (T − s) E|X(s)| ds ≤ kX(·)k2X . = (k + 1)! 0 (k + 1)!

k(Ak X)(·)k2X = E

Z

T

This implies that kAk k ≤

Lk T k+1 , (k + 1)!

k ≥ 1.

(2.8)

Thus, A is quasi-nilpotent. Using operators A and B, we can write the state equation (1.1) as X = ϕ + AX + Bu.

(2.9)

Since A is quasi-nilpotent, we have the existence and boundedness of (I − A)−1 . Moreover, one has the expansion: (I − A)−1 =

∞ X

Ak .

(2.10)

k=0

Hence, for any ϕ(·) ∈ X and u(·) ∈ U, state equation (1.1) admits a unique solution X(·) ∈ X given by X = (I − A)−1 (ϕ + Bu) ≡

∞ X

Ak (ϕ + Bu).

(2.11)

k=0

This proves our theorem. By the above result, we see that under (H1)–(H2), the cost functional J(u(·)) is well-defined on U. Consequently, we can state our optimal control problem as follows. Problem (LQ). Minimizing J(u(·)) over u(·) ∈ U. The following notions are concerned with Problem (LQ). Definition 2.1. (i) Problem (LQ) is said to be accessible if inf J(u(·)) > −∞.

u(·)∈U

(2.12)

(ii) Problem (LQ) is said to be (uniquely) solvable if there exists a (unique) u ¯(·) ∈ U such that J(¯ u(·)) = inf J(u(·)). u(·)∈U

(2.13)

Any control u ¯(·) ∈ U satisfying (2.13) is called an optimal control, and the cor¯ responding state process X(·) is called an optimal state process. We also call ¯ (X(·), u ¯(·)) an optimal pair.

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Our goals are: (a) Characterize the accessibility and solvability of Problem (LQ), and (b) in the case that Problem (LQ) is solvable, find or characterize optimal controls. 3. Quadratic Functionals in Hilbert Spaces In this section, we briefly look at some properties of quadratic functionals in Hilbert spaces which will be helpful for us to understand Problem (LQ) from a functional analysis point of view. Let H be a Hilbert space and Φ : H → H be a bounded self-adjoint operator. We denote R(Φ) and N (Φ) to be the range and kernel of Φ, respectively. Since Φ is

self-adjoint, N (Φ)⊥ = R(Φ) (and we always have Φ R(Φ) ⊆ R(Φ)). Thus, under

the decomposition H = N (Φ) ⊕ R(Φ), we have the following representation for Φ: ! 00 Φ= (3.1) b , 0Φ

b : R(Φ) → R(Φ) is self-adjoint. Now, we define the pseudo-inverse Φ† by where Φ the following: ! 0 0 † Φ = (3.2) b −1 , 0Φ with domain

D(Φ† ) = N (Φ) + R(Φ) ≡ {u0 + u1 u0 ∈ N (Φ), u1 ∈ R(Φ)} ⊇ R(Φ).

(3.3)

From the above, we can easily seen the following facts:

(i) Φ† is (closed, densely defined, and) self-adjoint; R(Φ) is closed if and only if Φ is bounded. (ii) By the definition of Φ† (see (3.2)), together with (3.3), one has that †

ΦΦ† Φ = Φ,

Φ† ΦΦ† = Φ† ,

(Φ† )† = Φ.

(3.4)

Thus, by (i), R(Φ† ) is closed since Φ is bounded. (iii) Although D(Φ† ) is not necessarily closed, the operator ΦΦ† : D(Φ† ) → H is an orthogonal projection onto R(Φ). Thus, we may naturally extend it, still denoted it by itself, to D(Φ† ) = H. Hence, ΦΦ† : H → R(Φ) ⊆ H is the orthogonal projection onto R(Φ). Note that since Φ is bounded, Φ† Φ is an orthogonal projection from H onto R(Φ† ) = N (Φ† )⊥ = N (Φ)⊥ = R(Φ). Therefore, in fact, we have ΦΦ† = Φ† Φ ≡ PR(Φ) ≡ orthogonal projection onto R(Φ).

(3.5)

(iv) The map Φ 7→ Φ† is not continuous (which can be seen even from onedimensional case).

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Now, let us consider a quadratic functional on H: J(u) = h Φu, u i +2 h v, u i,

u ∈ D(Φ) ⊆ H,

(3.6)

where Φ : D(Φ) ⊆ H → H is a self-adjoint linear operator and v ∈ H. The following result is concerned with the completing square and critical point(s) of the functional J(·). Note here that we do not assume positive (negative) semi-definite condition on Φ. Proposition 3.1. (i) There exists a u ˆ ∈ H such that J(u) = h Φ(u − u ˆ), u − u ˆ i − h Φˆ u, u ˆ i,

∀u ∈ H,

(3.7)

if and only if v ∈ R(Φ) ⊆ D(Φ† ) .

(3.8)

(ii) Any u ˆ ∈ H satisfies (3.7) if and only if it is a solution of the following equation: Φˆ u + v = 0,

(3.9)

which is equivalent to the following: u ˆ = −Φ† v + (I − Φ† Φ)e v,

(3.10)

for some ve ∈ H (in particular, u ˆ = −Φ† v is a solution). (iii) When (3.7) holds, it is necessary that J(u) = h Φ(u − u ˆ), u − u ˆ i − h Φ† v, v i,

∀u ∈ H.

(3.11)

Moreover, u ˆ is unique if and only if N (Φ) = {0}. Proof. (i) For any u ˆ ∈ H, on has J(u) ≡ h Φu, u i +2 h v, u i = h Φ(u − u ˆ), u − u ˆ i +2 h Φˆ u + v, u i − h Φˆ u, u ˆ i,

∀u ∈ H.

(3.12)

Hence, there exists a u ˆ ∈ H such that (3.7) holds if and only if (3.9) holds, which gives (3.8) (and the first part of (ii)). Conversely, if (3.8) holds, then there exists a u ˆ ∈ H such that (3.9) holds. Consequently, h Φ(u − u ˆ), u − u ˆ i − h Φˆ u, u ˆ i = h Φu, u i −2 h Φˆ u, u i + h Φˆ u, u ˆ i − h Φˆ u, u ˆi = h Φu, u i +2 h v, u i = J(u),

(3.13)

proving (3.7). (ii) We have proved the first part of (ii) (from (3.12)). The second part is straightforward.

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(iii) For any u ˆ ∈ H satisfying (3.7), one must have (3.9). Hence, h Φ(u − u ˆ), u − u ˆ i − h Φ† v, v i = h Φu, u i −2 h Φˆ u, u i + h Φˆ u, u ˆ i − h Φ† Φˆ u, Φˆ ui = h Φu, u i +2 h v, u i = J(u), (3.14) which proves (3.11). Finally, by (3.9), we see that u ˆ is unique if and only if N (Φ) = {0}. Note that (3.9) is equivalent to the following: 0 = Φˆ u+v ≡

1 ∇J(ˆ u). 2

(3.15)

Thus, u ˆ is actually a critical point of functional J(·). Hence, Proposition 3.1 characterizes critical points of the quadratic functional J(·). Equations (3.7) and (3.11) are completion of square for the functional J(·) (although Φ is not necessarily positive/negative semi-definite). Next, for any self-adjoint operator Φ, regardless whether it is bounded or unbounded, we have the following spectrum decomposition 12 Z Φ= λdPλ , (3.16) σ(Φ)

where σ(Φ) ⊆ lR is the spectrum of Φ, (which is a compact set if Φ is bounded, and it is unbounded if Φ is unbounded); and {Pλ λ ∈ σ(Φ)} is a family of projection measures. In the case that Φ ≥ 0,

(3.17)

one has from (3.16) that σ(Φ) ⊆ [0, ∞), and  Z    Φα = λα dPλ , ∀α ≥ 0,  σ(Φ) Z   † α α †  λ−α dPλ ,  (Φ ) = (Φ ) =

(3.18) ∀α > 0.

σ(Φ)\{0}

Now, we can consider minimization problem for functional J(·). Proposition 3.2. Let Φ : H → H be bonded and self-adjoint, and v ∈ H. (i) The following holds: inf J(u) > −∞,

u∈H

(3.19)

if and only if (3.17) holds and 1

v ∈ R(Φ 2 ).

(3.20)

In this case, 1

inf J(u) = −|(Φ† ) 2 v|2 .

u∈H

(3.21)

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(ii) There exists a u ˆ ∈ H such that J(ˆ u) = inf J(u),

(3.22)

u∈H

if and only if (3.17) and (3.8) hold; and in this case, all the conclusions in Proposition 3.1 hold. Proof. We need only to prove (i). First, let (3.19) hold. It is straightforward that one must have (3.17). Next, we prove (3.20) by contradiction. Suppose (3.20) does not hold. For any n ≥ 1, let Z dPλ v. vn = 1 ,n] σ(Φ)∩[ n

Then vn ∈ R(Φ), and Z † h v, Φ vn i =

1

λd|Pλ v|2 = |(Φ† ) 2 vn |2 → ∞,

1 σ(Φ)∩[ n ,n]

n → ∞.

Hence, letting un = −Φ† vn , we obtain 1

J(un ) = h Φun , un i +2 h v, un i = −|(Φ† ) 2 vn |2 → −∞,

n → ∞,

contradicting (3.19). Conversely, if (3.17) and (3.20) hold, then for any u ∈ H, one has 1

1

1

1

1

1

1

J(u) = |Φ 2 u|2 +2 h(Φ† ) 2 v, Φ 2 u i = |Φ 2 u+(Φ† ) 2 v|2 −|(Φ† ) 2 v|2 ≥ −|(Φ† ) 2 v|2 > −∞. (3.23) Hence, sufficiency follows. Finally, from the fact that 1

1

R((Φ† ) 2 ) ⊆ R(Φ 2 ) = R(Φ), we can always find a sequence un ∈ H so that (note (3.23)) 1

1

1

1

J(un ) = |Φ 2 un + (Φ† ) 2 v|2 − |(Φ† ) 2 v|2 → −|(Φ† ) 2 v|2 ,

n → ∞.

Thus, (3.21) follows. The above result tells us that the existence of minimum is strictly stronger than the finiteness of the infimum of the functional J(·), which have been described by 1 conditions (3.8) and (3.20), respectively. Note here that R(Φ) ⊆ R(Φ 2 ) when (3.17) holds. The following example shows the necessity of condition (3.20) in a concrete way. Example 3.1. Let H = `2 . For any u = {ai }∞ i=1 ∈ H, define Φu by Φu = {β i−1 ai }∞ i=1 ,

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where β ∈ (0, 1). Then Φ : H → H is bounded, self-adjoint, and positive definite (but not uniformly). Let v = {i−1 }∞ lim Φun i=1 ∈ H1 . Then v ∈ R(Φ) since v = n→∞ with 1 1 1 , ,··· , , 0, 0, · · · } ∈ H. un = {1, 2β 3β 2 nβ n−1 But, clearly, v ∈ / R(Φ). Now, consider a quadratic functional ∞ X 2ai J(u) = h Φu, u i +2 h v, u i = β i−1 a2i + . i i=1 Then by letting un as above, we see that J(−un ) = h Φun , un i −2 h v, un i n h n X X 2 i 1 1 = β i−1 2 2(i−1) − 2 i−1 = − → −∞, 2 β (i−1) i β i β i i=1 i=1

as n → ∞.

This means that inf J(u) = −∞.

u∈H

An interesting point here is that positive semi-definiteness of Φ does not even ensure the finiteness of the infimum of J(·). Now, we return to Problem (LQ). Denote ( QX = Q(·)X(·), SX = S(·)X(·), Ru = R(·)u(·),

∀u(·) ∈ U.

∀X(·) ∈ X ,

(3.24)

Then under (H1)–(H2) and with ϕ(·) ∈ L2F (0, T ; lRn ), the cost functional (1.2) can be written as follows: J(u) = h QX, X i +2 h SX, u i + h Ru, u i Q S∗ (I − A)−1 (ϕ + Bu) (I − A)−1 (ϕ + Bu) =h , i S R u u Q S∗ (I − A)−1 (I − A)−1 B ϕ (I − A)−1 (I − A)−1 B ϕ =h , i S R 0 I u 0 I u = h Φ2 u, u i +2 h Φ1 ϕ, u i + h Φ0 ϕ, ϕ i, (3.25) where   Φ = (I − A∗ )−1 Q(I − A)−1 ,   0 n o Φ1 = B ∗ (I − A∗ )−1 Q + S (I − A)−1 ,    Φ2 = B ∗ (I − A∗ )−1 Q(I − A)−1 B + S(I − A)−1 B + B ∗ (I − A∗ )−1 S ∗ + R. (3.26) Consequently, by Propositions 3.1 and 3.2, we obtain the following abstract result.

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Proposition 3.3. (i) If Problem (LQ) admits an optimal control, then Φ2 ≥ 0.

(3.27)

(ii) When (3.27) holds, Problem (LQ) admits an optimal control if and only if Φ1 ϕ ∈ R(Φ2 ).

(3.28)

In this case, u ¯ is an optimal control if and only if it is a solution of the equation: Φ2 u ¯ + Φ1 ϕ = 0,

(3.29)

u ¯ = −Φ†2 Φ1 ϕ + (I − Φ†2 Φ2 )v,

(3.30)

which is given by

for some v ∈ U, with the optimal cost functional J(¯ u(·)) = h(Φ0 − Φ∗1 Φ†2 Φ1 )ϕ, ϕ i .

(3.31)

Further, if Φ2 is invertible, the optimal control is unique. Although the above gives necessary and sufficient conditions under which Problem (LQ) is (uniquely) solvable, the conditions imposed a little too abstract, for example, it is by no means obvious how one can represent A∗ and B ∗ . Therefore, we need to make further efforts in our investigation.

4. BSVIEs and Duality Principle In this section, we recall/modify some results on BSVIEs from Ref. 33, which will be useful in studying Problem (LQ). To begin with, let us consider the following BSVIE introduced in Ref. 33: Z T Z T Y (t) = ψ(t) + g(t, s, Y (s), Z(s, t))ds − Z(t, s)dW (s), t ∈ [0, T ], (4.1) t

t

n

n×d

where g : [0, T ]2 × lR × lR the following definition.

n

→ lR is a given map. According to Ref. 33, we have

Definition 4.1. A pair of process (Y (·), Z(· , ·)) ∈ L2F (0, T ; lRn ) × L2 (0, T ; L2F (0, T ; lRn×d )) is called an adapted solution of (4.1) if for almost all t ∈ [0, T ], almost surely, (4.1) is satisfied in the Itˆ o sense. In Ref. 33, it was proved that under the condition that (y, z) 7→ g(t, s, y, z) is uniformly Lipschitz, there exists a unique adapted solution (Y (·), Z(· , ·)) to (4.1). Based on this, some further results were established (see Ref. 33).

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Before going further, let us look at a simple situation. For any Y (·) ∈ L2F (0, T ; lRn ) and Z(· , ·) ∈ L2 (0, T ; L2F (0, T ; lRn )), we call Z T ψ(t) = Y (t) + Z(t, s)dW (s), t ∈ [0, T ]. (4.2) t

2

Then ψ(·) ∈ L ((0, T ) × Ω), and ψ(·) is not necessary lF-adapted. (Y (·), Z(· , ·)) is an adapted solution of the following BSVIE: Z T Y (t) = ψ(t) − Z(t, s)dW (s), t ∈ [0, T ].

Clearly,

(4.3)

t

On the other hand, for ψ(·), by martingale representation theorem, there exists a b , ·) ∈ L2 (0, T ; L2 (0, T ; lRn )) such that unique Z(· F Z T b s)dW (s), ψ(t) = Eψ(t) + Z(t, t ∈ [0, T ]. (4.4) 0

If we define

∆ Yb (t) = Eψ(t) +

Z

t 0

b s)dW (s), Z(t,

t ∈ [0, T ],

b , ·)) is also an adapted solution of (5.14). Hence, we have then (Yb (·), Z(· Z Th i b s) − Z(t, s) dW (s), b Z(t, t ∈ [0, T ]. Y (t) − Y (t) =

(4.5)

(4.6)

t

By taking conditional expectation, we see that (note (4.5) and Eψ(t) = EY (t)) Z t b s)dW (s), b Z(t, t ∈ [0, T ]. (4.7) Y (t) = Y (t) = EY (t) + 0

Hence, we must have

b s), Z(t, s) = Z(t,

s ∈ [t, T ].

(4.8)

In such a sense, the adapted solution to (5.14) is unique. But, in general, (4.8) might not be true for s ∈ [0, t]. Hence, we might not have (comparing with (4.5) and (4.7)) Z t Y (t) = EY (t) + Z(t, s)dW (s), t ∈ [0, T ]. (4.9) 0

To further convince ourselves, let us look at the following example. Example 4.1. Example 4.2. Take d = 1, Y (t) = W (t), and Z(t, s) ≡ 2. Then Z T ψ(t) = Y (t) + Z(t, s)dW (s) = 2W (T ) − W (t), t ∈ [0, T ], t

b s) = 2 − and (trivially) (5.14) holds. On the other hand, (4.5) holds with Z(t, I[0,t] (s). Clearly, (4.8) holds, but b s), Z(t, s) 6= Z(t,

s ∈ [0, t],

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and Y (t) = W (t) 6= 2W (T ) − W (t) = EY (t) +

Z

T

Z(t, s)dW (s),

t ∈ [0, T ]. (4.10)

t

The above shows the following: (i) The uniqueness of adapted solution (Y (·), Z(· , ·)) to BSVIE (4.1) does not give the uniqueness of Z(t, s) for s ∈ [0, t], and (ii) in general, we do not have (4.9) for any adapted solution (Y (·), Z(· , ·)). However, from Ref. 33, we know that relation (4.9) played an important role in establishing duality relation. Hence, the following notion will be useful. Definition 4.2. A pair of process (Y (·), Z(· , ·)) ∈ L2F (0, T ; lRn ) × L2 (0, T ; L2F (0, T ; lRn×d )) is called an adapted M-solution of (4.1) if (4.1) is satisfied in the Itˆ o sense, and (4.9) holds. In the above, “M” in “M-solution” stands for “Martingale Representation”. Now, we state the following result concerning the existence and uniqueness of adapted M-solutions of BSVIE (4.1). Proposition 4.1. Suppose that (y, z) 7→ g(t, s, y, z) is uniformly Lipschitz continuous. Then for any ψ(·) ∈ L2 ((0, T ) × Ω), (4.1) admits a unique adapted M-solution (Y (·), Z(· , ·)). Proof. Let M be the set of all processes (y(·), z(· , ·)) ∈ L2F (0, T ; lRn ) × L2 (0, T ; L2F (0, T ; lRn×d )) satisfying Z t y(t) = Ey(t) + z(t, s)dW (s), t ∈ [0, T ]. 0

Then M is a (nontrivial) closed subspace of L2F (0, T ; lRn )×L2 (0, T ; L2F (0, T ; lRn×d )). Now, for any (y(·), z(· , ·)) ∈ M, let Z T ˆ = ψ(t) + ψ(t) g(t, s, y(s), z(s, t))ds, t ∈ [0, T ]. t

By martingale representation theorem, there exists a unique Z(· , ·) L2 (0, T ; L2F (0, T ; lRn×d )) such that Z T ˆ = E ψ(t) ˆ + ψ(t) Z(t, s)dW (s), t ∈ [0, T ]. 0

Next, we define ˆ + Y (t) = E ψ(t)

Z

t

Z(t, s)dW (s),

Then (Y (·), Z(· , ·)) ∈ M and Z T Z Y (t) = ψ(t) + g(t, s, y(s), z(s, t))ds − t

t ∈ [0, T ].

0

T

Z(t, s)dW (s), t

t ∈ [0, T ].

∈

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Hence, we obtain a map Φ : M → M, (y(·), z(· , ·)) 7→ (Y (·), Z(· , ·)). Then by a contraction mapping argument similar to that in Ref. 33, we can show that Φ admits a unique fixed point, which is the adapted M-solution that we are looking for. Note that unlike adapted solution, the uniqueness of adapted M-solution really gives the uniqueness of Z(· , ·) over [0, T ] × [0, T ], due to the additional requirement (4.9). In Example 4.1, we see that although (Y (·), Z(· , ·)) is merely an adapted solution b , ·)) is the adapted M-solution and (4.9) fails, we can modify Z(· , ·) so that (Y (·), Z(· (therefore (4.7) holds). Sometime, we might need some other restriction other than (4.9). Inspired by the above, we have the following result. Proposition 4.2. Let (y, z) 7→ g(t, s, y, z) be uniformly Lipschitz continuous. Then for any ψ(·) ∈ L2 ((0, T )×Ω) and ϕ(·) ∈ L2F (0, T ; lRn ), (4.1) admits a unique adapted solution (Y (·), Z(· , ·)) satisfying Z t Z(t, s)dW (s), t ∈ [0, T ]. (4.11) ϕ(t) = Eϕ(t) + 0

Proof. First of all, by martingale representation theorem, we can find ζ(· , ·) ∈ L2 (0, T ; L2F (0, T ; lRn )) such that Z t ϕ(t) = Eϕ(t) + ζ(t, s)dW (s), t ∈ [0, T ]. (4.12) 0

Note that since ϕ(·) is lF-adapted, ζ(t, s) = 0 for s ∈ [t, T ]. Next, consider the following BSVIE: Z T Z T Y (t) = ψ(t) + g(t, s, Y (s), ζ(s, t))ds − Z(t, s)dW (s), t ∈ [0, T ]. (4.13) t

t

By Proposition 4.1, the above admits a unique adapted solution (Y (·), Z(· , ·)). Note that in (4.13), only the values Z(t, s) with 0 ≤ s ≤ t ≤ T are used and by changing the values Z(t, s) for 0 ≤ s ≤ t ≤ T only, (4.13) remains unchanged. On the other hand, in the drift term, only ζ(t, s) with 0 ≤ s ≤ t ≤ T are used. Hence, it we redefine Z(t, s) = ζ(t, s),

0 ≤ s ≤ t ≤ T,

then (Y (·), Z(· , ·)) is still an adapted solution of (4.1) and (4.11) holds. Note of course that the adapted solution (Y (·), Z(· , ·)) obtained in Proposition 4.2 is not an adapted M-solution. Actually, (4.9) is replaced by (4.11). Also, we see that in the proof of Proposition 4.2, we have taken the advantage that the drift does not depending on Z(t, s) (only depending on Z(s, t)). In the case that the drift also depends on Z(t, s), we will still have a unique adapted M-solution. But, it is not clear if Proposition 4.2 holds.

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Now, we strengthen the standing assumptions (H1) for the coefficients of the state equation as follows. (H3) Suppose the following hold: Ai (· , ·) ∈ L∞ ((0, T )2 ; lRn×n ),

Bi (· , ·) ∈ L∞ ((0, T )2 ; lRn×m ),

0 ≤ i ≤ d. (4.14)

The following result gives representations for A∗ and B ∗ in terms of BSVIEs. Theorem 4.1. Suppose (H3) holds. Let A and B be defined by (2.4). Then for any ψ(·) ∈ L2F (0, T ; lRn ) and v(·) ∈ L2F (0, T ; lRm ), (A∗ ψ)(t) = η(t),

t ∈ [0, T ], a.s. ,

(4.15)

(B ∗ v)(t) = λ(t),

t ∈ [0, T ], a.s. ,

(4.16)

and

where (η(·), ζ(· , ·)) is the unique adapted M-solution to the following BSVIE:

η(t) =

Z

T t

h

A0 (s, t)T ψ(s)+

d X i=1

Z i Ai (s, t)T ζi (s, t) ds−

T

ζ(t, s)dW (s),

t ∈ [0, T ],

t

(4.17)

satisfying ψ(t) = Eψ(t) +

Z

t

ζ(t, s)dW (s),

t ∈ [0, T ],

(4.18)

0

and (λ(·), µ(· , ·)) is the unique adapted M-solution to the following BSVIE:

λ(t) =

Z

T t

h

T

B0 (s, t) v(s)+

d X

T

i

Bi (s, t) µi (s, t) ds−

i=1

Z

T

µ(t, s)dW (s),

t ∈ [0, T ],

t

(4.19)

satisfying v(t) = Ev(t) +

Z

t

µ(t, s)dW (s),

t ∈ [0, T ].

(4.20)

0

Proof. By Proposition 4.2, BSVIE (4.17) admits a unique adapted solution (η(·), ζ(· , ·)) ∈ L2F (0, T ; lRn ) × L2 (0, T ; L2F (0, T ; lRn )) satisfying (4.18). Conse-

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quently, for any X(·) ∈ L2F (0, T ; lRn ), one has E

Z

T

h(A∗ ψ)(t), X(t) i dt ≡ E 0

=E

Z

=E

Z

T 0 T 0

+E =E

Z

T 0

=E

Z

=E

Z

n Z

Z

h 0

T s

h ψ(t), (AX)(t) i dt 0

A0 (t, s)X(s)ds + 0

d Z X i=1

0

Z

0

d Z X

t

ζ(t, s)dW (s), 0

i=1

h A0 (s, t)T ψ(s), X(t) i dsdt +

t

Z

t

o Ai (t, s)X(s)dWi (s) i dt

h A0 (t, s)T ψ(t), X(s) i dtds h Eψ(t) +

T

T

t

T

Z

T

Z

h ψ(t),

Z

t

Ai (t, s)X(s)dWi (s) i dt 0

d X

E

i=1

T t

h

A0 (s, t)T ψ(s) +

T

h η(t) + 0

Z

d X i=1

Z

T 0

Z

t

h ζi (t, s), Ai (t, s)X(s) i dsdt 0

i Ai (s, t)T ζi (s, t) ds, X(t) i dt

T

ζ(t, s)dW (s), X(t) i dt = E t

Z

T

h η(t), X(t) i dt. 0

Since X(·) ∈ L2F (0, T ; lRn ) is arbitrary, and η(·) ∈ L2F (0, T ; lRn ), we obtain (4.15). Representation (4.16) can be proved similarly. Let us now consider the following FSVIE: X(t) = f (t) +

Z

t

A0 (t, s)X(s)ds + 0

d Z X i=1

t

Ai (t, s)X(s)dWi (s),

t ∈ [0, T ], (4.21)

0

with f (·) ∈ L2F (0, T ; lRn ). It is clear that our state equation (1.1) is a special case of the above with Z t d Z t X f (t) = ϕ(t) + B0 (t, s)u(s)ds + Bi (t, s)u(s)dWi (s), t ∈ [0, T ]. (4.22) 0

i=1

0

The following result is called a duality principle. Theorem 4.2. Let Ai (· , ·) (0 ≤ i ≤ d) satisfy (H3) and f (·) ∈ L2F (0, T ; lRn ), g(·) ∈ L2 ((0, T ) × Ω; lRn ). Let X(·) ∈ L2F (0, T ; lRn ) be the solution of FSVIE (4.21), and (Y (·), Z(· , ·)) ∈ L2F (0, T ; lRn )×L2 (0, T ; L2F (0, T ; lRn×d )) be the adapted solution to the following BSVIE: Y (t) = g(t) +

Z

T t

h

A0 (s, t)T Y (s) +

d X i=1

−

i Ai (s, t)T Zi (s, t) ds Z

t

(4.23)

T

Z(t, s)dW (s),

t ∈ [0, T ],

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satisfying Y (t) = EY (t) +

Z

t

Z(t, s)dW (s),

t ∈ [0, T ].

(4.24)

0

Then the following relation holds: Z T Z E h X(t), g(t) i dt = E 0

T

h f (t), Y (t) i dt.

(4.25)

0

We call (4.23)–(4.24) the adjoint equation of FSVIE (4.21), and call (4.25) a duality relation between (4.21) and (4.23)–(4.24). We point out that process g(·) appeared in the above is not required to be lF-adapted. The above result modifies that in Ref. 33. Proof. By Theorem 4.1, we see that (4.23) with (4.24) can be written as Y = g + A∗ Y. Since (I − A)−1 exists and bounded, we have Y = (I − A∗ )−1 g. Hence, h X, g i = h(I − A)−1 f, g i = h f, (I − A∗ )−1 g i = h f, Y i, proving (4.25). 5. A Maximum Principle and FBSVIEs In this section, we would like to seek some alternative conditions for the solvability of Problem (LQ). We begin with a necessary condition for optimality. Suppose ¯ (X(·), u ¯(·)) is an optimal pair of Problem (LQ). Then, by a standard variational technique, we have the following: For any u(·) ∈ U, nZ T h i o ¯ + S(t)T u ¯ + R(t)¯ E h Q(t)X(t) ¯(t), X(t) i + h S(t)X(t) u(t), u(t) i dt = 0, 0

(5.1)

where X(t) =

Z th 0

d Z th i i X A0 (t, s)X(s) + B0 (t, s)u(s) ds + Ai (t, s)X(s) + Bi (t, s)u(s) dWi (s)

≡ f (t) +

Z

t

A0 (t, s)X(s)ds + 0

d Z t X

i=1

0

Ai (t, s)X(s)dWi (s),

t ∈ [0, T ],

0

i=1

(5.2)

with f (t) =

Z

t

B0 (t, s)u(s)ds + 0

d Z X i=1

t

Bi (t, s)u(s)dWi (s), 0

t ∈ [0, T ].

(5.3)

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By the duality principle established in the previous section, we have the following result which can be regarded as a Pontryagin’s maximum principle. ¯ Theorem 5.1. Let (H2)-(H3) hold. Let (X(·), u ¯(·)) be an optimal pair of Problem (LQ). Then there exists adapted solutions (Y (·), Z(· , ·)), (λ(·), µ(· , ·)) to the following BSVIEs: Z

¯ + S(t)T u Y (t) = Q(t)X(t) ¯(t) + −

Z

T t

h

T

A0 (s, t) Y (s) +

d X i=1

i Ai (s, t)T Zi (s, t) ds

T

Z(t, s)dW (s), t

(5.4)

and λ(t) =

Z

h

T t

B0 (s, t)T Y (s) +

d X i=1

Z i Bi (s, t)T Zi (s, t) ds −

T

µ(t, s)dW (s),

(5.5)

t

such that ¯ + R(t)¯ S(t)X(t) u(t) + λ(t) = 0,

a.e. t ∈ [0, T ], a.s.

(5.6)

¯ We see that (1.1) (with (X(·), u(·)) replaced by (X(·), u ¯(·))) together with (5.4)– (5.5) is a system of coupled forward and backward stochastic Volterra integral equations. The coupling is given through (5.6). We call such a system an FBSVIE. By Proposition 4.1, BSVIEs (5.4) and (5.5) admit unique adapted

Proof. solutions

(Y (·), Z(· , ·)) ∈ L2F (0, T ; lRn ) × L2 (0, T ; L2F (0, T ; lRn×d )), (λ(·), µ(· , ·)) ∈ L2F (0, T ; lRm ) × L2 (0, T ; L2F (0, T ; lRm×d )), ¯ + S(·)T u respectively. By Theorem 4.1, with g(·) = Q(·)X(·) ¯(·), and f (·) given by (4.22), we have

E

Z

T 0

=E

Z

¯ + S(t)T u h X(t), Q(t)X(t) ¯(t) i dt T

h 0

=E

Z

=E

Z

T 0

t

B0 (t, s)u(s)ds + 0

Z th

d Z X i=1

h

Z

T t

t

Bi (t, s)u(s)dWi (s), Y (t) i dt 0

h B0 (t, s)T Y (t), u(s) i + h

0

T 0

Z

d X i=1

h

B0 (s, t)T Y (s) +

d X i=1

i Bi (t, s)T Zi (t, s), u(s) i dsdt

i Bi (s, t)T Zi (s, t) ds, u(t) i dt.

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Hence, (5.1) implies Z T ¯ + R(t)¯ 0=E h S(t)X(t) u(t) 0

+ =E =E

Z

Z

T t

h

T

h λ(t) + 0

Z

T

B0 (s, t)T Y (s) + Z

d X i=1

T

i Bi (s, t)T Zi (s, t) ds, u(t) i dt

(5.7)

¯ + R(t)¯ µ(t, s)dW (s) + S(t)X(t) u(t), u(t) i dt

t

¯ + R(t)¯ h λ(t) + S(t)X(t) u(t), u(t) i dt,

∀u(·) ∈ U[0, T ].

0

Consequently, (5.6) follows. Let us look at the above in a little point view, using the results from Section 3. To this end, let us denote X u (·) and X ϕ (·) to be the solution of (1.1) corresponding to (u(·), 0) and (0, ϕ(·)), respectively. Then X(·) = X u (·) + X ϕ (·). Now, by the definition of Φ2 , we have that for any u(·) ∈ U, (Φ2 u)(t) = B ∗ (I − A∗ )−1 [QX u + S T u] + SX u + Ru (t) = (B ∗ Y u )(t) + S(t)X u (t) + R(t)u(t) = λu (t) + S(t)X u (t) + R(t)u(t), (5.8)

with  Z Th d  i X   u u T T u   Y (t) = Q(t)X (t) + S(t) u(t) + A0 (s, t) Y (s) + Ai (s, t)T Ziu (s, t) ds    t  i=1   Z T u − Z (t, s)dW (s),   t    Z Th Z T d  i X   u T u T u  λ (t) = B (s, t) Y (s) + B (s, t) µ (s, t) ds − µu (t, s)dW (s),  0 i i  t

i=1

t

(5.9)

satisfying

 Z t   u u  Y (t) = EY (t) + Z u (t, s)dW (s),  Z 0t   u u   λ (t) = Eλ (t) + µu (t, s)dW (s),

t ∈ [0, T ].

(5.10)

0

Thus, Φ2 ≥ 0 is equivalent to the following: Z T E h λu (t) + S(t)X u (t) + R(t)u(t), u(t) i dt ≥ 0,

∀u(·) ∈ U,

(5.11)

0

On the other hand,

(Φ1 ϕ)(t) = B ∗ (I − A∗ )−1 QX ϕ (t) + S(t)T X ϕ (t).

(5.12)

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Hence, (Φ2 u + Φ1 ϕ)(t) = B ∗ (I − A∗ )−1 [QX + S T u] + SX + Ru (t)

(5.13)

= (B ∗ Y )(t) + S(t)X(t) + R(t)u(t) = λ(t) + S(t)X(t) + R(t)u(t),

with  Z Th d  i X   T T T   Y (t) = Q(t)X(t) + S(t) u(t) + A (s, t) Y (s) + A (s, t) Z (s, t) ds 0 i i    t  i=1   Z T − Z(t, s)dW (s),   t    Z Th Z T d  i X   T T  B0 (s, t) Y (s) + Bi (s, t) µi (s, t) ds − µ(t, s)dW (s),   λ(t) = t

i=1

t

(5.14)

satisfying

 Z t    Z(t, s)dW (s),  Y (t) = EY (t) + 0 Z t     λ(t) = Eλ(t) + µ(t, s)dW (s),

t ∈ [0, T ].

(5.15)

0

Note that the equation for (λ(·), µ(· , ·)) in (5.14) is different from the equation (5.5). But, as we explained in the previous section, we can redefine µ(· , ·) in (5.5) so that it takes the same form as that in (5.14). From the above, we obtain the following. Theorem 5.2. Problem (LQ) is solvable if and only if (5.11) holds and for the given ϕ(·) ∈ L2F (0, T ; lRn ), there exists a u(·) ∈ U ≡ L2F (0, T ; lRm ) such that λ(t) + S(t)X(t) + R(t)u(t) = 0,

t ∈ [0, T ],

(5.16)

with λ(·) given through (5.14)–(5.15). In this case, (X(·), u(·)) is an optimal pair. The above gives alternative conditions for the solvability of Problem (LQ). In some sense, conditions in Theorem 5.2 are a little less abstract than those presented in Section 3. Note that if R(t)−1 exists and uniformly bounded. Then (5.16) is equivalent to h i u(t) = −R(t)−1 S(t)X(t) + λ(t) ,

t ∈ [0, T ].

(5.17)

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Substituting the above into (1.1) and (5.14), we obtain the following:  Z tn o    X(t) = ϕ(t) + [A0 (t, s) − B0 (t, s)R(s)−1 S(s)]X(s) − B0 (t, s)R(s)−1 λ(s) ds    0    d Z tn  o X   −1 −1  [A (t, s) − B (t, s)R(s) S(s)]X(s) − B (t, s)R(s) λ(s) dWj (s), +  j j j    j=1 0   Y (t) = [Q(t) − S(t)T R(t)−1 S(t)]X(t) − S(t)T R(t)−1 λ(t)   Z Th Z T d  i X   T T  + A0 (s, t) Y (s) + Ai (s, t) Zi (s, t) ds − Z(t, s)dW (s),    t t  i=1    Z Th Z T  d i  X  T T   λ(t) = B (s, t) Y (s) + B (s, t) µ (s, t) ds − µ(t, s)dW (s), 0 i i  t

t

i=1

(5.18)

with constraints:

 Z t    Z(t, s)dW (s),  Y (t) = EY (t) + 0 Z t     λ(t) = Eλ(t) + µ(t, s)dW (s).

(5.19)

0

This is a coupled FBSVIE. The general solvability problem for the above equation is still under our careful investigation. We hope to report some further results related to this in the near future.

References 1. M. Ait Rami, J. B. Moore, and X. Y. Zhou, Indefinite stochastic linear quadratic control and generalized differential Riccati equation, SIAM J. Control Optim. 40, 1296–1311 (2001/02). 2. B. D. O. Anderson and J. B. Moore, Linear Optimal Control, Prentice-Hall, Englewood Cliffs, New Jersey, 1971. 3. S. A. Belbas, Iterative schemes for optimal control of Volterra integral equations, Nonlinear Anal. 37, 57–79 (1999). 4. S. A. Belbas and W. H. Schmidt, Optimal control of Volterra equations with impulses, Appl. Math. Computation 166, 696–723 (2005). 5. R. Bellman, I. Glicksberg, and O. Gross, Some Aspects of the Mathematical Theory of Control Processes, Rand Corporation, Santa Monica, CA, 1958. 6. M. Berger and V. Mizel, Volterra equations with Itˆ o integrals, I, II, J. Int. Eqs. 2, 187–245; and 319–337 (1980). 7. D. A. Carlson, Infinite-horizon optimal controls for problems governed by a Volterra integral equation with state-and-control-dependent discount factor, J. Optim. Theory Appl. 66, 311–336 (1990). 8. S. Chen, X. Li, and X. Y. Zhou, Stochastic linear quadratic regulators with indefinite control weight costs, SIAM J. Control Optim. 36, 1685–1702 (1998). 9. S. Chen and J. Yong, Stochastic linear quadratic optimal control problems with random coefficients, Chinese Ann. Math. Ser. B 21, 323–338 (2000).

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10. S. Chen and J. Yong, Stochastic linear quadratic optimal control problems, Appl. Math. Optim. 43, 21–45 (2001). 11. S. Chen and X. Y. Zhou, Stochastic linear quadratic regulators with indefinite control weight costs, II, SIAM J. Control Optim. 39, 1065–1081 (2000). 12. N. Dunford and J. T. Schwartz, Linear Operators, Part II, Spectral Theory, Selfadjoint Operators in Hilbert Space, John Wiley & Sons, Inc., New York, 1963. 13. D. Duffie and C.-F. Huang, Stochastic production-exchange equilibria, Research paper No. 974, Graduate School of Business, Stanford Univ, 1986. 14. R. E. Kalman, Contributions to the theory of optimal control, Bol. SOc. Mat. Mexicana 5, 102–119 (1960). 15. M. T. Kamien, and E. Muller, Optimal control with integral state equations, Rev. Economic Studies 43, 469–473 (1976). 16. A. M. Letov, Analytic design of regulators, Avtomat. i Telemekh, (1960), 436–446, 561–571, 661–669 (in Russian); English transl. in Automat. Remote Control, 21 (1960). 17. I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, I. Abstract parabolic systems; II. Abstract hyperbolic-like systems over a finite time horizon, Encyclopedia of Mathematics and its Applications, 74, 75, Cambridge University Press, Cambridge, 2000. 18. X. Li and J. Yong, Optimal Control Theory for Infinite-dimensional Systems, Birkh¨ auser, Boston, 1995. 19. J. Lin, Adapted solution of a backward stochastic nonlinear Volterra integral equation, Stochastic Anal. Appl. 20, 165–183 (2002). 20. J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1971. 21. N. G. Medhin, Optimal processes governed by integral equations, J. Math. Anal. Appl. 120, 1–12 (1986). 22. L. W. Neustadt and J. Warga, Comments on the paper “Optimal control of processes described by integral equations. I” by V. R. Vinokurov, SIAM J. Control 8, 572 (1970). 23. E. Pardoux and P. Protter, Stochastic Volterra equations with anticipating coefficients, Ann. Probab. 18, 1635–1655 (1990). 24. A. J. Pritchard and Y. You, Causal feedback optimal control for Volterra integral equations, SIAM J. Control Optim. 34, 1874–1890 (1996). 25. P. Protter, Volterra equations driven by semimartingales, Ann. Probab. 13, 519–530 (1985). 26. V. R. Vinokurov, Optimal control of processes described by integral equations, I, II, III, Izv. Vysˇs. Uˇcebn. Zaved. Matematika, 7 (62) (1967), 21–33; 8 (63) (1967), 16–23; 9 (64) (1967), 16–25; (in Russian) English transl. in SIAM J. Control 7 (1969), 324–336; 337–345; 346–355. 27. J. Willems, Least squares stational optimal control and the algebraic Riccati equation, IEEE Trans. Auto. Control 16, 621–634 (1971). 28. W. M. Wonham, On a matrix Riccati equation of stochastic control, SIAM J. Control 6, 681–697 (1968). 29. W. M. Wonham, Linear Multivariable Control: A Geometric Approach, 2nd Edition, Springer-Verlag, 1979. 30. H. Wu and X. Li, A linear quadratic problem with unbounded control in Hilbert spaces, Diff. Int. Eqs. 13, 529–566 (2000). 31. D. Yao, S. Zhang, and X. Y. Zhou, Stochastic linear-quadratic control via primal-dual semidefinite programming, SIAM Rev. 46, 87–111 (2004). 32. J. Yong, A leader-follower stochastic linear quadratic differential game, SIAM J. Control Optim. 41, 4, 1015–1041 (2002).

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33. J. Yong, Backward stochastic Volterra integral equations and some related problems, Stoch. Proc. Appl., to appear. 34. J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. 35. Y. You, Quadratic integral games and causal synthesis, Trans. Amer. Math. Soc. 352, 2737–2764 (2000).

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AN ADDITIVITY OF MAXIMUM EXPECTATIONS AND ITS APPLICATIONS

ZENGJING CHEN∗ Department of Mathematics Shandong University, Jinan 250100, P.R. China E-mail: [email protected] MATT DAVISON, MARK REESOR Department of Applied Mathematics The University of Western Ontario London, Canada YING ZHANG Department of Mathematics and Statistics Acadia University, Canada

In this paper, we explore an additivity of a class of maximum (minimum) expectations which come from the pricing of contingent claim in incomplete markets. We give examples to show this property can be used to calculate maximum (minimum) expectations and Choquet integral. Furthermore, we also explore its applications in partial differential equation (shortly PDE) and the pricing of the contingent claims in incomplete markets. Keywords: Maximum expectation, partial differential equation, backward stochastic differential equation; contingent claim, Malliavin calculus.

1. Introduction Fixed time horizon T > 0, let {Wt }0≤t≤T be a d-dimensional standard Brownian motion defined on a completed probability space (Ω, F, P ) and {Ft }0≤t≤T be the natural filtration generated by Brownian motion {Wt }0≤t≤T , that is, Ft = σ(Ws ; s ≤ t), we assume F = FT . To ease of exposition, in this paper, we assume d = 1 and Ω is a Wiener space, that is Ω := C0 (0, T ), the set of all continuous functions {ωt } defined on [0, T ] with w0 = 0 and for any ω ∈ Ω, Brownian motion Wt (ω) = ωt . We present the following notations that will be used in the rest of this

∗ Financial

support partly from NSF of China (10325106) and (10131030), FANEDD (2001059). 67

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paper. Let p ∈ (1, ∞) and set n o RT Lp (0, T ) := V : all Ft −adapted processes {Vt } such that E 0 |Vs |p ds < ∞ ; B 2 :=nL2 (0, T ) × L2 (0, T ); P := Q : all probability measures Q s.t. {E[ dQ dP |Ft ]} is a square integrable o martingale .

Lp (Ω, F, P ) := {ξ : all F−measurable random variables such that E|ξ|p < ∞} ;

We define maximum (minimum) expectations by E[ξ] := sup EQ [ξ],

E[ξ] := inf EQ [ξ];

(1)

E[ξ|Ft ] := ess inf EQ [ξ|Ft ].

(2)

Q∈P

Q∈P

and conditional expectations by E[ξ|Ft ] := ess sup EQ [ξ|Ft ],

Q∈P

Q∈P

In order to make maximum expectation E[ξ] be well-defined for any ξ ∈ Lp (Ω, F, P ), we further need some assumptions on P. Indeed, for any Q ∈ P, let Mt := E[ dQ dP |Ft ], because {Mt }0≤t≤T is a square integrable martingale, by Martingale Representation Theorem (see for example, Theorem 4.3.3, p. 53, Ref. 13), there exists a unique predictable process h ∈ L2 (0, T ) such that Z t Mt = 1 + hs dWs . 0

Let as :=

hs Ms ,

if E

RT 0

2

|as | ds < ∞, then Z t Mt = 1 + as Ms dWs . 0

Solving the above linear stochastic differential equation (shortly SDE), we obtain Mt is of the following form: 1

Mt = e − 2

Rt 0

|as |2 ds+

Rt 0

as dWs

, 0 ≤ t ≤ T.

Which implies that for Q ∈ P, there exists a process {at } such that form: RT RT 2 1 dQ = e− 2 0 as ds+ 0 as dWs . dP

dQ dP

is of the

(3)

We rewrite Q as Qa and call Qa the probability measure generated by {at }. Hence, P actually is the set of probability measures generated by some processes {a t } via (3). In this paper, we further assume there exists a deterministic positive function RT {k(t)} with 0 k 2 (s)ds < ∞ such that a R R − 12 0T as 2 ds+ 0T as dWs a dQ =e , |at | ≤ k(t), t ∈ [0, T ] . (∗) P= Q : dP

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Immediately, the maximum (minimum) expectations defined in (1) and (2) are welldefined. Indeed, for any Qa ∈ P and ξ ∈ Lp (Ω, F, P ), by Schwartz inequality and the fact that 1

E(e− 2

RT 0

|qas |2 ds+

RT 0

qas dWs

) = 1, ∀q ∈ (1, ∞),

we get a

|EQa ξ| ≤ E|ξ|| dQ dP | a

1

1

q q ≤ (E|ξ|p ) p (E| dQ dP | ) 1

1

≤ (E|ξ|p ) p (Ee− 2 1

q−1 2

1

q

= (E|ξ|p ) p e

≤ (E|ξ|p ) p e p where

1 p

+

1 q

RT

RT 0

RT

0

0

|qas 2 |ds+

|k(s)|2 ds

|k(s)|2 ds

RT 0

qas dWs + q

2 −q 2

RT 0

|as |2 ds

1

)q

< ∞,

= 1. Thus p

1

q

|E[ξ]| = | sup EQ ξ| ≤ (E|ξ| ) p e p Q∈P

RT 0

|k2 (s)|ds

< ∞,

which implies E[ξ] and E[ξ|Ft ] are well-defined for any ξ ∈ Lp (Ω, F, P ), so are E[ξ] and E[ξ|Ft ]. In this paper, we only consider maximum expectation E[·] as minimum expectation E[·] can be treated by the fact that E[ξ] = −E[−ξ],

E[ξ|Ft ] = −E[−ξ|Ft ], ∀ξ ∈ Lp (Ω, F, P ).

(∗∗)

To simplify notations, let us write E[·] and E[·|Ft ] simply as E[·] and E[·|Ft ] respectively. Obviously, by the definition of E[·], E[·] is a nonlinear operator on Lp (Ω, F, P ) and E[ξ + η] ≤ E[ξ] + E[η], ∀ξ, η ∈ Lp (Ω, F, P ), moreover, the equality holds if and only if P = {P }. We are interesting in finding some conditions on ξ and η under which the following relations hold. E[ξ + η] = E[ξ] + E[η], or E[ξ + η] < E[ξ] + E[η]. We will indicate the applications of the above results in Section 3. The maximum (minimum) expectations defined in (1) and (2) actually come from the pricing of contingent claim in incomplete markets. As we know, for given contingent claim ξ, which usually depends on the price of stocks, if the market is a complete market, then there exists a unique equivalent martingale measure Q such that the pricing of this contingent claim at time t is given by conditional expectation EQ [ξ|Ft ]. However, if the market is an incomplete market, such probability measure

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Q is not unique. Thus it is impossible to price and hedge this claim. In that case, an investor hopes to know the superpricing E[ξ] and subpricing E[ξ] (cf. El Karoui and Quenez10 ; Karatzas8; Cvitanic6 ; F¨ ollmer and Kramkov7). Hence, to explore the pricing of contingent claim in incomplete markets, it is important to study the properties of maximum expectations E[ξ] and minimum expectations E[ξ]. 2. Main Result In this paper, we will use backward stochastic differential equations (BSDE) and Malliavin calculus to study the properties of maximum expectations. We refer readers to Refs. 14 and 9 for BSDEs and Refs. 11 and 12 for Malliavin calculus. Let us now give the following Lemmas: Lemma 2.1. If ξ ∈ L2 (Ω, F, P ) and (yt , zt ) is the unique solution of BSDE, yt = ξ +

Z

T

k(s)|zs |ds − t

Z

T

zs dWs ,

(4)

t

where k(t) is defined in (∗), then E[ξ|Ft ] = yt ;

E[ξ] = y0 .

Proof. For any n > 0, let k (n) (t) = k(t) ∧ n and v R R (n) − 12 0T |vs |2 ds+ 0T vs dWs (n) v dQ , |v(t)| ≤ k (t) . =e P := Q : dP Obviously, P (n) ⊂ P (n+1) ⊂ · · · ⊂ P and P (n) ↑ P. Since |k (n) (t)| ≤ n, thus BSDE(5) below satisfies uniform Lipschitz condition. By Theorem 2.1 of Ref. 9, for each n > 0 , there exists a unique solution (yt (n) , zt (n) ) ∈ B 2 such that yt

(n)

=ξ+

Z

T

k

(n)

(s)|zs

(n)

|ds −

0

Z

T

zs (n) dWs .

(5)

0

Moreover, by Proposition 35.2 of Ref. 15, there exists a constant c > 0 such that Z T 2 E |zs (n) | ds ≤ c E|ξ 2 |. 0

(n)

Let E [ξ|Ft ] := ess supQ∈P (n) EQ [ξ|Ft ], by Proposition 3.2.2 of Ref. 10, or Lemma 1 in Ref. 4, E (n) [ξ|Ft ] = yt (n) . Note that E (n) [ξ|Ft ] ↑ E[ξ|Ft ], as n → ∞, it remains to prove that the limit of {yt (n) } is the solution of BSDE (4).

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In fact, for any n, m ≥ 1, by Proposition 35.2 of Ref. 15, there exists a constant C > 0 such that RT 2 2 E[sup0≤t≤T |yt (n) − yt (m) | + 0 |zs (n) − zs (m) | ds] h R i2 T (n) (n) ≤ C E 0 (|k (n) (s)|zs | − k (m) (s)|zs |)ds RT R T (n) ≤ C 0 |k (n) (s) − k (m) (s)|2 dsE 0 |zs |2 ds R T ≤ CcE|ξ|2 0 |k (n) (s) − k (m) (s)|2 ds → 0, as n, m → ∞, which implies that (y (n) , z (n) ) is a Cauchy sequence of B 2 , thus there exists (y, z) ∈ B 2 such that (y (n) , z (n) ) → (y, z), as n → ∞. Since R 2 T E 0 |k(s)|zs | − k (n) (s)|zs (n) ||ds 2 R RT T ≤ E 0 k(s)|zs (n) − zs |ds + E 0 |k (n) (s) − k(s)||zs (n) |ds RT RT RT RT . 2 2 ≤ 0 k 2 (s)dsE 0 |zs (n) − zs | ds + 0 |k (n) (s) − k(s)| dsE 0 |zs (n) |2 ds R R T (n) RT 2 2 2 T ≤ 0 k (s)dsE 0 |zs − zs | ds + cEξ 2 0 |k (n) (s) − k(s)| ds → 0, as n → ∞. Thus, Z

T

k

(n)

(s)|zs

(n)

|ds →

0

Z

T

k(s)|zs |ds,

as n → ∞

0

in L2 (Ω, F, P ). Set n → ∞ on both sides of BSDE (5), then (y, z) is the solution of BSDE(4). Thus yt = E[ξ|Ft ]. In particular, let t = 0, we get y0 = E[ξ]. Using the same method and the fact (∗∗), we can get Corollary 2.1. Assume ξ ∈ L2 (Ω, F, P ), then E[ξ|Ft ] is the unique solution of BSDE: Z T Z T yt = ξ − k(s)|zs |ds − zs dWs . 0

t

RT Lemma 2.2. Let ξ ∈ L2 (Ω, F, P ) be of the form ξ = f ( 0 σs dWs ), where f (x) is a continuous differentiable function, {σt } is a deterministic function. Let (yt , zt ) be the solution of the following BSDE Z T Z T yt = ξ + k(s)|zs |ds − zs dWs . (6) t

t

(i) If f is increasing (resp. decreasing), then σt zt ≥ 0, (resp. σt zt ≤ 0). (ii) Assume f is strictly increasing (resp. decreasing), if σt 6= 0, then zt 6= 0, a.e. t ∈ [0, T ].

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Proof. The proof of (i): For any ε > 0, ( |z|, gε (z) := 1 z2 2 (ε + ε ),

if |z| ≥ ε; if |z| < ε.

Obviously, gε (z) → |z|, as ε → 0. Let (yt (ε) , zt (ε) ) be the solution of BSDE Z T Z T Z T yt (ε) = f ( σs dWs ) + k(s)gε (zs )ds − zs (ε) dWs . t

0

t

(ε)

(ε)

Thus, by Proposition 35.2 of Ref. 15, (yt , zt ) → (y, z) in B 2 as ε → 0, where (y, z) is the solution of BSDE (6). Thanks to Proposition 5.3 of Ref. 9, we have for ε > 0, zt (ε) = Dt yt (ε) . where Dt yt (ε) denotes the Malliavin derivative of yt (ε) . Moreover, Dt yt (ε) satisfies the following linear BSDE Z T Z T Z T Dt yt (ε) = Dt f ( σs dWs ) + k(s)gε 0 (zs (ε) )Dt zs (ε) ds − Dt zs (ε) dWs , t

0

t

where gε (z) denotes the derivative of gε (z) with respect to z. Solving the above linear BSDE, we have Z T (ε) Dt yt = EQ [Dt f ( σs dWs )|Ft ], 0

0

dQ dP 0

− 12

RT

2

0

ε

2

RT

0

(ε)

[k (s)gε (zs )] ds+ 0 k(s)gε (zs )dWs 0 . =e where Let f be the derivative of f with respect to x, by the definition of Malliavin derivative, we have Z T Z T Dt f ( σs dWs ) = f 0 ( σs dWs )σt , 0

thus zt (ε) = Dt yt (ε) = EQ [f 0 (

0

Z

T

σs dWs )σt |Ft ] = σt EQ [f 0 ( 0

Z

T

σs dWs )|Ft ],

(7)

0

which implies if f 0 (x) ≥ 0, ∀x ∈ R, then σt zt (ε) ≥ 0 and σt zt (ε) ≤ 0 for f 0 (x) ≤ 0, ∀x ∈ R. It then follows by the fact z (ε) → z in L2 (0, T ) that we complete the proof of (i). The proof of (ii):R Without loss of generalization, we assume f 0 > 0, from T (ε) (i), σt zt = σt2 EQ [f 0 ( 0 σs dWs )|Ft ], note that |gε0 | ≤ 1, thus Q ∈ P, by strict RT (ε) Comparison Theorem 35.5 of Ref. 15, σt zt ≥ σt2 E[f 0 ( 0 σs dWs )|Ft ] > 0, the proof of (ii) is complete. RT RT p Let ξ, η ∈ L (Ω, F, P ) be of the forms ξ := f ( 0 σs dWs ), η := h( 0 vs dWs ), where {σt } and {vt } are two deterministic functions, f and h are two functions

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with continuous differentiation, under the above assumptions, We have the following main result. Theorem 2.1. Suppose that ξ and η are the random variables defined above, f and h are both increasing (decreasing) functions, (i) If σt vt ≥ 0, t ∈ [0, T ], then E[ξ + η] = E[ξ] + E[η]. (ii) If f and h are strictly increasing (decreasing) functions and σt vt < 0, t ∈ [0, T ], then E[ξ + η] < E[ξ] + E[η]. Proof. We turn our proof into three steps. Step 1. We assume f and h are bounded by some constant N > 0, i.e. |f | ≤ N, |h| ≤ N. Let yt = E[ξ|Ft ], y t = E[η|Ft ]. By Lemma 2.1, {yt } and {yt } are the solutions of BSDEs respectively, Z T Z T yt = ξ + k(s)|zs |ds − zs dWs (8) t

t

and yt = η +

Z

T

k(s)|z s |ds − t

Z

T

z s dWs .

(9)

t

By Lemma 2.2(i), we have σt zt ≥ 0, and vt z t ≥ 0,

a.e. t ∈ [0, T ].

It then follows by σt vt ≥ 0 that we have z t zt ≥ 0, which implies |z t | + |zt | = |z t + zt |,

a.e. t ∈ [0, T ].

Adding BSDE (8) with BSDE (9), we have Z T Z yt + y t = ξ + η + k(s)|zs + z s |ds − t

(10)

T

(zs + z s )dWs , t

which means (yt + y t , zt + z t ) is the solution of BSDE Z T Z T yt = ξ + η + k(s)|zs |ds − zs dWs , t

t

by Existence and Uniqueness Theorem of BSDE (Theorem 2.1 of Ref. 9) and Lemma 1, yt + yt = E[ξ + η|Ft ].

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That is, E[ξ|Ft ] + E[η|Ft ] = E[ξ + η|Ft ].

(11)

Step 2. For sufficient large N > 0 and N = 1, 2, · · · , let us denote (n)

fN (x) := xe−nd

(N )

(x)

,

(n)

hN (x) := xe−nd

(N )

(x)

(12)

,

where d(N ) (x) is defined by d

(N )

(x) :=

0, (x − N )1+α ,

|x| ≤ N ; |x| > N, 0 < α < 1.

(n)

(n)

It is easy to check that fN (x) → xI(|x|≤N ) and hN (x) → xI(|x|≤N ) as N → ∞, (n)

(n)

here and below IA is an indicator function. Obviously fN and hN are bounded and continuous differentiable. By Step 1, Z T Z T Z T (n) (n) (n) E[fN ( σs dWs ) + hN ( vs dWs )|Ft ] = E[fN ( σs dWs )|Ft ] 0

0

0

(n) + E[hN (

Z

T

vs dWs )|Ft ]. 0

(n) R T (n) R T Note that fN ( 0 σs dWs ) → ξ and hN ( 0 vs dWs ) → η as n, N → ∞ in Lp (Ω, F, P ) . We have

E[ξ + η|Ft ] = E[ξ|Ft ] + E[η|Ft ]. In particular, let t = 0, E[ξ + η] = E[ξ] + E[η]. Step 3. If f and g are strictly increasing (decreasing), by Lemma 2.2(ii) and the assumptions, we have σt vt < 0. Thus, z t zt < 0 , ∀t ∈ [0, T ]. Moreover, Equality (10) becomes |z t | + |zt | > |z t + zt |,

t ∈ [0, T ],

(13)

this with strict Comparison Theorem 35.5 of Ref. 15, we have E[ξ|Ft ] + E[η|Ft ] > E[ξ + η|Ft ]; t ∈ [0, T ]. Let t = 0, the proof is complete. Using the same method as the proof of Theorem 2.1, we can get the following Corollary: RT Corollary 2.2. Suppose that ξ = f ( 0 σs dWs ) is defined in Theorem 2.1, let a1 ≤ a2 ≤ · · · < an be a sequence, then E[

n X i=1

I(ξ≥ai ) ] =

n X i=1

E[Iξ≥ai ) ].

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3. Applications To illustrate the applications of the above main results, we give the following examples: 3.1. Compute maximum expectation In general, it is not easy to calculate maximum expectations, but for special case, we can calculate maximum expectations by using main results in this paper, let us give the following examples. RT RT Example 3.1. Compute supQ∈P EQ e−2 0 k(s)dWs − e2 0 k(s)dWs . In fact, let f (x) := e−2x and h(x) := −e2x , then f and h are monotone, by Theorem 2.1, RT RT RT RT supQ∈P EQ e−2 0 k(s)dWs − e2 0 k(s)dWs = E[e−2 0 k(s)dWs ] + E[−e2 0 k(s)dWs ] RT

= E[e−2

By Lemma 2.1, E[e−2 BSDE at time t = 0,

RT 0

k(s)dWs

yt = e−2

RT 0

0

k(s)dWs

] − E[e2

RT 0

k(s)dWs

].

] is the value of (yt ), the solution of the following

k(s)dWs

+

Z

T

k(s)|zs |ds − t

Z

T

zs dWs . t

Solving the above BSDE, we get Rt Rt (yt , zt ) = e−2 0 k(s)dWs , 2k(t)e−2 0 k(s)dWs . RT

Thus E[e−2 0 k(s)dWs ] = y0 = 1. Similarly, by Corollary 2.1, E[e2 value of {yt }, the solution of the following BSDE, at time t = 0, Z T Z T R 2 0T k(s)dWs yt = e − k(s)|zs |ds − zs dWs . t

RT 0

k(s)dWs

] is the

t

Solving the above BSDE, we obtain Rt Rt (yt , zt ) = e2 0 k(s)dWs , −2k(t)e2 0 k(s)dWs , and E[e2

RT 0

k(s)dWs

] = 1. Thus E[e−2

RT

k(s)dWs

0

− e2

RT 0

k(s)dWs

] = 1 − 1 = 0.

Remark 3.1. From this example, we get the following interesting facts: (1) Since RT 2 RT Ee−2 0 k (s)ds−2 0 k(s)dWs = 1, thus Ee−2

RT 0

k(s)dWs

= e2

which depends on k, but by Example 3.1, E[e−2

RT 0

k2 (s)dWs

RT 0

k2 (s)ds

,

] = 1,

which does not depend on k. R RT −2 0T k(s)dWs (2) Since E[e ] = E[e2 0 k(s)dWs ] = 1, thus for any Q ∈ P, EQ [e−2

RT 0

k(s)dWs

] ≤ EQ [e2

RT 0

k(s)dWs

].

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3.2. Compute Choquet integral For any A ∈ F, let V (A) := supQ∈P Q(A). Obviously V (·) is a capacity and V (A) = E[IA ]. Choquet5 introduced the following integral (shortly Choquet expectation) which generalizes the usual mathematical expectations, Z 0 Z ∞ C[ξ] := (V (ξ ≥ t) − 1)dt + V (ξ ≥ t)dt. (14) 0

−∞

Example 3.2. Compute C[e−2 Let ξ := e−2

RT 0

k(s)dWs

RT 0

k(s)dWs

].

, for sufficient large N and n = 1, 2, · · · , and n

(n) ξN

:=

2 X iN i=0

2n

, I( iNn ≤ξ∧N < (i+1)N ) n 2

2

(n)

then ξN → ξ as N, n → ∞ in L2 (Ω, F, P ). Note that ξ ∧ N ≤ N and P2n iN P2n iN iN I = I − I (i+1)N (i+1)N iN n n (ξ∧N ≥ 2n ) i=0 2 i=0 2 ( 2n ≤ξ∧N < 2n ) (ξ∧N ≥ 2n ) P2n iN P2n +1 (i−1)N = i=1 2n I(ξ∧N ≥ iNn ) − i=2 2n I(ξ∧N ≥ iN 2 2n ) P2n iN P2n (i−1)N = i=1 2n I(ξ∧N ≥ iNn ) − i=2 2n I(ξ∧N ≥ iNn ) 2 2 n N P2 = 2n i=1 I(ξ∧N ≥ iNn ) . 2

Applying Corollary 2.2, n

E[

n

2 X iN i=0

2 N X I E[I(ξ∧N ≥ iNn ) ] (i+1)N ] = iN 2 2n ( 2n ≤ξ∧N < 2n ) 2n i=1

We have, from the definition (14) and applying the fact that V (ξ ∧ N ≥ t) = E[I(ξ∧N ≥t) ], C[e−2

RT 0

k(s)dWs

]= = = = =

RT V e−2 0 k(s)dWs > t dt RN limN →∞ 0 V (ξ ∧ N ≥ t)dt P 2n limN →∞ limn→∞ i=0 2Nn V (ξ ∧ N ≥ iN 2n ) P 2n N limN →∞ limn→∞ i=0 2n E[I(ξ∧N ≥ iNn ) ] 2 P2n iN limN →∞ limn→∞ E[ i=0 2n I( iNn ≤ξ∧N ≤ (i+1)N ] ) n R∞ 0

2

= limN →∞ E[ξ ∧ N ] = E[e−2 According to Example 3.1, E[e−2

RT 0

k(s)dWs

RT 0

2

k(s)dWs

] = 1, thus C[e−2

RT 0

].

k(s)dWs

] = 1.

Remark 3.2. This example also shows that under some assumptions on ξ, the maximum expectation of ξ is equal to Choquet expectation.

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3.3. Application to PDE Suppose that σt , bt , K(t), Φ are continuous and bounded functions. Let UΦ be the solution of the following PDE with respect to terminal value Φ, ( ∂U 1 2 ∂2U ∂U ∂U ∂t + 2 σt ∂x2 + bt ∂x = k(t)| ∂x σt x|; (15) U (T, x) = Φ(x), x ∈ R+ , 0 ≤ t ≤ T. Obviously, In general, UΦ1 +Φ 6= UΦ1 + UΦ2 . However, we have Example 3.3. Assume Φ1 and Φ2 are two continuous bounded functions, UΦ1 and UΦ2 are the solution of PDE (15) corresponding to Φ = Φ1 and Φ = Φ2 . If Φ is an increasing or decreasing function, then UΦ1 +Φ2 = UΦ1 + UΦ2 , that is, the solution of PDE (15) is linear with respect to Φ. In fact, by Theorem 12.3 of Ref. 1, UΦ (t, x) = yt t,x , where {ys t,x }0≤s≤T is the solution of BSDE yt t,x = Φ(XT t,x ) +

Z

T

k(s)|zs t,x |ds −

t

Z

T

zs t,x dWs .

t

And {Xs t,x } is the solution of SDE

dXs t,x = bt Xs ds + σs Xs dWs Xt = x, t ≤ s ≤ T.

By Lemma 1 and Proposition 4.2 of Ref. 9, yt t,x = E[Φ(XT t,x )|Ft ] = E[Φ(XT t,x )]. That is, UΦ (x, t) = E[Φ(XTt,x )]. Thus if Φ1 and Φ2 are increasing (decreasing) functions with continuous differentiation, applying Theorem 2.1, we have E[Φ1 (XTt,x ) + Φ2 (XTt,x )] = E[Φ1 (XTt,x )] + E[Φ2 (XTt,x )]. That is, UΦ1 +Φ2 = UΦ1 + UΦ2 .

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3.4. Application to European Option As explained in introduction, maximum expectations defined in this paper come from the pricing of contingent claim in incomplete market, for given contingent claims ξ and η, the superpricing of contingent claims given the maximum expectation E[ξ] and E[η], since E[ξ + η] ≤ E[ξ] + E[η], thus, in an incomplete market, it is better for an investor to buy ξ and η together than to buy them separately. Let us now further consider the application of our result in incomplete markets. If given two contingent claims XT and YT which satisfy the following SDEs dXt = at Xt dt + σt Xt dWt ; X0 = x > 0, 0 ≤ t ≤ T. and

dYt = bt Yt + vt Yt dWt ; Y0 = y > 0, 0 ≤ t ≤ T.

Where at , bt , σt , vt are continuous bounded functions. the above linear YTR are of Rthe forms XT = R T SDEs, we get XT and − R Solving T T 2 1 T f ( t σs dWs ) and YT = h( 0 vs dWs ). where f (x) = e 0 at dt− 2 0 σt dt+x , h(x) = RT R T 2 1 e− 0 bt dt− 2 0 vt dt+x . The maximal pricing for investors to buy XT and YT together is E[YT + XT ]. However, by Theorem 2.1, if σt vt ≥ 0, then E[YT + XT ] = E[YT ] + E[XT ], which implies that the maximal price that the investors buy XT and YT together is equal to buy them separately. However, if σt vt < 0, then E[YT + XT ] < E[YT ] + E[XT ]. In this case, the investors should buy the two contingent claims together to hedge risk. References 1. E. Barles and E. Lesigne, SDE, BSDE and PDE, Pitman Research Notes in Mathematics Series No. 364, 47–80 (1997). 2. Z. Chen, A property of backward stochastic differential equations, C.R. Acad. Sci. Paris No. 1, 483–488 (1998). 3. Z. Chen and L. Epstein, Ambiguity, risk and asset return in continuous time, Econometrica 70, 1403–1443 (2002). 4. Z. Chen and S. Peng, A general downcrossing inequality for g-martingales, Statist. Probab. Lett. 46, no. 2, 169–175 (2000). 5. G. Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble) 5, 131–295 (1955).

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6. J. Cvitanic, Minimizing expected loss of hedging in incomplete and constrained markets, Preprint, Columbia Univ. New York (1998). 7. H. F¨ ollmer and D. Kramkov, Optimal decomposition under constraints, Probab. Theory & Rel. Fields 109, 1–25 (1997). 8. I. Karatzas, Lectures in mathematical finance, Providence, American Mathematical Society (1997). 9. N. El Karoui, S. Peng, and M. Quenez, Backward stochastic differential equations in finance, Math. Finance No. 1, 1–71 (1997). 10. N. El Karoui, M. Quenez, Dynamic programming and pricing contingent claims incomplete markets, SIAM J. Control Optim. 33, 29–66 (1995). 11. D. Nualart, The Malliavin Calculus and Related Topics, New York and Berlin, Springer-Verlag (1995). 12. B. ∅ksendal, An introduction to Malliavin Calculus with application to economics, Working paper No. 3/96, Norwegian School of Economics (1997). 13. B. ∅ksendal, Stochastic differential equations, 5th Edition, Springer (1991). 14. E. Pardoux and S. Peng, Adapted solution of a Backward stochastic differential equation, Systems and Control Letters 14, 55–61 (1990). 15. S. Peng, BSDE and related g–expectation, Pitman Research Notes in Mathematics Series No. 364, 141–159 (1997). 16. S. Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer type, Probab. Theory & Rel. Fields 113, 473–499 (1999).

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STOCHASTIC CONTROL AND BSDES WITH QUADRATIC GROWTH

MARCO FUHRMAN Dipartimento di Matematica, Politecnico di Milano piazza Leonardo da Vinci 32, 20133 MILANO, Italy E-mail: [email protected] YING HU IRMAR, Universit´e Rennes 1 Campus de Beaulieu, 35042 RENNES Cedex, France E-mail: [email protected] GIANMARIO TESSITORE Dipartimento di Matematica, Universit` a di Parma Parco Area delle Scienze, 53/A - 43100 Parma, Italy E-mail: [email protected]

In this talk, we study a stochastic optimal control problem where the drift term of the equation has a linear growth on the control variable, the cost functional has a quadratic growth and the control process takes values in a closed set. This problem is related to some BSDE with quadratic growth. We prove that the optimal feedback control exists and the optimal cost is given by the initial value of the solution of the related backward stochastic differential equation.

1. Introduction This talk is based upon the paper 6 . The interested readers can see the full paper for all the proofs of the results presented here. In this talk, we consider a controlled equation of the form:

dXt = b(t, Xt ) dt + σ(t, Xt ) [dWt + r(t, Xt , ut ) dt], X0 = x.

t ∈ [0, T ],

6

(1)

In the equation, W is a Rd -valued Wiener process, defined on a complete probability space (Ω, F, P) with respect to a filtration F := (Ft )t≥0 satisfying the usual conditions; the unknown process X takes values in Rn ; x is a given element of Rn ; u is the control process, which is assumed to be an Ft -adapted process taking values in a given nonempty closed set K ⊂ Rm . The control problem consists in minimizing 80

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a cost functional of the form J =E

Z

T

g(t, Xt , ut ) dt + E φ(XT ).

(2)

0

We suppose that r has a linear growth in u, g has quadratic growth in x and u, and φ has quadratic growth in x. The main novelty of the present work, in comparison with the existing literature is that, on the one hand we neither assume K nor r to be bounded, on the other hand we consider a degenerate control problem (since nothing is assumed on the image of σ). Moreover we also allow φ to have linear growth. Nonlinear Backward Stochastic Differential Equations (BSDEs) were first introduced by Pardoux and Peng 9 . They have found applications in stochastic control, see, e.g., Refs. 3 and 10. To have an overview of recent applications of backward stochastic differential equations techniques to control problems being by several aspects more general than the one considered here but involving “bounded control image” assumptions see for instance Ref. 4 or Ref. 7 and references therein. The special “unbounded” case corresponding to the assumptions K = Rm and g = 21 |u|2 + q(t, x) is treated in Ref. 5 by an ad-hoc exponential transform. We notice that in Ref. 5 φ is allowed to take value +∞. Finally this same special case (in which the Hamiltonian is exactly the square of the norm of the gradient) was treated in Ref. 8 by analytic techniques, under non-degeneracy assumptions and in an infinite dimensional framework. The difficulty here is that the Hamiltonian corresponding to the control problem has quadratic growth in the gradient and the terminal cost is not bounded. Thus the backward stochastic differential equation corresponding to the problem has quadratic growth in the Z variable and unbounded terminal value. To treat such equation we have to apply the localization procedure recently introduced by Briand and Hu in Ref. 2. We notice that for such BSDEs no general uniqueness results are known: we replace uniqueness by the selection of a maximal solution. Moreover the usual application of Girsanov technique is not allowed (since the Novikov condition is not guaranteed) and we have to develop specific arguments both to prove the fundamental relation, see §4, and to obtain the existence of a (weak) solution to the closed loop equation, see §5. Our main result is to prove that the optimal feedback control exists and the optimal cost is given by the value Y0 of the maximal solution (Y, Z) of the backward stochastic differential equation (BSDE) with quadratic growth and unbounded terminal value mentioned above. Moreover we show that we can construct an optimal feedback in terms of the process Z. Finally we prove that if we fix a particular optimal feedback law then the solution of the corresponding closed loop equation is unique, see Proposition 5.2. The talk is organized as follows: in the next section, we describe the control problem; in section 3, we study the related BSDE; in section 4, we establish the fundamental relation between the optimal control problem and BSDE; and the last section is devoted to the proof of the existence of optimal feedback control.

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2. The Controlled Problem We consider here the optimal control problem given by a state equation of the form: dXt = b(t, Xt ) dt + σ(t, Xt ) [dWt + r(t, Xt , ut ) dt], t ∈ [0, T ], (3) X0 = x, and by a cost functional of the form Z T J =E g(t, Xt , ut ) dt + E φ(XT ).

(4)

0

We work under the following assumptions. Hypothesis A. (1) The process W is a Wiener process in Rd , defined on a complete probability space (Ω, F, P) with respect to a filtration (Ft ) satisfying the usual conditions. (2) The set K is a nonempty closed subset of Rm . (3) The functions b : [0, T ] × Rn → Rn , σ : [0, T ] × Rn → Rn×d , r : [0, T ] × Rn × K → Rd , g : [0, T ] × Rn × K → R, φ : Rn → R, are Borel measurable. (4) For all t ∈ [0, T ], x ∈ Rn , r(t, x, ·) and g(t, x, ·) are continuous functions from K to Rd and from K to R respectively. (5) There exists a constant C > 0 such that for every t ∈ [0, T ], x, x0 ∈ Rn , u ∈ K it holds: |b(t, x) − b(t, x0 )| ≤ C|x − x0 |,

|b(t, x)| ≤ C(1 + |x|),

(5)

|σ(t, x) − σ(t, x0 )| ≤ C|x − x0 |,

(6)

|σ(t, x)| ≤ C,

(7)

|r(t, x, u) − r(t, x0 , u)| ≤ C(1 + |u|)|x − x0 |,

(8)

|r(t, x, u)| ≤ C(1 + |u|),

(9)

0 ≤ g(t, x, u) ≤ C(1 + |x|2 + |u|2 ),

(10)

0 ≤ φ(x) ≤ C(1 + |x|2 ).

(11)

(6) There exist R > 0 and c > 0 such that for every t ∈ [0, T ], x ∈ Rn , and every u ∈ K satisfying |u| ≥ R g(t, x, u) ≥ c|u|2 .

(12)

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We will say that an (Ft )-adapted stochastic process {ut , t ∈ [0, T ]} with values in K is an admissible control if it satisfies: Z T E |ut |2 dt < ∞. (13) 0

This square summability requirement is justified by (12): a control process which is not square summable would have infinite cost. Next we show that for every admissible control the solution to (3) exists. Proposition 2.1. Let u be an admissible control. Then there exists a unique continuous, (Ft )-adapted process X satisfying E supt∈[0,T ] |Xt |2 < ∞ and, P-a.s, Z t Z t Z t Xt = x+ b(s, Xs ) ds+ σ(s, Xs ) dWs + σ(s, Xs ) r(s, Xs , us ) ds, t ∈ [0, T ]. 0

0

0

Proof. The proof of Proposition 2.1 relies on an approximation procedure. The stochastic control problem associated with (3)-(4) consists in minimizing the cost functional J(x, u) among all the admissible controls. 3. The Forward-Backward System Let us consider again the functions b, σ, g, φ satisfying the assumptions in Hypothesis A. We define the Hamiltonian function ψ(t, x, z) = inf [g(t, x, u) + z · r(t, x, u)], u∈K

t ∈ [0, T ], x ∈ Rn , z ∈ Rd ,

(14)

where · denotes the usual scalar product in Rd . We collect some immediate properties of the function ψ. Lemma 3.1. The map ψ is a Borel measurable function from [0, T ] × Rn × Rd to R. There exists a constant C > 0 such that −C(1 + |z|2 ) ≤ ψ(t, x, z) ≤ g(t, x, u) + C|z|(1 + |u|),

∀u ∈ K.

(15)

Moreover the infimum in (14) is attained in ball of radius C(1 + |x| + |z|) that is ψ(t, x, z) =

min

[g(t, x, u)+z·r(t, x, u)],

u∈K,|u|≤C(1+|x|+|z|)

t ∈ [0, T ], x ∈ Rn , z ∈ Rd , (16)

and ψ(t, x, z) < g(t, x, u) + z · r(t, x, u) if |u| > C(1 + |x| + |z|).

(17)

Finally, for every t ∈ [0, T ] and x ∈ Rn , z → ψ(t, x, z) is continuous on Rd . Next we take an arbitrary complete probability space (Ω, F, P◦ ) and a Wiener process W ◦ in Rd with respect to P◦ . We denote by (Ft◦ ) the associated Brownian filtration, i.e. the filtration generated by W ◦ and augmented by the P◦ -null sets of F; (Ft◦ ) satisfies the usual conditions.

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We introduce the forward equation: dXt = b(t, Xt ) dt + σ(t, Xt ) dWt◦ , X0 = x,

t ∈ [0, T ],

(18)

whose solution is a continuous (Ft◦ ) adapted process, which exists and is unique by classical results. Next we consider the associated backward equation dYt = −ψ(t, Xt , Zt ) dt + Zt dWt◦ , t ∈ [0, T ], (19) YT = φ(XT ). The solution of (19) exists in the sense specified by the following proposition. Proposition 3.1. Assume that b, σ, g, φ satisfy Hypothesis A. Then there exist Borel measurable functions v : [0, T ] × Rn → R,

ζ : [0, T ] × Rn → Rd

with the following property: for arbitrary complete probability space (Ω, F, P ◦ ) and Wiener process W ◦ in Rd , denoting by X the solution of (18), the processes Y, Z defined by Yt = v(t, Xt ),

Zt = ζ(t, Xt )

satisfy E

◦

2

sup |Yt | < ∞,

E

◦

t∈[0,T ]

Z

T

|Zt |2 dt < ∞, 0

Y is continuous, nonnegative, and finally, P -a.s, Z T Z T ◦ Zs dWs = φ(XT ) + ψ(s, Xs , Zs ) ds, Yt + ◦

t

t ∈ [0, T ].

t

Proof. We apply the localization procedure together with a priori bounds, which is introduced by Briand and Hu 2 .

4. The Fundamental Relation In this section we revert to the notation introduced in the first section. Proposition 4.1. For every admissible control u and for the corresponding trajectory X starting at x we have Z T J(u) = v(0, x) + E [−ψ(t, Xt , ζ(t, Xt )) + ζ(t, Xt ) · r(t, Xt , ut ) + g(t, Xt , ut )] dt. 0

Proof. We apply an approximation procedure together with the Girsanov theorem.

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Corollary 4.1. For every admissible control u and any initial datum x we have J(u) ≥ v(0, x) and the equality holds if and only if the following feedback law holds P-a.s. for a.e. t ∈ [0, T ] ψ(t, Xt , ζ(t, Xt )) = ζ(t, Xt ) · r(t, Xt , ut ) + g(t, Xt , ut ), where X is the trajectory starting at x and corresponding to control u. 5. Existence of Optimal Controls: The Closed Loop Equation Let us consider again the functions b, σ, g, φ satisfying the assumptions in Hypothesis A. We recall the definition of the Hamiltonian function t ∈ [0, T ], x ∈ Rn , z ∈ Rd .

ψ(t, x, z) = inf [g(t, x, u) + z · r(t, x, u)], u∈K

(20)

Lemma 5.1. There exists a Borel measurable function γ : [0, T ] × Rn × Rd → K such that ψ(t, x, z) = g(t, x, γ(t, x, z)) + z · r(t, x, γ(t, x, z)),

t ∈ [0, T ], x ∈ Rn , z ∈ Rd . (21)

Moreover there exists a constant C > 0 such that |γ(t, x, z)| ≤ C(1 + |x| + |z|).

(22)

Proof. It suffices to apply the Filippov theorem 1 . Next we address the problem to find a weak solution to the so-called closed loop equation. We define u(t, x) = γ(t, x, ζ(t, x)),

t ∈ [0, T ], x ∈ Rn ,

where ζ has been introduced in Proposition 3.1. The closed loop equation is dXt = b(t, Xt ) dt + σ(t, Xt ) [dWt + r(t, Xt , u(t, Xt )) dt], t ∈ [0, T ], (23) X0 = x. By a weak solution we mean a complete probability space (Ω, F, P) with a filtration (Ft ) satisfying the usual conditions, a Wiener process W in Rd with respect to P and (Ft ), and a continuous (Ft )-adapted process X with values in Rn satisfying, P-a.s., Z T |u(t, Xt )|2 dt < ∞, (24) 0

and We note that by (9) it also follows that R T such that (23) holds. |r(t, X , u(t, X ))| dt < ∞, P-a.s., so that (23) makes sense. t t 0

Proposition 5.1. Assume that b, σ, g, φ satisfy Hypothesis A. Then there exists a weak solution of the closed loop equation, satisfying in addition Z T E |u(t, Xt )|2 dt < ∞. (25) 0

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Corollary 5.1. By Corollary 4.1 it immediately follows that if X is the solution to (23) and we set u]s = u(s, Xs ) then J(x, u] ) = v(0, x) and consequently X is an optimal state, u]s is an optimal control and u is an optimal feedback. Next we prove uniqueness in law for the closed loop equation. We remark that the condition (24) is part of our definition of a weak solution. Proposition 5.2. Assume that b, σ, g, φ satisfy Hypothesis A. Fix γ : [0, T ] × Rn × Rd → K satisfying (21) (and consequently (22)) and let u(t, x) = γ(t, x, ζ(t, x)). Then the weak solution of the closed loop equation (23) is unique in law. References 1. J.-P. Aubin, H. Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, vol. 2. Birkh¨ auser Boston Inc., Boston, MA, 1990. 2. P. Briand, Y. Hu, BSDE with quadratic growth and unbounded terminal value. Preprint no. 05-07, IRMAR, Universit´e Rennes 1, 2005. 3. N. El Karoui, S. Peng, M. C. Quenez, Backward stochastic differential equations in finance. Math. Finance 7, 1-71 (1997). 4. N. El Karoui, S. Hamad`ene, BSDEs and risk-sensitive control, zero-sum and nonzerosum game problems of stochastic functional differential equations. Stochastic Process. Appl. 107, 145–169 (2003). 5. M. Fuhrman, A class of stochastic optimal control problems in Hilbert spaces: BSDEs and optimal control laws, state constraints, conditioned processes. Stochastic Process. Appl. 108, 263-298 (2003). 6. M. Fuhrman, Y. Hu, G. Tessitore, On a class of stochastic optimal control problems related to BSDEs with quadratic growth. Preprint. 7. M. Fuhrman, G. Tessitore, Existence of optimal stochastic controls and global solutions of forward-backward stochastic differential equations. SIAM J. Control Optim. 43, 813–830 (2004). 8. F. Gozzi, Global regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz nonlinearities. J. Math. Anal. Appl. 198, 399-443 (1996). 9. E. Pardoux, S. Peng, Adapted solution of a backward stochastic differential equation. Systems and Control Lett. 14, 55-61 (1990). 10. J. Yong, X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York, 1999.

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A FUNDAMENTAL THEOREM OF ASSET PRICING IN CONTINUOUS TIME WITH SQUARE INTEGRABLE PORTFOLIOS∗

HANQING JIN and XUN YU ZHOU Department of Systems Engineering and Engineering Management The Chinese University of Hong Kong {hqjin, xyzhou}@se.cuhk.edu.hk

This paper studies a continuous-time market with multiple stocks whose prices are governed by geometric Brownian motions, and admissible investment portfolios are defined via certain square integrability condition. It is proved that, when the investment opportunity set is deterministic (albeit possibly time varying), such a market being arbitrage free is equivalent to the existence of a square integrable (in time) market price of risk, and as a result equivalent to the existence of an equivalent martingale measure. Counterexamples are given to show that these equivalent results are no longer true in a market with a stochastic investment opportunity set.

Keywords: Continuous-time financial market, arbitrage opportunity, FTAP, martingale, square integrability, equivalent martingale measure.

1. Introduction Arbitrage-free is a basic assumption and a starting point for various problems (e.g., portfolio selection, option pricing, etc.) associated with a financial market. The so-called fundamental theorem of asset pricing (FTAP) is about an equivalent condition of the absence of arbitrage, which, in addition to economically characterizing this fundamental property of a market, should also provide an easily verifiable mathematical criterion. This, in turn, would make some powerful mathematical tools (such as martingale theory and functional analysis) available to analyze the market. In a dynamic multi-period market, it is well known that arbitrage-free is equivalent to the existence of an equivalent martingale measure [see, e.g., Harrison and Kreps,10 Dalang, Morton and Willinger,4 and F¨ ollmer and Schied9 (p. 217, Theorem 5.17)]. This is a very elegant and neat result. When it comes to the continuous-time setting, however, the validity of FTAP becomes a subtle issue. As elaborated in the very informative introduction section of Ref. 5, in a continuous∗ Supported

by the RGC Earmarked Grants CUHK 4175/03E and CUHK418605. The authors are indebted to Jia-an Yan, Chinese Academy of Sciences, for his helpful comments on an earlier version of the paper. 87

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time market arbitrage-free is only “essentially” equivalent to the existence of an equivalent martingale measure. The precise meaning of the word “essential” depends on the specific mathematical model under consideration and, in particular, on how the class of admissible portfolios is defined (more on this later), which has led to a few variants of the continuous-time FTAP in the literature. In Ref. 5, a very general bounded or locally bounded semimartingale model for (discounted) securities prices is investigated, and a condition stronger than no-arbitrage, called no free lunch with vanishing risk (NFLVR), is introduced. It is proved that NFLVR is equivalent to the existence of an equivalent martingale or local martingale (depending on whether the underlying securities prices are bounded or locally bounded) measure. In Ref. 6, a sequel to Ref. 5, unbounded securities price processes are considered, and similar equivalent condition is given with the local martingale replaced by the so-called sigma-martingale. While the results in these two papers5, 6 are very deep, the technical analyses therein are extremely involved, largely due to the generality of the model. Moreover, the equivalent conditions derived are not for no-arbitrage itself; and the existence of the equivalent local martingale/sigmamartingale is a rather stiff mathematical condition, which seems to be very hard to verify and use. A more specific and widely used continuous-time market model is the so-called geometric Brownian motion (GBM), up to the presence of some additional features such as a dividend and/or consumption; see, e.g., Refs. 17 and 13. In this model, the financial market consists of a risk-free bond and multiple risky stocks whose price processes are driven by a (multi-dimensional) Brownian motion. In Ref. 13 (p. 12, Theorem 4.2), it is stated that a necessary condition for the absence of arbitrage is the existence of a market price of risk; yet no further properties of this market price of risk are specified. Consequently this necessary condition is far too weak to be sufficient. Sufficient conditions are indeed presented in Ref. 13, namely the market price of risk is square integrable in time almost surely and the pricing kernel is a martingale. These conditions are relatively easier to check; unfortunately they are not if-and-only-if conditions for the absence of arbitrage. This drawback is nicely overcome in Ref. 15, where a necessary and sufficient condition for no-arbitrage is derived for a Brownian motion driven market with an invertible volatility matrix, via a pure probabilistic approach. [The results were later extended to the semimartingale setting in Ref. 19.] There is a very important point we need to note when studying FTAP. By its very definition whether a market is arbitrage free or not depends critically on the set of admissible investment strategies (portfolios) adopted. The admissible portfolios under consideration in Refs. 5, 6, 13 and 15 are the so-called tame strategies, namely, the ones whose corresponding wealth processes are almost surely bounded from below (the lower bounds may be different with different portfolios). As painstakingly argued in Refs. 5 and 13, imposing a lower bound on the wealth is to rule out the notorious doubling strategy that will lead to a sure win even on a finite time hori-

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zon. However, if one must rule out the doubling strategy a priori, there are many ways to do so. Being unbounded from below is just one property of the doubling strategy; one could certainly impose different restrictions to prevent the doubling strategy from being a legitimate strategy. For instance, square integrable portfolios will also exclude the doubling strategy (see Remark 2.4). It is important to note that tame portfolios eliminate many meaningful investment strategies from consideration. As an easy example, in a Black–Scholes market with one bond and one stock, shorting one share of the stock is a perfectly reasonable portfolio, which is however not tame. In this paper, we consider a GBM market where admissible portfolios are those that are square integrable — without any uniform boundedness restriction. The square integrability of portfolios is widely adopted in both multi-period and continuous-time setups; see, e.g., Refs. 14 and 7. Yet to our best knowledge an equivalent no-arbitrage condition is still absent for this class of admissible portfolios associated with a GBM model. Our direct motivation to solve this open problem, though, arises from the recent extensive study on the continuous-time version of Markowitz’s Nobel-prize-winning mean–variance portfolio selection problem [see, e.g., Refs. 20, 16, 12 and 1]. In these works, the no-arbitrage property of the underlying market is a prerequisite. One should note that the very nature of the mean–variance model requires an admissible portfolio to be square integrable, for otherwise the variance of the corresponding terminal wealth would not even be defined. Therefore, a characterization of the no-arbitrage condition with square integrable portfolios will lay a foundation to the continuous-time mean–variance portfolio selection model. The main result of this paper is that, when the set of the market coefficients [aka investment opportunity set in Merton’s terminology 17 ] is deterministic (yet possibly time varying), arbitrage-free is equivalent to the existence of a square integrable (in time) market price of risk. As an immediate consequence, arbitrage-free is equivalent to the existence of an equivalent martingale measure. Therefore, we have a version of continuous-time FTAP that is as clean and neat as that in the discrete-time case. The result is derived via exploring some property of the space L2 (0, T ; IRm ) (Lemma 4.1), together with a series of delicate analyses on functions that are out of this space. Furthermore, we show through two counterexamples that these equivalent results cannot be extended to the case with a stochastic investment opportunity set in either directions. Clearly our results are not covered by those in Refs. 5 and 6 or Ref. 15 in that different sets of investment strategies are being considered, and we do have an equivalent condition for arbitrage-free, rather than for NFLVR. Meanwhile we believe that our approach is new and different from those in existing literature. In addition our proofs are constructive in that we show explicitly arbitrage policies when the (mathematical) equivalent condition fails.

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The remainder of this paper is organized as follows. The market model is set up in section 2, and some preliminary results are given in section 3. Section 4 presents the main result (FTAP) of the paper, for the case when the investment opportunity set is deterministic. Section 5 contains two counterexamples demonstrating that the main results are no longer true with a stochastic investment opportunity set. Section 6 is devoted to the issue of market completeness. Finally, section 7 concludes the paper.

2. The Financial Market Throughout this paper T > 0 is a fixed, finite terminal time, (Ω, F, P, {Ft }0≤t≤T ) is a filtered complete probability space on which a standard m-dimensional Brownian motion W (t) ≡ (W 1 (t), · · · , W m (t))T with W (0) = 0 is defined. It is assumed that the filtration {Ft : 0 ≤ t ≤ T } is generated by the Brownian motion and augmented by all the P -null sets. For −∞ ≤ a < b ≤ +∞ and 1 ≤ p < +∞ we denote by Lp (a, b; IRd ) the set of all IRd -valued, measurable functions f (·) such Rb that a |f (t)|p dt < +∞, by LpF (a, b; IRd ) the set of all IRd -valued, Ft -progressively Rb measurable stochastic processes f (·) = {f (t) : 0 ≤ t ≤ T } such that E a |f (t)|p dt < +∞, and by LpFT (IRd ) the set of all IRd -valued, FT -measurable random variables d d ∞ η such that E|η|p < +∞. Moreover, L∞ (a, b; IRd ), L∞ F (a, b; IR ), and LFT (IR ) respectively denote the spaces of essentially bounded functions, Ft -progressively measurable processes, and random variables. Furthermore, we use AT to denote the transpose of a matrix A. Finally, a.s. (the abbreviation of “almost surely”) signifies that the corresponding statement holds true with probability 1 (with respect to P ). Consider a market where n+1 assets are traded continuously. One of the assets is the bond whose price process S0 (t) is subject to the following (stochastic) ordinary differential equation: dS0 (t) = r(t)S0 (t)dt, S0 (0) = s0 > 0,

(1)

where the interest rate process r(·) is a scalar-valued, Ft -progressively measurable stochastic process. The other n assets are stocks whose price processes S i (t), i = 1, · · · , n, satisfy the following stochastic differential equations:

dSi (t) = Si (t) bi (t)dt +

m X j=1

σij (t)dW j (t) , Si (0) = si > 0,

(2)

where bi (·) and σij (·), the appreciation and volatility rate processes, respectively, are scalar-valued, Ft -progressively measurable stochastic processes. The set {r(·), bi (·), σij (·), i = 1, · · · , n, j = 1, · · · , m} is called the investment opportunity set.

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Consider an agent, with an initial capital x0 at time 0, has a total wealth x(t) at time t > 0, while the dollar amount invested in stock i, i = 1, · · · , n, is πi (t). Assuming that the trading of shares takes place continuously in a self-financing fashion, i.e., n X πi (t) dSi (t), dx(t) = S (t) i=0 i

then we have the following wealth equation  n o Pn   dx(t) = r(t)x(t) + b (t) − r(t) π (t) dt i i  i=1   Pm Pn + j=1 i=1 σij (t)πi (t)dW j (t),      x(0) = x .

(3)

(4)

0

The process π(·) := (π1 (·), · · · , πn (·))T is called a (self-financing) portfolio or trading strategy. Set B(t) := (b1 (t) − r(t), · · · , bn (t) − r(t))T , σ(t) := (σij (t))n×m .

(5)

With this notation, equation (4) becomes dx(t) = [r(t)x(t) + π(t)T B(t)]dt + π(t)T σ(t)dW (t), x(0) = x0 .

(6)

A standing assumption imposed throughout this paper is Standing Assumption. r ∈ L∞ F (0, T ; IR). Now we define an “allowable” portfolio. Definition 2.1. A self-financing portfolio π(·) is said to be admissible (on [0, T ]) n if it -valued, Ft -progressively measurable R t stochastic process, with R Tis an IR E 0 |π(s)T σ(s)|2 ds < +∞ and E supt∈[0,T ] | 0 π(s)T B(s)ds|2 < +∞.

Remark 2.1. As elaborated in the introduction section the definition of admissible portfolios is vital in studying a market. The class of admissible portfolios defined here is mathematically and economically general enough, yet not too (mathematically) general to bury the true economical essence or significance into sheer technicality. Specifically, in Definition R2.1 we work within the framework of square T integrability. The requirement that E 0 |π(s)T σ(s)|2 ds < +∞ is to make the corresponding stochastic integral in (6) R Twell defined and a martingale (we could weaken it, as in many existing works, to 0 |π(s)T σ(s)|2 ds < +∞ a.s. so that the integral is a local martingale; but then we would end up only with some routine technical complication – e.g. evoking a stopping timeR argument – rather than essential t difference). The requirement that E supt∈[0,T ] | 0 π(s)T B(s)ds|2 < +∞ (which is weaker than that π(·)T B(·) be square integrable) is a weak condition to ensure that the terminal wealth is a square integrable random variable, which is necessary

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in studying the contingent claim problem. Moreover, this condition ensures that any admissible portfolio stopped by an Ft -stopping time is still admissible. This is certainly motivated by the observation that investors may stop their investment according to a pre-determined stopping rule. Remark 2.2. When the investment opportunity set is deterministic and there exists a function θ(·) ∈ L2 (0, T, IRm ) such that σ(t)θ(t) = B(t), then it is easy to show, viaRthe Cauchy–Schwartz inequality, that a portfolio π(·) is admissible if and T only if E 0 |π(t)T σ(t)|2 dt < +∞.

Remark 2.3. The Standing Assumption that r ∈ L∞ F (0, T ; IR) is for notational RT convenience only. It can be weakened to 0 |r(s)|ds < +∞ a.s., in which case one only needs R t to modify the definition of an admissible R Tportfolio π(·) to such that E supt∈[0,T ] | 0 S0 (s)−1 π(s)T B(s)ds|2 < +∞ and E 0 |S0 (s)−1 π(s)T σ(s)|2 ds < +∞. Definition 2.2. An admissible portfolio π(·) is called an arbitrage opportunity on [0, t] ⊆ [0, T ] if there exists an initial x0 ≤ 0 and a time s ∈ [0, t], so that the corresponding wealth process x(·) satisfies P (x(s) ≥ 0) = 1 and P (x(s) > 0) > 0. Moreover, a market is called arbitrage-free on [0, t] if there exists no arbitrage opportunity on [0, t]. Definition 2.3. A European contingent claim ξ ∈ L2FT (IR) is said to be replicable if there exists an initial wealth x0 and an admissible portfolio π(·) such that the corresponding wealth process x(·) satisfies x(T ) = ξ. Moreover, a market is called complete (on [0, T ]) if any contingent claim ξ ∈ L2FT (IR) is replicable. Remark 2.4. In Ref. 13 (pp. 8–9, Example 2.3), an example is given where a continuous-time analogue of a doubling strategy is explicitly constructed to rationalize the usage of the tame portfolios. In the example, r(·) = 0, b(·) = 0 and σ(·) = 1. TheR constructed strategy is π(t) = (T − t)−1/2 1τ ≥t , where V t τ := inf{t ∈ [0, T ) : 0 (T − u)−1/2 dW (u) = α} T and α > 0 a given number. It is shown in Ref. 13 that π(·) is a self-financed portfolio with the corresponding wealth process, starting from zero wealth, satisfying x(T ) = α (hence an arbitrage opportunity), and π(·) is not a tame portfolio. However, such a portfolio is also ruled out by our Definition 2.1 even though we do not impose any uniform lower boundedness. Indeed, if π(·) is admissible according to Definition R t 2.1, then the corresponding wealth process with zero initial wealth is x(t) = 0 π(s)dW (s), which RT would be a martingale. This contradicts to the fact that 0 π(t)dW (t) = α > 0. 3. Some Preliminary Results In this section we present some preliminary results that are useful in proving the main results of this paper.

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Theorem 3.1. If the market is arbitrage-free on [0, T ], then there exists an F t progressively measurable stochastic process θ(·) such that σ(t)θ(t) = B(t), a.e. t ∈ [0, T ], a.s.. Proof: By a measurable selection argument [see Ref. 12 (Lemma A.3)] we can prove that there exists a progressively measurable process θ0 (·) such that θ0 (t) ∈ argminx∈IRm |σ(t)x − B(t)|2 . Define π0 (t) := σ(t)θ0 (t) − B(t). Then σ(t)T π0 (t) = 0 and π0 (t)T B(t) = π0 (t)T [σ(t)θ0 (t) − π0 (t)] = −|π0 (t)|2 . Define    −π0 (t)/|π0 (t)|2 if π0 (t) 6= 0, π1 (t) :=  0 if π0 (t) = 0. Then π1 (t)T σ(t) = 0 and π1 (t)T B(t) = 1π0 (t)6=0 (t) ≤ 1, implying that π1 (·) is an admissible portfolio. Let x(·) be the wealth process corresponding to π1 (·) with the initial wealth x(0) = 0. Then dx(t) = r(t)x(t)dt + π1 (t)T [B(t)dt + σ(t)dW (t)] = r(t)x(t)dt + 1π0 (t)6=0 (t)dt. Hence x(T ) =

Z

T

e 0

RT t

r(s)ds

1π0 (t)6=0 (t)dt ≥ emin{rT,0}

Z

T 0

1π0 (t)6=0 (t)dt ≥ 0, a.s.,

where r is a lower bound of the essentially bounded process r(·). If there is no Ft -progressively measurable stochastic process θ(·) satisfying σ(t)θ(t) = B(t), then RT P { 0 1π0 (t)6=0 (t)dt > 0} > 0. Thus, P {x(T ) > 0} ≥ P {

Z

T

0

1π0 (t)6=0 (t)dt > 0} > 0,

which suggests that π0 (·) is an arbitrage opportunity.

Q.E.D.

The converse of Theorem 3.1 requires stronger conditions on θ(·). Define ρ(t) := e−

Rt 0

|θ(s)|2 /2ds−

Rt 0

θ(s)T dW (s)

, t ∈ [0, T ].

(7)

TheoremR3.2. If there exists an Ft -progressively measurable stochastic process θ(·) T such that 0 |θ(s)|2 ds < +∞ a.s., σ(t)θ(t) = B(t), a.e. t ∈ [0, T ], a.s., and ρ(·) is a martingale with E|ρ(T )|2 < +∞, then the market is arbitrage free on [0, T ]. Moreover, in this case for any admissible portfolio π(·), the corresponding wealth RT process x(·) satisfies x(t) = ρ(t)−1 E(e− t r(s)ds x(T )ρ(T )|Ft ) ∀t ∈ [0, T ].

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Proof: For any admissible portfolio π(·), denote by x(·) the corresponding wealth process starting from 0. Then Itˆ o’s formula yields d[¯ x(t)ρ(t)] = [ρ(t)¯ π (t)T σ(t) − ρ(t)¯ x(t)θ(t)T ]dW (t), where x ¯(t) := x(t)/S0 (t) and π ¯ (t) := π(t)/S0 (t). Therefore x ¯(·)ρ(·) is a local martingale. On the other hand, Z t Z t |¯ x(t)| = | π ¯ (s)T B(s)ds + π ¯ (s)T σ(s)dW (s)| 0 0 Z t Z t ≤ sup | π ¯ (s)T B(s)ds| + sup | π ¯ (s)T σ(s)dW (s)| =: X. t∈[0,T ]

t∈[0,T ]

0

0

2

Then EX < +∞ by the admissibility of π(·). It follows then from the assumption on ρ(·) that |¯ x(t)ρ(t)| ≤ Xρ(t) ≤ X supt∈[0,T ] ρ(t) ∈ L1FT (IR), implying that x ¯(t)ρ(t) isRa martingale (via the dominated convergence theorem). Hence T x(t) = ρ(t)−1 E(e− t r(s)ds x(T )ρ(T )|Ft ), and the market is arbitrage free. Q.E.D. Recall that an equivalent martingale measure (on [0, T ]) is a probability measure Q so that 1) Q is equivalent to P , and 2) the discounted stock prices, {Si (t)/S0 (t) : t ∈ [0, T ]}, i = 1, · · · , n, are martingales under Q. Proposition 3.1. If there exists an arbitrage opportunity π(·) on [0, T ] with RT T 2 |π(t) σ(t)| dt < N a.s. for some N ∈ IR, then there is no equivalent martin0 gale measure on [0, T ] in the market. Proof: Suppose there is an equivalent martingale measure Q under which S¯i (t) := Si (t)/S0 (t), i = 1, R T· · · , n, are martingales on [0, T ]. Let π(·) be an arbitrage opportunity π(·) with 0 |π(t)T σ(t)|2 dt < N a.s. as assumed, and x(·) the corresponding wealth process with initial wealth 0. Then it follows from (3) that the discounted wealth process x ¯(·) := x(·)/S0 (·) satisfies n X πi (t) ¯ d¯ x(t) = dSi (t). S (t) i=1 i

Furthermore, for any t ∈ [0, T ], Z tX n X n πi (s) πj (s) =

0 i=1 j=1 Z tX n X n

Si (s) Sj (s)

S0 (s)−2 πi (s)πj (s)

0 i=1 j=1

≤

−2rT s−2 0 e

dhS¯i , S¯j i(s)ds

Z

n X

σik (s)σkj (s)ds

k=1 t

π(s)T σ(s)σ(s)0 π(s)ds 0

−2rT ≤ s−2 N. 0 e

It follows from the theory of stochastic integration with respect to a vector-valued local martingale [see, e.g., Ref. 11] that x ¯(·) is a local martingale under Q.

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On the other hand, d¯ x(t) = π ¯ (t)T B(t)dt + π ¯ (t)T σ(t)dW (t), x ¯(0) = 0, Rt where π ¯ (t) := π(t)/S0 (t). It follows that h¯ x, x ¯i(t) = 0 |¯ π (s)T σ(s)|2 ds; thus −2 −2rT Q Q N < +∞, where E is the expectation under probabilE h¯ x, x ¯i(T ) < s0 e ity Q. This shows that x ¯(·) is a martingale under Q [see, e.g., Ref. 18 (p. 129, Proposition 1.23)], which contradicts to π(·) being an arbitrage opportunity. Hence an equivalent martingale measure does not exist. Q.E.D. Corollary 3.1. If there exists an equivalent martingale measure on [0, T ] in the market, then there exists an Ft -progressively measurable stochastic process θ(·) such that σ(t)θ(t) = B(t), a.e. t ∈ [0, T ], a.s.. Proof: If the conclusion is not true, then by the proof of Theorem 3.1 there is an arbitrage opportunity π1 (·) with π1 (t)T σ(t) ≡ 0. Proposition 3.1 then applies to yield that there is no equivalent martingale measure. Q.E.D. 4. FTAP: Deterministic Investment Opportunity Set Theorem 3.2 gives a necessary condition for a general market with stochastic parameters to be arbitrage-free. To our best knowledge there is no equivalent condition available in literature for a continuous-time market with square integrable portfolios. However, if the investment opportunity set is deterministic, then we will derive elegant equivalent conditions including FTAP. Hence, throughout this section we impose the following assumption. Assumption (D) r(·), bi (·) and σij (·) are all deterministic. The key result of this paper follows. Theorem 4.1. Under Assumption (D), the market is arbitrage-free on [0, T ] if and only if there exists a function θ(·) ∈ L2 (0, T ; IRm ) such that B(t) = σ(t)θ(t) a.e. t ∈ [0, T ]. Before we give a rigorous proof of this result, let us sketch its main idea. The “if” part is fairly straightforward by applying Theorem 3.2. The “only if” part is much deeper, the proof of which is one of the main contributions of this paper. The basic idea is as follows. If the market is arbitrage-free, then by Theorem 3.1 there is a measurable function θ(·) satisfying σ(t)θ(t) = B(t). There might be many such functions; so we take the “minimal” one by setting θ0 (t) := argminσ(t)θ=B(t) |θ|2 . Our target is then to prove that θ0 (·) ∈ L2 (0, T ; IRm ). We will prove it by connamely, we need to construct an arbitrage opportunity π(·) whenever Rtradiction, T 2 |θ (t)| dt = +∞. 0 0 We complete the construction (precisely speaking, the construction is for an auxiliary, simpler version of the wealth equation (6), see (9) below, which is nonetheless sufficient) for three different cases: Case 1: There exists a sub-interval [a, b] ⊆ [0, T ]

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Rb Rb such that a |θ0 (s)|2 ds = +∞ and t |θ0 (s)|2 ds < +∞ for any t ∈ (a, b). In other RT words, the infinity of the integration 0 |θ0 (t)|2 dt = +∞ is due to the ill-behavior, so to speak, of θ0 (·) in the right neighborhood R b of a certain pint a; Case R t 2: There exists a sub-interval [a, b] ⊆ [0, T ] such that a |θ0 (s)|2 ds = +∞ and a |θ0 (s)|2 ds < +∞ for any t ∈ (a, b). In this case, θ0 (·) is ill behaved in the left neighborhood of a cerRT tain pint b; and Case 3: Neither (i) nor (ii) occurs but overall 0 |θ0 (t)|2 dt = +∞. Notice that the last case may indeed happen. It turns out the constructions of arbitrage opportunities (bear in mind that the constructed portfolios must be admissible per Definition 2.1) for these three cases are very different, and all delicate. Yet, all the constructions are based on the following lemma of functional analysis. Lemma 4.1. A measurable function f ∈ L2 (0, T ; IRm ) if and only R T if 2 RT m T 2 | 0 f (t) g(t)dt| < +∞ for any g ∈ L (0, T ; IR ). Moreover, if 0 |f (t)| dt = +∞, then there exists g(·) ∈ L2 (0, T ; IRm ), which can be represented as g(·) R T = k(·)f (·) for a measurable function k(·) with k(t) ≥ 0 a.e. t ∈ [0, T ], such that 0 f (t)T g(t)dt = +∞. To prove this lemma we first need the following technical result. Lemma 4.2. Let {an : n = 1, 2, · · · } be a nonnegative sequence with a1 > 0 and P∞ Pn n=1 an = +∞, and let bn := i=1 ai . Then ∞ X an = +∞, b n=1 n

Proof: Observe

ai bi−1

=

bi bi−1

and

∞ X an < +∞. b2 n=1 n

(8)

bi − 1 ≥ ln bi−1 for any integer i ≥ 2. Hence

∞ ∞ X X ai ≥ (ln bi − ln bi−1 ) = +∞, ∀n ≥ 2. b i=n i−1 i=n

Now, if for any integer N > 0 there exists n > N such that abnn > 12 , then clearly P ∞ an n=1 bn = +∞. Therefore we may assume that there exists an integer N > 0 such that for any n > N , abnn ≤ 12 . In this case, for any n > N , bn−1 ≥ 12 bn , implying an an bn ≥ 2bn−1 and, thus, ∞ ∞ X an 1 X an ≥ = +∞. bn 2 bn−1

n=N +1

n=N +1

This proves the first equality of (8). For the second inequality of (8), notice that

Thus,

P∞

n=1

an bn − bn−1 bn − bn−1 1 1 = ≤ = − , ∀n ≥ 2. b2n b2n bn bn−1 bn−1 bn P∞ an 1 1 1 2 n=2 ( bn−1 − bn ) = a1 < +∞. b2 ≤ a1 + n

Q.E.D.

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Proof of Lemma 4.1. The if ” part is obvious. To prove the “if” part, it R T “only 2 suffices to show that if 0 |f (t)| dt = +∞ with a measurable function f on [0, T ], RT then there exists g(·) ∈ L2 (0, T ; IRm ) such that 0 f (t)T g(t)dt = +∞. To this end, for any integer n ≥ 1, denote In := {t ∈ [0, T ] : |f (t)| ∈ [n − 1, n)}, I∞ := {t ∈ [0, T ] : |f (t)| = +∞}, and d(t) := min{n ∈ N : |f (t)| < n}. Clearly d(t) = n if t ∈ In . On the other hand, if µ(I∞ ) > 0, where µ is the Lebesgue measure, then g := |ff | 1f 6=0 is the desired function. So we assume hereafter that µ(I∞ ) = 0. R Define an := In |f (t)|2 dt, n = 1, 2, · · · . Without loss of generality, suppose Pn f (t) a1 > 0. Define bn := i=1 ai and g(t) := bd(t) , t ∈ [0, T ]. Then we have the following identities Z T Z T ∞ Z ∞ Z ∞ X X X |f (t)|2 |f (t)|2 an |f (t)|2 dt = dt = dt = , |g(t)|2 dt = 2 2 2 bd(t) bd(t) bn b2 0 0 n=1 In n=1 In n=1 n and Z T

Z

∞ Z ∞ Z ∞ X X X |f (t)|2 |f (t)|2 |f (t)|2 an dt = dt = dt = . b b b b n d(t) d(t) 0 0 n=1 In n=1 In n=1 n RT RT P∞ Since 0 |f (t)|2 dt ≡ n=1 an = +∞, Lemma 4.2 applies to yield 0 |g(t)|2 dt < +∞ RT and 0 f (t)T g(t)dt = +∞. Finally, the second assertion of the lemma is evident from the constructive proof above. Q.E.D.

f (t)T g(t)dt =

T

Remark 4.1. Lemma 4.1 appears to be a standard result in functional analysis. Yet, surprisingly, we have searched this result in major texts in functional analysis to no avail. Hence we supply a proof here. Remark 4.2. Lemma 4.1 is valid if the interval [0, T ] is replaced by any [a, b] ⊆ [0, T ]. Indeed, based on its proof the lemma can be extended to any measure space (S, G, µ) with µ(S) < +∞. Next let us consider the following auxiliary, “controlled” Itˆ o process dx(t) = u(t)T θ(t)dt + u(t)T dW (t), x(0) = 0,

(9)

where θ(·) is a given IRm -valued measurable function with |θ(t)| < +∞ a.e. t ∈ [0, T ], and u(·) ∈ L2F (0, T ; IRm ) can be regarded as a control. Rb Lemma 4.3. If there is an interval [a, b] ⊆ [0, T ] such that a |θ(t)|2 dt = +∞ and Rb |θ(t)|2 dt < +∞ for any t ∈ (a, b), then there exists u(·) ∈ L2F (0, T ; IRm ) with RtT |u(t)|2 dt < N, a.s. for some N ∈ IR, u(t)T θ(t) ≥ 0, a.s., a.e. t ∈ [0, T ] and 0 Rt E supt∈[0,T ] | 0 u(s)T θ(s)ds|2 < +∞, so that the corresponding Itˆ o process in (9) satisfies x(T ) ≥ 0 a.s., and P (x(T ) > 0) > 0. Moreover, u(t) = α(t)θ(t) for some Ft -progressively measurable process α(·) ≥ 0.

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Proof: By Lemma 4.1, there exists a nonnegative measurable function k(·) on m 2 T 2 [a, b] such R b thatT g(·) := k(·)θ(·) satisfies g(·) ∈ L (a, b; IR ), g(·) θ(·) ≡ k(·)|θ(·)| ≥ 0, and a g(t) θ(t)dt = +∞. Extend k(·) to [0, T ] (still denoted by k(·)) by setting zero value outside of [a, b]R and let g(·) := k(·)θ(·). T Define f (t) := exp{− t g(s)T θ(s)ds}. Then by the assumption on θ(·) one has f = 0 on [0, a], f > 0 on (a, b], f = 1 on (b, T ], and f 0 (t) = f (t)g(t)T θ(t) ≥ 0 ∀t ∈ (0, T ] (hence f Ris non-decreasing). V t / [ 41 , 1]} b, and u(t) := Set y(t) := 21 + 0 g(s)T dW (s), τ := inf{t ≥ 0 : y(t) ∈ RT RT f (t)g(t)1t<τ . Then τ ≥ a, P (τ > a) > 0, and 0 |u(t)|2 dt ≤ 0 |f (t)g(t)|2 dt ≤ RT |g(t)|2 dt < +∞. Moreover, u(t)T θ(t) = f (t)k(t)|θ(t)|2 1t<τ ≥ 0, and 0 Z t Z t Z t |u(s)T θ(s)|ds ≤ |f (s)g(s)T θ(s)|ds = f 0 (s)ds = f (t) ≤ 1, a.s., ∀t ∈ [0, T ], 0

0

0

Rt

T

2

which yields E supt∈[0,T ] | 0 u(s) θ(s)ds| ≤ 1. Furthermore, for any t ∈ [0, T ], Z t Z t Z t f (t)y(t) = d[f (s)y(s)] = y(s)f (s)g(s)T θ(s)dt + f (s)g(s)T dW (s). (10) 0

0

0

Now, for x(·) of (9) under u(·), one has Z T Z T x(T ) = u(t)T θ(t)dt + u(t)T dW (t) 0 Z0 τ Z τ T = f (t)g(t) θ(t)dt + f (t)g(t)T dW (t) 0 0 Z τ Z τ T f (t)g(t)T dW (t) y(t)f (t)g(t) θ(t)dt + ≥ 0

0

= f (τ )y(τ ) ≥ 0, a.s., and the last inequality can be replaced by a strict inequality with a positive probability (recall that P (τ > a) > 0). Finally, u(t) = α(t)θ(t) with α(t) := k(t)f (t)1t<τ ≥ 0. Q.E.D. Rb Lemma 4.4. If there is an interval [a, b] ⊆ [0, T ] such that a |θ(s)|2 ds = +∞, yet Rt 2 a |θ(s)| ds < +∞ for any t ∈ (a, b), then the conclusion of Lemma 4.3 holds.

Proof: As in the proof of Lemma 4.3, there exists a measurable k(·) ≥ 0 such that k(t) = 0 ∀t ∈ R[0, a) ∪ (b, T ] andR g(·) := k(·)θ(·) satisfies g(·) ∈ L2 (0, T ; IRm ), T b g(·)T θ(·) ≥ 0, and 0 g(t)T θ(t)dt ≡ a g(t)T θ(t)dt = +∞. Define τ := inf{t ≥ 0 : Rt R V t g(s)T θ(s)ds − 1 ≥ − 0 g(s)T dW (s)} b. Then τ ≥ a a.s.; moreover, it follows 0 Rt from a |θ(s)|2 ds < +∞ ∀t ∈ (a, b) that τ > a a.s.. On the other hand, τ < b a.s. by Rb Rτ virtue of the fact that 0 g(t)T θ(t)dt = +∞. As a consequence, 0 g(s)T θ(s)ds−1 = Rτ RT RT − 0 g(s)T dW (s). Now define u(t) := g(t)1t≤τ . Then 0 |u(t)|2 dt ≤ 0 |g(t)|2 dt <

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RT Rτ Rτ +∞, u(t)T θ(t) ≥ 0, 0 u(s)T θ(s)ds = 0 g(s)T θ(s)ds = 1 − 0 g(s)T dW (s), imRt RT plying that E supt∈[0,T ] | 0 u(s)T θ(s)ds|2 = E( 0 u(s)T θ(s)ds)2 < +∞. Finally, x(T ) = x(τ ) = 1, a.s.. So u(·) is the desired process. Q.E.D. RT Proposition 4.1. If 0 |θ(t)|2 dt = +∞, then there exists u(·) ∈ L2F (0, T ; IRm ) RT with 0 |u(t)|2 dt < N, a.s. for some N ∈ IR, u(t)T θ(t) ≥ 0, a.s., a.e. t ∈ [0, T ] and Rt E supt∈[0,T ] | 0 u(s)T θ(s)ds|2 < +∞, so that the corresponding Itˆ o process in (9) satisfies x(T ) ≥ 0 a.s., and P (x(T ) > 0) > 0. Moreover, u(t) = α(t)θ(t) for some Ft -progressively measurable process α(·) ≥ 0. Proof: If the conclusion is not true, then we claim the following Z a1 Z T There exists a1 ∈ (0, T ) such that |θ(t)|2 dt = +∞, |θ(t)|2 dt = +∞. (11) a1

0

R a0 Indeed, if (11) is false, then take b := inf{a0 ≥ 0 : 0 R|θ(t)|2 dt = +∞}. We a must have 0 < b < T (if b = 0, then for any a ∈ (0, T ), 0 |θ(t)|2 dt = +∞, and RT hence a |θ(t)|2 dt < +∞ since (11) is not true. This, together with the fact that RT 2 0 |θ(t)| dt = +∞, would lead to the conclusion of the theorem in view of Lemma 4.3. On the other hand, if b = T , then the conclusion of the theorem is valid by virtue of Lemma 4.4). Now, to invalidate (11) we must have either Z b Z T |θ(t)|2 dt = +∞ and |θ(t)|2 dt < +∞. (12) b

0

or Z

b 2

|θ(t)| dt < +∞ and 0

Z

T

|θ(t)|2 dt = +∞.

(13)

b

Ra First suppose (12) is the case. By the definition of b we have 0 |θ(t)|2 dt < +∞ for any a ∈ (0, b). Hence it follows from Lemma 4.4 that the conclusion of the theorem holds true which by the R a is a contradiction. Next, if (13) is true. Then Ragain T definition of b, 0 |θ(t)|2 dt = +∞ for any a ∈ (b, T ). Consequently, a |θ(t)|2 dt < +∞ for any a ∈ (b, T ), for otherwise (11) would be true. However, this along with the second equality of (13) yields that the conclusion of the theorem is valid due to Lemma 4.3. We have now proved and we can apply trick to [a1 , T ] to find an R a(11), R Tthe same 2 2 2 a2 ∈ (a1 , T ) such that a1 |θ(t)| dt = +∞ and a2 |θ(t)| dt = +∞. Continuing with this, find a strictly increasing sequence {ai , i = 1, 2, · · · } on (0, T ) such that R ai we can 2 |θ(t)| dt = +∞. Denote c := limi→+∞ ai > a1 . ai−1 On any interval (ai , ai+1 ), it follows from Lemma 4.1 (and Remark 4.2) that there a measurable k(·) ≥ 0 such that g(·) = k(·)θ(·) ∈ L2 (aRi , ai+1 ; IRm ) and R ai+1 exists a g(t)T θ(t)dt = +∞. By scaling properly we can assume that aii+1 |g(t)|2 dt ≤ ai

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2−i . On the other hand, define    g(t), if g(t)T θ(t) ≤ k, gk (t) :=   0, if g(t)T θ(t) > k,

k = 1, 2, · · · .

Evidently gk (t) → g(t), a.e. t ∈ [0, T ] as k → +∞. Noticing Rthat each gk (t)T θ(t) a is nonnegative, we apply Fatou’s lemma to obtain limk→+∞ aii+1 gk (t)T θ(t)dt ≥ R ai+1 g(t)T θ(t)dt = +∞. In particular, we can take k sufficiently large to ensure ai R a that aii+1 gk (t)T θ(t)dt > 1. To summarize, we have proved that on (ai , ai+1 ) there exists a measurable function, denoted by gi (·), with the following properties: Z ai+1 Z ai+1 2 −i |gi (t)| dt ≤ 2 , 1 < gi (t)T θ(t)dt < +∞. (14) ai

ai

Rt RT Denote h(t) := i=1 gi (t)1t∈(ai ,ai+1 ] . Then 0 |h(t)|2 dt ≤ 1, 0 h(s)T θ(s)ds < Rt +∞ for any t < c, and meanwhile 0 h(s)T θ(s)ds−→+∞ as t → c. Set τ := inf{t ≥ Rt Rt 0 : 0 h(s)T θ(s)dt − 1 ≥ − 0 h(s)T dW (s)}. The preceding properties of h(·) imply that 0 < τ < c, a.s.. Hence employing exactly the same approach as in the proof of Lemma 4.4 we conclude that u(t) := h(t)1t<τ is the desired process. Q.E.D. P∞

Now we are in the position to prove Theorem 4.1. Proof of Theorem 4.1. We first assume that r(·) = 0. If there is θ(·) ∈ L2 (0, T ; IRm ) such that B(t) = σ(t)θ(t) a.e. t ∈ [0, T ], then by Novikov’s criterion ρ(·) defined by (7) is a martingale. Moreover, E|ρ(T )|2 = e

RT 0

|θ(s)|2 ds

1

Ee− 2

RT 0

|2θ(s)|2 ds−

RT 0

2θ(s)T dW (s)

=e

RT 0

|θ(s)|2 ds

< +∞.

So we can apply Theorem 3.2 to conclude that the market is arbitrage free on [0, T ]. Conversely, if the market is arbitrage free on [0, T ], then by Theorem 3.1, there exists a measurable θ(·) such that σ(t)θ(t) = B(t) a.e. t ∈ [0, T ]. Set θ0 (t) := argminσ(t)θ=B(t) |θ|2 . Then θ0 (·) is a well-defined measurable function satisfying σ(t)θ0 (t) = B(t). Moreover, since θ0 (t) point-wisely minimizes |θ|2 subject to σ(t)θ = B(t), there is a measurable Lagrange multiplier function λ(·) such that θ0 (t) = σ(t)T λ(t), a.e. t ∈ [0, T ]. We now claim that θ0 (·) ∈ L2 (0, T ; IRm ). If not, then it follows from Proposition 4.1 that there exist a measurable function α(·) ≥ 0 such that under u(·) := α(·)θ0 (·), the following equation dx(t) = u(t)T θ0 (t)dt + u(t)T dW (t), x(0) = 0

(15)

admits a solution satisfying x(T ) ≥ 0 a.s., and P (x(T ) R> 0) > 0. Moreover, RT t |u(t)|2 dt < N, a.s. for some N ∈ IR, and E supt∈[0,T ] | 0 u(s)T θ0 (s)ds| < +∞. 0 Define π(t) := α(t)λ(t). Then π(t)T σ(t) = α(t)λ(t)T σ(t) = α(t)θ0 (t)T = u(t)T ,

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and π(t)T B(t) = α(t)λ(t)T B(t) = α(t)λ(t)T σ(t)θ0 (t) = u(t)T θ0 (t). Together with (15) this means that π(·), being an admissible portfolio, is indeed an arbitrage opportunity, which leads to a contradiction. Finally, if r(·) 6≡ 0, then (¯ x(·), π ¯ (·)) := (x(·)/S0 (·), π(·)/S0 (·)) satisfies a wealth equation with r(·) = 0. In view of the Standing Assumption that r(·) is essentially bounded, the result follows readily. Q.E.D. We finally arrive at our FTAP. Theorem 4.2. Under Assumption (D), the following three statements are equivalent. (a) The market is arbitrage free on [0, T ]. (b) There is an equivalent martingale measure on [0, T ]. (c) There exists θ(·) ∈ L2 (0, T ; IRm ) such that B(t) = σ(t)θ(t) a.e. t ∈ [0, T ]. Proof: Theorem 4.1 shows that that (a) and (c) are equivalent. Now, if (c) holds, then define a new probability measure Q by Rt Rt 2 T dQ := e− 0 |θ(s)| /2ds− 0 θ(s) dW (s) , t ∈ [0, T ]. dP Ft

Girsanov’s theorem along with Novikov’s condition yields that Q is an equivalent martingale measure; hence (b). Conversely, suppose (b) holds. By Corollary 3.1, there exists θ(·) such that σ(t)θ(t) = B(t) a.e. t ∈ [0, T ]. Defining θ0 (·) := argminσ(t)θ=B(t) |θ|2 , we claim that θ0 (·) ∈ L2 (0, T ; IRm ). Indeed, if this is not true, then employing an exactly the same argument as that in the proof of Theorem R T 4.1 we conclude that there exists an arbitrage opportunity π(·) on [0, T ] with 0 |π(t)T σ(t)|2 dt < N, a.s. for some N ∈ IR. It follows from Proposition 3.1 then that there exists no equivalent martingale measure in the market, which is a contradiction. This proves (c). Q.E.D. Theorem 4.2 (especially the equivalence between (a) and (c)) provides very convenient and easy way to check whether or not a market (with a deterministic investment opportunity set) is arbitrage free. Here are a couple of examples.

Example 4.1. Consider a market where there is a bond and a stock. Suppose r(t) = 0, b(t) = 0.1, σ(t) = 0.1tα (α > 0), and T = 1 (year). For any α > 0, the volatility rate of the stock starts off very small, and gradually increases and reaches maximum at the terminal time. Now, the only θ(·) (up to the exception on a null-set) satisfying σ(t)θ(t) = B(t) is θ(t) = t−α (t > 0). Thus according to Theorem 4.2 the market is arbitrage-free if and only if θ(·) is square integrable, or α ∈ (0, 12 ). This shows that the variability of the market crucially depends on the value of α and, in particular, α = 12 is the critical number.

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Example 4.2. For a market with multiple stocks and time-invariant market coefficients, the arbitrage-free condition boils down to the solvability of the algebraic equation σθ = B (with θ being the unknown). There are of course many convenient algebraic criteria to check the latter. 5. Counterexamples: Stochastic Investment Opportunity Set Given the results in the previous section, it is tempting to extend them to the case with a stochastic investment opportunity set. Unfortunately, this cannot be done. In this section, we show, via two counterexamples, that with a stochastic investment opportunity set Theorem 4.1 is no longer true in either directions if we replace θ ∈ L2 (0, T, IRm ) by θ ∈ L2F (0, T, IRm ). Example 5.1. In this example we show that arbitrage-free does not ensure the existence of θ ∈ L2F (0, T, IRm ) satisfying B(t) = σ(t)θ(t). Consider a market with one bond and one stock, with r(t) = 0, B(t) = 1, and σ(t) = 1t 0 such that R t¯ ¯ ¯ x(t) = x0 + 0 π(s)B(s)ds + 0 π(s)σ(s)dW (s) satisfies P (x(t) ≥ 0) = 1, P (x(t¯) > 0) > 0. Since the market coefficients are deterministic constants up to t0 , there is no arbitrage opportunity on [0, t0 ]; so t¯ > t0 . On the other hand, for almost all ω, the following equation Z t Z t 1 x(t) = x(t0 , ω) + π(s)ds + π(s)dW (s), t ∈ (t0 , T ] ξ(ω) t0 t0 can be regarded as a wealth equation associated with a market, which has deterministic coefficients, on the probability space (Ω, F, P (·|Ft0 )(ω)) and a time horizon [t0 , T ]. So we can apply Theorem 3.2 to obtain x(t0 ) = E(x(t¯)e−ξ

2

(t¯−t0 )/2−ξ(W (t¯)−W (t0 ))

|Ft0 ) a.s..

It follows that P (x(t0 ) ≥ 0) = 1, P (x(t0 ) > 0) > 0, which is impossible because there is no arbitrage opportunity on [0, t0 ]. Example 5.2. This example demonstrates that the existence of θ ∈ L2F (0, T, IRm ) satisfying B(t) = σ(t)θ(t) needs not lead to the absence of arbitrage. Consider a 5dimensional Bessel process R(·) starting from 1 (namely, R(·) is the radial part of a 5-dimensional Brownian motion with R(0) = 1), z(t) := R(t)2 , and v(t) := R(t)−1 . By the theory of Bessel process [see, e.g., Ref. 18, p. 439], p ¯ (t) (16) dz(t) = 5dt + 2 z(t)dW

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¯ (·). On the other hand, inf 0≤t<+∞ R(t) has for some standard Brownian motion W a probability distribution function F (y) = y 3 , 0 ≤ y ≤ 1 [see Ref. 2 (p. 373, 1.2.4)]; thus Z 1 1 3 1 1 2 ≤E = dy = 3. Ev(t) = E 2 R(t)2 (inf 0≤t<+∞ R(t))2 y 0 Moreover, it is evident that ER(t)2 = 1 + 5t. So we have proved that Z 1 Z 1 2 E R(t) dt + E v(t)2 dt < +∞. 0

(17)

0

Now, construct a market with one bond and one stock, and a terminal time T = 2. The stock is driven by the following Brownian motion   W ˜ (t), if t < 1, W (t) =  W ¯ (t − 1) + W ˜ (1), if 1 ≤ t ≤ 2,

˜ (·) is a standard Brownian motion independent of W ¯ (·). Let the market where W coefficients be r(t) = 0, σ(t) = 2, B(t) = 1t<1 + 7v(t − 1)1t≥1 , t ∈ [0, 2]. Then θ(·) := B(·)/σ(·) = 12 1·<1 + 27 v(t − 1)1·≥1 ∈ L2F (0, 2; IR) by virtue of (17). Take π(t) := R(t − 1)1t≥1 , t ∈ [0, 2]. It is straightforward to verify, thanks to (17), that π(·) is an admissible portfolio. Nevertheless, the corresponding terminal wealth with initial x(0) = 0 is R2

R(t − 1)v(t − 1)dt + 2 R1p ¯ (t) = 7 + 2 0 z(t)dW

x(2) = 7

1

R2 1

R(t − 1)dW (t)

= 7 + z(1) − z(0) − 5 ≥ 1,

where the last equality was due to (16). So π(·) is an arbitrage opportunity. 6. Completeness Completeness (or the lack thereof) is another fundamental property of a market. It turns out that we can fully characterize the completeness based on the FTAP obtained in section 4. Theorem 6.1. Under Assumption (D), if the market is arbitrage free, then the market is complete on [0, T ] if and only if Rank(σ(t)) = m, a.e. t ∈ [0, T ]. Proof: First of all, it follows from Theorem 4.1 that there exists θ(·) ∈ L2 (0, T ; IRm ) such that σ(t)θ(t) = B(t), a.e. t ∈ [0, T ]. As a result, by Ref. 3 the following backward stochastic differential equation (BSDE) dx(t) = u(t)T [θ(t)dt + dW (t)], x(T ) = ξ

(18)

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admits a unique solution pair (x(·), u(·)) ∈ L2F (0, T ; IR) × L2F (0, T ; IRm ), where ξ ∈ L2FT (IR) is arbitrarily given. To prove the theorem, it suffice to prove for the case when r(·) = 0 in view of the Standing Assumption. If Rank(σ(t)) = m, a.e. t ∈ [0, T ], then there exists an IRn -valued Ft -progressively measurable process satisfying σ(t)T π(t) = u(t), a.e. t ∈ [0, T ], a.s. [the existence of such a process can be proved by a measurable selection theorem as in Ref. 12, Lemma A.3]. Substituting it in (18) we have that (x(·), π(·)) satisfies the wealth equation. Moreover, is admisR t π(·) m T 2 T sible since σ(·) π(·) = u(·) ∈ LF (0, T ; IR ) and E supt∈[0,T ] | 0 π(s) B(s)ds|2 = Rt RT RT E supt∈[0,T ] | 0 u(s)T θ(s)ds|2 ≤ 0 |θ(s)|2 ds E 0 |u(s)|2 ds < +∞. Thus ξ is replicable by an admissible portfolio. Conversely, suppose that the market is complete. For any u ∈ Qm , where Qm is the set of all the m-dimensional rational vectors, let y(·) satisfy dy(t) = uT [θ(t)dt + dW (t)], y(0) = 0. Then y(T ) ∈ L2FT (IR), and hence is replicable; that is, there exists an admissible portfolio π(·) such that dx(t) = π(t)T σ(t)[θ(t)dt + dW (t)], x(T ) = y(T ). Comparing the two preceding equations and by the uniqueness of the BSDE solution we conclude that σ(t)T π(t) = u, a.e. t ∈ [0, T ]. We have proved that {σ(t)T v : v ∈ IRm } ⊇ Qm , a.e. t ∈ [0, T ]. Since {σ(t)T v : v ∈ IRm } is a closed set, we conclude {σ(t)T v : v ∈ IRm } ⊇ IRm , a.e. t ∈ [0, T ]. Hence Rank(σ(t)) = m, a.e. t ∈ [0, T ]. Q.E.D. Remark 6.1. Again, the preceding proof depends heavily on Assumption (D). An equivalent condition of the market completeness with a stochastic investment opportunity set (with the admissible portfolios defined as in this paper) remains an open problem as far as we are concerned. 7. Concluding Remarks In this paper we have derived a continuous-time FTAP, in the form of if and only if conditions, for the diffusion market model. We would like to reiterate that the results are not superseded or covered by existing ones [such as those of Refs. 5, 6, 13 and 15 since their admissible portfolios are defined differently (see elaboration in the introduction section). We have emphasized that the form of FTAP is crucially dependent of the definition of admissible strategies. No only so. In fact many other market properties (e.g., market completeness and certain optimality) also depend largely on how one defines the admissible portfolios. We feel that the validity of an equivalent FTAP condition testifies the sensibility of the set of admissible strategies being chosen.

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Indeed, an equivalent condition, if available, would suggest that the underlying admissibility of strategies is exactly, no more and no less, what is required to make a market viable, and hence is a sensible choice of admissibility in studying most other problems in this market. It certainly remains an outstanding open problem to derive an equivalent arbitrage-free condition for a market with a stochastic investment opportunity set and square integrable portfolios. As the counterexamples in section 5 suggest, such a condition, if indeed available, must be quite different from the ones in this paper. References 1. Bielecki, T.R., H. Jin, S.R. Pliska and X.Y. Zhou: “Continuous-Time Mean– Variance Portfolio Selection with Bankruptcy Prohibition,” Mathematical Finance, 15, 213–244 (2005). 2. Borodin, A.N., and P. Salminen: Handbook of Brownian Motion – Facts and Formulae, 2nd Edition, Birkh¨ auser Verlag, Basel (2002). 3. Chen, Z.J.: “Existence and Uniqueness for BSDE with Stopping Time,” Chinese Science Bulletin, 43, 2379–2382 (1998). 4. Dalang, R.C., A. Morton and W. Willinger: “Equivalent Martingale Measures and No-arbitrage in Stochastic Securities Market Model,” Stochastics and Stochastics Reports, 29, 185–201 (1990). 5. Delbaen, F. and W. Schachermayer: “A General Version of the Fundamental Theorem of Asset Pricing,” Mathematische Annalen, 306, 463–520 (1994). 6. ——: “The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes,” Mathematische Annalen, 312, 215–250 (1998). 7. Duffie, D. and C.F. Huang: “Multiperiod Security Markets with Differential Information,” Journal of Mathematical Economics, 15, 283–303 (1986). 8. Elliott, R.J. and P.E. Kopp: Mathematics of Financial Markets, New York: Springer-Verlag (1999). ¨ llmer, H. and A. Schied: Stochastic Finance: An Introduction in Discrete Time, 9. Fo Berlin/New York: Walter de Gruyter (2002). 10. Harrison, M.J. and D.M. Kreps: “Martingales and Arbitrage in Multiperiod Securities Markets,” Journal of Economic Theory, 20, 381–408 (1979). 11. Jacod, J.: “Int´egrales Stochastiques par Rapport a ` une Semimartingale Vectorielle et Changements de Filtration,” S´ eminaire de Probabilite XIV, Lecture Notes in Mathematics, 784, Berlin: Springer-Verlag, 161–172 (1980). 12. Jin, H. and X.Y. Zhou: “Continuous-time Markowitz’s Problems in an Incomplete Market, with Constrained Portfolios,” Working paper, The Chinese University of Hong Kong (2005).

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13. Karatzas I. and S.E. Shreve: Methods of Mathematical Finance, New York: Springer-Verlag (1998). 14. Kreps D.M.: “Arbitrage and Equilibrium in Economics with Infinitely Many Commodities,” Journal of Mathematical Economics, 8, 15–35 (1981). 15. Levental S. and A.V. Skorohod: “A Necessary and Sufficient Condition for Absence of Arbitrage with Tame Portfolios,” The Annals of Applied Probability, 5, 906–925 (1995). 16. Lim, A.E.B.: “Quadratic Hedging and Mean–Variance Portfolio Selection with Random Parameters in an Incomplete Market,” Mathematics of Operations Research, 29, 132–161 (2004). 17. Merton, R.C.: Continuous-Time Finance, Oxford: Blackwell (1992). 18. Revuz, D. and M. Yor: Continuous Martingales and Brownian Motion, 3rd Edition, Berlin: Springer-Verlag (1999). 19. Strasser, E.: “Characterization of Arbitrage-Free Markets,” The Annals of Applied Probability, 15, 116–124 (2005). 20. Zhou X.Y. and D. Li: “Continuous Time Mean–Variance Portfolio Selection: A Stochastic LQ Framework,” Applied Mathematics and Optimization, 42, 19–33 (2000).

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Chap19-MaJin

INDIFFERENCE PRICING OF UNIVERSAL VARIABLE LIFE INSURANCE

JIN MA∗ and YUHUA YU Department of Mathematics Purdue University West Lafayette, IN 47907-1395, USA {majin, yyu}@math.purdue.edu

In this paper we study the pricing problem for a class of Universal Variable Life (UVL) insurance products, using the idea of “Principle of Equivalent Utility”. The main features of the UVL products include the varying (death) benefit based on both tradable and non-tradable investment incomes and “multiple decrement” cases. We formulate the pricing problem as stochastic control problems, and derive the corresponding HJB equations for the value functions. In the case of exponential utilities, we obtain the explicit pricing formulae in terms of the global, bounded solutions of a class of semilinear parabolic PDEs with exponential growth. The general insurance models with multiple decrements and random time benefit payments are discussed as well. Keywords: Principle of equivalent utility, indifference pricing, HJB equations.

1. Introduction In this paper we are interested in the pricing of the so-called Universal Variable Life Insurance (UVL for short) products, initiated in as early as the 1950’s. The main features of such a product include, but not limited to: allowing a low risk cash account and linking the death benefit to the investment returns, and the flexibility on the premiums. Due to the additional investment growth potential as opposed to the traditional life insurance, the UVL insurance became an attractive financial product and has grown substantially in the recent years. However, the dependence of the death benefit in a UVL contract on various uncertainties, especially the investment performance and the possible withdrawal/addition activities of the insured, makes the pricing of such products more complicated than those of traditional life insurance products as well as the usual contingent claims in finance theory. In this paper we apply the so-called “indiffernece pricing” method, first introduced by Hodges and Neuberger in Ref. 5 to solve the pricing problme. Such a method was studied in a recent paper by YoungZariphopoulou 18 for life insurance with constant death benefit, and by Musuiela∗ This

author is supported in part by nsf grant #0204332. 107

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Zariphopoulou 11,12 , for indifferece pricing with nontradable assets. We refer also to Bielecki-Jeanblanc-Rutkowski 1 , Frittelli 3 , Owen 14 , and Rouge-El Karoui 16 for general framework of the indifference pricing as well as the dynamic utility optimization that are close to our problem. However, our UVL model allows some special features that does not seems to be covered by existing literatures. This will particularly be the case when the multiple decrements are considered. Following a more general strategy without using the knowledge of the arbitrage price of the death benefit at any time t, as was done in Ref. 18, we derive the pricing formula for simple UVL models. We also show that such a strategy will actually produce the same solution if the benchmark asset is tradable. In the case of exponential utility function, we provide a detailed analysis assuming the general benefit functions which could depend on the non-tradable assets. It turns out that in this case the method of “separation of variables” (see, e.g., Ref. 18) still works well, but result will eventually depend on the solvability of a semilinear (reaction-diffusion) PDE with exponential growth on the reaction term. We prove that in our case blow-up does not occur, and the global, bounded solution exists, and the closed form solutions of the pricing problems then follow. We will also give a brief description of the general insurance model and present the form of the corresponding HJB equation, which in this case is a system of fully nonlinear partial differential-difference equations of parabolic type. The rest of the paper is organized as follows. In section 2 we formulate the problems and set up the mathematical bases. In section 3 we study the simple UVL model, in which the death benefits are payable at the end of a fixed term. Two types of the strategies, subject to whether the death benefits depend on the non-tradable assets, will be studied separately. In section 4 we treat the special case where the utility function is exponential. In section 5 we discuss the general insurance models involving multiple decrements. 2. Problem Formulation Throughout this paper we assume that all the randomness comes from a common complete probability space (Ω, F, P ). We assume that the probability space is rich ˜ = {(Bt , B ˜t ) : t ≥ 0}, enough to carry a d + 1-dimensional Brownian motion (B, B) which will be thought of as the source of the randomness of a financial market where all the investments under consideration will be made. For notational clarity ˜ ˜ we shall denote FB = {FtB : t ≥ 0}, FB = {FtB : t ≥ 0}, to be the natural filtrations ˜ respectively, with usual P -augmentation so that they satisfy generated by B and B, the usual hypotheses (cf. e.g., Ref. 15). Throughout this paper we denote | · | to be the norm of a generic Euclidean space, and h·, ·i to be its inner product. Further, for any real-valued function Φ(t, x, y, ...), where x, y could be vectors, we denote Φt , Φx , ..., etc., to be the partial derivatives (gradients) with respect to the corresponding variables (vectors). The higher order derivatives are denoted similarly when there is no danger of confusion.

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We begin by giving a detailed mathematical description of the life models on which all our future discussion will be based. We refer to the ubiquitous Ref. 2 for most of the ideas and notations below. Let T (x) be the “future life-time random variable”, where x is the current age of the insured. That is, the time to death from the present time. We assume that ˜ The “survival function” for T (x) it is independent of the Brownian motion (B, B). 4

4

is given by Gx (t) = P {T (x) > t}, t ≥ 0. Further, denote Xt = 1{T (x)≤t} , t ≥ 0, 4

and let FX = {FtX }t≥0 with FtX = σ{Xs , 0 ≤ s ≤ t}. We define a new filtration ˜

˜

F = FB ⊗ FB ⊗ FX = {FtB ∨ FtB ∨ FtX : t ≥ 0}. Clearly, under such a setting B is still an F-Brownian motion, and the random time T (x) becomes an F-stopping time. Furthermore, following the conventional actuarial terms we define the “survival probability” of the life (x) by t px = P {T (x) > t}, and denote t qx = 1 −t px . The “force of mortality” is then defined by h qx+t . (1) λx (t) = lim h→0 h In this paper we assume that the death benefit takes the following form: bt = g(t, St1 , · · · , Std , Zt ) = g(t, St , Zt ),

∀t ∈ [0, T ]

(2)

where g : [0, ∞) × IRd+1 7→ (0, ∞) is some function, and Sti , i = 1, · · · , d and Z are the prices of d + 1-risky assets at time t, and St0 is the value of a riskless asset at time t. We assume that the risky assets S i ’s are liquid in a given market, but Z is a non-tradable asset. We note that the benefit function can also be path dependent (see 9 ), but the explicit solution will become less likely. We shall assume that the dynamics of the prices S 1 , · · · , S d , Z, and S 0 are described by the following stochastic differential equations (SDEs): for t ≥ 0,  0 dSt = rt St0 dt, S00 = s0 ;    d  X ij j dSti = Sti {µit dt + σt dBt }, S0i = si , i = 1, · · · , d; (3)   j=1   Z ˜t }, Z0 = z. dZt = Zt {µZ ˜ t dB t dt + hσt , dBt i + σ

˜ is another Brownian motion, independent of B. We shall assume that where B the investment can be made in the market S = (S 0 , S 1 , · · · , S d ), and we denote πti , i = 1, · · · , d to be the amount of money invested in the i-th stock. We will contend ourselves only to those (portfolio) processes π = {πt : t ≥ 0} that are ˜ FB ⊗ FB -adapted, and that Z T E |πt |2 dt < ∞. (4) 0

We denote A to be the set of all such portfolios, and call it the set of “admissible strategies” as usual.

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Let Wt be the total investment income (wealth) at time t, we then assume that Pd all the rest of the money πt0 = Wt − i=1 πti is put into the money market. Assuming that the portfolio π is “self-financing”, it is known (cf. e.g., Ref. 6) that the wealth 4 ˜= process W satisfies the following SDE: denoting 1 (1, · · · , 1)T , ˜ dWt = rt Wt dt + hπt , µt − rt 1idt + hπt , σt dBt i.

(5)

We are now ready to formulate the main problem of the paper. Let u be a given utility function, that is, it is a non-decreasing, concave function. Assume that the insurance company measures its performance by the simple rule of “expected utility”, and consider the contract in which the death benefit is payable at the end of a prescribed terminal time T . In other words, we assume that the “cost functional” takes the form: J(t, w, s, z; π) = Et,w,s,z {u(WTπ − g(T, ST , ZT )XT )}, 4

(6)

where W π is the solution to (5) with given portfolio π, and Et,w,s,z { · } denotes the conditional expectation E{ · |Wt = w, St = s, Zt = z}, for t ≥ 0, w ∈ IR, s ∈ IRd+ , and z ≥ 0. Note that, if the death benefit is not paid (e.g., either the insurance is not sold, or death does not occur before time T ), then g(T, ST , ZT )XT = 0, P -a.s., and the cost functional becomes J 0 (t, w; π) = Et,w {u(WTπ )}, 4

(7)

and the optimization problem is reduced to a standard utility maximization problem in finance. Another special case is when g = g(ST ), namely the death benefit does not involve any non-tradable asset. In this case we denote the cost functional to be 4 b w, s; π) = J(t, Et,w,s {u(WTπ − g(T, ST )XT )}.

(8)

The “value functions” of the optimization problems with the cost functionals (6)–(8) are, respectively: V 0 (t, w) = sup J 0 (t, w; π);

(9)

π∈A

b w, s; π); V (t, w, s) = sup J(t,

(10)

π∈A

U (t, w, s, z) = sup J(t, w, s, z; π).

(11)

π∈A

The idea of “indifference pricing” can be described as follows. For any given benefit function g : IRd+1 7→ IR+ and any given initial state (S0 , Z0 , W0 ) = (s, z, w), we define the “fair price” at time t of a UVL insurance with death benefit g(ST , ZT ) payable at time T > t to be a lump-sum p∗ ≥ 0 such that p∗ = inf{p : V 0 (t, w) ≤ U (t, w + p, s, z),

∀(t, w, s, z)}.

(12)

9

It can be shown that (see ), under mild regularity conditions on the value function U , the fair price p∗ = p∗ (s, z) defined by (12) can be determined by solving the equation V 0 (t, w) = U (t, w + p∗ , s, z),

∀(t, w).

(13)

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We note that the equation (13) is also called the “principle of equivalent utility”, initiated in Ref. 5; and it should be self-evident now why the method is named “indifference pricing”. Throughout this paper we shall make use of the following Standing Assumptions: (H1) All the market parameters µ, σ, and r are deterministic, continuous functions of t. Furthermore, there exists a c0 > 0, such that ξ T σt σtT ξ ≥ c0 |ξ|2 ,

∀ξ ∈ IRn , t ∈ [0, T ].

(H2) The death benefit function g : [0, T ] × IRd × IR 7→ IR+ is bounded, and continuously differentiable, with bounded derivatives. 3. The HJB Equations Let us begin by considering the simplest UVL insurance models, in which the only uncertainty for the termination (or decrement) is death, derive the corresponding HJB equations via two different pricing strategies, depending on the structure of the death benefits. To simplify presentations, we shall assume d = 1 throughout the section. However, we should note that all the analysis can be generalized to higher dimensional cases without substantial difficulties. Since the benefit is paid at fixed time, we will drop T in the benefit function g. To begin with, let us consider an intermediate stochastic control problem with the cost functional: 4 ˜ w, s, z; π) = J(t, Et,w,s,z {u(WTπ − g(ST , ZT ))}.

(14)

This can be thought of as that the death has occurred before T , hence XT ≡ 1. The utility maximization problem is then reduced to a standard stochastic control problem. Denoting the value function as ˜ w, s, z; π), ˜ (t, w, s, z) = sup J(t, U

(15)

π∈A

˜ is at least the unique viscosity solution to the following then it is well-known that U the HJB equation (cf, Ref. 17):  2 1 2 2˜ Z  ˜ ˜ ˜ ˜  0 = Ut + max σ π Uww + π σ sUws + σ zσ Uwz + (µ − r)Uw ]   π 2     1 2  ˜ss + 1 (˜ ˜zz + σσ Z sz U ˜sz σ 2 + σ Z )z 2 U + σ 2 s2 U (16) 2 2   Z  ˜s + µ z U ˜z ;  +rwU˜w + µsU     ˜ U (w, T, s, z) = u(w − g(s, z)). To derive the HJB equation for the original optimization problem, we first argue heuristically. By virtue of the Bellman Principle (of dynamic programming, cf.

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Ref. 17), and using the total probability formula we have, for any h > 0 and any admissible portfolio π ∈ A, π U (w, t, s, z) ≥ Et,x,s,z {U (t + h, Wt+h , St+h , Zt+h )} π = Et,w,s,z {U (Wt+h , t + h, St+h , Zt+h )|T (x) > t + h}P {T (x) > t + h} π +Et,w,s,z {U (Wt+h , t + h, St+h , Zt+h )|T (x) ≤ t + h}P {T (x) ≤ t + h} π = Et,w,s,z {U (Wt+h , t + h, St+h , Zt+h )}h px+t π ˜ +Et,w,s,z {U(W , t + h, St+h , Zt+h )}h qx+t .

(17)

t+h

Here we used the fact that the future-life-time random variable T (x) is independent of the processes (W π , S, Z); and that, given T (x) ≤ t + h, the optimization problem (11) on [t + h, T ] is the same as (15). We remark that the argument above actually reflect a pricing strategy: when death occurs before t + h became known, then one should simply carry out the optimization problem knowing that the terminal wealth will be deducted by the amount g(ST , ZT ) at time T . Such a strategy is quite generic, hence particularly reasonable when the non-tradable assets are present, since there seems to no simple way to hedge the risk g(ST , ZT ) a priori. ˜ are smooth. Applying Itˆ Suppose now that both value functions U and U o’s ˜ (t, Wt , St , Zt ), respectively, and noting that h px+t + formula to U (t, Wt , St , Zt ) and U h qx+t = 1, one shows that (suppressing variables) ˜ w, s, z)]h qx+t [U (t, w, s, z) − U(t, Z t+h 1 ≥ Et,w,s,z { (Ut + (rW + π(µ − r)Uw ) + sUs µ + zUz µZ + σ 2 (Uww π 2 + s2 Uss ) 2 t 1 2 2 + z Uzz (˜ σ 2 + σ Z ) + sUws πσ 2 + zUwz πσσ Z + zsUsz σσ Z )du}h px+t (18) 2 Z t+h +Et,w,s,z { ...du}h qx+t . t

In the above the integrand of the last integral is similar to that of the first one ˜ Now we divide both sides of (18) by h and let h → 0. with U being replaced by U. Noting that 0 px+t = 1 and h qx+t /h → λx (t), we obtain the following HJB equation for U :  1   [(µ − r)πUw + σ 2 π 2 Uww + (Uws Sσ 2 + Uwz Zσ Z σ)π]   0 = Ut + max π 2     1 1 2 +rwUw + sUs µ + zUz µZ + σ 2 s2 Uss + z 2 Uzz (˜ σ2 + σZ ) (19) 2 2      +szUsz σσ Z + λx (t)(U˜ − U )    U (T, w, s, z) = u(w),

˜ satisfies (16). where U

The Special Case: g(s, z) ≡ g(s).

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First note that in this case the non-tradable asset Z does not appear in the previous argument, thus (19) is reduced to  1 1   0 = Ut + max[(µ − r)πUw + σ 2 π 2 Uww + (σ 2 sUws )π] + σ 2 s2 Uss   π 2 2 (20) ˜ +rwU + sµU + λ (t)( U − U ) = 0,  w s x    U (T, w, s) = u(w), and (16) becomes:  ˜ww + (σ 2 sU ˜ws )π] + 1 s2 σ 2 U ˜ss ˜t + max[(µ − r)π U˜w + 1 σ 2 π 2 U  0 = U π 2 2 ˜s +rwU˜w + µsU  ˜ U (T, w, s) = u(w − g(s))

(21)

We should point out that in this case one can actually use a more specific strategy if the market is complete. In fact, considering the benefit payment as a contingent claim, one can easily identify its current market price with which the payment amount g(ST ) can be replicated at time T . Simply setting aside the current price of the benefit payment, one can then proceed the optimization problem as if there is no insurance risk involved at all. Such a strategy was used in Ref. 18 in the case g ≡ 1. In what follows we give a brief sketch of the argument to modify the result to our case, and we show that the two strategies will actually produce the same result. First recall value function V 0 defined by (9). Assuming that all the market parameters r, µ, and σ are deterministic, continuous functions then it is well-known that the value function V 0 is C 1,2 ([0, T ) × IRd ), and it satisfies the following HJB equation: (cf. e.g., 18 ): ( 1 0 0 = Vt0 + max{ |σ T π|2 Vww + π(µ − r)Vw0 } + rwVw0 , (22) π 2 V 0 (T, w) = u(w). Next, we consider the value function V , defined by (10). Applying the Bellman Principle and the total probability formula again, we can show that a counterpart of (17) holds: V (t, w, s) ≥ Et,w,s {V (t + h, Wt+h , St+h )}h px+t +Et,w,s {V 0 (t + h, Wt+h − c(t + h, St+h ))}h qx+t ,

(23)

where c(t, s) is the market price of the contingent claim g(ST ). We note that in the above we deducted c(t + h, St+h ) from the wealth at t + h and then carried out the “future” optimization problem without the insurance risk. Thus the value function ˜ in (17)!). becomes V 0 (t + h, Wt+h − c(t + h, St+h )) (compared to U Now repeating the same argument as before, assuming that V and V 0 and c are all smooth, and using the fact that c(·, ·) satisfies the Black-Scholes PDE, it is fairly

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easily checked that V satisfies the following HJB equation (suppressing all variables for V ):  1   0 = Vt + max{ σ 2 π 2 Vww + (µ − r)πVw + σ 2 πsVws } + rwVw + µsVs π 2 (24) + 21 σ 2 s2 Vss + λx (t)(V 0 (t, w − c(t, s)) − V )   V (T, w, s) = u(w)

At this point it should be clear that this strategy will not work in general since the arbitrage price c(t, s) is not uniquely determined if the payoff contains the nontradable asset ZT ! One should also note that the two different strategies yield almost the same HJB equations (20) and (24), the only difference is that the last ˜ (t, w, s) in (20) is replaced by V 0 (t, w − c(t, s)) in (24). The following result term U clarifies this point even further. Theorem 3.1. Assume (H1), and assume that the benefit function g ≡ g(s). Then, it holds that V (t, w, s) ≡ U (t, w, s), for all (t, w, s) ∈ [0, T ] × IR+ × IR+ .

Proof. Comparing (24) and (19), it is clear that we need only verify that ˜ (t, w, s) = V 0 (t, w − c(t, s)), ∀(t, w, s). U To this end, let us first recall from (5) that for the given initial state w and the portfolio π, the wealth process satisfies the SDE Z t Z t Wt = w + (ru Wu + hπu , µu − ru 1i)du + hπu , σu dBu i, (25) 0

0

w,π

and we denote the solution by W to specify the dependence on w and π. It is by now well-known (see, e.g., 8 ) that under (H1) the Black-Scholes price c(·, ·) for the contingent claim g(ST ) satisfies the Black-Scholes PDE:  ∂c 1 2 2 ∂ 2 c  ∂c + rt s + σ s − rt c(t, s) = 0, (26) ∂t ∂s 2 t ∂s2  c(T, s) = g(s), 4

and the process Yt = c(t, St ), t ≥ 0, can be expressed as the unique solution to the following “backward stochastic differential equation” (BSDE): dYt = rt Yt dt + hπt0 , µt − rt 1idt + hπt0 , σt dBt i, (27) YT = g(ST ), where π 0 ∈ A is the “hedging strategy” of the claim g(ST ). 4

Let us now define a mapping T : A 7→ A by T (π) = π 0 = π + π 0 , π ∈ A. Then clearly T is one-to-one mapping, so that T (A) = A. Furthermore, the linearity of 0 the equation (25) and the uniqueness of the solution to SDE implies that Wtw,π = w−c(0,s),π Yt + W t , for all t ∈ [0, T ] . In particular, at terminal time T , this becomes w−c(0,s),π

WT

0

= WTw,π − g(ST ),

∀π ∈ A.

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˜ we have Finally, by the definition of the value functions V 0 and U w−c(0,s),π

V 0 (0, w − c(0, s)) = sup E0,w {u(WT

)}

π∈A

w,T (π)

= sup E0,w {u(WT π∈A

− g(ST ))}

0

˜ (0, w, s), = sup E0,w {u(WTw,π − g(ST ))} = U π 0 ∈A

proving the theorem. 4. The Case of Exponential Utility In this section we consider a special type of utility function—the “exponential utility”. Such a utility function has been widely used in practice, especially in actuarial mathematics. In fact, the premium principle given by (13) has been known to have certain very desirable properties if and only if the utility function is exponential (cf. e.g., Ref. 4). We should note that while the discussion in this section could be considered as a generalized version of Example 4.1 of Ref. 18, the generality of our benefit function g makes the problem a little more involved. Let us be more specific. In what follows we shall assume that the utility function takes the form 1 4 u(w) = − e−αw , w ∈ IR. (28) α Then, recall from 18 that in this particular case the HJB equation (22) has the following explicit solution. (µ − r)2 1 exp{−αwer(T −t) − (T − t)} (29) α 2σ 2 Our discussion will depend heavily on some classical results in non-linear PDEs. In particular, our solution will not be possible if certain reaction-diffusion equation does not have global solution. But fortunately all the PDEs involved in our discussion can be shown to be well-posed, therefore the solution process can be carried out to the end. We shall discuss two cases of benefit functions separately. V 0 (t, w) = −

A. The case g = g(s). As we mentioned before, in this case the contingent claim g(ST ) is hedgeable, and its price c(·, ·) satisfies the Black-Scholes PDE (26), the optimal price of the UVL is relatively easier to obtain. We have the following theorem. Theorem 4.1. Assume (H1), and assume that utility function u takes the form (28). Suppose also that the benefit function g ≡ g(s), s ≥ 0, and that the force of mortality λx (t), t ≥ 0, are both bounded and deterministic. Then, the solution to (24) can be written as V (t, w, s) = V 0 (t, w) exp{αc(t, s)er(T −t) − h(t, s)},

(30)

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where c(·, ·) is the classical solution to the Black-Scholes equation (26), and h(·, ·) is the classical solution to the following reaction-diffusion system: ( 1 0 = ht + srhs + σ 2 s2 hss − λx (t)(eh − 1) (31) 2 h(T, s) = αg(s) Furthermore, the optimal premium of the UVL insurance is given by p = c(0, s) −

h(0, s) −rT e . α

(32)

Proof. First note that the solution to the HJB equation (24) is unique (at least in the viscosity sense), we need only show that a classical solution to (24) exists. To this end, recall that the value function V 0 (t, w), given explicitly by (29), is a smooth function, concave in w, and it satisfies the PDE:  2 (µ − r)2 Vw0  + rwVw0 0 = Vt0 − 2V 0 (33) 2σ ww  0 V (T, w, s) = u(w).

We shall look for a classical solution of (24) with the special form: V (t, w, s) = V (t, w)Φ(t, s). Note that any solution of such a form will be concave (and C 2 ) in w, thus we can solve the maximum in (24) by choosing 0

π∗ = −

(µ − r)Vw + σ 2 sVws , σ 2 Vww

and the equation (24) becomes  ((µ − r)Vw + σ 2 sVws )2 1   0 = V − + rwVw + µsVs + σ 2 s2 Vss  t  2 2σ Vww 2 0 +λ (V (t, w − c(t, s)) − V )  x    V (T, w, s) = u(w).

(34)

Plugging in V = V 0 Φ we obtained from (34) that

2 n o (µ − r)2 Vw0 1 0 = Vt0 − + rwVw0 + V 0 (Φt + sµΦs + σ 2 s2 Φss ) 2 0 2σ Vww 2 2 V0 s2 σ 2 Φ2s +λ(t) V 0 (t, w − c(t, s)) − V 0 Φ) − w0 ((µ − r)sΦs − 0 Φ Vww 2Vww

In the above, the first {· · · } vanishes because of (33). Also, using the explicit form (29) of V 0 , and with some straightforward computation we deduce that Φ must satisfy the following PDE:  1 Φ2  Φt + rsΦs + σ 2 s2 Φss − s + λx (t) exp{c(t, s)αer(T −t) } − Φ = 0, (35) 2 Φ  Φ(T, s) = 1.

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We now show that the equation (35) has a classical solution. To see this, we first consider the transformation: h(t, s) = c(t, s)αer(T −t) − ln Φ(t, s), (t, s) ∈ [0, T ] × IR. Then it is readily seen that (suppressing variables): ht = ct αer(T −t) − cαrer(T −r) −

Φt ; Φ

hs = cs αer(T −t) −

Φs ; Φ

Φss (Φs )2 + . Φ Φ2 Using these relations one easily verifies that h satisfies the PDE:  1   0 = αer(T −t) (ct + rscs + σ 2 s2 css − rc) − ht − srhs 2 − 21 σ 2 s2 hss + λx (t)(eh − 1)   h(T, s) = αc(T, s) hss = css αer(T −t) −

(36)

Since c(·, ·) solves the Black-Scholes PDE (26), (36) becomes (31). Furthermore, if we make a change of variables: v = log s and τ = T − t in (31), and denote b h(τ, v) = h(T − τ, ev ), then b h satisfies the following “reaction-diffusion” equation: ( 1 1 hτ − (r − σ 2 )hv − σ 2 hvv + λx (T − τ )(eh − 1) = 0 (37) 2 2 h(0, v) = αg(ev ) We should note that the equation (37) is a semilinear parabolic PDE with exponential growth, which in general may have a finite time blow-up. But by examing the particular form of the nonlinear term carefully and applying the results in Ladyzenskaja7 we can show the equation (37) does have a bounded classical solu4 tion b h (see 9 for details). Therefore h(t, s) = b h(T − t, log s) is a solution to (31), and

V (t, w, s) = V 0 (t, w)Φ(t, s) = V 0 (t, w) exp{αc(t, s)er(T −t) − h(t, s)} is a classical solution to (24), proving the first part of the theorem. To conclude the proof, we recall that by the Principle of Equivalent Utility, the optimal premium is defined by: V 0 (0, w) = V (0, w + p, s). By virtue of Theorem 4.1, this relation becomes: V 0 (0, w) = V 0 (0, w + p) exp{αc(0, s)erT − h(0, s)}.

(38)

Now the conclusion follows from the explicit form (29) of V 0 (0, w), and some fairly simple calculations. We leave the details to the interested reader, and the proof is now complete. We remark that if the force of mortality λx (t) ≡ 0, that is, the death never occurs, then it is easily checked that h(t, s) = αc(t, s)er(T −t) satisfies (31), thus p ≡ 0 by (32), as it should be. B. The case g = g(s, z).

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In this case the explicit formula for the price of UVL price is a little more complicated, due to the presence of the non-tradable asset Z in the death benefit (hence the contingent claim g(ST , ZT ) is no longer hedgable in general). But we can still proceed along the same line as before, and possibly with slightly different notations, we have the the following theorem. Theorem 4.2. Assume that the utility function is of the form u(w) = − α1 e−αw . Assume also that the benefit function g(·, ·) and the force of mortality λx (·) are both smooth and bounded. Then the optimal premium can be written as p(t, s, z) =

1 −r(T −t) e h(T − t, log s, log z), α

(39)

where h is a bounded, classical solution to the PDE  1 2 2 1 1 2  ˜ hy2 − σ 2 hy1 y1 − (˜ σ + σ z 2 )hy2 y2 − σσ z hy1 y2  hτ − σ   2 2 2    1 ˜2 + σz 2  µ−r z σ − r − σ 2 h y1 − µ z − σ − h y2 2 σ 2   ˜   −λx (T − τ )(eh−h − 1) = 0;     h(0, y1 , y2 ) = 0,

˜ is a bounded, classical solution to the PDE: and h  z˜ 2 z2 ˜ ˜ 2 − 1 σ2 h ˜ y y − 1 (˜ ˜  h − 1σ ˜2 h y2 1 1  2 2 σ + σ )hy2 y2 − σσ hy1 y2  τ 2  1 2 ˜ µ−r z σ ˜ 2 + σ z 2 ˜ z − r − hy2 = 0; σ h − µ − σ − y 1  2 σ 2   ˜ h(0, y1 , y2 ) = αg(ey1 , ey2 )

(40)

(41)

Proof. The idea of the proof is similar to that of Theorem 4.1. But this time we seek solutions of (16) and (19) with u(w) = − α1 e−αw such that they are of the special forms ˜ −t,ln s,ln z) ˜ (t, w, s, z) = V 0 (t, w)eh(T U ,

and U (t, w, s, z) = V 0 (t, w)eh(T −t,ln s,ln z) ,

respectively. Following the same arguments as in Theorem 4.1, one shows that ˜ s, z) and h(τ, s, z) will have to solve the reaction-diffusion equations (41) and h(τ, (40), respectively, after a change of variables τ = T − t, y1 = ln s, and y2 = ln z. Thus it remains to verify that (40) and (41) both have (bounded) classical solutions. But this can be done in a similar way as in the proof of the previous theorem with some modifications. We again refer to Ref. 9 for details. Finally, the explicit solution for the price p (39) follows from the principle of equivalent utility and solving the equation V 0 (t, w) = U (t, w + p, s, z). The proof is now complete.

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5. Some Remarks on General Insurance Models The results of previous sections can be extended to a more general life insurance model (see, e.g., Ref. 13). For example, we can consider the cases of “multiple decrement” with status-dependent payments, as well as the benefit be payable at a random time (such as “moment of death”), and so on. In these cases the the “state process” {Xt }t≥0 should modified as follows. Suppose that the policy starts at time 0 for a person aged x, and assume that X is a Markov chain with a finite state space {0, 1, ..., m}, representing the numerical code of the status at time t. We specify i = 1 to be a (absorbing) “cemetary state”, representing “death”, and X0 = 0 be the initial state. Now denote Iti = 1{Xt =i} , and define the counting process: Ntij = #{transitions of X from state i to j during [0, t]}. 4

(42)

Also, for each t we define a stopping time τt = inf{s ≥ t : Xs 6= Xt }; and τti = [1{Xτt =i} ]−1 τt , for i = 0, ..., m. Using these stopping times we can define the conditional probabilities: for i, j ∈ {0, ..., m}, 4 ¯is = tp

P {τs > t|Xs = i};

4 ¯sij = tq

P {τsj = τs ≤ t|Xs = i},

s ≤ t.

(43)

Similar to (1), we can define the “force of decrement of status i due to cause j” as 4 ¯ ij = λ lim t

¯tij t+h q

h→0

i, j = 0, 1, · · · m.

,

h

(44)

¯ ij is nothing but the transition It is not hard to show (see 9 for details ) that λ t intensity of the Markov chain X, defined as λij t = lim 4

h↓0

ij t+h qt

,

h

i 6= j,

(45)

4

where t qsij = P {Xt = j|Xs = i} is the transition probability. We note that if m = 1, then there are only two states: life or death. In this case the state process X becomes the one in the previous section, and the general life model reduces to the simple life model with τ01 = T (x). With a slight abuse of notation, we shall use 4

˜

the same notation F for the filtration F = {Ft }t≥0 = {FtX ∨ FtB ∨ FtB }t≥0 . Assume now in the general life insurance problem there are two types of payments: one takes the form of “annuity” and the other of “insurance”. More precisely, the (cumulative) payment process is XZ t XZ t At = Iui bi (u, Su , Zu )du + bij (u, Su , Zu )dNuij , t ≥ 0, (46) i

0

i6=j

0

where bi (t, s, z) is the rate of net payment of life annuity at state i, given St = s, Zt = z; and bij (t, s, z) is the rate of net payment of life insurance upon transition from state i to state j, given St = s, Zt = z. Given a payment process A, the wealth process (5) should then be modified to the following: dWtπ = rWtπ dt + πtT (µt − rt 1)dt + πtT σt dBt − dAt ,

t ≥ 0.

(47)

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Since the non-tradability of the asset Z will not play any essential role in such a setting, one can simply consider it as the (d + 1)-th component of the vector S = (S 1 , · · · , S d , S d+1 ). and write the SDE for the risky assets as dSt = D[St ]{µt dt + σt dBt },

t ≥ 0,

(48)

where D[s] denotes the diagonal matrix diag[s1 , ..., sd+1 ], and B is a d+1-dimensional Brownian motion. For a given portfolio π and the initial data (t, w, s), we denote W t,w,π (resp. S t,s ) to be the solution to (25) (resp. (48)), such that Wt = w (resp. St = s). We now introduce a family of value functions in accordance with all the status. For all (t, w, s) ∈ [0, T ] × IR × IRd+1 + , and k = 0, · · · , m, U k (t, w, s) = sup E{u(WTπ )|Wt = w, St = s, Xt = k}. 4

(49)

π∈Au

Then, by a standard procedure of dynamic programming, one can more or less conjecture that the HJB equation for the family {U k ; k = 1, · · · , m} should be a system of partial differential equations, but the exact form of these PDEs does not seem to be clear without a careful examination. To obtain the HJB equation, let us denote, for ϕ ∈ C 1,2 ([0, T ] × IRd+2 ), X X L[ϕ] = Lt,w,s [ϕ] = µit si ϕsi (t, w, s) + σtik σtjk si sj ϕsi sj (t, w, s) (50) i

i,j,k

2 = hϕs (t, w, s), D[s]µt i + tr{D[s](σt σtT )D[s](Dss ϕ(t, w, s))}.

Further, for (t, w, s) ∈ [0, T ] × IR × IRd+1 , (ϕ, ψ, p) ∈ IR × (−∞, 0) × IRd+1 , and k = 0, 1, · · · , m, we define  4 1   Hk (t, w, s, ϕ, ψ, p; π) = |σt π|2 ψ + [hπ, µt − rt 1i + rt w − bk (t, s)]ϕ   2  (51) +hπ, σt σtT D[s]pi     k 4  H (t, w, s, ϕ, ψ, p) = supπ Hk (t, w, s, ϕ, ψ, p; π).

Note out that the quadratic nature (in π) of the Hamiltonian in (51) and the unrestricted choice of π implies that H k < ∞ if and only if ψ ∈ (−∞, 0). We have the following theorem, whose proof can be found in Ref. 9. Theorem 5.1. Under some standard conditions on the coefficients, and assume that for all k = 0, 1, ..., m the value functions U k ∈ C 1,2,2 ([0, T ] × IR × IRn ). Then, for each k, U k is strictly concave in w, and U = (U 0 , U 1 , · · · , U m ) satisfies the following system of HJB equations: X kj  k k  0 = Utk + L[U k ]+H k (t, w, s, Uwk , Uww , Uws )+ λt (U j (t, w − bkj , s) − U k ), (52) j6=k  k U (w, T, s) = u(w), k = 0, 1, · · · , m.

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Finally, we remark here that although Theorem 5.1 gives only a sufficient condition, it shows a significant difference between this HJB equation and the traditional ones. A rigorous proof of the theorem is by no means trivial, due to the generality of the utility function (see Ref. 9). A more detailed study of the value function and the fact that the value function is indeed a viscosity solution to the HJB equation will be studied in another forthcoming paper 10 . References 1. Bielecki, T.R., Jeanblanc, M., and Rutkowski, M., Hedging of defaultable claims, Paris-Princeton Lectures on Mathematical Finance (2004). 2. Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J., Actuarial Mathematics, The Society of Actuaries, USA (1997). 3. Frittelli, M., Introduction to a theory of value coherent with the no-arbitrage principle, Finance and Stochastics 4(3), 275-297 (2000). 4. Gerber, H.U., An Introduction to Mathematical Risk Theory, Huebner Foundation for Insurance Education, Wharton School, University of Pennsylvania, Phiadelphia (1979). 5. Hodges, S.D. and Neuberger, A., Optimal Replication of Contingent Claims under Transaction Costs, Rev. Futures Markets 8, 222-239 (1989). 6. Karatzas, I. and Shreve, S.E., Brownian Motion and Stochastic Analysis, SpringerVerlag (1988). 7. Ladyzenskaja, O.A., Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society (1968). 8. Ma, J. and Yong, J. Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Mathematics, 1702, Springer (1999). 9. Ma, J. and Yu, Y. Principle of Equivalent Utilility and Universal Variable Life Insurance Pricing (2005), preprint. 10. Ma, J. and Yu, Y. UVL Insurance Pricing Problems and Systems of Partial Differential-Difference Equations, working paper. 11. Musiela, M. and Zariphopoulou, T. An Example of Indifference Prices under Exponential Preferences, Finance Stoch. 8, no. 2, 229-239 (2004). 12. Musiela, M. and Zariphopoulou, T. Indifference Prices of Early Exercise Claims, Mathematics of finance, 259–271, Contemp. Math., 351 (2004) Amer. Math. Soc., Providence, RI. 13. Norberg, R., Hattendorff’s Theorem and Thiele’s Differential Equation Generalized, Scand. Actuarial J. 1, 2-14 (1992). 14. Owen, M.P., Utility based optimal hedging in incomplete markets, Annals of Applied Probability 12, 691-709 (2002). 15. Protter, P., Stochastic integration and differential equations: A new approach, Springer-Verlag, Berlin (1990). 16. Rouge, R. and El Karoui, N., Pricing via utility maximization and entropy, Mathematical Finance 10, 259-276 (2000). 17. Yong, J and Zhou, X.Y., Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer, New York (1999). 18. Young, V.R. and Zariphopoulou, T., Pricing Dynamic Insurance Risks Using the Principle of Equivalent Utility, Scand. Actuarial J. 4, 246-279 (2002).

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Chap20-PengShige

GΓ -EXPECTATIONS AND THE RELATED NONLINEAR DOOB-MEYER DECOMPOSITION THEOREM

SHIGE PENG∗ and MINGYU XU School of Mathematics and System Science, Shandong University, 250100, Jinan, China [email protected] (S. Peng), [email protected] (M. Xu)

In this paper, we define a nonlinear expectation via the BSDE with constraint (BSDE with sigular coefficient), then we introduce definitions of super and sub martingale under this nonlinear expectation and prove their nonlinear Doob-Meyer type decomposition

1. Introduction The objective of backward stochastic differential equation (BSDE in short) with generator g and with constraint Γ is to find the smallest (g–) supersolution (Y, Z, A), where A is an increasing process by which (Y, Z) is forced to remain inside of a given subset Γ ⊂ R × Rd. This problem was studied by El Karoui, Kapoudjian, Pardoux, Peng and Quenez, 16 for the constraint only on Y called reflected BSDE, and then initialed an extensive study (see Refs. 9, 19, 22, 24, 28, 29, 41, 42 among many others) for single and double barriers of reflected BSDEs. El Karoui & Quenez 17 and then Cvitanic & Karatzas, 7,8 Karatzas & Kou 25 may be considered as a strong motivation in finance to BSDE with constraint on Z. Cvitanic, Karatzas and Soner 10 studied this problem of BSDE for the case where the constraint Γ of Z is convex and the corresponding generator g is concave. Peng 38 studied constraints on (Y, Z) where g is a general Lipschitz function. As we will show in this paper, this result can applied to a very general situation where the corresponding Γ was assumed to be only a closed subset of R × Rd . In this paper we will work with these weaker conditions. This will provide a wide space of freedom to treat different types of situations. Typically, in the situation of differential games, the generator g is neither convex nor concave (see Refs. 20, 21 and 23). Recently, special constraint BSDEs (reflected BSDEs) are extended to the high dimensional case (see Refs. 32 and 35). ∗ The

author would like to acknowledge the partial support from the natural science foundation of china, grant no. 10131040. 122

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An interesting point of view is that this smallest Γ–constrained g-supersolution is, in fact, the solution of the BSDE with a singular coefficient gΓ defined by gΓ (t, y, z) = g(t, y, z)1Γ(y, z) + (+∞) · 1ΓC (y, z). Recent developments of continuous time finance requires a nonlinear version of conditional expectations. In 1997, the first author has introduced a Brownian filtration (Ft )t≥0 consistent nonlinear expectation Eg [X] : X ∈ L2 (Ω, FT , P ) → R call g–expectation, which is defined by y0X , where (ytX , ztX )0≤t≤T is the solution of the BSDE with a given Lipschitz function g(t, y, z), as its coefficient and with the above X as its terminal condition. Here we assume g satisfies Lipschitz condition in (y, z) as well as g(t, y, 0) ≡ 0. When g is a linear function in (y, z), this g–expectation Eg [·] is just a Girsanov transformation. But it becomes a nonlinear functional once g is nonlinear in (y, z), i.e., Eg [·] is a constant preserving monotonic and nonlinear functional defined on L2 (Ω, FT , P ). A significant feature of this nonlinear expectation is that, just like the classical notion of the conditional expectation, the g–expectation of X under Ft , noted by Eg [X|Ft ] ∈ L2 (Ω, Ft , P ), can be still characterized by Eg [X1A ] ≡ Eg [Eg [X|Ft ]1A ], ∀A ∈ Ft . In fact, it is proved that the unique element in L2 (Ω, Ft , P ) satisfying the above classical criterion is Eg [X|Ft ] = ytX . This stricken fact gives us a hint: many beautiful and powerful properties in the modern probability might still hold true without the linearity assumption. For example the essentially central notions of martingales, sub and supermartingales do not need the linearity. We then can ask if the well–known submartingale decomposition theorem is still hold true. In fact we have to introduce an intrinsic proof for those nonlinear decomposition theorem. (see Refs. 6, 38, 39, 40, and 44). Similarly as the above g–expectation, we can also define the corresponding g Γ – expectations by the smallest solution of BSDE with gΓ as well as the corresponding gΓ –supermartingales and submartingales. We then prove a gΓ –supermartingale decomposition theorem, which is a nonlinear version of Doob–Meyer decomposition theorem. The gΓ –submartingale decomposition can not be obtained by the above mentioned gΓ –supermartingale decomposition theorem. We shall obtain this decomposition theory in a quite different way. Recently a profound link between super–replication, risk measures (see Refs. 1 and 18) nonlinear expectations are being explored (see Refs. 3, 39, and 45). We hope that the results of this paper will be proved to be useful in this direction. We also refer to Refs. 2, 4, 5, 13, 14, 24, 30, and 31 for interesting research works in this domain. This paper is organized as follows. In the next section we list our main notations and main conditions required. In Section 3 we shall introduce the gΓ –expectations

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and related definitions. In Section 4, we will give the results of the existence and uniqueness of the solution of reflected BSDE with constraints. In Section 5, we present and prove the the nonlinear decomposition theorem of the related gΓ – supermartingales and gΓ –submartingales. Some useful results such as monotonic limit theorem, are listed in Appendix. 2. gΓ -Solution: the Smallest Supersolution of BSDE Constrained in Γ Let (Ω, F, P ) be a probability space, and B = (B1 , B2 , · · · , Bd )T be a d-dimensional Brownian motion defined on [0, ∞). We denote by {Ft ; 0 ≤ t < ∞} the natural filtration generated by this Brownian motion B : Ft = σ{{Bs ; 0 ≤ s ≤ t} ∪ N }, where N is the collection of all P −null sets of F. The Euclidean norm of an element x ∈ Rm is denoted by |x|. We also need the following notations for p ∈ [1, ∞): • Lp (Ft ; Rm ) :={Rm -valued Ft –measurable random variables X s.t. E[|X|p ] < ∞}; • LpF (0, t;RRm ) :={Rm –valued and Ft –adapted processes ϕ defined on [0, t], t s. t. E 0 |ϕs |p ds < ∞}; • DpF (0, t; Rm ) :={Rm –valued and RCLL Ft –progressively measurable processes ϕ on [0, t], s.t. E[sup0≤s≤t |ϕs |p ] < ∞}; • SpF (0, t; Rm ) :={continuous processes in DpF (0, t; Rm )}; • ApF (0, t) :={increasing processes in DpF (0, t; R) with A(0) = 0}. When m = 1, they are denoted by Lp (Ft ), LpF (0, t), DpF (0, t) and SpF (0, t), respectively. We are mainly interested in the case p = 2. In this section, we consider BSDE on the interval [0, T ], with a fixed T > 0. We are given a function g(ω, t, y, z) : Ω × [0, T ] × R × Rd → R, which always plays the role of the coefficient of our BSDEs. g satisfies the following assumption: there exists a constant µ > 0, such that, for each y, y 0 in R and z, z 0 in Rd , we have (i) g(·, y, z) ∈ L2F (0, T ), (ii) |g(t, ω, y, z) − g(t, ω, y 0 , z 0 )| ≤ µ(|y − y 0 | + |z − z 0 |).

(1)

The constraint of our BSDE is Γ(t, ω) : Ω×[0, T ] → C(R × Rd ), where C(R × Rd ) is the collection of all closed non–empty subsets of R × Rd which is Ft –adapted, namely, (i) (y, z) ∈ Γt (ω) iff dΓt (ω) (y, z) = 0, t ∈ [0, T ], a.e.; (ii) dΓ· (y, z) is Ft –adapted process, for each (y, z) ∈ R × Rd ,

(2)

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where dΓ (·) is a distant function of Γ: dΓt (y, z) :=

inf

(y 0 ,z 0 )∈Γt

(|y − y 0 |2 + |z − z 0 |2 )1/2 ∧ 1.

dΓt (y, z) is a Lipschitz function: for each y, y 0 in R and z, z 0 in Rd , we always have |dΓt (y, z) − dΓt (y 0 , z 0 )| ≤ (|y − y 0 |2 + |z − z 0 |2 )1/2 . Remark 2.1. The constraint discussed in Peng38 is Γt (ω) = {(y, z) ∈ R1+d : Φ(ω, t, y, z) = 0}.

(3)

Here Φ(ω, t, y, z) : Ω × [0, T ] × R × Rd → [0, ∞) is a given nonnegative measurable function, and satisfies integrability condition and Lipschitz condition. In this paper we always consider the case Φ(t, y, z) = dΓt (y, z). We are then within the framework of supersolution of BSDE of the following type: Definition 2.1. (g–supersolution, cf. El Karoui, Peng and Quenez15 (1997) and Peng38 (1999) ) A process y ∈ D2F (0, T ) is called a g–supersolution if there exist a predictable process z ∈ L2F (0, T ; Rd) and an increasing RCLL process A ∈ A2F (0, T ) such that Z T Z T yt = y T + g(s, ys , zs )ds + AT − At − zs dBs , t ∈ [0, T ]. (4) t

t

Here z and A are called martingale part and increasing part, respectively. y is called a g–solution if At ≡ 0. y is called a Γ–constrained g–supersolution if y and its corresponding martingale part z satisfy (yt , zt ) ∈ Γt , (or dΓt (yt , zt ) = 0), dP × dt a.s. in Ω × [0, T ].

(5)

Remark 2.2. We observe that, if y ∈ D2F (0, T ) is a g–supersolution, then the pair (z, A) in (4) are uniquely determined since the martingale part z is uniquely determined. Occasionally, we also call (y, z, A) a g–supersolution. By Peng38 , (see Appendix Theorem 6.2), if there exists at least one Γ– constrained g–supersolution, then the smallest Γ–constrained g–supersolution exists. In fact, a Γ-constraint g-supersolution can be considered as a solution of the BSDE with a singular coefficient gΓ defined by gΓ (t, y, z) = g(t, y, z)1Γt (y, z) + (+∞) · 1ΓCt (y, z). So we define the smallest Γ–constrained g–supersolution by gΓ –solution.

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Definition 2.2. (gΓ –solution) y is called the gΓ –solution on [0, T ] with a given terminal condition X if it is the smallest Γ–constrained g–supersolution with yT = X: for t ∈ [0, T ], Z T Z T yt = X + g(s, ys , zs )ds + AT − At − zs dBs , (6) t

t

(yt , zt ) ∈ Γt , dAt ≥ 0. Remark 2.3. (y, z, A) is the gΓ -solution does not implies that the increasing pro¯ with cess is also the smallest one, i.e. for any Γ–constrained g–supersolution (¯ y , z¯, A) ¯ the same terminal X, we may have At ≥ At . An example is as follows: consider the case [0, T ] = [0, 2], X = 0, g = 0 and Γt = {(y, z) : y ≥ 1[0,1] (t)}. It’s easy to see that the smallest solution is yt = 1[0,1) (t) with zt = 0, At = 1[1,2] (t). Obviously y t = 1[0,2) (t) with z t = 0, At = 1{t=2} (t) is another Γ–constrained g–supersolution with the same terminal condition yT0 = 0. But we have At > At on the interval [1, 2). 3. F-Consistent Nonlinear Expectations 3.1. gΓ –expectations We now introduce a notion of F–consistent nonlinear expectations via gΓ –solutions. We make the following assumption: there exists a large enough constant C0 such that g(t, y, 0) ≤ C0 + µ|y|, ∀y ≥ C0 , and (y, 0) ∈ Γt , ∀y ≥ C0

(7)

We need the terminal conditions to be in the following linear subspace of L2 (FT ): L2+,∞ (FT ) := {ξ ∈ L2 (FT ), ξ + ∈ L∞ (FT ). Proposition 3.1. We assume (1) and (7). Then for each X ∈ L2+,∞ (FT ), the smallest Γ–constrained g–supersolution with terminal condition y T = X exists. Furthermore, we have yt ∈ L2+,∞ (FT ), t ∈ [0, T ]. Proof. We consider

y0 (t) = ( X + ∞ ∨ C0 )eµ(T −t) + C0 (T − t) + (X − X + ∞ ∨ C0 )1{t=T } .

It is the solution of the following backward equation: Z T y0 (t) = X + (C0 + µ|y0 (s)|)ds + A0 (T ) − A0 (t), t

0

where A is an increasing process: A0 (t) := (kX + k∞ ∨ C0 − X)1t=T . But y0 (·) can be expressed to: Z T Z T y0 (t) = X + g(s, y0 (s), 0)ds + [c + µ|y0 (s)| − g(s, y0 (s), 0)]ds + A0 (T ) − A0 (t). t

t

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Thus the triple defined on [0, T ] by (y1 (t), z1 (t), A1 (t)) := (y0 (t), 0,

Z

t

[c + µ|y0 (s)| − g(s, y0 (s), 0)]ds + A0 (t)) 0

is a Γ–constrained g–supersolution with y1 (T ) = X. According to Theorem 6.2, the gΓ –solution with y(T ) = X exists. We also have (yt )+ ∈ L∞ (FT ) since yt ≤ y1 (t) = y0 (t). We now introduce the notion of gΓ –expectation Definition 3.1. We assume that g(t, 0, 0) = 0 and (0, 0) ∈ Γt , for each 0 ≤ t ≤ T < ∞. Then consider X ∈ L2 (FT ) with X + ∈ L∞ (FT ), let (y, z, A) be the gΓ – Γ solution defined on [0, T ] with terminal condition yT = X. We set Egt,T [X] := yt . The system Γ [·] : L2+,∞ (FT ) → L2+,∞ (Ft ), Egt,T

0≤t≤T <∞

is called gΓ -expectation. We have Proposition 3.2. A gΓ -expectation is an F-consistent evaluation (cf. Peng39 ), i.e., it satisfies the followings: for each 0 ≤ t ≤ T < ∞ and X, X 0 ∈ L2+,∞ (FT ), Γ Γ (A1) Egt,T [X] ≤ Egt,T [X 0 ], if X ≤ X 0 ; gΓ (A2) ET,T [X] = X; Γ Γ Γ (A3) Egs,t [Egt,T [X]] = Egs,T [X], 0 ≤ s ≤ t ≤ T ; gΓ gΓ (A4) 1D Et,T [X] = Et,T [1D X], ∀D ∈ Ft . Moreover, (A5) if Γ1t ⊇ Γ2t

g1

and g 1 (t, y, z) ≤ g 2 (t, y, z), ∀(t, y, z),

Γ1 then Et,T [X] ≤

g2

Γ2 Et,T [X].

Proof. (A1) and (A5) is a direct consequence of the comparison theorem 6.4 of the smallest Γ–constrained g–supersolution. (A2) is obvious. It is easy the check that, if (ys )0≤s≤T is the gΓ –solution on [0, T ] with yT = X, then (ys )0≤s≤t is also the gΓ –solution on [0, t] with the fixed terminal condition yt . The proof of (A3) is similar. To prove (A4), we multiply 1D on the two sides of the equation, for t ≤ s ≤ T , Z T Z T ys = X + g(r, yr , zr )dr + AT − As − zr dBr , s

s

dΓs (ys , zs ) ≡ 0. Since g(s, 0, 0) ≡ 0, and dΓs (0, 0) ≡ 0, we have Z T Z 1D ys = 1 D X + g(r, 1D yr , 1D zr )dr + 1D AT − 1D As − s

dΓs (1D ys , 1D zs ) ≡ 0.

T

1D zr dBr , s

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Thus it is easy to check that (1D ys , 1D zs )t≤s≤T must be the gΓ –solution on [s, T ] with yT 1D as the terminal condition, which implies (A4). Definition 3.2. We assume that g(t, y, 0) = 0,

and (y, 0) ∈ Γt , ∀y ∈ R, t ∈ [0, T ].

(8)

For each EgΓ [X] := y0 . The functional EgΓ [·] : L2+,∞ (FT ) → R is called a constant preserving (gΓ )expectation. Moreover, we define conditional (gΓ )-expectation: Γ EgΓ [X|Ft ] := Egt,T [X] = yt ,

the operator EgΓ [·|Ft ] : L2+,∞ (F T ) → L2+,∞ (Ft ). Proposition 3.3. A conditional gΓ -expectation is an F-consistent nonlinear expectation, i.e. for X, X 0 ∈ L2+,∞ (FT ) and 0 ≤ t ≤ T < ∞, it satisfies, 0 (A1’) EgΓ [X|Ft ] ≤ EgΓ [X 0 |Ft ], if X ≤ X ; (A2’) EgΓ [Y |Ft ] = Y , if Y ∈ L2+,∞ (Ft ); (A3’) EgΓ [EgΓ [X|Ft ]|Fs ] = EgΓ [X|Fs ], 0 ≤ s ≤ t ≤ T ; (A4’) 1D EgΓ [X|Ft ] = EgΓ [1D X|Ft ], for D ∈ Ft . Moreover, we have (A5’) if Γ1t ⊇ Γ2t and g 1 (t, y, z) ≤ g 2 (t, y, z), for ∀(t, y, z), then Eg1 1 [X|Ft ] ≤ Γ Eg2 2 [X|Ft ]. Γ

Proof. (A1’), (A3’), (A4’) and (A5’) are direct consequences of Proposition 3.2. (A2’) is proved as follows, from (8) we know that the triple (ys , zs , As ) ≡ (Y, 0, 0) is the g–solution on [t, T ] with the terminal condition yT = Y . By comparison theorem of BSDE, this triple must be the smallest Γ–constrained g–supersolution. Thus (A2’) holds. Now we introduce the definition of gΓ -martingale, gΓ -supermartingale and gΓ submartingale, by gΓ -expectation. Definition 3.3. A process Y ∈ D2F (0, T ) is called a gΓ -supermartingale (resp. gΓ -submartingale) on [0, T ], if for each 0 ≤ s ≤ t ≤ T , we have Yt ∈ L2∞,+ (Ft ) and Γ Egs,t [Yt ] ≤ Ys , (resp. ≥ Ys ).

It is called a gΓ –martingale if it is both a gΓ –super and gΓ –submartingale. Remark 3.1. We can also define (gΓ )-martingale, (gΓ )-supermartingale, (gΓ )submartingale with respect to conditional (gΓ )-expectation EgΓ [·|Ft ].

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A gΓ -supermartingale has the following interesting property. Proposition 3.4. A process Y is a gΓ -supermartingale on [0, T ], if and only if for all m ≥ 0, it is a (g + mdΓ )-supermartingale on [0, T ]. Γ Proof. We fix t ∈ [0, T ], and set yst = Egs,t [Yt ], 0 ≤ s ≤ t. Let (y t , z t , At ) be the smallest Γ–constrained g–supersolution on [0, t]: Z t Z t t t t t t ys = Yt + g(r, yr , zr )dr + At − As − zrt dBr ,

s

s

dΓs (yst , zst ) = 0, s ∈ [0, t]. Consider the following penalization equation Z t Z t Z t yst,m = Yt + g(r, yrt,m , zrt,m )dr + m dΓr (yrt,m , zrt,m )dr − zrt,m dBr . s

s

s

We observe that the above (y t , z t , At ) also satisfies Z t Z t Z t t t t t t t t ys = Yt + g(r, yr , zr )dr + m dΓr (yr , zr )dr + At − As − zrt dBr . s

s

From comparison theorem, we get y Γ Eg+md [Yt ] s,t

t,m

≤

s

t

≤ y on [0, t]. Thus

Γ Egs,t [Yt ]

≤ Ys , ∀m ≥ 0.

It follows that Y is a (g + mdΓ )-supermartingale on [0, T ]. Conversely, if for each Γ m ≥ 0, Y is a (g + mdΓ )-supermartingale on [0, T ] i.e. Eg+md [Yt ] = yst,m ≤ Ys . s,t t,m When we let m → ∞, by the monotonic limit theorem 6.2, y· converges to Eg·,tΓ [Yt ], Γ which is the smallest g–supersolution with constrain Γ. We thus have Egs,t [Yt ] ≤ Ys , on [0, T ]. This implies that Y is a gΓ -supermartingale.

3.2. gΓ -supersolutions and gΓ -subsolutions Siminarly as g-super(sub)solutions, we can define gΓ -super(sub)solutions. Definition 3.4. (gΓ –super(sub)solution) A process y ∈ D2F (0, T ) is called a gΓ –supersolution (resp.gΓ –subsolution), if there exist a predictable process z ∈ L2F (0, T ; Rd) and an increasing RCLL process A ∈ A2F (0, T ) (resp. K ∈ A2F (0, T )), such that t ∈ [0, T ], Z T Z T yt = y T + g(s, ys , zs )ds + AT − At − zs dBs , (9) t

(resp. yt = yT +

Z

t

T

g(s, ys , zs )ds + AT − At − (KT − Kt ) − t

Z

T

zs dBs .) t

And (yt , zt ) ∈ Γt , (or dΓt (yt , zt ) = 0), dP × dt a.s. in Ω × [0, T ].

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Proposition 3.5. A gΓ -supersolution y on [0, T ] with yt ∈ L2+,∞ (Ft ), t ∈ [0, T ] is a gΓ -supermartingale. Proof. Since y is a Γ–constrained g–supersolution on [0, T ] and, for each t ∈ [0, T ] Γ and s ∈ [0, t], Egs,t [yt ] is the smallest Γ–constrained g–supersolution with the termiΓ nal condition with yt at t. We then have Egs,t [yt ] ≤ ys . Now we give two simple example for the gΓ -super(sub)martingale by the reflected BSDEs. Example 3.1. Fix T < ∞ and set Γt = [Lt , ∞) × Rd , where L ∈ L2F (0, T ) 2 satisfies E[ess sup0≤t≤T (L+ t ) ] < ∞. Our problem is in fact the reflected BSDE Γ with one lower obstacle L. Then we can introduce a gΓ -expectation: Egt,T [·] : 2 2 L (FT ) → L (Ft ). Let (y, z, A) be the solution of reflected BSDE with terminal condition y T ∈ L2 (FT ), coefficient g and lower obstacle L, where L ∈ L2F (0, T ) + satisfies E[ess sup0≤t≤T (Lt )2 ] < ∞ and Lt ≥ Lt , a.e.a.s.. So (y t )0≤t≤T is a gΓ -supermartingale. In fact for 0 ≤ s ≤ t ≤ T , let (y, z, A) be the solution of BSDE(y t , gΓ ): ∀r ∈ [s, t], Z t Z t yr = y t + g(u, yu , zu )du + At − As − zu dBu r

r

dΓr (yr , zr ) = 0, a.e.,a.s., Γ it follows Egs,t [y t ] = ys . While (y, z, A) is the solution of reflected BSDE with one

d lower obstacle L, i.e. y s = Eg,Γ s,t [y t ], where Γt = [Lt , ∞) × R , since Lt ≥ Lt , a.e.a.s., we have for t ∈ [0, T ],

Γt ⊇ Γ t , thanks to (A5), g,Γ Γ Egs,t [yt ] ≤ Es,t [y t ] = y s . Γ Example 3.2. As the above example we first introduce a gΓ -expectation: Egt,T [·] : 2 2 L (FT ) → L (Ft ), by reflected BSDE with one lower obstacle L, i.e. set Γt = 2 [Lt , ∞) × Rd , where L ∈ L2F (0, T ) such that E[ess sup0≤t≤T (L+ t ) ] < ∞. − 2 2 Let U ∈ LF (0, T ) with E[esssup0≤t≤T (Ut ) ] < +∞, such that there exists a Rt process Xt = X0 + A0t − Kt0 + 0 Zs0 dBs , 0 ≤ t ≤ T with Z 0 ∈ L2F (0, T ), A0 , K 0 ∈ D2F (0, T ), such that A0 and K 0 are increasing with A00 = K00 = 0 and such that

Lt ≤ Xt ≤ Ut , a.e., a.s.. Then by the existence result in Peng & Xu41 , for y T ∈ L2 (FT ), satisfying LT ≤ y T ≤ UT , the reflected BSDE with two obstacle L and U admits the unique solution (y, z, A, K) on [0, T ]. So (y t )0≤t≤T is a gΓ -submartingale. In fact for 0 ≤ s ≤ t ≤ T ,

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n ∈ N, we consider the BSDE with upper reflected obstacle U penalizing from the lower obstacle L on the interval [s, t]: Z t Z t Z t y nr = yt + g(u, ynu , z nu )du + n dΓu (y nu , z nu )du − (K t − K s ) − z nu dBu . r

r

r

Compare it with the BSDE penalizing only from lower obstacle L on the interval [s, t]: Z t Z t Z t n n n n n g(u, yu , zu )du + n dΓu (yu , zu )du − yr = y t + zun dBu , r

ynr

r

r

yrn ,

≤ s ≤ r ≤ t. By theorem 6.2 and theorem 6.1, we get the we deduce convergence result of the sequences: y nr % y r and yrn % yr , s ≤ r ≤ t. So Γ Egs,t [y t ] = ys ≥ ys .

4. gΓ -Reflected BSDEs In this section we consider the smallest g–supersolution with constraint Γ and reflected by a lower obstacle L or a upper obstacle U . We assume that the two reflected obstacles L and U are Ft -adapted processes satisfying L, U ∈ L2F (0, T )

and

− 2 ess sup L+ t , sup Ut ∈ L (FT ). 0≤t≤T

(10)

0≤t≤T

The problem is formulated as follows: Definition 4.1. A gΓ –solution reflected by the lower obstacle L is a quadruple of ¯ satisfying processes (y, z, A, A) ¯ (i) (y, z, A, A) ∈ D2F (0, T ) × L2F (0, T ; Rd ) × (A2F (0, T ))2 solves the BSDE Z T Z T zs dBs , (11) yt = X + g(s, ys , zs )ds + AT − At + AT − At − t

t

dΓt (yt , zt ) = 0, yt ≥ Lt ,

dP × dt a.s..

(ii) A and A¯ are increasing processes and the generalized Skorohod reflecting condition is satisfied: for each L∗ ∈ D2F (0, T ) such that yt ≥ L∗t ≥ Lt ,; dP × dt a.s., we have Z T (12) (ys− − L∗s− )dAs = 0, a.s., 0

(iii) y is the smallest g–supersolution, i.e., for any quadruple (y ∗ , z ∗ , A∗ , A¯∗ ) satisfying (i) and (ii), we have yt ≤ yt∗ , Our first main result is:

∀t ∈ [0, T ], a. s..

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Theorem 4.1. We assume (1), (2) and (10). For a given terminal condition X ∈ L2 (FT ), we assume that there exists a triple (y ∗ , z ∗ , A∗ ) ∈ D2F (0, T ) × L2F (0, T ) × A2F (0, T ), such that dA∗ ≥ 0 and Z T Z T yt∗ = X + g(s, ys∗ , zs∗ )ds + (A∗T − A∗t ) − zs∗ dBs , (13) t ∗ ∗ (yt , zt )

t

d

∈ Γt ∩ [Lt , ∞) × R .

¯ reflected by L in the sense of Definition 4.1 (i)-(iii) Then the gΓ –solution (y, z, A, A) exists. The smallest g–supersolution constrained by Γ and reflected by the upper obstacle U is relatively more complicate than the case of the lower obstacle. Definition 4.2. The gΓ –solution reflected by the upper obstacle U is a quadruple of processes (y, z, A, K) satisfying (i) (y, z, A, K) ∈ D2F (0, T ) × L2F (0, T ; Rd) × (A2F (0, T ))2 with dA ≥ 0 and dK ≥ 0 such that Z T Z T yt = X + g(s, ys , zs )ds + AT − At − (KT − Kt ) − zs dBs , (14) t

t

dΓt (yt , zt ) = 0, a.s. a.e., V[0,t] [A − K] = V[0,t] [A + K], where V[0,t] [η] denotes the total variation of a process η on [0, t]. (ii) yt ≤ Ut , dP × dt a.s., the generalized Skorohod reflecting condition is satisfied: Z T ∗ (Ut− − yt− )dKt = 0, a.s., ∀U ∗ ∈ D2F (0, T ), s.t. yt ≥ Ut∗ ≥ Ut , dP × dt a.s.. 0

(iii) For any other quadruple (y ∗ , z ∗ , A∗ , K ∗ ) satisfying (i) and (ii), we have yt ≤ yt∗ ,

0 ≤ t ≤ T, a.s.

The roles of dA and dK have a similar interpretation as in the lower obstacle case. Theorem 4.2. We assume that (1) holds for g and (2) holds for the constraint Γ, U is a process in L2F (0, T ), satisfying (10). We also assume that exists a quadruple (y ∗ , z ∗ , A∗ , K ∗ ) ∈ D2F (0, T ) × L2F (0, T ) × (A2F (0, T ))2 , such that dA∗t ≥ 0, dKt∗ ≥ 0 and Z T Z T yt∗ = X + g(s, ys∗ , zs∗ )ds + (A∗T − A∗t ) − (KT∗ − Kt∗ ) − zs∗ dBs , (15) t

t

dΓt (yt∗ , zt∗ ) = 0, a.s. a.e. Z T ∗ yt∗ ≤ Ut , (yt− − Ut− )dKt∗ = 0, a.s.. 0

Then the gΓ –solution (y, z, A, K) reflected from upper obstacle U in the sense of Definition 4.2 (i)-(iii) exists. Moreover, If U is continuous (Ut− ≥ Ut ), then K is also a continuous process.

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The proofs can be found in Peng and Xu42 . 5. Nonlinear Decomposition Theorem for gΓ -Supermartigales and gΓ -Submartingales 5.1. Nonlinear decomposition theorem for gΓ -supermartingale We have the following gΓ -supermartingale decomposition theorem. Theorem 5.1. We assume that the coefficent g and the constraint Γ satisfy (1), (2) and (7). Let Y be a RCLL gΓ -supermartingale on [0, T ]. Then there exists a unique RCLL increasing process A ∈ A2F (0, T ), such that Y is a Γ–constrained g–supersolution, namely, Z T Z T yt = YT + g(s, ys , zs )ds + AT − At − zs dBs , t

t

dΓt (yt , zt ) = 0, a.e. a.s.. Proof. For each fixed m ≥ 0, m ∈ N, we consider the solution (y m , z m , Am ) ∈ D2F (0, T ) × L2F (0, T ; Rd) × A2F (0, T ) of the the following reflected BSDE, with the gΓ –supermartingale as the lower obstacle: Z T Z T m m m m m m yt = YT + g (s, ys , zs )ds + AT − At − zsm dBs , (16) t

dAm ≥ 0, ytm ≤ Yt ,

Z

t

T

(Yt − ytm )dAm t = 0,

0

where g m (t, y, z) := (g + mdΓ )(t, y, z). By Proposition 3.4, this gΓ –supermartingale is also a g m –supermartingale for each m. It follows from the g–supermartigale decomposition theorem (see Peng38 ) that ytm ≡ Yt . Thus z m is invariant with m: z m ≡ Z ∈ L2F (0, T ; Rd) and the above equation (16) can be written Yt = Y T +

Z

T t

m (g + mdΓ )(s, Ys , Zs )ds + Am T − At −

Z

T

Zs dBs . t

Consequently, for all m ≥ 0, notice that Am is increasing, we have Z T 0≤m dΓ (s, Ys , Zs )ds "

0

≤ Y0 − Y T +

Z

T

Zs dBs − 0

Z

T

g(s, Ys , Zs )ds 0

#+

∈ L2 (FT ).

RT From this it follows immediately 0 dΓ (s, Ys , Zs )ds = 0. Thus Am is also invariant with m: Am = A ∈ D2F (0, T ). We thus complete the proof.

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5.2. Nonlinear decomposition theorem for gΓ -submartingale We now consider the decomposition theorem of a given gΓ -submartingale Y ∈ D2F (0, T ). We make following assumptions: (H) There exists a quadruple (y ∗ , z ∗ , A∗ , K ∗ ) ∈ D2F (0, T ) × L2F (0, T ) × (A2F (0, T ))2 , such that Z T Z T ∗ ∗ ∗ ∗ ∗ ∗ ∗ yt = YT + g(s, ys , zs )ds + (AT − At ) − (KT − Kt ) − zs∗ dBs , t

t

dΓt (yt∗ , zt∗ ) = 0, a.s. a.e., Z T ∗ yt∗ ≤ Yt , (yt− − Yt− )dKt∗ = 0, a.s.. 0

We have the following decomposition theorem for RCLL gΓ -submartingales. Theorem 5.2. We assume that the coefficient g and the constraint Γ satisfy (1), (2) and (7). Let Y ∈ D2F (0, T ) be a gΓ -submartingale on [0, T ] such that Yt− ≥ Yt , ∀t ∈ [0, T ], a.s..

(17)

We suppose also (H) is satisfied. Then there exists a unique continuous increasing process K with E[KT2 ] < ∞, such that the triple (Y − K, Z, A) ∈ D2F (0, T ) × L2F (0, T ; Rd) × A2F (0, T ) is the smallest g–supersolution with terminal condition constrained in d Yt − K t ∈ Γ K t = {(y, z) ∈ R × R : (y − Kt , z) ∈ Γt }, t ∈ [0, T ],

where we set g K (t, y, z) := g(t, y + Kt , z), (t, y, z) ∈ [0, T ] × R × Rd . Proof. Consider the BSDE(X, g) with constraint Γ and reflecting upper obstacle Y . From Theorem 4.2, we know that there exists a process (y, Z, A, K) ∈ D2F (0, T )× L2F (0, T ; Rd) × (A2F (0, T ))2 Z T Z T yt = YT + g(s, ys , Zs )ds + AT − At − (KT − Kt ) − Zs dBs , (18) t

t

(yt , Zt ) ∈ Γt , dA ≥ 0, dK ≥ 0 , Z T yt ≤ Yt , (ys− − Ys− )dKs = 0, a.s., 0

and K is continuous in t ∈ [0, T ], notice that Yt− ≥ Yt . To prove y ≡ Y , we only need one direction: y ≥ Y . For each δ > 0, define a stopping time σ δ := inf{t, yt ≤ Yt − δ} ∧ T.

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If there exists a δ > 0, such that P (σ δ < T ) > 0. We can define the following stopping time τ := inf{t ≥ σ δ : yt ≥ Yt }. It’s clear that σ δ ≤ τ ≤ T . Since y − Y is RCLL, we have yt < Yt on the interval [σ δ , τ ) and yτ = Yτ . Thus Kt = Kσ , for t ∈ [σ δ , τ ). Since Kτ − = Kτ , we have Γ Kσδ = Kτ . It follows that yt∧τ,τ = Egt∧τ,τ [Yτ ], t ∈ [σ δ , τ ] and thus yσδ = EgσΓδ ,τ [Yτ ] ≥ Yσδ , a.s. But this formula contradicts P (σ δ (ω) < T ) > 0. We thus conclude that Yt ≤ yt , for each t ∈ [0, T ], a.s..

6. Appendix We recall monotonic limit theorem introduced in Peng38 and a generalized version in Peng & Xu41 . We consider the following sequence of Itˆ o’s processes, for i ∈ N, Z t Z t yti = y0i + gsi ds − Ait + Kti + zsi dBs , t ∈ [0, T ]. (19) 0

0

Here g i ∈ L2F (0, T ) and Ai , K i ∈ D2F (0, T ) are given increasing processes. We assume (i) (yti ) increasingly converges to y ∈ L2F (0, T ) with E[ sup0≤t≤T (yt+ )2 ] < ∞; (ii) (gti , zti ) are weakly converges to (g 0 , z) in L2F (0, T ; R × Rd ); (iii) Ai is continuous and increasing with Ai0 = 0 and E[(AiT )2 ] < ∞; (20) Futhermore for K i , we assume (iv) Ktj − Ksj ≥ Kti − Ksi , ∀0 ≤ s ≤ t ≤ T, a.s. , ∀i ≤ j; (v) For each t ∈ [0, T ], Ktj % Kt , with E[KT2 ] < ∞.

(21)

An easy consequence is (i) E[sup0≤t≤T |yti |2 ] ≤ C; RT (ii) E 0 |yti − yt |2 ds → 0,

(22)

The generalized monotonic limit theorem given in Peng & Xu41 is as follows. Theorem 6.1. We assume (20) and (21). Then the limit y of the sequence {y i }∞ i=1 has a form Z t Z t yt = y 0 + gs0 ds − At + Kt + zs dBs , (23) 0

A2F (0, T )

0

such that A, K ∈ are increasing processes. Here, for each t ∈ [0, T ], At i ∞ 2 (resp. Kt ) is the weak (resp. strong) limit of {Ai }∞ i=1 (resp.{K }i=1 ) in L (Ft ).

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136 p d Furthermore, for any p ∈ [0, 2), {z i }∞ i=1 strongly converges to z in LF (0, T ; R ), i.e., Z T lim E |zsi − zs |p ds = 0. (24) i→∞

0

If furthermore A is a continuous process, then we have Z T lim E |zsi − zs |2 ds = 0. i→∞

(25)

0

This theorem was originally obtained in Peng38 Z t Z t i i i i yt = y 0 + gs ds − At + zsi dBs , , t ∈ [0, T ], i = 1, 2, · · · . 0

(26)

0

Since this result will be used in this paper, situation, we state it as follows: Theorem 6.2. We assume the same conditions as in Theorem 6.1. Then the limit y of the sequence {y i }∞ i=1 given in (26) has a form Z t Z t 0 gs ds − At + zs dBs , 0 ≤ t ≤ T, yt = y 0 + 0

0

where A ∈ A2F (0, T ) is an increasing process. Here, for each t ∈ [0, T ], At is the 2 i ∞ weak limit of {Ai }∞ i=1 in L (Ft ). Furthermore, for any p ∈ [0, 2), {z }i=1 strongly p d converges to z in LF (0, T, R ), i.e., Z T lim E |zsi − zs |p ds = 0, p ∈ [0, 2). (27) i→∞

0

If furthermore (At )t∈[0,T ] is continuous, then we have Z T lim E |zsi − zs |2 ds = 0. i→∞

(28)

0

A more general situation was considered in Peng38 of the smallest g– supersolution with constraint Γ, when Γ is Γt (ω) = {(y, z) ∈ R1+d : Φ(ω, t, y, z) = 0}. where Φ is a nonnegative Lipschitz function and Φ(·, y, z) ∈ L2F (0, T ), for each (y, z) ∈ R × Rd . Under the following assumption, the existence of the smallest solution is proved. The following theorem of the existence of the smallest solution was obtained in Peng38 . Theorem 6.3. We assume that the function g satisfies (1) and the constraint Γ satisfies (2). We assume that there is at least one Γ–constrained g–supersolution y 0 ∈ D2F (0, T ): Z T Z T yt0 = X 0 + g(s, ys0 , zs )ds + A0T − A0t − zs0 dBs , (29) t

t

A ∈ A2F (0, T ) , (yt0 , zt0 ) ∈ Γt , t ∈ [0, T ], a.s. a.e.

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Then, for each X ∈ L2 (FT ) with X ≤ X 0 , a.s., the gΓ –solution y ∈ D2F (0, T ) with the terminal condition yT = X (defined in Definition 2.2) exists. Moreover, this gΓ –solution is the limit of a sequence of g n –solutions with g n = g + ndΓ , i.e., Z T Z T ytn = X + (g + ndΓ )(s, ysn , zsn )ds − zsn dBs , (30) t

t

where the convergence is in the following sense: Z T ytn % yt , with lim E[|ytn − yt |2 ] = 0, lim E |zt − ztn |p dt = 0, p < 2, n→∞ n→∞ 0 Z t n n n At := (g + ndΓ )(s, ys , zs )ds → At weakly in L2 (Ft ),

(31) (32)

0

where z and A are the corresponding martingale part and increasing part of y, respectively. The proof is a consequence of theorem 6.2. Remark 6.1. If the constraint Γ is of the following form Γt = (−∞, Ut ] × Rd , where Ut ∈ L2 (Ft ), then the gΓ –solution with terminal condition yT = X exists, if and only if dΓt (Yt , Zt ) ≡ 0, a.s. a.e., where (Y, Z) is the solution of the BSDE −dYt = g(t, Yt , Zt )dt − Zt dBt , t ∈ [0, T ], YT = X. Indeed, if dΓt (Yt , Zt ) ≡ 0, a.s. a.e., then by the comparison theorem of BSDE, Y itself is the gΓ –solution with YT = X. On the other hand, assume that (y, z, A) is the gΓ –solution with yT = X. Again by the comparison theorem, we have yt ≥ Yt . Thus dΓt (Yt , Zt ) ≡ 0, since 0 ≡ dΓt (yt , zt ) = (yt − Ut )− ≥ (Yt − Ut )− ≡ dΓt (Yt , Zt ). We also present Theorem 6.4. (Comparison Theorem of Γ–Constrained BSDE) We assume that g 1 , g 2 satisfy (1) and Γ1 , Γ2 satisfy (2). We also assume that ∀(t, y, z) ∈ [0, T ] × R × Rd , X 1 ≤ X 2 , g 1 (t, y, z) ≤ g 2 (t, y, z), Γ1t ⊇ Γ2t .

(33)

For i = 1, 2, let Y i ∈ D2F (0, T ) be the gΓi i –solution with terminal condition YTi = X i . Then we have Yt1 ≤ Yt2 , for t ∈ [0, T ], a.s. Moreover, if for each y 1 , y 2 ∈ R, z 1 , z 2 ∈ Rd with y 1 ≤ y 2 , , t ∈ [0, T ], we have dΓ1t (y 1 , z 1 ) − dΓ2t (y 2 , z 2 ) ≤ 0, (resp. ≥ 0) then we have, for each 0 ≤ s ≤ t ≤ T , A1t − A1s ≤ A2t − A2s , (resp. A1t − A1s ≥ A2t − A2s ).

(34)

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The proof is obtained easily by penalisation method and the convergence result. Remark 6.2. The comparison theorem for reflected BSDE with one lower obstacle is also a corollary of Theorem 6.4. In fact, in this case, the distance functions dΓ1t (y, z) = dΓ2t (y, z) = (y − Lt )− satisfy which the first condition of (34). Acknowledgment. The first author thanks to Freddy Delbaen for a fruitful discussion, after which we have understood a point of view risk measure of gΓ – solution (see definition 2.2). References 1. P. Artzner, F. Delbaen, J. M. Eber and D. Heath, Coherent measures of risk, Math. Finance, 9, 203–228, (1999). 2. M. Alario-Nazaret, J. P. Lepeltier and B. Marchal, Dynkin games. Lecture Notes in Control and Inform. Sci. 43, 23-42. Springer, Berlin (1982). 3. P. Barrieu and N. El Karoui, Optimal Derivatives Design under Dynamic Risk Measures, Contemporary Mathematics, (2004). 4. A. Bensoussan and A. Friedman, Non-linear variational inequalities and differential games with stopping times. J. Funct. Anal. 16, 305-352 (1974). 5. J. M. Bismut, Sur un probleme de Dynkin. Z. Wahrsch. Verw. Gebiete 39, 31-53 (1977). 6. F. Coquet, Y. Hu, J. Memin and S. Peng, Filtration–consistent nonlinear expectations and related g–expectations, Probab. Theory Relat. Fields, 123, 1–27 (2002). 7. J. Cvitanic and I. Karatzas, Convex duality for constrained porfolio optimization. Ann. Appl. Probab. 2, 767–818 (1992). 8. J. Cvitanic and I. Karatzas, Hedging contingent claims with constrained portfolios. Ann. Appl. Probab. 3, 652-681 (1993). 9. J. Cvitanic and I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games. Ann. Probab. 24, 2024–2056 (1996). 10. J. Cvitanic, I. Karatzas and M. Soner, Backward stochastic differential equations with constraints on the gain-process, The Annals of Probability, 26, No. 4, 1522–1551 (1998). 11. C. Dellacherie and P. A. Meyer, Probabilit´es et Potentiel, I-IV. (Hermann. Paris) (1975). 12. C. Dellacherie and P. A. Meyer, Probabilit´es et Potentiel, V-VIII. (Hermann. Paris) (1980). 13. E. B. Dynkin and A. A. Yushkevich, Theorems and Problems in Markov Processes. Plenum Press, New York (1968). 14. N. El Karoui, Les aspects probabilistes du contrˆ ole stochastique. Ecole d’´et´e de SaintFlour, Lecture Notes in Math. 876. (Springer, Berlin), 73-238 (1979). 15. N. El Karoui, S. Peng, and M. C. Quenez, Backward stochastic differential equations in Finance. Math. Finance, 7, 1-71 (1997). 16. N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez, Reflected Solutions of Backward SDE and Related Obstacle Problems for PDEs, Ann. Probab. 25, no 2, 702–737 (1997). 17. N. El Karoui and M. C. Quenez, Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33, 29–66 (1995). 18. H. F¨ ollmer and A. Schied, Convex measures of risk and trading constraints, preprint (2002).

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19. S. Hamadene, Reflected BSDE’s with Discontinuous Barrier and Application. Stochastics and Stochastics Reports, 74, no 3-4, 571–596 (2002). 20. S. Hamadene and J.-P. Lepeltier, Zero-sum stochastic differential games and backward equations. Syst. Control Lett. 24, No.4, 259-263 (1995). 21. S. Hamadene and J.-P. Lepeltier, Backward equations, stochastic control and zerosum stochastic differential games. Stochastics Stochastics Rep. 54, No.3-4, 221-231 (1995). 22. S. Hamad`ene, J.-P. Lepeltier, and A. Matoussi, Double barrier backward SDEs with continuous coefficient. Backward Stochastic Differential Equations, Pitman Research Notes in Math. Series, No.364, (El Karoui and Mazliak edit.), 161-177 (1997). 23. S. Hamadene, J.–P. Lepeltier, and S. Peng, BSDEs with Continuous Coefficients and Stochastic Differential games, Backward Stochastic Differential Equations, Pitman Research Notes in Math. Series, No.364, (El Karoui and Mazliak edit.), 115–128 (1997). 24. S. Hamad`ene and J.-P. Lepeltier, Reflected BSDE’s and mixed game problem, Stochastics Processes Appl. 85, 177-188 (2000). 25. I. Karatzas and S. G. Kou, Hedging American contingent clains with constrained portfolios, Finance and Stochastics, 2, 215–258 (1998). 26. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York (1991). 27. A. N. Kolmogorov, Foundations of Theory of Probability, Engl. trans., Chelsea, New York (1933–1956). 28. J.-P. Lepeltier and J. San Martin, Backward SDE’s with two barriers and continuous coefficient. An existence result. Journal of Applied Probability, vol. 41, no. 1. 162-175 (2004). 29. J.-P. Lepeltier and M. Xu, Penalization method for Reflected Backward Stochastic Differential Equations with one r.c.l.l. barrier. Statistics and Probability Letters, 75, 58-66 (2005). 30. H. Morimoto, Dynkin games and martingale methods. Stochastics, 13, 213-228 (1984). 31. J. Neveu, Discrete-Parameter Martingales. North-Holland, Amsterdam (1975). 32. E. Pardoux and A. Gegout-Petit, Equations diff´erentielles stochastiques r´etrogrades r´efl´echies dans un convexe. Stoch. Stoch. Reportes 57, 111-128 (1996). 33. E. Pardoux and S. Peng, Adapted solutions of Backward Stochastic Differential Equations. Systems Control Lett. 14, 51-61 (1990). 34. E. Pardoux and S. Peng, Backward Stochastic Differential Equations and Quasilinear Parabolic Partial Differential Equations. In: B. Rozovskii and R.Sower, eds, Stocastic Differential Equations and their Applications, Lecture Notes in Control and Inform. Sci. 186. (Springer, Berlin), 200-217 (1992). 35. E. Pardoux and A. Rascanu, Backward stochastic differential equations with subdifferential operator and Related variational inequalities. Stoch. Processes and Appl. 76, 191-215 (1998). 36. S. Peng, A Generalized Dynamic Programming Principle and Hamilton-JacobiBellmen equation, Stochastics, 38, 119–134, (1992). 37. S. Peng, Backward SDE and Related g–Expectation, in Backward Stochastic Differential Equations, Backward Stochastic Differential Equations, Pitman Research Notes in Math. Series, No.364, (El Karoui and Mazliak edit.), 141–159 (1997). 38. S. Peng, Monotonic limit theory of BSDE and nonlinear decomposition theorem of Doob-Meyer’s type. Probab. Theory and Related Fields, 113 , 473–499 (1999). 39. S. Peng, Nonlinear Expectations, Noninear Evaluations and Risk Measures, Lecture Notes in CIME–EMS, Bressanone, Italy July, 2003, LNM 1856, Springer (2004).

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40. S. Peng, Dynamical consistent nonlinear evaluations and expectations, preprint (2003). 41. S. Peng and M. Xu, Smallest g–Supermartingales and Related Reflected BSDEs with Single and Double L2 Barriers, Annuals of Institute of Henri Poincare, Vol. 41, 3, 605-630 (2003). 42. S. Peng and M. Xu, Reflected BSDE with Constraints and the Related Nonlinear Doob-Meyer Decomposition Theorem, preprint (2005). 43. S. Peng and F. Yang, Duplicating and Pricing Contingent Claims in Incomplete Markets, Pacific Economic Review, 4(3) 237–260, (1999). 44. M. Royer, BSDEs with jumps and related non linear expectations, preprint of Institut de Recherche Mathematique de Rennes, Universite de Rennes I (2003). 45. E. G. Rosazza, Some examples of risk measures via g–expectations, preprint (2003). 46. A. V. Shorokhod, Studies in the Theory of the Random Processes, Addison Wesley, New-York (1965).

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Chap21-TangShanjian

ˆ INTEGRALS AND NONCONVEXITY PHENOMENON ON ITO’S ON STOCHASTIC ATTAINABLE SETS

SHANJIAN TANG Department of Finance and Control Sciences, School of Mathematical Science, Fudan University, Shanghai 200433, CHINA E-mail: [email protected]

Two examples are given, to illustrate the unusual nonconvexity phenomenon of the closed range R of the vector-valued measure for Itˆ o’s integrals, and of the closed attainable sets A(t) ( at some time t ) for stochastic control systems. Keywords: Itˆ o’s integral, vector-valued measures, stochastic control, attainable sets. AMS(MOS) subject classifications: 93E20, 46G10, 28B05.

1. Introduction It is well-known that the vector-valued measure defined by Lebesgue’s integrals for lRn − valued functions have closed and convex ranges. The result was fist proved by Lyapunov 8 in 1940, and its many other proofs can be found in the literature ( see Lindenstrauss 7 , Yorke 14 , and Tardella 12 ). The finiteness of the space where the associated functions take their values turns out to be crucial. In 1946, Lyapunov 9 pointed out that the assertion may fail to hold for Bochner’s integrals. In 1969, however, Uhl 13 proved that the closure of the range of the vector-valued measure defined via Bochner’s integrals is convex. Consider a complete probability space (Ω, F, Ft , P ) with the filtration {Ft ; t ∈ [0, 1]} satisfying the so-called “usual conditions”. Let L2 (Ω, F, P ; lRn ) denote the space of lRn -valued square integrable random variables defined on (Ω, F, P ). Let {W (t); t ∈ [0, 1]} be a one-dimensional (Ft )−adapted standard Brownian motion, and let L2F (0, 1; lRn ) denote the space of lRn -valued (Ft )-adapted processes f (·) such that E

Z

1

|f (t)|2 dt < ∞.

0

Let χC (·) denote the characteristic function of the subset C of [0, 1]. For ∀ f (·) ∈ L2F (0, 1; lR), with the help of Itˆ o’s integrals, we can define the following vector-valued 141

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measure on the measurable space ([0, 1], B([0, 1])) : Z Z 1 m(C) =: f (s) dW (s) =: χC (s)f (s) dW (s), ∀ C ∈ B([0, 1]). C

0

It is clear that m(·) takes its values in the vector space L2 (Ω, F, P ; lR). A natural question is asked: Does the measure m(·) necessarily have a convex range ? If not, is the closure of its range necessarily convex ? In Section 2, we observe the nonconvexity phenomenon of both the range R of the measure m(·) and its closure R, and illustrate it by one example. It is also well-known that the vector-valued measure theory ( including Lyapunov’s convexity theorem and Uhl’s approximate convexity result ) has wide applications in the theory of optimal control and the calculus of variations. Early in 1959, LaSalle 3 , for the first time, applied the theory to study the time optimal control problem for lumped parameter systems, and revealed that the associated attainable sets are both closed and convex for arbitrary nonempty bounded control domain U ( possibly nonconvex and non-closed). Later, Halkin 1 used it to prove the Pontryagin’s maximum principle. Li and Yao 4,5 , Li and Yong 6 extended its application to optimal control problems for distributed parameter systems. Li and Yao 4 pointed out that, for the infinite-dimensional optimal control problems, LaSalle’s well-known bang-bang principle may be incorrect, and its following weaker formulation, however, still holds: the closure of the associated attainable set is convex. For stochastic control systems with control not entering into the diffusion coefficients, Hu 2 proved that the closure of the associated attainable set is convex. In Section 3, we observe the nonconvexity phenomenon of both the attainable set and its closure when the diffusion term depends explicitly on the control variable. This is surprised. The unexpected new phenomenon is illustrated via one example, and some conditions are also given to ensure the convexity of the closure of the attainable set when the diffusion contains the control. The nonconvexity phenomenon of the closed attainable set sheds some new insight on general optimal stochastic control problems with the terminal state constrained in some closed and convex subset of L2 (Ω, F1 , P ; lR). It is known that the convexity result on attainable sets is of fundamental importance in the theory of Pontryagin’s maximum principle. Thus, the nonconvexity phenomenon almost suggests, from our point of view, that the corresponding general maximum principle, should not be expected. One might think that, in view of Peng’s technique of the second-order expansion (see Peng 10 ), the difficulty caused by the terminal constraints can be attacked by using Ekeland’s variational principle, and the corresponding maximum principle can be derived immediately. The idea is conventional, but fails to do. Now, we might conjecture that the failure of applying Ekeland’s variational principle is a consequence of the observed nonconvexity phenomenon. Some related works should be mentioned. The terminal constraint discussed in Tang and Li 11 is a statistical one, different from what is concerned above. Hu 2 derived a maximum principle for optimal control problems with terminal constraints,

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where he assumed that the diffusion does not contain the control variable. When the control variable appears in the diffusion, to formulate a maximum principle for the optimal stochastic control problems with the above-concerned terminal constraint, it seems necessary to impose some conditions ensuring the convexity of the closed attainable sets. Detailed discussion on the problem will be given in later publications. Finally, from the nonconvexity phenomenon on the stochastic attainable sets, we also see that, the value function of optimal stochastic control problem should not, in general, be expected to be equal to the one of its relaxed version. 2. Nonconvexity phenomenon on Itˆ o’s integrals Given f (·) ∈ L2F (0, 1; lR), consider the following vector valued-measure m(·) on ([0, 1], B([0, 1])): R1 R m(C) =: C f (s) dW (s) =: 0 χC (s)f (s) dW (s), (1) ∀ C ∈ B([0, 1]). We are to show that the range R of the measure m(·) is not convex in general. This can be illustrated by considering the case of f (s) ≡ 1. For the case of f (s) ≡ 1, we have {0, W (1)} ⊂ R,

1 W (1) ∈ Co(R) =: the convex hull of R. 2

But, for ∀ C ∈ B([0, 1]), we have = = = = = with

E|m(C) − 12 W (1)|2 R E| C dW (s) − 21 W (1)|2 R1 R E| C dW (s) − 0 21 dW (s)|2 R1 E| 0 u(s) dW (s)|2 R1 E 0 |u(s)|2 ds R1 E 0 41 ds = 14 6= 0,

u(s) =

1 1 χC (s) − χ[0,1]\C (s). 2 2

This shows 1 1 W (1) ∈ / R, W (1) ∈ / R =: the closure of R. 2 2 Hence, both the range R and its closure R are nonconvex. The above nonconvexity phenomenon shows that Ito’s integrals do not have the effect of exact ( or approximate ) convexification, which Lebesgue’s (or Bochner’s) ones have.

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3. Nonconvexity phenomenon on stochastic attainable sets Consider the following linear stochastic control system dx(t) = [A1 (t)x(t) + B1 (t)u1 (t)] dt + [A2 (t)x(t) + B2 (t)u2 (t)]dW (t), x(0) = x0 ∈ lRn .

(2)

Here, Ai (·) : [0, 1] → lRn×n , Bi (·) : [0, 1] → lRn×m , and ui (·) : [0, 1] → Ui ⊂ lRm , i = 1, 2, are square integrable. Denote by Uad the totality of U1 × U2 -valued (Ft )-adapted processes u(·) such that Z 1 E |u(t)|2 dt < ∞. (3) 0

The associated attainable set A(t) at t is defined as A(t) =: {x(t; u(·)) : ∀ u(·) ∈ Uad .}.

(4)

We are to show that A(1) is not convex in general. Consider the following case: n = m = 1, A1 (t) = A2 (t) = 0, B1 (t) = 0, B2 (t) = 1, U2 = {−1, 1}.

(5)

A(1) ⊃ {−W (1), W (1)},

(6)

0 ∈ Co(A(1)) =: the convex hull of A(1),

(7)

E|y|2 = 1, ∀ y ∈ A(1).

(8)

We have

The last relation implies that 0∈ / A(1), 0 ∈ / A(1) =: the closure of A(1). In view of (7) , we conclude that both the attainable set A(1) (at t = 1) and its closure A(1) are nonconvex. Just as the convexity results on deterministic attainable sets are consequences of the effects of the exact or approximate convexification for Lebesque’s or Bochner’s integrals, the above nonconvexity phenomenon is a consequence of the nonconvexity phenomenon on Itˆ o’s integrals. Careful readers may see that the above nonconvexity phenomenon is caused by the nonconvexity of U2 . If the convexity of U2 is assumed, one can show that the closure of A(t) is convex This is a natural sufficient condition for the convexity of the closed stochastic attainable sets A(t) at time t ∈ [0, 1].

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4. Conclusion The nonconvexity phenomenon is a new feature both for Itˆ o’s integrals and for stochastic control problems. It distinguishes Itˆ o’s integrals from Lebesgue’s or Bochner’s ones, and distinguishes stochastic control problems from deterministic (both finite- and infinite- dimensional ) ones. Acknowledgments The author would like to thank Professors Xunjing Li and Jiongmin Yong for their helpful comments. Thanks are also due to Professor Shige Peng for his discussion with the author. Postscripts This short paper was completed in 1994. The main results were presented in the CIMPA school, held in the summer of the year 1994. The paper was initially submitted to SIAM Journal on Control and Optimization. Although the paper received some good comments in some aspects from reviewers, it was finally rejected by the journal. Since that failure, the paper has been buried until now. The main results of the paper were later deepened and extended by Jiongmin Yong in his paper, entitled with “some results on the reachable sets of linear stochastic systems ” (Proc. 37th CDC, pages 1335–1340). They were also exposed in the book by Jiongmin Yong and Xunyu Zhou, entitled with “Stochastic Controls: Hamiltonian Systems and HJB Equations ” (Springer, 1999). The paper shows very well how the author’s scientific thinking at that time was greatly impacted by the works of Professor Xunjing Li and his colleagues. References 1. H. Halkin, Lyapunov’s theorem on the range of a vector measure and Pontryagin’s maximum principle, Arch. Rational Mech. Anal. 10, 296-304 (1962). 2. Y. Hu, Maximum principle of optimal control for Markov processes, Acta Mathematica Sinica 33, 43-56 (1990). 3. J. P. Lasalle, The time optimal control problem, Contributions to the theory of nonlinear oscillations, vol.5, Princeton Univ. Press, Princeton, N.J., 1-24 (1959). 4. X. Li and Y. Yao, Time optimal control of distributed parameter systems, Scientia Sinica 24, 455-465 (1981). 5. X. Li and Y. Yao, Maximum principle of distributed parameter systems with time lags, Distributed Parameter Systems, Lecture Notes in Control and Information Sciences, Vol. 75, Springer-Verlag, NY, 410-427 (1985). 6. X. Li and J. Yong, Necessary conditions for Optimal control of distributed parameter systems, SIAM J. Control and Optimalization 29, 895-908 (1991). 7. J. Lindenstrauss, A short proof of Lyapunov’s convexity theorem, J. Math. Mech. 15, 971-972 (1966).

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8. A. Lyapunov, Sur les fonctions-vecteurs compl`etement additives, Bull. Acad. Sci. URSS Ser. Math. 4, 465-478 (1940). 9. A. Lyapunov, ibid., T. 10, 277-279 (1946). 10. S. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control 28, 966-979 (1990). 11. S. Tang and X. Li, Necessary conditions for optimal controls of stochastic systems with random jumps, SIAM J. Control 32, no. 5., 1447-1475(1994). 12. F. Tardella, A new proof of the Lyapunov convexity theorem, SIAM J. Control and Optimization 28, 478-481 (1990). 13. J. J. Jr. Uhl, The range of a vector-valued measure, Proc. Amer. Math. Soc. 23, 158-163 (1969). 14. J. A. Yorke, Another proof of the Lyapunov convexity theorem, SIAM J. Control and Optimization 9, 351-353 (1971).

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Chap22-ZhangXu

UNIQUE CONTINUATION AND OBSERVABILITY FOR STOCHASTIC PARABOLIC EQUATIONS AND BEYOND∗

XU ZHANG Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China; and Key Laboratory of Systems and Control, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, China E-mail: [email protected]

In this paper a unified approach is presented to address the unique continuation and observability of stochastic parabolic equations, and deterministic parabolic and hyperbolic equations, developed by the author and his collaborators. It is found that the unique continuation and observability properties of these equations can be derived by means of a universal approach, which is based on the global Carleman estimate via an identity for a stochastic parabolic-like operator.

1. Introduction Let T > 0 be given. Let (Ω, F, {Ft }t≥0 , P ) be a complete filtered probability space on which a 1 dimensional standard Brownian motion {w(t)}t≥0 is defined. Let H be a Fr´echet space. We denote by L2F (0, T ; H) the Fr´echet space consisting of Z T |Y (t)|2H dt < ∞ (here all H-valued {Ft }t≥0 -adapted processes Y (·) such that E 0

and henceforth | · |H stands for any norm in H), with the canonical norms; by L∞ echet space consisting of all H-valued {Ft }t≥0 -adapted bounded F (0, T ; H) the Fr´ processes, with the canonical norms; and by L2F (Ω; C([0, T ]; H)) the Fr´echet space consisting of all H-valued {Ft }t≥0 -adapted continuous processes Y (·) such that E max |Y (t)|2H < ∞, with the canonical norms. t∈[0,T ]

We begin with the following abstract stochastic evolution system ( dx = [Ax + B(t)x]dt + C(t)xdw(t), t ∈ (0, T ), x(0) = x0 .

(1)

∗ The work is supported by the FANEDD of China (Project No: 200119), the NCET of China under grant NCET-04-0882, the NSF of China under grants 10371084 and 10525105, the Grant MTM2005-00714 of the Spanish MEC, and the SIMUMAT projet of the CAM (Spain).

147

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In the above, x = x(t) is the state which takes values in a Hilbert space X, P − a.s., x0 ∈ F0 is any given initial state, A is the generator of some C0 −semigroup {eAt }t≥0 on X, and B(·), C(·) ∈ L2F (0, T ; L(X)) (Here, L(X) denotes the Banach space of linear continuous operators from X into itself). Many stochastic Partial Differential Equations (PDEs in short) enter in this context. For instance, the stochastic parabolic equation, hyperbolic equation, plate equation, Schr¨ odinger equation, Maxwell equations, and Lam´e system, etc. We now fix an observation space V , which is another Hilbert space. Also, fix a linear observation operator G : L2F (Ω; C([0, T ]; X)) 7→ L2F (0, T ; V ). The unique continuation property of solutions to (1) is formulated as follows: Gx(·) = 0 in L2F (0, T ; V ) ⇒ x0 = 0. The observability (estimate) of solutions to (1) can be stated as: Find (if possible) a constant C > 0 such that |x(T )|L2 (Ω,FT ,P ;X) ≤ C|Gx(·)|L2F (0,T ;V ) . In the deterministic cases, the observability property is usually stronger than the unique continuation one. But, things are not completely clear in the stochastic situation. It is easy to see that, in some sense and to some extend, the observability estimate is a quantitative version of the unique continuation property. Unique continuation and observability properties are important problems not only in PDEs itself, but also in some application problems such as controllability (see Refs. 12, 18, 24, 25), optimal control (see Ref. 11), inverse problems (see Ref. 7) and so on. Numerous studies on unique continuation and observability estimate for deterministic PDEs can be found in Refs. 6, 18, 24, 25 and the rich references cited therein. It would be quite interesting to extend the deterministic unique continuation and observability estimate to the stochastic ones, but there are many things which remain to be done, and some of which seem to be challenging. There are two classical tools in the study of the unique continuation for deterministic PDEs. One is Holmgren-type uniqueness theorem, another is Carleman-type estimate (see Ref. 6). Note however that the solution of a stochastic equation is generally non-analytic in time even if the coefficients of the equation are constants. Therefore, one cannot expect a Holmgren-type uniqueness theorem for the unique continuation for stochastic equations except for some very special cases. On the other hand, the usual approach to employ Carleman-type estimate for the unique continuation needs to localize the problem. The difficulty for the stochastic situation consists in the fact that one cannot simply localize the problem as usual because the usual localization technique may change the adaptedness of solutions, which is a key feature in the stochastic setting. In order to establish the observability estimate for PDEs, several powerful methods have been introduced. For example, for the parabolic equations, one uses the Carleman estimates (see Ref. 2), and/or the spectral method (Ref. 14); for the hyperbolic equations, one uses the multiplier method (see Ref. 12), the microlocal

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analysis method (see Ref. 1), the Carleman estimates (see Ref. 19), and/or the spectral method (see Ref. 14). However, these methods are quite different and are very difficult to combine. Especially, these methods depend very strongly on the nature of the principal part of the related differential operator. This, in turn, leads to a fundamental problem, that is, from the philosophical point of view, it would be natural to expect to establish a unified theory, in some sense and to some extent, on the observability of parabolic and hyperbolic equations. It is clear that the Carleman estimate approach is one of the common methods applied to the observability problems of parabolic and hyperbolic equations. We remark that Carleman estimate is simply a weighted energy method. However, at least formally, the usual Carleman estimate used to derive the observability inequality for parabolic equations is quite different from that for hyperbolic ones. Sometimes it is desired to develop a unified Carleman estimate method for different types of PDEs. Stimulated by the works 10,13 , the author and his collaborators have developed a unified global Carleman estimate method, based on a fundamental point-wise estimate on the related principal operator, to derive boundary and/or internal observability inequalities for hyperbolic equations with various different lower order terms, and/or boundary conditions (see Refs. 8, 18, 19 and 20). This method has the advantage, among others, to give an explicit estimate on the observability constant with respect to suitable Sobolev space norms of the coefficients in the equations, which is crucial for some control problems. This method can also be applied to other equations, say the plate equations (see Ref. 21), the Schr¨ odinger equations (see Ref. 9), and even a very general PDEs of second order (see Ref. 4). Recently in a joint work with S. Tang (see Refs. 16, 17), the author successfully extended this method to the stochastic parabolic equations. The purpose of this paper is to apply a key identity (see Theorem 2.3 in Section 2) established in Refs. 16 and 17 to give a unified treatment on the unique continuation property and observability estimate of the stochastic parabolic equations, and deterministic parabolic and hyperbolic equations. We would like to emphasize that all of the unique continuation and observability results presented in this paper follow from this identity. The rest of this paper is organized as follows. Section 2 is devoted to the unique continuation. First, we recall the classical unique continuation property for deterministic parabolic equations and the main ingredients for its proof. Next, we state our main unique continuation theorem for stochastic parabolic equations. Further, we outline the key points in the proof of our unique continuation result. Finally, we present a list of open problems in the field of stochastic unique continuation. Section 3 is devoted to the observability. First, we show how to establish, in one shot, the observability estimate for the stochastic parabolic equations, and deterministic parabolic and hyperbolic equations. Then, we present a list of open problems related to the stochastic observability estimate, which may serve as a guide for

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the further unified treatment for the observability problems of the stochastic and deterministic PDEs. 2. Unique continuation 2.1. Review on unique continuation for deterministic parabolic equations Let T > 0, G ⊂ lRn (n ∈ lN) be a given bounded domain with a C 2 boundary ∂G, and G0 6= G be a given open subset of G. 4

4

Put Q = (0, T ) × G,

4

Σ = (0, T ) × ∂G and Q0 = (0, T ) × G0 . Throughout this section, we assume 2,∞ that aij ∈ W 1,∞ (0, T ; Wloc (G)) satisfy aij = aji (i, j = 1, 2, · · · , n) and for any open subset G1 of G, there is a constant s0 = s0 (G1 ) > 0 so that n X

aij ξ i ξ j ≥ s0 |ξ|2 ,

∀ (t, x, ξ) ≡ (t, x, ξ 1 , ξ 2 , · · · , ξ n ) ∈ (0, T ) × G1 × lRn .

i,j=1

In the sequel, we shall use C to denote a generic positive constant which may change from line to line. We consider the following deterministic parabolic equation: yt −

n X

(aij yxi )xj = h a, ∇y i +by

in Q.

(2)

i,j=1

Here a and b are suitable coefficients, h ·, · i stands for the scalar product in lR n . The following unique continuation result for deterministic parabolic equations is classical (see Ref. 15, for example): Theorem 2.1. Let a ∈ L∞ (0, T ; L∞ lRn )) and b ∈ L∞ (0, T ; L∞ loc (G; loc (G)). Then T 2 2 1 any weak solution y ∈ C([0, T ]; Lloc (G)) L (0, T ; Hloc (G)) of (2) vanishes identically in Q provided y = 0 in Q0 . The proof of Theorem 2.1 is based on the following two tools: • Carleman estimate for deterministic parabolic operators: Lemma 2.1. Let K be a nonempty open subset of Q. Then there exist two positive constants c0 and τ0 such that for all τ ≥ τ0 and any u ∈ C0∞ (K), it holds 2 Z Z n X 2τ φ 2τ φ 3 2 2 ij (3) e e τ |u| + τ |∇u| dtdx ≤ C (a uxi )xj dtdx, ut − K K i,j=1 where

∆

φ ≡ φ(t, x1 , · · · , xn ) = e−c0 xn .

(4)

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• Holmgren coordinate transform: The transform is defined as follows   t˜ = t,     0 x ˜ = x0 ,   2  4 (2t − T )  x + |x0 |2 + xn , ˜n = f (t, x) = T2 where x = (x0 , xn ).

(5)

The fundamental inequality (3) in Lemma 2.1 and transform (5) allow to propagate the zero set of any weak solution y of (2), say from xn = 0, to some neighborhood of this hyperplane. We remark that, the role of t and x0 is completely symmetric in the above lemma and transform. As we shall see, this is the key obstacle to extend Theorem 2.1 to the stochastic situation. 2.2. Unique continuation for stochastic parabolic equations We consider now the following stochastic parabolic equation: dz −

n X

(aij zxi )xj dt = [h a, ∇z i +bz]dt + czdw(t)

in Q.

(6)

i,j=1

Here a, b and c are suitable coefficients. The main result of this section is stated as follows: n ∞ ∞ ∞ Theorem 2.2. Let a ∈ L∞ F (0, T ; Lloc (G; lR )), b ∈ LF (0, T ; Lloc (G)), and 1,∞ ∞ 2 c ∈ LF (0, T ; Wloc (G)). Then any weak solution z ∈ LF (Ω; C([0, T ]; T 1 L2loc (G))) L2F (0, T ; Hloc (G)) of (6) vanishes identically in Q × Ω, P -a.s. provided z = 0 in Q0 × Ω, P -a.s.

It is easy to see that Theorem 2.1 is a special case of Theorem 2.2. The above result is a unique continuation theorem for stochastic parabolic equations. There are numerous references on the unique continuation for deterministic parabolic equations. However, to my best acknowledge, very little is known for its stochastic counterpart. 2.3. Key points in the proof of Theorem 2.2 We now present the key points in the proof of Theorem 2.2. We remark that, as for the proof of Theorem 2.2, for the space variable x in equation (6), we may proceed as in the classical argument. However, for the time variable t, due to the adaptedness requirement, we will have to treat it separately. • Carleman estimate for stochastic parabolic operators: The proof of Theorem 2.2 is based on the following key exponentially weighted energy identity for a stochastic parabolic-like operator established in Refs. 16 and 17:

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Theorem 2.3. (Refs. 16 and 17) Let m ∈ lN, bij = bji ∈ C 1,2 ([0, T ] × lRm ) (i, j = 1, 2, · · · , m), and ` ∈ C 1,3 ((0, T ) × lRm ), Ψ ∈ C 1,2 ((0, T ) × lRm ). Assume u is a C 2 (lRm )-valued semimartingale. Set θ = e` and v = θu. Then for any x ∈ lRm and ω ∈ Ω (a.s. dP ), Z T h X m m ih i X 2 θ − (bij vxi )xj + Av du − (bij uxi )xj dt 0

+2

Z

i,j=1

T

0

i,j=1

m h X m X

i,j=1

0 0

0 0

2bij bi j `xi0 vxi vxj0 − bij bi j `xi vxi0 vxj0

i0 ,j 0 =1

Z T X m Ψ xi 2 i v dt + 2 (bij vxi dv)xj +Ψbij vxi v − bij A`xi + 2 xj 0 i,j=1 Z T X m n X m h 0 i 0 0 0 =2 2bij bi j `xi0 − bij bi j `xi0 0

i,j=1

xj 0

i0 ,j 0 =1

xj 0

Z

T o bij t + Ψbij vxi vxj dt + Bv 2 dt 2 0 Z Th X m m ih X i +2 − (bij vxi )xj + Av − (bij vxi )xj + (A − `t )v dt

−

0

+ −

Z

bij vxi vxj

i,j=1 T

0

where

i,j=1

m X

n

θ2 A −

T Z + Av 2 − 0

m h X

T

θ 0

i,j=1 m X 2

bij duxi duxj

i,j=1

bij `xi `xj + (bij `xi )xj

i,j=1

io

(du)2 ,

 m X  4 ij   A=− (bij `xi `xj − bij  xj `xi − b `xi xj ) − Ψ,   i,j=1

m m h i  X X  4  ij  B = 2 AΨ − (Ab ` ) − A − (bij Ψxj )xi .  x x t i j  i,j=1

i,j=1

It is notable that we only assume the symmetry of bij in the above theorem. Now, for any nonnegative and nonzero function ψ ∈ C 3 (G), any k ≥ 2, and any (large) parameters λ > 1 and µ > 1, put ` = λα,

α(t, x) =

eµψ(x) − e2µ|ψ|C(G) , tk (T − t)k

ϕ(t, x) =

eµψ(x) . tk (T − t)k

(7)

By Theorem 2.3, we have a crucial Carleman-type estimate for stochastic parabolic operators as follows: Theorem 2.4. (Ref. 22) Let ψ ∈ C 3 (G) satisfy minx∈G |∇ψ(x)| > 0. Then there is some µ0 > 0 such that for all µ ≥ µ0 , one can find two constants C = C(µ) > 0

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and λ1 = λ1 (µ) so that for all u ∈ L2F (Ω; C([0, T ]; L2 (G))) f ∈ L2F (0, T ; L2 (G)) and g ∈ L2F (0, T ; H 1(G)) with du −

n X

(aij uxi )xj dt = f dt + gdw(t)

T

L2F (0, T ; H02(G)),

in Q,

i,j=1

and all λ ≥ λ1 , it holds Z Z λ3 µ 4 E ϕ3 θ2 u2 dxdt + λµ2 E ϕθ2 |∇u|2 dxdt Q

Q

Z n n Z X 2 2 ≤C E θ f dxdt + E θ2 bij gxi gxj dxdt Q

+2E

Z

Q

Q

i,j=1

(8)

n X o θ2 bij (`xi `xj + `xi xj ) g 2 dxdt . i,j=1

Theorem 2.4 can be regarded as a stochastic version of Lemma 2.1. Note however that the time variable t in (8) is treated globally. This is the key feature of our stochastic Carleman estimate. • Selection of weighted function: In the proof of Theorem 2.2, we choose ψ as ψ = ψ(x) = 1 − xn . In this way, α in (7) plays a similar role as that of φ in (4). • Partial Holmgren coordinate transform: The desired transform is defined as follows ( 0 x ˜ = x0 ,

x ˜n = |x0 |2 + xn .

(9)

Note that, unlike (5), the time variable t does not appear in (9)! We refer to Ref. 22 for a detailed proof of Theorem 2.2. We remark that one can also use Theorem 2.3 to derive some unique continuation property for the hyperbolic equations. But, we shall not pursue this possibility in this paper. Instead, in the next section, we shall use this theorem to derive an observability estimate result for the hyperbolic equations. 2.4. Open problems The field of stochastic unique continuation is full of open problems! In what follows, we only present a very limited list of these ones.

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• Strong unique continuation for stochastic parabolic equations. Does the conclusion in Theorem 2.2 remain true under a weaker assumption in which the condition “z = 0 in Q0 × Ω, P -a.s.” is replaced by Z E z 2 dxdt = O(rm ), ∀ m ∈ N, |t−t0 |+|x−x0 |2
as r → 0, for some point (t0 , x0 ) ∈ Q? • Unique continuation for stochastic hyperbolic equations, or even more general stochastic PDEs. Almost nothing is known in this respect. We refer to Ref. 23 for some preliminary recent result in this respect for the stochastic wave equation. • Unique continuation for backward stochastic PDEs, or even forwardbackward stochastic PDEs. Nothing is known in this respect. • Application of unique continuation properties of stochastic PDEs, backward stochastic PDEs and forward-backward stochastic PDEs. Almost nothing has been done in this respect. Nevertheless, applications to control and inverse problems can be expected, but remains to be done. Applications to some finance problem is also possible. 3. Observability 3.1. Observability estimates for four classes of equations in one shot Throughout this section, we assume that aij ∈ W 2,∞ (G) satisfy aij = aji (i, j = 1, 2, · · · , n) and for some constant s1 > 0, n X

aij ξ i ξ j ≥ s1 |ξ|2 ,

∀ (x, ξ) ≡ (x, ξ 1 , ξ 2 , · · · , ξ n ) ∈ G × lRn .

i,j=1

In the sequel, we fix a n∗ ∈ lR so that  ∗  n ≥ 2, if n = 1, n∗ > 2, if n = 2,  ∗ n ≥ n, if n ≥ 3.

Fix suitable coefficients a, b and c. We are concerned with the observability estimates for the following stochastic parabolic equation  n X    dy − (aij yxi )xj dt = [h a, ∇y i +by]dt + cydw(t) in Q,    i,j=1 (10)  y=0 on Σ,      y(0) = y0 in G,

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(deterministic) parabolic equation  n X    yt − (aij yxi )xj = h a, ∇y i +by in Q,    i,j=1  y=0      y(0) = y0

(11)

on Σ, in G,

and (deterministic) hyperbolic equation  n X    y − (aij yxi )xj = h a, ∇y i +by + cyt in Q, tt    i,j=1  y=0      y(0) = y0 ,

(12)

on Σ,

yt (0) = y1

in G.

That is, we expect to find a constant C = C(a, b, c) > 0 such that solutions of systems (10)–(12) satisfy respectively |y(T )|L2 (Ω,FT ,P ;L2 (G)) ≤ C|y|L2F (0,T ;L2 (G0 )) , |y(T )|L2 (G) ≤ C|y|L2 (Q0 ) ,

∀ y0 ∈ L2 (Ω, F0 , P ; L2 (G)),

∀ y0 ∈ L2 (G),

(13) (14)

and ∀ (y0 , y1 ) ∈ H01 (G) × L2 (G). (15) We shall see that the observability estimates of the above three equations can be established by means of the global Carleman estimate based on the key identity m X for stochastic parabolic-like operator “du − (bij uxi )xj dt” derived in Refs. 16 |(y(T ), yt (T ))|H01 (G)×L2 (G) ≤ C|y|H 1 (0,T ;L2 (G0 )) ,

i,j=1

and 17, which we have recalled in Theorem 2.3. Indeed, starting from Theorem 2.3, it is established in Ref. 17 the following observability estimate results for system (10). n ∞ ∞ n Theorem 3.1. (Ref. 17) Let a ∈ L∞ F (0, T ; L (G; lR )), b ∈ LF (0, T ; L (G)), and ∞ 1,∞ c ∈ LF (0, T ; W (G)). Then there is a constant C = C(a, b, c) > 0 such that all solutions y of system (10) satisfy (13). Moreover, the constant C may be bounded as ∗

2

C(a, b, c) = CeCr1 , 4

n ∞ with r1 = |a|L∞ + |b|L∞ n∗ (G)) + |c|L∞ (0,T ;W 1,∞ (G)) . F (0,T ;L (G;lR )) F F (0,T ;L

Next, when Theorem 2.3 degenerates into the deterministic situation, it gives

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Theorem 3.2. Let m ∈ lN, bij = bji ∈ C 1,2 ([0, T ] × lRm ) (i, j = 1, 2, · · · , m), and ` ∈ C 1,3 ((0, T )×lRm ), Ψ ∈ C 1,2 ((0, T )×lRm ). Assume u ∈ C 2 (lRm ), and set θ = e` and v = θu. Then for any (t, x) ∈ (0, T ) × lRm , it holds m m h X ih i X 2θ − (bij vxi )xj + Av ut − (bij uxi )xj

+2

i,j=1 m h X

i,j=1

i,j=1

m X

0 0

0 0

2bij bi j `xi0 vxi vxj0 − bij bi j `xi vxi0 vxj0

i0 ,j 0 =1

m X Ψ xi 2 i v +2 (bij vxi vt )xj +Ψbij vxi v − bij A`xi + 2 xj i,j=1

=2

m n X m h X

i,j=1

h +2 − +

i0 ,j 0 =1 m X ij

0 0 2bij bi j `xi0

xj 0

(b vxi )xj + Av

i,j=1

m X

i,j=1

bij vxi vxj + Av 2

t

ih

−

0 0 − bij bi j `xi0 m X

xj 0

i

−

o bij t + Ψbij vxi vxj + Bv 2 2

(bij vxi )xj + (A − `t )v

i,j=1

i

,

where  m X  4 ij   A=− (bij `xi `xj − bij  xj `xi − b `xi xj ) − Ψ,   i,j=1

m m h i  X X  4  ij  B = 2 AΨ − (Ab ` ) − A − (bij Ψxj )xi . x x t  i j  i,j=1

i,j=1

Now, starting from Theorem 3.2, one can show the following known observability estimate results for system (11). Theorem 3.3. (Refs. 3 and 2) Let a ∈ L∞ (Q; lRn ) and b ∈ L∞ (0, T ; Lp (G)) for some p ∈ [n, ∞]. Then there is a constant C = C(a, b) > 0 such that all solutions y of system (11) satisfy (14). Moreover, the constant C may be bounded as C(a, b) 1 1 2 = exp C 1 + + T |b|L∞(0,T ;Lp (G)) + |b|L3/2−n/p + (1 + T )|a| . n ∞ (0,T ;Lp (G)) L∞ (Q;lR ) T Finally, by taking the functions in Theorem 3.2 to be independent of variable t, it gives Theorem 3.4. Let m ∈ lN, bij = bji ∈ C 2 (lRm ) (i, j = 1, 2, · · · , m), and ` ∈ C 3 (lRm ), Ψ ∈ C 2 (lRm ). Assume u ∈ C 2 (lRm ), and set θ = e` and v = θu. Then

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for any x ∈ lRm , it holds 2θ

m hX

(bij vxi )xj − Av

i,j=1 m X

+2

i,j=1

m i X

(bij uxi )xj

i,j=1

m h X 0 0 0 0 2bij bi j `xi0 vxi vxj0 − bij bi j `xi vxi0 vxj0 i0 ,j 0 =1

Ψ xi 2 i v +Ψbij vxi v − bij A`xi + 2 xj m n X m h X 0 0 0 0 − bij bi j `xi0 =2 2bij bi j `xi0

where

i,j=1 i0 ,j 0 =1 m X +2 (bij vxi )xj i,j=1

xj 0

xj 0

i

o + Ψbij vxi vxj + Bv 2

2 − Av ,

 m X  4 ij   A = − (bij `xi `xj − bij  xj `xi − b `xi xj ) − Ψ,   i,j=1

m m h i X  X  4  ij  B = 2 AΨ − (Ab ` ) − (bij Ψxj )xi . xi xj   i,j=1

i,j=1

We see that only the symmetry condition is assumed for bij in the above. Hence, Theorem 3.4 is applicable to hyperbolic and ultra-hyperbolic operators, especially n X to the operator “utt − (aij uxi )xj ”. As a consequence of Theorem 3.4, we will i,j=1

show an observability estimate for system (12). For this, we introduce the following condition. (H1) There exists a function d(·) ∈ C 2 (G) satisfying the following: i) For some constant µ0 > 0, n n X n h X

i,j=1

≥ µ0

0

0

0 0

ij 2aij (ai j dxi0 )xj0 − aij d xi0 xj 0 a

i0 ,j 0 =1

n X

ij

a ξi ξj ,

io

ξi ξj (16)

n

∀ (x, ξ) ∈ G × lR ;

i,j=1

ii) The function d(·) does not have any critical point in G, i.e., min |∇d(x)| > 0. x∈G

Denote by ν = ν(x) = (ν1 , ν2 , · · · , νn ) the unit outward normal vector of G at x ∈ ∂G. For the function d(·) satisfying Condition (H1), we introduce the following

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set: n X n o 4 Γ0 = x ∈ ∂G aij νi dxj > 0 .

(17)

i,j=1

It is easy to check that, if d(·) ∈ C 2 (G) satisfies (16), then for any given constants α ≥ 1 and β ∈ lR, the function 4 ˆ = dˆ = d(x) αd(x) + β

(18)

still satisfies Condition (H1) with µ0 replaced by αµ0 , the set Γ0 (defined by (17)) remaining unchanged. Hence, without loss of generality, we may assume that      (16) holds with µ0 ≥ 4, n 1 X ij   a (x)dxi (x)dxj (x) ≥ max d(x) ≥ min d(x) > 0,  4 x∈G x∈G

(19)

∀ x ∈ G.

i,j=1

In what follows, put 4

R1 = max x∈G

n o 4 T0 = 2 inf R1 d(·) satisfies (19) .

p d(x) ,

We further introduce the assumption on the observation domain G0 : (H2) There is a constant δ > 0 such that G0 = Oδ (Γ0 )

\

G, o where Oδ (Γ0 ) = x ∈ lRn |x − x0 | < δ for some x0 ∈ Γ0 . n

Now, following Refs. 5 and 3, based on Theorem 3.4, we can establish the following observability estimates for system (12):

Theorem 3.5. Assume that conditions (H1)–(H2) hold. Let a ∈ L∞ (Q; lRn ), b ∈ L∞ (0, T ; Lp (Ω)) for some p ∈ [n, ∞], and c ∈ L∞ (Q). Then, for any T > T0 , there is a constant C = C(a, b, c) > 0 such that all solutions y of system (12) satisfy (15). Moreover, the constant C may be bounded as 1 2 2 C(a, b, c) = exp C 1 + |b|L3/2−n/p + |a| + |c| . n ∞ (0,T ;Lp (G)) L∞ (Q;lR ) L∞ (Q) 3.2. Open problems The field of observability estimate for stochastic PDEs is full of open problems as well! In what follows, we only present a very limited list of these ones. • Similar identity as that in Theorem 2.3 for stochastic parabolic + Schr¨ odinger-like operator: du −

m X

i,j=1

[(bij +

√ −1cij )uxi ]xj dt.

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Nothing is known in this respect. We refer to an interesting recent paper partial result in the deterministic situation.

4

for some

• Similar identity as that in Theorem 2.3 for stochastic wave-like operator: m X √ dut − [(bij + −1cij )uxi ]xj dt. i,j=1

Almost nothing is published in this respect. We refer to Ref. 23 for some preliminary recent result in this respect for the stochastic wave equation. • Similar identity as that in Theorem 2.3 for general stochastic operator of second order: m X √ √ a, b, c ∈ lR. [(bij + −1cij )uxi ]xj dt, adut + (b + c −1)du − i,j=1

Nothing is known in this respect. The main goal is to derive the observability estimate results for all stochastic and deterministic PDEs of second order in one shot! References 1. C. Bardos, G. Lebeau, J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control and Optim., 30, pp. 1024–1065 (1992). 2. A. Doubova, E. Fern´ andez-Cara, M. Gonz´ alez-Burgos, E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM J. Control Optim., 41, 798–819 (2002). 3. T. Duyckaerts, X. Zhang, E. Zuazua, On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, to appear. 4. X. Fu, A weighted identity for partial differential operators of second order and its applications, C. R. Math. Acad. Sci. Paris, 342, pp. 579–584 (2006). 5. X. Fu, J. Yong, X. Zhang, Exact controllability for the multidimensional semilinear hyperbolic equations, in submission. 6. L. H¨ ormander, The Analysis of Linear Partial Differential Operators (III–IV), Springer-Verlag, Berlin, (1985). 7. V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, Berlin, (1998). 8. I. Lasiecka, R. Triggiani, X. Zhang, Nonconservative wave equations with purely Neumann B.C.: Global uniqueness and observability in one shot, Contemp. Math., 268, pp. 227–326 (2000). 9. I. Lasiecka, R. Triggiani, X. Zhang, Global uniqueness, observability and stabilization of nonconservative Schr¨ odinger equations via pointwise Carleman estimates. Part I: H 1 (Ω)-estimates, J. Inverse and Ill-Posed Problems, 11, pp. 43–123 (2004). 10. M. M. Lavrent’ev, V. G. Romanov, S. P. Shishat.skii, Ill-posed problems of mathematical physics and analysis, Translated from the Russian by J. R. Schulenberger, Translations of Mathematical Monographs, 64, American Mathematical Society, Providence, RI, (1986).

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11. X. Li, J. Yong, Optimal control theory for infinite-dimensional systems, Systems & Control: Foundations & Applications, Birkh¨ auser Boston, Inc., Boston, MA, (1995). 12. J. L. Lions, Contrˆ olabilit´e exacte, perturbations et syst´emes distribu´es, tome 1, RMA No: 8, Masson, Paris (1988). 13. M. Kazemi, M. V. Klibanov, Stability estimates for ill-posed Cauchy problems involving hyperbolic equations and inequalities, Appl. Anal., 50, pp. 93–102 (1993). 14. D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open problems, SIAM Rev., 20, pp. 639–739 (1978). 15. J.-C. Saut, B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66, pp. 118–139 (1987). 16. S. Tang, X. Zhang, Carleman inequality for backward stochastic parabolic equations with general coefficients, C. R. Math. Acad. Sci. Paris, 339, pp. 775–780 (2004). 17. S. Tang, X. Zhang, Null controllability for forward and backward stochastic parabolic equations, preprint. 18. X. Zhang, Exact Controllability of the Semilinear Distributed Parameter System and Some Related Problems, Ph. D. Thesis, Fudan University, Shanghai, China (1998). 19. X. Zhang, Explicit observability estimate for the wave equation with potential and its application, Royal Soc. London Proc. Ser. A Math. Phys. Eng. Sci., 456, pp. 1101– 1115 (2000). 20. X. Zhang, Explicit observability inequalities for the wave equation with lower order terms by means of Carleman inequalities, SIAM J. Control and Optim., 39, pp. 812– 834 (2001). 21. X. Zhang, Exact controllability of the semilinear plate equations, Asymptot. Anal., 27, pp. 95–125 (2001). 22. X. Zhang, Unique continuation for stochastic parabolic equations, in submission. 23. X. Zhang, Work in progress. 24. E. Zuazua, Some problems and results on the controllability of partial differential equations, in Progress in Mathematics, vol: 169, Birkh¨ auser Verlag, Basel/Switzerland, pp. 276–311 (1998). 25. E. Zuazua, Controllability and observability of partial differential equations: some results and open problems, in Handbook of Differential Equations: Evolutionary Differential Equations, vol. 3, Elsevier Science, pp. 527–621 (2006).

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DESIGN OF DYNAMIC HIGH-GAIN OBSERVERS FOR A CLASS OF MIMO NONLINEAR SYSTEMS∗

HAO LEI, JIANFENG WEI and WEI LIN Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, Ohio 44106, U.S.A. E-mail: [email protected]

Under the boundedness and observability conditions, a globally convergent observer is designed for a class of multi-output nonlinear systems which covers the block-triangular observer forms studied previously in the literature. The result of this paper incorporates and generalizes the earlier work on the observer design for single-output observable systems. Extensions to detectable systems and nonlinear systems with control inputs are also considered. Two examples are given to illustrated the validity of proposed method.

1. Introduction We consider in this paper the problem of global observer design for a multi-output nonlinear system in the observable canonical form x˙ i1 = xi2 x˙ i2 = xi3 .. . x˙ i,ki −1 = xi,ki

(1)

x˙ i,ki = fi (x),

i = 1, 2, · · · , p T T y = y 1 , y2 , · · · , yp = x11 , x21 , · · · , xp1

where x = (x1 , x2 , · · · , xp )T , xi = (xi1 , xi2 , · · · , xiki )T , ki ’s are suitable integers Pp satisfying i=1 ki = n. Without loss of generality, suppose 1 ≤ k1 ≤ k2 ≤ · · · ≤ kp ≤ n. In Ref. 4, Gauthier and Bornard illustrated that under a uniform observability condition, the autonomous system z˙ = f (z) y = h(z) ∗ This

(2)

work was supported in part by the NSF under grants DMS-0203387 and ECS-0400413, and in part by the AFRL Grant FA8651-05-C-0110. 163

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is transformed into the canonical form (1) by the following change of coordinates x = Φ(z) k

= (h1 (z), · · · , Lkf 1 h1 (z); · · · ; hp (z), · · · , Lf p hp (z))T where z ∈ IRn and y ∈ IRp are the system state and output, respectively. The vector fields f : IRn → IRn and h : IRn → IRp are smooth, with n ≥ p ≥ 1. For the autonomous system (2), a common approach for the observer design is to find a change of coordinates and an output injection so that (2) can be transformed into the so-called observer form. The approach was first introduced by Krener and Isidori 11 and Bestle and Zeitz 1 , in the single-output case (i.e., p = 1), and then was generalized to the multi-output case by Krener and Respondek 13 , Xia and Gao 22 and to discrete-time nonlinear systems by Lin and Byrnes 19 . More recent extensions can be found in the papers by Kazantzis and Kravvaris 8 , and Krener and Xiao 14 . The observer form based design method was further extended by Rudolph and Zeitz 20 to multi-output autonomous systems with a block triangular observer form, which essentially requires fi (x) in (1) to have certain triangular structure. In the work 21 , an explicit form of nonlinear observer was presented by Shim et al. for a class of multi-output multi-input (MIMO) nonlinear systems in a block triangular form. However, it is required that the bounds of the control inputs and system states be known. The nonlinearities of the systems are assumed to be Lipschitz with a known Lipschitz constant. In the paper by Krener and Kang 12 , a stepby-step, local observer design method was developed for MIMO nonlinear control systems which are also in a block-triangular form. An interesting feature of the paper 12 is that the observer gains are nonlinear functions of the estimated states and recursively designed. In this work, we consider the observer design for the observable canonical form (1) which does not have a block-triangular structure, because the nonlinearities fi (·)’s in (1) depend on the entire system states and all the sub-blocks of system (1) are coupled each other. To remove the block-triangular structure restriction in the previous work, we make the following assumption in this paper. Assumption 1.1. For every x(0) = x0 ∈ IRn , the corresponding solution trajectory x(x0 , t) of the observable system (1) uniquely exists and is globally bounded on [0, +∞). That is, there is an unknown constant C ≥ 0 depending on the initial condition x0 , such that |xij (x0 , t)| ≤ C, i = 1, · · · , p; j = 1, · · · , ki ; ∀t ∈ [0, ∞). Assumption 1.1 is a mild condition for autonomous systems (without control), because it covers an important class of dynamic systems such as the Van der Pol equation and Duffing oscillator 6,14 — both of them are unstable at the origin but nevertheless have globally bounded solution trajectories from any initial condition.

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On the other hand, the boundedness condition excludes the class of nonlinear systems with unbounded solutions or having a finite escape time, and hence is somewhat restrictive. This is, however, a trade-off for removing the block-triangular structure assumption. With the aid of Assumption 1.1, a universal-like global observer can be designed for the multi-output autonomous system (1). Following the spirit of our recent work 17 , we propose, in section 2, an adaptive observer scheme in which a delicate rescaling technique is employed to deal with the inter-coupling terms fi (x)0 s in (1) that consist of the entire system states. Due to the lack of the bound information of the solution trajectories, a saturation technique 9 is used in the construction of multivariable observers but the saturation threshold is tuned by a universal control law instead of being a prescribed constant. As done in the single-output case, the observer gain needs to be tuned adaptively. As a result, the proposed observer is a dynamic system with dimension of n + 2. In addition to the main result presented in section 2, we present in section 3 an extension of the global observer design scheme for a class of detectable systems. In section 4, the problem of global observer design is discussed for a class of systems with control inputs. Illustrative examples are given in section 5. Conclusions are drawn in section 6. 2. A Global Observer for Multi-Output Autonomous Systems In this section, we show that under Assumption 1.1, a globally observable system (2), or equivalently, (1) permits a globally convergent observer, no matter how complicatedly the sub-blocks are coupled each other. Moreover, it is possible to explicitly design a universal-like high-gain observer whose gains are adaptively updated. To make the presentation easy to follow, we give a constructive design procedure in section 2.1, while in section 2.2 the bounded analysis and proof of convergence are included. 2.1. Design of Dynamic High-Gain Observers To introduce the main result of this paper, we first recall the definition of a unit saturation function. Definition 2.1.

A unit saturation function sat(s) is defined as  if s > 1 1 sat(s) = s if |s| ≤ 1  −1 if s < −1

(3)

From the definition, it is not difficult to show that Lemma 2.1.

Given real numbers s1 , s2 and m > 0, suppose that |s1 | ≤ m. Then, s2 (4) |s1 − msat( )| ≤ |s1 − s2 |. m

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Define the mapping satm : IRn → [−m, m]n for any m > 0 as satm (x) := msat(

x1 x2 xn ), msat( ), · · · , msat( ) , m m m

∀x ∈ IRn .

Now, we are ready to state the main theorem of the paper. Theorem 2.1. For the multi-output system in observer canonical form (1), suppose the Assumption 1 holds. Then, there exists a global observer. In particular, a globally convergent observer can be constructed as x ˆ˙ i1 = xˆi2 + (M N )ai1 (yi − x ˆi1 ) 2 x ˆ˙ i2 = xˆi3 + (M N ) ai2 (yi − x ˆi1 ) .. . x ˆ˙ i,ki −1 = xˆi,ki + (M N )ki −1 ai,ki −1 (yi − x ˆi1 ) ki ˙x ˆi,ki = fi satN (ˆ x) + (M N ) ai,ki (yi − x ˆi1 ) p X 2 yi − x ˆi1 N˙ = γ , N (0) = 1 kp −ki +1 (M N ) i=1 M˙ = −M + ∆(N ),

M (0) = 1

(5)

where aij > 0, i = 1, · · · , p, j = 1, · · · , ki are the coefficients of the Hurwitz polynoP i aij ski −j , γ ≥ 1 is a prescribed constant, and ∆(N ) ≥ 1 is mials pi (s) = ski + kj=1 a smooth function which can be determined explicitly. Moreover, all the states of the closed-loop system (1)-(5) are well-defined and bounded on [0, ∞), and, lim [x(x0 , t) − x ˆ(x0 , x ˆ0 , t)] = 0,

t→∞

∀(x0 , x ˆ0 ) ∈ IRn × IRn .

Remark 2.1. (5) is a universal-like high-gain observer that is motivated by Refs. 23, 7 and 16. Different from the traditional high-gain observer 5,10 , the observer gain of (5) is composed of two parts. One is the moving saturation level N(t) which needs to be tuned in a manner similar to the one in Refs. 16 and 17. The other one is M (t), which is used to recover the offset of fi (satN (ˆ x)) from fi (x), to be updated through a linear ODE driven by a nonlinear function of N (t). The introduction of non-constant gains N (t) and M (t) enables us to deal with issue of the unknown bound of the solution trajectories of the observable system (1) or (2). It should be mentioned that ∆(N ) in the observer (5) can be calculated directly based on the observable system (1), in particular, by the nonlinear functions fi (x)’s. To make this point clear, we introduce the following technical lemma whose proof can be carried out in a fashion similar to that of Lemma 2.4 in Ref. 18, and hence is omitted here.

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Lemma 2.2. Let g : IRn → IR be a C 1 real-valued function. Then, there exist two smooth functions α, β : [0, +∞) → [1, +∞), such that ∀x, z ∈ IR n , |g(x) − g(z)| ≤ α(||x||)β(||z||)

n X

|xi − zi |.

(6)

i=1

Using the inequality (6), fi (x) − fi (satN (ˆ x)) can be estimated as follows. By Assumption 1, ||x(x0 , t)|| ≤ C, ∀t ≥ 0. Since ||satN (ˆ x)|| ≤ N , by Lemma 2.2, for each i = 1, 2, · · · , p, there exist two smooth positive functions αi (·) and βi (·) such that p X ki X fi (x) − fi (satN (ˆ xij − N sat( xˆij ) x)) ≤ αi (C)βi (N ) N i=1 j=1

Denote α(C) =

Pp

i=1

αi (C), β(N ) =

Pp

i=1

(7)

βi (N ), then one can simply choose

∆(N ) = β 2 (N ) ≥ 1.

(8)

In the next subsection, it will be shown that such a choice of ∆(N ) suffices to ensure the dynamic system (5) being a globally convergent observer of system (1). To sum up, a global observer for the observable system (1) with bounded solution trajectories can be constructed in three steps: Step 1. Pick a suitable γ > 0 and choose constants aij > 0, i = 1, · · · , p, j = Pki −1 ki −j a is Hurwitz; 1, 2, · · · , ki , such that pi (s) = ski + j=1 ij s Step 2. Use inequality (7) to estimate fi (x) − fi (satN (ˆ x)) and find β(N ) ≥ 1. 2 Then, compute ∆(N ) = β (N ); Step 3. With the obtained parameters γ, aij ’s and ∆(N ), design the observer (5). 2.2. Analysis of Boundedness and Convergence To show the proposed observer design method works, we start with analyzing the error dynamics. Let eij = xij − x ˆij , i = 1, 2, · · · , p; j = 1, 2, · · · , ki , be the estimate errors and denote L = M N . Then, the error dynamics is given by e˙ i1 = ei2 − Lai1 ei1 .. . e˙ i,ki −1 = ei,ki − L

ki −1

(9)

ai,ki −1 ei1

e˙ i,ki = fi (x) − fi (satN (ˆ x)) − Lki ai,ki ei1 ,

i = 1, · · · , p

Now, introduce the following rescaling transformation εij =

eij , Lkp −ki +j

i = 1, 2, · · · , p; j = 1, 2, · · · , ki .

(10)

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By construction (5), N (t) ≥ 1 and M (t) ≥ 1, hence L(t) = M (t)N (t) ≥ 1, ∀t ≥ 0. In the new coordinates, the error dynamics (9) can be expressed in the following compact form ε˙i = LAi εi +

L˙ 1 b f (x) − f sat (ˆ x ) − Di ε i i0 i i N L kp L

(11)

where for each 1 ≤ i ≤ p, 

εi1  εi2  εi =  .  ..

εi,ki

bi0

  0  ..    = .  0 1





−ai1  ..  . Ai =   −ai,k −1

  , 

i

−ai,ki

 0 ..  .  1

1 ··· .. . . . . 0 ··· 0 ···

0

ki ×ki



, ki ×1

kp − k i + 1 0 ···  0 kp − ki + 2 · · ·  Di =  .. .. ..  . . . 0

0

0 0 .. .

· · · kp

    

ki ×ki

(I). Proof of Boundedness To show the boundedness of all the signals, in what follows we shall construct a Lyapunov function and use a contradiction argument. By the choice of aij ’s, Ai ’s are Hurwitz matrices. Therefore, by Ref. 15 (see inequality (6)) there exist positive definite matrices Pi = PiT and real constants c2 > c1 > 0, such that ATi P + P Ai ≤ −I,

c 1 I ≤ D i P i + P i Di ≤ c 2 I

Now, consider the Lyapunov function V (ε) = the error dynamics (11). A direct calculation gives

Pp

i=1

Vi (εi ) =

(12) Pp

T i=1 εi Pi εi

1 L˙ x)) V˙ i ≤−Lkεi k2 − εTi (Di Pi + Pi Di )εi + 2εTi Pi bi0 kp fi (x) − fi (satN (ˆ L L Observe that L˙ M˙ N˙ = + , L M N M˙ = −M + β 2 (N ), Then, it is deduced from (13) that

N˙ = γ

p X

ε2i1 ≥ 0

i=1

N ≥ 1 and M ≥ 1.

for

(13)

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V˙ ≤ −L −

p X

kεi k2 +

i=1 p X

˙ M M

p 2 X T εi Pi bi0 fi (x) − fi (satN (ˆ x)) p L i=1

εTi (Di Pi + Pi Di )εi

i=1

i c3 α(C)β(N ) X X ˆij X xij − N sat( x ) · kεi k ≤ −L kεi k + Lp N i=1 i=1 i=1 j=1

p X

+c2

p X

p

p

k

2

p

kεi k2 − c1

i=1

≤ −M (N − c2 + c1

β 2 (N ) X kεi k2 M i=1

p β 2 (N ) X ) kεi k2 M 2 i=1

p ki p x ˆij X c3 α(C)β(N ) X X xij − N sat( ) · + kεi k Lp N i=1 j=1 i=1

(14)

where c3 is a suitable positive constant. Using the inequality (14), one can show that starting from any initial condition (x(0), xˆ(0)) ∈ IRn × IRn and M (0) = N (0) = 1, all the states of the closed-loop system (1)-(5) are well defined and globally bounded on [0, +∞). By Assumption 1, x(t, x0 ) of the observable system (1) is bounded ∀x0 ∈ IRn . Hence, the boundedness of all signals follows immediately if one can show that the error signal e(t) = x(t) − xˆ(t) and the observer gains (M (t), N (t)) are well-defined and globally bounded on [0, +∞). To this end, we employ a contradiction argument. Consider the error dynamics (11) or, equivalently, (9) and assume that it has a solution X(t) := (N (t), M (t), e(t)) which is neither well defined nor globally bounded on [0, +∞). Then, there is a maximal time interval [0, tf ) on which X(t) is well defined. In addition, lim ||(N (t), M (t), e(t))|| = +∞.

t→tf

(15)

That is, tf > 0 is a finite escape time of the dynamic system (5)-(11). (i). We prove that N (t) cannot escape at t = tf . If N (t) has a finite escape time tf , limt→tf N (t) = +∞. By construction, N˙ ≥ 0 and N (t) is a monotone nondecreasing function. Thus, there exists a time t∗1 ∈ [0, tf ) such that N (t) ≥ C ≥ |xij (t)|, This, together with Lemma 2.1, yields ˆij xij − N sat( x ) ≤ |eij |, N

t ∈ [t∗1 , tf ).

t ∈ [t∗1 , tf ).

(16)

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Using (16) and εij = eij /Lkp −ki +j with L ≥ 1, we have i c3 α(C)β(N ) X X ˆij X xij − N sat( x ) · kεi k Lp N i=1 i=1 j=1

p

≤ c3 α(C)β(N ) ≤α ¯ (C)β(N )

p

k

p ki p X 1 XX |e | · kεi k ij Lp i=1 j=1 i=1

p X

kεi k2

i=1

p

M 2 c1 β 2 (N ) X ≤ α ¯ (C) + kεi k2 , c1 M i=1

t ∈ [t∗1 , tf )

where α ¯ (C) > 0 is a real constant depending on C. In view of the estimation above, it is deduced from (14) that p

α ¯ 2 (C) X kεi k2 , V˙ ≤ −M N − c2 − c1 i=1

t ∈ [t∗1 , tf ).

Since limt→tf N (t) = +∞, there is a t∗2 ∈ [t∗1 , tf ) such that N (t) ≥ γ + c2 +

α ¯ 2 (C) , c1

t ∈ [t∗2 , tf ).

Using the last two inequalities and noting that M (t) ≥ 1, we arrive at V˙ ≤ −γ

p X

2

kεi k ≤ −γ

i=1

p X

˙ ε2i1 = −N,

∀t ∈ [t∗2 , tf ).

(17)

i=1

Consequently, +∞ = N (tf ) ≤ V (ε(t∗2 )) + N (t∗2 ) = constant,

(18)

which is a contradiction. Therefore, N (t) is well-defined and bounded on [0, tf ]. (ii). We show that M (t) is well-defined and bounded on [0, tf ]. Indeed, by the boundedness of N (t), there exists a real constant d > 1 such that ∆(N (t)) ≤ d, t ∈ [0, tf ]. Hence, for ∀t ∈ [0, tf ], M˙ = −M + ∆(N ) ≤ −M + d and M (t) ≤ exp(−t) + exp(−t)

Z

t

d exp(τ )dτ 0

≤ exp(−t) + d exp(−t)(exp(t) − 1) = exp(−t) + d − d exp(−t) ≤ d+1

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which implies that M (t) is bounded on [0, tf ]. (iii). With the boundedness property of M (t) and N (t), we can prove that e(t) is well-defined and bounded on [0, tf ]. To show this claim, we rescale system (9) by introducing the transformation eij ∗k p L −ki +j

ξij =

,

L∗ = M N ∗ ,

i = 1, · · · , p, j = 1, 2, · · · , ki

(19)

where N ∗ > 0 is a constant to be determined later. In the ξ-coordinates, the error dynamic system (9) can be expressed as M˙ Di ξ i ξ˙i = L∗ Ai ξi + L∗ ai ξi1 − Lkp −ki +1 Γi ai ξi1 − M 1 +bi0 ∗kp fi (x) − fi (satN (ˆ x)) L

(20)

where  ξi1   ξi =  ...,  , 

ξi,ki



 ai1   ai =  ...  , 

1 0  Γi =  .  ..

ai,ki

0 ··· N N∗ · · · .. . . . .

0 0 .. .

0 0 · · · ( NN∗ )ki −1

    

and Ai , Di , b0 are defined as in (11). For the rescaled system (20), consider the Lyapunov function W (ξ) = Pp T i=1 ξi Pi ξi with Pi satisfying (12). Then, there exist 0 < µ ≤ ν such that νW (ξ) ≥

p X

kξi k2 ≥ µW (ξ)

(21)

i=1

Taking derivative of W (ξ) along (20) gives ˙ ≤ −L∗ W

p X

kξi k2 + 2L∗

i=1

−2

p X

p X i=1

Lkp −ki +1 ξiT Pi Γi ai ξi1 +

i=1

≤ −(L∗ − c2 )

p X

kξi k2 − c1

i=1

+

2 L∗kp

−2L

p M˙ X ξiT Pi ai ξi1 − ξ T (Pi Di + Di Pi )ξi M i=1 i

p X i=1

p X i=1

2 L∗kp

∆(N ) M

p X i=1

p X i=1

ξiT Pi bi0 fi (x) − fi (satN (ˆ x))

kξi k2 + 2L∗

p X

ξiT Pi ai ξi1

i=1

ξiT Pi bi0 (fi (x) − fi (satC (ˆ x))) + fi (satC (ˆ x)) − fi (satN (ˆ x))

ξiT Pi Γi ai ξi1

(22)

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Using the boundedness of N (t) and M (t) and Lemma 2.2, we can obtain the following estimations: 2 |2L∗ ξiT Pi ai ξi1 | ≤ kξi k2 + (L∗ kPi ai k)2 ξi1

|2LξiT Pi Γi ai ξi1 |

2

≤ kξi k + L

2

2 kPi Γi ai k2 ξi1

(23) ≤ θi0 kξi k

2

(24)

where θi0 is positive constant related to Pi and the bound of M and N . Moreover, T 2ξi Pi bi0 1 fi (x) − fi (satC (ˆ x)) ∗k p L p ki 1 XX |eij | ≤ k2Pi bi0 kαi (C)βi (C)kξi k ∗kp L i=1 j=1 ≤ k2Pi bi0 kαi (C)βi (C)kξi k

p X

kξi k ≤ θi1

i=1

p X

kξi k2

(25)

i=1

2ξiT Pi bi0 f (sat (ˆ x )) − f (sat (ˆ x )) i C i N ∗k L p 2 ≤ θi2 kξi k ≤ θi2 kξi k2 + 1

(26)

where θi1 ’s are positive constants depending on C, while θi2 ’s are positive constants depending on C and the bound of N (t). Substituting (23)-(26) into (22) results in p p p p X X X X ∗ 2 2 ˙ W≤−(L −c2 −1− θi1 ) kξi k2+ [(θi0 + θi2 )kξi k2 ]+ (L∗ kPi ai k)2 ξi1 +p i=1

i=1

i=1

i=1

Pp 2 Denote θ0 = max{θi0 }, θ1 = i=1 θi1 , θ2 = max{θi2 }, and choose N ∗ = max{2 + P2 c2 + j=0 θj , N (tf )}. By (21), there exists a positive real number µ1 such that ˙ ≤− W

p X

kξi k2 + µ1

i=1

≤ −µW (ξ) + µ1

p X

ε2i1 + p

i=1

p X

ε2i1 + p

i=1

From (27) it follows that p X d(exp µtW ) ≤ exp µt(µ1 ε2i1 + p) dt i=1

(27)

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Hence, for any t ∈ [0, tf ], W (ξ(t)) ≤ exp −µtW (ξ(0)) + µ1 ≤ W (ξ(0)) + µ1

Z

0

Z

p tf X

Z

t

exp−µ(t − s) 0

ε2i1 ds +

i=1

p X

ε2i1 ds +

i=1

p µ

p µ

tf

p 1 ˙ N ds + γ µ 0 µ1 p = W (ξ(0)) + (N (tf ) − 1) + γ µ < +∞

≤ W (ξ(0)) + µ1 (

Therefore it is concluded that W (ξ) is bounded on [0, tf ], so is ξ(t). In view of (19) and (10), both e(t) and ε(t) are bounded on [0, tf ] as well. Summing up the above argument, we arrive at the conclusion that all signals N (t), M (t), e(t) are well-defined and bounded on [0, tf ], which contradicts to the assumption (15). Therefore, the dynamics system (5)-(11) have no finite escape time over [0, +∞) and the boundedness property holds. (II). Proof of Convergence Finally, we show that limt→∞ e(t) = 0. To begin with, observe that boundedness of all the states guarantees that the ω-limit set Ω of the dynamic systems composed of (1), (5) and (9) is nonempty, closed and invariant. Besides, note that both εi1 and ε˙i1 are bounded on [0, +∞). Moreover, εi1 ∈ L2 as N (t) is globally bounded. By the Barbalat’s Lemma, we have lim εi1 = lim ei1 = 0.

t→∞

t→∞

(28)

Therefore, it can be concluded that the ω-limit set Ω has the following property: Ω ⊆ {(e1 , e2 , · · · , ep , x, xˆ, M, N ) | ei1 = 0, i = 1, · · · , p}. Since ei,j+1 = e˙ i,j + Li aij ei1 , i = 1, · · · , p; j = 1, 2, · · · , ki − 1, by the invariance of Ω, we conclude that ei,2 = 0, ei,3 = 0, · · · , ei,ki = 0, i = 1, 2, · · · , p on Ω inductively. Therefore, lim e(t) = 0.

t→∞

This completes the proof of Theorem 2.1. Remark 2.2. It is worth pointing out that the dynamic update law of M can be modified as M˙ = −σM + ∆(N ), σ > 0, ∆(N ) ≥ σ without affecting the argument in the above proof. A bigger σ makes the convergence of M faster and the gain L = M N smaller, however, the convergence of the estimation slower.

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Using Theorem 2.1, it is easy to obtain a corollary which is devoted to the design of a global observer for observable systems in a lower-triangular form: z˙i1 = zi2 + fi1 (z1 ) z˙i2 = zi3 + fi2 (z1 , z2 ) .. .

(29)

z˙i,ki −1 = zi,ki + fi,ki −1 (z1 , z2 , · · · , zki −1 ) z˙i,ki = fi,ki (z) y = z1 Pp where 1 < k1 ≤ · · · ≤ kp and i=1 ki = n, zi = (z1i , z2i , · · · , zp,i )T , if 1 ≤ i ≤ k1 ; zi = (zli , z2i , · · · , zp,i )T , if kl < i ≤ kp , l = 1, · · · , p − 1; z = (z1 · · · , zkp ) are states and y = z1 = (z11 , z21 , · · · , zp1 )T ∈ IRp are the outputs. fij (·), i = 1, · · · , p, j = 1, · · · , ki are smooth functions with fij (0, · · · , 0) = 0. Due to the lower-triangular structure, one can explicitly construct a global change of coordinates x = Ψ(z) which renders system (29) globally diffeomorphic to system (1). As a consequence, we have the following conclusion. Corollary 2.1. Assume that all the solution trajectories of the lower-triangular system (29) from any initial condition are well-defined and bounded on [0, +∞). Then, a globally convergent observer exists and can be explicitly constructed. 3. Observer Design for a Class of Detectable Systems This section is devoted to the design of global observers for a class of detectable nonlinear systems. Consider a class of autonomous systems of the form η˙ = Au η + Ψ(y) x˙ i1 = xi2 .. . x˙ i,ki −1 = xi,ki x˙ i,ki = fi (x),

(30) i = 1, 2, · · · , p

y = (y1 , y2 , · · · , yp )T = (x11 , x21 , · · · , xp1 )T where η ∈ IRn−r and x ∈ IRr are the system states, y ∈ IRp are the outputs, and Pp 1 < k 1 ≤ k2 ≤ · · · ≤ k p , i=1 ki = r, Ψ(y) is a continuous function and fi (·)’s are a smooth functions vanishing at origin. Clearly, the state η ∈ IRn−r is unobservable from the output y. This is because η has no influence on the system output. However, if the matrix Au is Hurwitz, one can still design a global observer for the autonomous system (30) under the condition that the x-subsystem is bounded.

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Theorem 3.1. Suppose the x-subsystem of (30) satisfies the bounded assumption in the sense of Assumption 1, and Au is a Hurwitz matrix. Then, a global observer can be constructed for the system (30) in the following way: ηˆ˙ = Au ηˆ + Ψ(y) x ˆ˙ i1 = xˆi2 + (M N )ai1 (yi − x ˆi1 ) .. . x ˆ˙ i,ki −1 = xˆi,ki + (M N )ki −1 ai,ki −1 (yi − x ˆi1 ) x ˆ˙ i,ki = fi satN (ˆ x) + (M N )ki ai,ki (yi − x ˆi1 ) N˙ = γ

p X i=1

2 yi − x ˆi1 , (M N )kp −ki +1

M˙ = −M + ∆(N ),

N (0) = 1 M (0) = 1

(31)

where aij > 0, i = 1, · · · , p, j = 1, · · · , ki are the coefficients of the Hurwitz polynoP i aij ski −j , γ ≥ 1 is a prescribed constant. mials hi (s) = ski + kj=1 The observer (31) guarantees that all the states of the closed-loop system (30)-(31) are well-defined and bounded on [0, ∞). In addition, limt→∞ [η(η0 , t) − ηˆ(η0 , t)] = 0, limt→∞ [x(x0 , t) − x ˆ(ˆ x0 , t)] = 0, ∀(η0 , x0 ) ∈ Rn , (ˆ η0 , x ˆ0 ) ∈ IRn . The proof of Theorem 3.1 is similar to that of Theorem 2.1, and hence left to the reader as an exercise. Remark 3.1. Theorem 3.1 suggests that, in terms of observer design, the observability is not a necessary condition. This is similar to the linear case, i.e., an unobservable yet detectable system still permits the existence of an observer. Remark 3.2. Theorem 3.1 remains true if the unobservable sub-system is replaced by

η˙ = ϕ(y)(Au ηˆ + Ψ(y)),

ϕ(y) > 0.

(32)

In this case, one can still design a global observer using a manner similar to the one suggested in Theorem 3.1, with a slight modification.

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4. MIMO Nonlinear Control Systems We now discuss briefly the observer design problem for the multi-output multi-input (MIMO) nonlinear system x˙ i1 = xi2 + gi1 (y, u) x˙ i2 = xi3 + gi2 (x, u) .. . x˙ i,ki −1 = xi,ki + gi,ki −1 (x, u) x˙ i,ki = fi (x) + gi,ki (x, u) y = (x11 , x21 , · · · , xp1 )T

(33)

where xi = (xi1 , xi2 , · · · , xi,ki )T , x = (x1 , · · · , xp )T ∈ IRn , u ∈ IRm and y ∈ IRp are Pp the system state, input and output, respectively, k1 ≤ k2 ≤ · · · ≤ kp , i=1 ki = n. The functions gi,j (·) and fi (·) are smooth with gi,j (0, 0) = 0 and fi (0) = 0. We assume that the function gij (x, u) := gij (y, x, u), with x = (y T , xT )T and x = col(xij , i = 1, · · · , p; j = 2, · · · , ki ) ∈ Rn−p , satisfies the following condition. Assumption 4.1.

For i = 1, · · · , p and j = 2, · · · , ki ,

|gij (y, x, u) − gij (y, x ˆ , u)| ≤ c(x, u)

j i X X s=1 l=2

|xsl − x ˆsl |

where x ˆ = col(ˆ xij , i = 1, · · · , p; j = 2, · · · , ki ) ∈ Rn−p , and c(·, ·) ≥ 0 is a smooth function. Assumption 4.2. For any control input u(t) in the compact set U ⊂ R m and any initial condition x0 ∈ Rn , the corresponding solution trajectory xu (x0 , t) of the controlled system (33) is well-defined over the interval [0, +∞) and x u (x0 , t) is globally bounded, i.e. ||xu (x0 , t)|| ≤ C. • In Ref. 21, a nonlinear observer was presented for a class of MIMO nonlinear systems. The system studied there is of a block triangular form. Moreover, it is required that the bounds of the system input and state be known. The system nonlinearities are assumed to satisfy a Lipschitz condition with a known Lipschitz constant. • In Ref. 12, a step-by-step local observer design method was proposed for a class of multi-input multi-output nonlinear control systems. The systems under consideration are also in a block-triangular form, and the observer gains are nonlinear functions of the estimated states. Due to the local design feature, the boundedness condition is automatically satisfied.

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• Assumption 4.2 basically requires that all the solution trajectories do not blow up under bounded control. It contains, for instance, boundedinput/bounded-state (BIBS) systems. It should be noticed that a key feature of the proposed observer does not need the bound information of the solution trajectories. Under the two assumptions above, we can design a global observer for the MIMO system (33) by following the spirit of observer design method in section 2. Theorem 4.1. For the MIMO nonlinear control system (33), suppose Assumptions 4.1 and 4.2 hold. Then, a global observer can be designed for the controlled systems (33) as xˆ˙ i1 = x ˆi2 + (M N )ai1 (yi − x ˆi1 ) + gi1 (y, u) 2 ˙xˆi2 = x ˆi3 + (M N ) ai2 (yi − xˆi1 ) + gi2 (y, x ˆ , u) .. .

x ˆ˙ i,ki = fi satN (ˆ x) + (M N )ki ai,ki (yi − x ˆi1 ) + gn (y, x ˆ, u) p 2 X yi − xˆi1 N˙ = γ , N (0) = 1 k −k +1 (M N ) p i i=1 M˙ = −M + ∆(N ),

M (0) = 1

(34)

where aij > 0, i = 1, · · · , p, j = 1, · · · , ki are the coefficients of the Hurwitz polyPki aij ski −j , γ ≥ 1 is a prescribed constant, and ∆(N ) ≥ 1 is a nomials ski + j=1 smooth function which can be determined explicitly. Moreover, all the states of the closed-loop system (33)-(34) are well-defined and bounded on [0, ∞). In addition, lim [x(x0 , t) − x ˆ(x0 , x ˆ0 , t)] = 0,

t→∞

∀(x0 , x ˆ0 ) ∈ IRn × IRn .

The proof of this theorem can be carried out by modifying suitably the argument of Theorem 2.1. The boundedness property of x and u has to be used while the bound can be unknown. 5. Two Examples In this section, we give two examples to illustrate the applications of the observer design methods proposed in this paper. Example 5.1. Consider the two-output observable autonomous systems x˙ 11 = x12 x˙ 12 = −x11 − x312 + x23 + x321 x˙ 21 = x22 x˙ 22 = x23 x˙ 23 = y=

−3x221 x22 − x22 − x12 (y1 , y2 )T = (x11 , x21 )T

(35)

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This system is of the form (29). Choosing the Lyapunov function V (x) = 12 [x211 + + 12 x421 + x222 + (x321 + x23 )2 ], one can see that the derivative of V (x) along the trajectories of (35) satisfies V˙ = −x412 ≤ 0, which implies that the system is stable but not asymptotically stable. Hence, all the solutions trajectories of (35) are globally bounded, and the design method proposed in Theorem 2.1 can be applied. To find the function ∆(N ), we first compute β1 (N ) and β2 (N ) from f1 (x) = −x11 − x312 + x23 + x321 and f2 (x) = −3x221 x22 − x22 − x12 . By the mean value theorem, there is a ξ ∈ IRn between x and satN (ˆ x), such that x212

|f1 (x) − f1 (satN (ˆ x))| =|

3 2 X X ∂f1 x ˆ1i xˆ2i ∂f1 )) + )| (ξ)(x1i − N sat( (ξ)(x2i − N sat( ∂ξ N ∂ξ N 1i 2i i=1 i=1

2 2 ≤ (2 + 3ξ12 + 3ξ21 )·

2 X

|x1i − N sat(

i=1

3 X xˆ2i xˆ1i )| + |x2i − N sat( )| N N i=1

2 X

3 X xˆ1i xˆ2i ≤ (4 + 6(C + N ) ) · |x1i − N sat( )| + |x2i − N sat( )| N N i=1 i=1 2

≤ 6(C 2 + 1)(N 2 + 1)

2 X i=1

3

|x1i − N sat(

X xˆ1i xˆ2i )| + |x2i − N sat( )| . N N i=1

2

Thus, β1 (N ) = N + 1. Similarly, it is deduced from f2 (x) that β2 (N ) = N 2 + 1. Hence, β(N ) = β1 (N ) + β2 (N ) = 2N 2 + 2 and ∆(N ) = β 2 (N ) = 4(N 2 + 1)2 . Choose a11 = a12 = a21 = a23 = 1, a22 = 3, γ = 8. Then, the observer for the autonomous system (35) can be designed as xˆ˙ 11 = x ˆ12 + M N (x11 − x ˆ11 ) x ˆ xˆ12 xˆ23 x ˆ21 11 xˆ˙ 12 = −N sat( ) − N 3 sat3 ( ) + N sat( ) + N 3 sat3 ( ) N N N N +M 2 N 2 (x11 − x ˆ11 ) ˙xˆ21 = x ˆ22 + M N (x21 − x ˆ21 ) xˆ˙ 22 = x ˆ23 + 3M 2 N 2 (x21 − x ˆ21 ) x ˆ x ˆ xˆ21 xˆ12 21 22 xˆ˙ 23 = −3N 3 sat2 ( )sat( ) − N sat( ) − N sat( ) + M 3 N 3 (x21 − x ˆ21 ) N N N N 8 N˙ = (x11 − x ˆ11 )2 + M 2 N 2 (x21 − x ˆ21 )2 (36) 4 4 M N M˙ = −M + 4(N 2 + 1)2

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Figure 1 illustrates the transient response of the observer and the system (35) starting from the initial conditions (x011 , x012 , x021 , x022 , x023 , x ˆ011 , xˆ012 , x ˆ021 , x ˆ022 , xˆ023 ) = (2, 2, 2, −2, 4, 3, 1, 1, −2, 2). 6

30

4

∧ x11

0 −2

0

10 x12 0.2

0.4

0.6

0.8

0

1

3

0

0.2

0.6

0.8

1

x21

−10

1.5 0

x22

0

2

0.2

0.4

0.6

0.8

−20

1

x∧22 0

200

0.2 e11

50

∧ x23

100

0 x23

e21

−50

0

0

0.2

0.4

Figure 1.

0.6

0.8

1

0.4

0.6

0.8

1

0.4

0.6

0.8

1

e12 e22 e23

−100 −100

0.4

10 ∧ x21

2.5

1

∧ x12

20

x11

2

0

0.2

First 1 Observation of a 5-dimension 2-output system

Example 5.2. Consider the observation of a point-mass satellite model (see, for instance, Ref. 2 or 3): ρ˙ = v v˙ = ρω 2 − θ1

1 + θ 2 u1 ρ2

φ˙ = ω

(37)

1 ω˙ = − (2vω + θ2 u2 ) ρ in which (ρ, φ) denotes the position of the satellite in polar coordinates on the plane, v is the radial velocity, ω is the angular velocity and u1 , u2 are the radial and tangential thrust, respectively. We assume that the measurable signals are y1 = ρ,

y2 = φ.

Consider the case when the parameters θ1 = 4 and θ2 = 1, while the control inputs u1 = 4/ρ2 −ρ−v and u2 = φρ . Then, it is easy to verify that the system states are globally bounded, by using the Lyapunov function V = 12 (ρ2 + v 2 + φ2 + ρ2 ω 2 )

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whose derivative is V˙ = −v 2 ≤ 0. The state trajectories of the closed-loop system are shown in Fig. 2. 3

2

2.5

1.5

2

1

1.5

0.5

1

0

0.5

−0.5

0

0

20

40

60

80

100

−1

4

4

3

3

2

2

1

1

0

0

−1

−1

−2

−2

−3

0

20

40

60

80

100

−3

0

20

40

60

80

100

0

20

40

60

80

100

Figure 2. First 2 State trajectories of point-mass satellite model from initial point (ρ0 , v 0 , φ0 , ω 0 ) = (2, −1, 3, 1)

For the closed-loop system, we can design a global observer of the form ρˆ˙ = vˆ + 6M N (ρ − ρˆ), ρˆ(0) > 0 ρˆ ω ˆ ρˆ vˆ vˆ˙ = N 3 sat( )sat2 ( ) − N sat( ) − N sat( ) + 9(M N )2 (ρ − ρˆ) N N N N ˙ ˆ φˆ = ω ˆ + M N (φ − φ) ˆ

φ sat( N ) vˆ ω ˆ ˆ )sat( ) + ) + (M N )2 (φ − φ) ρˆ ρˆ N N N sat( N ) sat( N ) 5 2 ˆ2 , N˙ = (ρ − ρ ˆ ) + (φ − φ) N (0) = 1 2 (M N ) M˙ = −M + (N 2 + 1)2 , M (0) = 1

ω ˆ˙ = −

1

(2N 2 sat(

(38)

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Figure 3 illustrates the simulation results of the closed-loop system and the observer (38) starting from the initial conditions (ρ0 , v 0 , φ0 , ω 0 , ρˆ0 , vˆ0 , φˆ0 , ω ˆ 0) = (2, −1, 3, 1, 4, 2, 1, 2)

4

2

3.5

1

2.5

−1

2

−2

∧ 3 ρ

0

ρ

1.5

−3

1 0.5

v

∧ v

−4 0

1

2

3

4

4

−5

0

1

2

3

4

2

3

4

8 φ

6

3

2

∧ ω

4

∧ φ

2 ω

0 1 −2 0

0

1

2

Figure 3.

3

4

−4

0

1

First 3 Observation of point-mass satellite model

6. Concluding Remarks Under the global boundedness and observability conditions, it has been shown that a globally convergent observer can be explicitly designed for the multi-output autonomous system (1) or (2) without requiring a block-triangular structure nor imposing restrictions on the coupling relations between each sub-block. The constructed observer is of high-gain type but different from the traditional one 10 in the sense that the observer gains here are composed of two time-varying components M (t) and N (t), both of them must be adaptively updated in order to deal with the issue of the unknown bound of the solution trajectories. The gain update law is reminiscent from the recent work 16 on universal output feedback control of nonlinear systems with unknown parameters. The proposed observer design technique was also extended to a class of detectable systems and multi-input/multi-output (MIMO) nonlinear systems with bounded solution trajectories, such as boundedinput/bounded-state systems.

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

D. Bestle and M. Zeitz, Int. J. Contr., 38, 419 (1983). R. Brockett, Finite Dimensional Linear Systems, (1970). P. C. De and A. Isidori, IEEE Trans. Automat. Contr., 46, 853 (2001). J. P. Gauthier, and G. Bornard, IEEE Trans. Automat. Contr., 26, 922 (1981). J. P. Gauthier, H. Hammouri and S. Othman, IEEE Trans. Automat. Contr., 37, 875 (1992). J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1983). A. Ilchmann, Lecture Notes in Control and Information Sciences, 189, (1993). N. Kazantzis and C. Kravaris, Syst. Contr. Lett., 34, 241 (1998). H. K. Khalil and F. Esfandiari, IEEE Trans. Automat. Contr., 38, 1412 (1993). H. K. Khalil and A. Saberi, IEEE Trans. Automat. Contr., 32, 1031 (1987). A. J. Krener and A. Isidori, Syst. & Contr. Lett., 3, 47 (1983). A. Krener and W. Kang, SIAM J. Contr. Optimiz., 42, 155 (2003). A. J. Krener and W. Respondek, SIAM J. Contr. Optimiz., 23, 197 (1985). A. J. Krener and M. Xiao, SIAM J. Contr. Optimiz., 41, 932 (2002). P. Krishnamurthy and F. Khorrami, Proc. of the 41st IEEE CDC, 1503 (2002). H. Lei and W. Lin, Proc. of the 16th IFAC World Congress, (2005). Also, to appear in Automatica (2006). H. Lei, J. Wei and W. Lin, Proc. of the 44th IEEE CDC, 1911 (2005). W. Lin and R. Pongvuthithum, IEEE Trans. Automat. Contr.,47, 1356 (2002). W. Lin and C. I. Byrnes, Syst. Contr. Lett., 25, 31 (1995). J. Rudolph and M. Zeitz, Syst. & Contr. Lett., 23, 1 (1994). H. Shim, Y.I. Son and J.H. Seo, Proc. of the ACC, 3077 (1999). X. Xia and W. B. Gao, SIAM J. Contr. Optimiz., 27, 199 (1989). J. C. Willems and C. I. Byrnes, Lecture Notes in Control and Information Sciences, 62, (1984).

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Chap25-LouHongwei

SOME COUNTEREXAMPLES IN EXISTENCE THEORY OF OPTIMAL CONTROL∗

HONGWEI LOU School of Mathematical Sciences, Fudan University, Shanghai 200433, China and Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, China E-mail: [email protected]

Though there should exist many optimal control problems admitting no solution, rare counterexamples were shown in the literature. By using relaxed controls, we will give some counterexamples in existence of optimal control.

1. Introduction Relaxed control (see Gamkrelidze [10], McShane [18] and Warga [24]) is an important tool to study the existence of optimal controls when Cesari type conditions are no longer satisfied. In this aspect, there are many positive results. Among them, we mention the papers by Artstein [1], Balder [2]–[4], Berliocchi–Lasry [5], Cesari [6], Colombo–Goncharov [7], Flores-Baz´ an–Perrotta [9], Lou [12]–[13], Marcellini [16], Mariconda [17], Olech [19] and Raymond [20]–[22], Suryanarayana [23]. Nevertheless, rare results were shown about the non-existence of optimal controls for optimal control problems. Concerning the non-existence results, the following counter example is a typical one, which is mentioned in many books, see for example, [25], Ch. 3, p. 246. Other similar examples can be found in [6], Ch. 9, p. 321 and [8], Ch. 2, p. 51. Example 1.1. Let U = [−1, 1] (or U = {−1, 1}), n o U = v(·) : [0, 1] → U v(·) measurable , dy(t) = u(t), dt ∗ This

t ∈ [0, 1],

work was supported by the National Science Foundation of China (No. 10371024) 183

(1)

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and I(y(·), u(·)) =

Z

1 0

y 2 (t) − u2 (t) dt.

(2)

Let P denote the set of all pairs (y(·), u(·)) satisfying (1) and u(·) ∈ U . The optimal control problem is that Problem (C) Does there exist a pair (¯ y (·), u ¯(·)) ∈ P such that I(¯ y (·), u ¯(·)) =

inf

I(y(·), u(·)).

(y(·),u(·))∈P

(3)

One can prove that Problem (C) admits no solution. To see this, let  k−1  [ j j  1   ), 1, t ∈ [ , +   k k 2k j=0 uk (t) = k−1 [ j  1 j 1    −1, t ∈ [ + , + ),   k 2k k k j=0

and yk (·) be the solution of (1) corresponding to uk (·) and the initial condition yk (0) = 0. Then (yk (·), uk (·)) ∈ P and |yk (t)| ≤

1 , 2k

∀ t ∈ [0, 1].

Consequently, we have I(yk (·), uk (·)) =

Z

1 0

1 2 − 1. yk2 (t) − u2k (t) dt ≤ 2k

On the other hand, for any (y(·), u(·)) ∈ P, Z 1 I(y(·), u(·)) ≥ −u2 (t) dt ≥ −1. 0

Thus combining (4) and (5) we get inf

(y(·),u(·))∈P

I(y(·), u(·)) = −1.

If for some (¯ y (·), u ¯(·)) ∈ P, I(¯ y (·), u ¯(·)) = −1. Then, by the definition of I and U , we must have y¯(t) = 0,

|¯ u(t)| = 1,

a.e. t ∈ [0, 1].

Since y¯(t) = 0,

a.e. t ∈ [0, 1]

implies u ¯(t) = 0, a.e. t ∈ [0, 1]

(4)

(5)

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we get a contradiction. That is, Problem (C) has no solution. The proof in the above example deeply depends on that U (and I) is symmetric. If we replacing U = [−1, 1] by [a, 1] with a < 0, a 6= −1, then to our best knowledge, there is no result concerning that for what kind of a, I(·, ·) admits no minimizer. By using the method of this paper, we can treat the case a 6= −1. In this paper, we will consider the generalized cases of Example 1. Let a ≤ 1 and n o Ua = v(·) : [0, 1] → [a, 1] v(·) measurable . Let Pa be the set of all pairs (y(·), u(·)) satisfying (1) and u(·) ∈ Ua . Our problem is that Problem (Ca ) Does there exist a pair (¯ y (·), u ¯(·)) ∈ Pa such that I(¯ y (·), u ¯(·)) =

inf

I(y(·), u(·)).

(y(·),u(·))∈Pa

(6)

We will solve Problem (Ca ) by using relaxed controls. The method we use in this paper can be used to treat more general cases. 2. Relaxation We introduce the relaxed problem corresponding to Problem (Ca ). Let Ua = [a, 1]. We denote by M+1 the set of all probability measures in Ua , by Ra the set of all measurable M+1 -valued functions on [0, 1]. Element in Ra is called relaxed control. The relaxed state equation and cost functional are Z dy(t) = vσ(t)(dv), t ∈ [0, 1], (7) dt Ua and I(y(·), σ(·)) =

Z

1

dt 0

Z

Ua

y 2 (t) − v 2 (t) σ(t)(dv).

(8)

Let RPa denote the set of all pairs (y(·), σ(·)) satisfying (7) and σ(·) ∈ Ua . Now, we state the optimal relaxed control problem corresponding to Problem (Ca ). Problem (Ra ) Does there exist a pair (¯ y (·), σ ¯ (·)) ∈ RPa such that I(¯ y (·), σ ¯ (·)) =

inf

(y(·),σ(·))∈RPa

I(y(·), σ(·)).

(9)

We mention that Ua can be imbedded into Ra by identifying each u(·) ∈ Ua with the Dirac measure-valued function δu(·) ∈ Ra . Moreover, I(δu(·) ) defined by (8) coincides with I(u(·)) defined by (2). It is known that Ua is dense in Ra , i.e.,

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for any σ(·) ∈ Ra , there exists a sequence uk (·) in Ua , such that Z 1 Z 1 Z h(t, uk (t))dt → dt h(t, v)σ(t)(dv), 0

0

Ua

∀ h ∈ L1 ([0, 1]; C(Ua )) see for example, [25]. By the density of Ua in Ra , one can easily get that inf

(y(·),σ(·))∈RPa

I(y(·), σ(·)) =

inf

I(y(·), u(·)).

(y(·),u(·))∈Pa

(10)

Thus, a solution to Problem (Ca ) must be a solution to Problem (Ra ). Furthermore, if (¯ y (·), σ ¯ (·)) ∈ Ra is a solution to Problem (Ra ) and σ ¯ (t) = δu¯(t) , a.e. t ∈ [0, 1], then (¯ y (·), u ¯(·)) ∈ Ua and it must be a solution to Problem (Ca ). By Theorem IV.2.1 in Warga [25], p. 272, we have Lemma 2.1. Problem (Ra ) admits at least one solution. Moreover, it is easy to prove that Proposition 2.1. Let (¯ y (·), σ ¯ (·)) ∈ RPa be a solution to Problem (Ra ). Define  ¯  dψ(t) = 2¯ y (t), in [0, 1], (11) dt  ψ(0) ¯ ¯ = ψ(1) = 0,

and

H(y, v, ψ) ≡ vψ + v 2 − y 2 .

(12)

Then we have

n o ¯ ¯ supp σ ¯ (t) ⊆ v ∈ Ua H y¯(t), v, ψ(t) = max H y¯(t), w, ψ(t) w∈Ua  ¯  {1}, if ψ(t) > −(1 + a), ¯ < −(1 + a), = {a}, if ψ(t) a.e. t ∈ [0, 1].  ¯ {1, a}, if ψ(t) = −(1 + a),

(13)

¯ ¯ The condition ψ(0) = ψ(1) = 0 follows from that y(0) and y(1) are free. By Proposition 2.1, we see that σ ¯ (t) is a Dirac measure for almost all ¯ t ∈ {ψ¯ 6= −(1 + a)} ≡ {t ∈ [0, 1] ψ(t) 6= −(1 + a)}. On the other hand, when a 6= 0, we will prove in the next section that for almost ¯ = −(1 + a)}, t ∈ {ψ¯ = −(1 + a)} ≡ {t ∈ [0, 1] ψ(t)

σ ¯ (t) should not be a Dirac measure. Thus, if for some solution (¯ y (·), σ ¯ (·)) ∈ RPa to Problem (Ra ), the corresponding set {ψ¯ = −(1+a)} has zero measure, then Problem

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(Ca ) has at least one solution. If a 6= 0 and for any solution (¯ y (·), σ ¯ (·)) ∈ RPa to Problem (Ra ), the corresponding set {ψ¯ = −(1 + a)} has positive measure, then Problem (Ca ) has no solution. 3. Main Result In this section, we will get our main result by analyzing the necessary condition (13) that a solution to Problem (Ra ) satisfies. Theorem 3.1. Let a < 1. Then Problem (Ca ) admits no solution if and only if a ∈ (− 34 , − 34 ). ¯ be defined Proof. Let (¯ y (·), σ ¯ (·)) ∈ RPa be a solution to Problem (Ra ) and ψ(·) by (11). By the discussion in the end of Section 2, σ ¯ (t) is a Dirac measure for almost all t ∈ {ψ¯ 6= −(1 + a)}. On the other hand, as solutions to ordinary differential ¯ and y¯(·) are absolutely continuous in [0, 1]. Thus, ψ(·) ¯ and y¯(·) equations, both ψ(·) are differentiable for almost all t ∈ [0, 1]. Consequently, by (11), a.e. t ∈ {ψ¯ = −(1 + a)},

0 = 2¯ y(t), and 0= Therefore,

Z

Z

v¯ σ (t)(dv),

a.e. t ∈ {¯ y = 0}.

Ua

v¯ σ (t)(dv) = 0,

a.e. t ∈ {ψ¯ = −(1 + a)}.

(14)

Ua

(i) Case I: a ∈ [0, 1). In this case, (14) implies supp σ ¯ (t) = {0},

a.e. t ∈ {ψ¯ = −(1 + a)},

(in particularly, {ψ¯ = −(1 + a)} has zero measure if a > 0). Thus, σ ¯ (t) is a Dirac measure for almost all t ∈ [0, 1]. That is, Problem (Ca ) has at least one solution. (ii) Case II: −1 ≤ −a < 0. We claim that {ψ¯ < −(a + 1)} is empty. Otherwise ¯ ¯ E ≡ {ψ¯ < −(a + 1)} is a nonempty open set. Since ψ(0) = ψ(1) = 0, and −(a + 1) ≥ 0, we have ¯ ∂E = −(a + 1). ψ|

(15)

By (13), supp σ ¯ (t) = {a},

a.e. t ∈ E.

(16)

Therefore by (11) and (¯ y (·), σ ¯ (·)) ∈ RPa we get that −

¯ d2 ψ(t) d¯ y(t) = −2 = −2a > 0, dt2 dt

a.e. t ∈ E.

(17)

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Combining (15) and (17), we have ¯ ≥ −(1 + a), ψ(t)

a.e. t ∈ E.

(18)

a.e. t ∈ [0, 1].

(19)

This is a contradiction. Thus, we prove that for almost all t ∈ [0, 1], ¯ ≥ −(1 + a), ψ(t) On the other hand, by (13), supp σ ¯ (t) ⊆ {a, 1},

a.e. t ∈ {ψ¯ = −(1 + a)}.

(20)

a.e. t ∈ {ψ¯ = −(1 + a)}.

(21)

Thus, (14) and (20) implies supp σ ¯ (t) = {a, 1}, Since ¯ d¯ y(t) d2 ψ(t) =2 =2 2 dt dt

Z

v¯ σ (t)(dv) ≤ 2,

a.e. t ∈ [0, 1],

(22)

Ua

¯ ¯ and ψ(0) = ψ(1) = 0, we have ¯ ≥ (x − 1 )2 − 1 , ψ(t) 2 4 Therefore, if a ∈ [− 43 , 0), then ¯ > − 1 ≥ −(1 + a), ψ(t) 4

a.e. t ∈ [0, 1].

(23)

a.e. t ∈ [0, 1].

And consequently, supp σ ¯ (t) = {1},

a.e. t ∈ [0, 1].

That is, Problem (Ca ) has a solution. In fact, we can see that Problem (Ca ) admits a unique solution (¯ y (·), u ¯(·)) with 1 y¯(t) = x − , 2

u ¯(t) = 1,

a.e. t ∈ [0, 1].

If a ∈ [−1, − 43 ), then the measure of {ψ¯ = −(1 + a)} should not be zero. Otherwise, by (19) and (13), we must have supp σ ¯ (t) = {1},

a.e. t ∈ [0, 1].

(24)

Thus ¯ = (x − 1 )2 − 1 , ψ(t) a.e. t ∈ [0, 1]. 2 4 This contradict to {ψ¯ = −(1 + a)} having zero measure. Finally, by (21), we get that Problem (Ca ) admits no solution when a ∈ [−1, − 43 ). (iii) Case III: a < −1. The results of this case can be gotten form Case II by the fact of y(·) u(·) I(y(·), u(·)) = a2 I( , ). a a

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More precisely, we have that Problem (Ca ) admits a solution if a ∈ (−∞, − 43 ] and admits no solution if a ∈ (− 43 , −1). Thus, we have proved that Problem (Ca ) admits no solution if and only if a ∈ (− 34 , − 34 ). References 1. Z. Artstein, On a variational problem, J. Math. Anal. Appl., 45, 404–415 (1974). 2. E. J. Balder, On a useful compactification for optimal control problems, J. Math. Anal. Appl., 72, 391–398 (1979). 3. E. J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Optim., 22, 570–598 (1984). 4. E. J. Balder, New existence results for optimal controls in the absence of convexity: the importance of extremality, SIAM J. Control Optim., 32, 890-916 (1994). 5. H. Berliocchi, and J. M. Lasry, Int´egrandes normales et measures param´etr´ees en calcul des variations, Bull. Soc. Math. France, 101, pp. 129–184 (1973). 6. L. Cesari, “Optimization Theory and Applications, Problems with Ordinary Differential Equations”, Springer-Verlag, New York (1983). 7. G. Colombo and V. V. Goncharov, Existence for nonconvex optimal problem with nonlinear dynamics, Nonlinear Anal. T. M. A., 24, 795–800 (1995). 8. H. O. Fattorini, Relaxed controls in semilinear infinite dimensional systems, in “Int. Ser. Num. Math.,” Vol. 100, pp. 115-128, Birkh¨ auser, Basel (1991). 9. F. Flores-Baz´ an and S. Perrotta, Nonconvex variational problems related to a hyperbolic equation, SIAM J. Control Optim., 37, 1751–1766 (1999). 10. R. Gamkrelidze, “Principle of Optimal Control Theory”, Plenum Press, New York (1978). 11. X. Li, and J. Yong, “Optimal Control Theory for Infinite Dimensional Systems”, Birkh¨ auser, Boston (1995). 12. H. Lou, Existence of optimal controls for semilinear elliptic equations without Cesari type conditions, ANZIAM Journal, 45, 115-131 (2003). 13. H. Lou, Existence of optimal controls for semlinear parabolic equations without Cesaritype conditions, Appl. Math. Optim., 47, 121-142 (2003). 14. H. Lou, Maximum principle of optimal control for degenerate quasi-linear elliptic equations, SIAM J. Control Optim., 42, 1-23 (2003). 15. H. Lou, Existence of optimal controls in the absence of Cesari-type conditions for semilinear elliptic and parabolic systems, J. Optim. Theory Appl., 125, 367-391 (2005). 16. P. Marcellini, Alcune osservazioni sull’exixtenza del minimo di integrali del calcolo delle variazioni senza ipotesi di convessit` a, Rend. Mat., 13, 271–281 (1980). 17. C. Mariconda, A generalization of the Cellina-Colombo theorem for a class of nonconvex variational problems, J. Math. Anal. Appl., 175, 514-552 (1993). 18. E. C. McShane, Relaxed controls and variational problems, SIAM J. Control, 5, 438485 (1967). 19. C. Olech, Integrals of set-valued functions and linear optimal control problems, Colloque sur la Th´eorie Math´ematique du Contrˆ ole Optimal, 109–125 (1970). 20. J. P. Raymond, Conditions n´ecessaires et suffisantes d’existence de solutions en calcul des variations, Ann. Inst. H. Poincar´e, Analyse non lin´eaire, 4, 169–202 (1987). 21. J. P. Raymond, Existence theorems in optimal control theory without convexity assumptions, J. Optim. Theory Appl., 67, 109–132 (1990). 22. J. P. Raymond, Existence theorems without convexity assumptions for optimal control

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problems governed by parabolic and elliptic systems, Appl. Math. Optim., 26, 39–62 (1992). 23. M. B. Suryanarayana, Existence theorems for optimization problem concerning linear, hyperbolic partial differential equations without convexity conditions, J. Optim. Theory Appl., 19, 47-61 (1976). 24. J. Warga, Relaxed variational problems, J. Math. Anal. Appl., 4, 111-128 (1962). 25. J. Warga, “Optimal Control of Differential and Functional Equations”, Academic Press, New York (1972).

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Chap26-QianChunjiang

A GENERALIZED FRAMEWORK FOR GLOBAL OUTPUT FEEDBACK STABILIZATION OF INHERENTLY NONLINEAR SYSTEMS WITH UNCERTAINTIES

JASON POLENDO AND CHUNJIANG QIAN∗ Department of Electrical and Computer Engineering University of Texas at San Antonio San Antonio, Texas 78249, USA [email protected], [email protected]

In this paper, we introduce a generalized framework for global output feedback stabilization of a class of uncertain, inherently nonlinear systems of a particularly complex nature since their linearization around the equilibrium is not guaranteed to be either controllable or observable. Based on a subtle, yet nontrivial observer/controller construction and homogeneous domination design, this new framework not only unifies the output feedback stabilization results existing in Refs. 1, 2, and 3, but also leads to more results which have yet to be achieved, establishing this methodology as a useful tool for the global output feedback stabilization of inherently nonlinear triangular systems with uncertainties.

1. INTRODUCTION A formidable problem in the nonlinear control literature is the global stabilization of a nonlinear dynamic system by output feedback. Such a problem formulation is intrinsically practical in that only partial sensing is utilized to feedback state information, which is an efficient and cost-effective solution in many applications while a necessity in others. Unfortunately, the existing output feedback stabilization schemes have been very limited in what types of nonlinearities could be handled, an issue made more complex due to the lack of a true “separation principle” for nonlinear systems. In this work, we investigate lower-triangular systems, such as x˙ 1 = xp21 + φ1 (d(t), x1 ) x˙ 2 = xp32 + φ2 (d(t), x1 , x2 ) .. . x˙ n = u + φn (d(t), x1 , . . . , xn ) y = x1 ,

(1)

∗ Chunjiang

Qian’s research was supported in part by the U.S. National Science Foundation under grant ECS-0239105. Jason Polendo’s research was supported by the University of Texas at San Antonio Doctoral Fellowship and the NASA Harriett G. Jenkins Predoctoral Fellowship. 191

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where x = (x1 , · · · , xn )T ∈ IRn , u ∈ IR and y ∈ IR are the system state, input and output, respectively. For i = 1, · · · , n, φi (·) is an unknown nonlinear function with bounded disturbance d(t), and pi ∈ IR≥1 odd := {q ∈ IR : q ≥ 1 and q is a ratio of odd integers} with pn = 1. When pi = 1, the results on global output feedback stabilization of system (1) were based on quite restrictive conditions imposed on the nonlinear terms φi (·), mainly attributed to finite escape time phenomena (see Ref. 4). These stabilization results include systems where the nonlinear function is only dependent on the output (see Refs. 5, 6, 7, 8, and 9); or is Lipschitz or linear in the unmeasurable states (see Refs. 10, 11, 12, 13, and 14). A pertinent feedback domination design was proposed in Ref. 15 which considered a lower-triangular (strict-feedback) linear growth condition with a constant growth rate. In Refs. 16 and 17, this condition was extended to the case when the growth rate is a polynomial of the output, while Refs. 18 and 19 lifted the polynomial growth rate restriction to allow the growth rate to be a smooth function of the output. Until recently, though, there was no systematic way of dealing with systems whose dynamics are highly nonlinear in the unmeasured states. In Ref. 1, however, the nonlinearity restriction was relaxed to globally stabilize more general systems such as (1) by output feedback under a less restrictive polynomial growth condition. This result of Ref. 1 has made it possible to achieve global output feedback stabilization of systems such as x˙ 1 = x2 x˙ 2 = x3 x˙ 3 = x4 + d(t)x3 ln(1 + x23 ) x˙ 4 = u + d(t)x32 + x22 sin x4 y = x1 , with a bounded disturbance d(t). However, when pi are of higher order, i.e. the system (1) has uncontrollable/unobservable linearization, the global output feedback stabilization solutions are few, where the state feedback problem was resolved in Ref. 20 for the stabilization of systems such as (1) with pi ≥ 1 being odd integers and φi (·) bounded by a lower-triangular C 1 function vanishing at the origin. Compared to the state feedback stabilization advancement, the output feedback control problem for these high-order systems presents a much more difficult challenge. In Ref. 18, the problem was handled for a planar system, with an odd integer power, using a smooth output feedback controller. Later, this result was extended to n-dimensions in Ref. 3 to a class of nonlinear systems, under the conditions that pi are all the same odd integer and φi (·) is assumed to be bounded by uniformly powered growth of the states in lower-triangular form. For more general nonlinear systems with different pi , it is well known that a smooth design will not be sufficient to adequately stabilize the system in all instances. Hence a non-smooth design method was introduced in Ref. 2, where global output feedback stabilization was achieved under a certain

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growth condition, which is detailed later. Note that the methods in Refs. 2, 18, and 3 are quite different from each other and their results do not overlap, in a general sense, not even when considering the exact same system such as x˙ 1 = x32 , x˙ 2 = u. This is obvious given that Refs. 18 and 3 yield a smooth controller of different forms, while a non-smooth control law is obtained in Ref. 2. An interestingly unresolved problem is if we can find a generalized framework to unify the existing solutions, simultaneously covering both Refs. 3 and 2, while also containing Ref. 1 when pi = 1. To handle this issue, we employ the concept of homogeneous domination (see Ref. 1) to cover a larger class of inherently nonlinear systems. In doing so, we allow pi to be any odd real numbera such that pi ≥ 1 and the pi ’s are not necessarily equivalent. Within this present work we establish a new generalized framework, built upon the idea of homogeneity, to globally stabilize these inherently nonlinear systems, with pi ∈ IR≥1 odd and φi (·) bounded by a polynomial growth condition, using only the system output as feedback. Such a formalism will allow for more complex nonlinearities than those seen in Refs. 1, 2, and 3, and in fact, generalizes the homogeneous domination approach introduced in Ref. 1. This design methodology will allow for the general stabilization of systems such as x˙ 1 = x32 ,

x˙ 2 = u + d(t)xq2 ,

y = x1 ,

(2)

with a bounded disturbance d(t). When q = 1, system (2) was stabilized via output feedback by the techniques described in Ref. 2, and when q = 3, output feedback stabilization of (2) was achieved in Ref. 18, however, when q = 2, the global output feedback stabilization of (2) is unresolved. Nevertheless, we will show that (2) can now be controlled via output feedback for, though not limited to, q = 2 by methods described herein. The main contribution of this paper is two-fold: 1) it encompasses the aforementioned results as special cases, as described in Section 4, and 2) this work simultaneously leads to new results in the global output feedback stabilization of inherently nonlinear triangular systems. In particular, this paper will illustrate: • The unification of the existing global output feedback stabilization literature for inherently nonlinear systems (see Refs. 2 and 3), as well as formalizing the generalization of the homogeneous domination approach (see Ref. 1); • This design methodology can offer smoother performance than the previous non-smooth schemes (see Ref. 3); • We extend these existing results by providing solutions to a family of lowertriangular systems of a form whose output feedback stabilization were illfitted for the current literature; a We

say that a real number is odd when it is in the set IR odd = {q ∈ IR : q is a ratio of odd integers}.

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2. GLOBAL STABILIZATION BY HOMOGENEOUS STATE FEEDBACK During the past two decades, the analysis of nonlinear dynamic systems has been studied from the viewpoint of homogeneity and homogeneous systems. Utilizing these notions has allowed the undertaking of the concepts of controllability and controller design for nonlinear systems to be realizable (see Refs. 21, 22, 23, 24, and 25). Of particular importance in this paper, concerning homogeneity, are the definitions of homogeneous systems with weighted dilation and the homogeneous norm (see Refs. 21, 24, 25, and 1), where the particular work of Ref. 1 is inspiration for the methodology discussed in this paper. With the help of the notion of homogeneity, we propose in this section a design method for a homogeneous state feedback stabilizer for (1) under the following assumption: There is a constant τ ≥ 0 such that for i = 1, · · · , n, ri +τ ri +τ ri +τ |φi (d(t), x1 , . . . , xi )| ≤ c |x1 | r1 + |x2 | r2 + · · · + |xi | ri ,

Assumption 2.1.

(3)

for a constant c > 0 and ri defined as r1 = 1,

ri pi−1 = τ + ri−1 ,

i = 2, . . . , n + 1, with pn = 1.

(4)

For simplicity, we assume the degree of homogeneity, τ ∈ IR odd . Under this assumption, and taking into account the odd, not necessarily equivalent, powers of (1), we know that the homogeneous weights, ri , will always be in IRodd . Note that an equivalent result will be achieved for the case when the ri are not odd. The following theorem shows that Assumption 2.1 guarantees a homogenous state feedback controller for (1). Theorem 2.1. Under Assumption 2.1 there exists a homogeneous state feedback controller such that the nonlinear system (1) is guaranteed globally asymptotically stableb . Proof. The inductive proof relies on the simultaneous construction of a C 1 Lyapunov function which is positive definite and proper, as well as a homogeneous stabilizer at each iteration. Initial Step. Let µ ∈ IRodd and µ ≥ max{ri pi−1 }2≤i≤n+1 , where ri is defined as in (4). Note that by definition, 2µ − τ − ri−1 ≥ 2ri pi−1 − ri pi−1 = ri pi−1 > 0. Choose Z x1 2µ−τµ−r1 µ V1 = s r1 − 0 ds. 0

b Under

certain conditions global strong stability may be the only achievable result since the system may only be continuous. See Refs. 18, 2 and the references therein for details of these conditions and global strong stability.

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The time derivative of V1 along the trajectory of (1) is 2µ−τ −r1 ∂V1 V˙ 1 = x˙ 1 = x1 r1 [xp21 + φ1 (d(t), x1 )] . ∂x1

(5)

By Assumption 2.1, 2µ−τ −r1 r1

V˙ 1 ≤ x1

h

xp21 + x1

(τ +r1 )/r1

i c .

1 Then, the virtual controller x∗p defined by 2

(τ +r1 )/r1

1 x∗p = −x1 2

r p1 /r1

(n + c) = −x12

r p1 /r1

(n + c) := −x12

β1 ,

yields 2µ−τ −r1 r1

2µ

V˙ 1 (x1 ) ≤ −nx1r1 + x1

1 xp21 − x∗p . 2

(6)

Inductive Step. Suppose at step k − 1, there is a C 1 Lyapunov function Vk−1 : IRk−1 → IR, which is positive definite and homogeneous with respect to (4), ∗pk−1 1 and a set of C 0 virtual controllers x∗1 , x∗p , defined by 2 , · · · , xk µ

x∗1 1 x∗p 2

∗pk−1

xk

ξ1 = x1r1 − x1∗

=0 = .. .

µ r2

r p /µ −ξ12 1 β1

r p

k k−1 = −ξk−1

/µ

ξ2 = x 2 − .. . βk−1

µ r1

,

µ

r x2∗ 2

µ r

ξk = xk k − x∗k

µ rk

(7) ,

with appropriate constants β1 > 0, · · · , βk−1 > 0, such that 2µ−τ −rk−1 p ∗p 2 (xkk−1 − xk k−1 ). V˙ k−1 ≤ −(n + 2 − k) ξ12 + · · · + ξk−1 + ξk−1 µ

(8)

It is clear that (8) reduces to the inequality (6) when k = 2 under the definitions of (7). We claim (8) also holds at step k. To prove this, we set Z xk 2µ−τµ−rk µ µ Wk = s rk − x∗k rk ds x∗ k

and consider the Lyapunov function Vk : IRk → IR, as Vk (x1 , · · · , xk ) = Vk−1 (x1 , · · · , xk−1 ) + Wk (x1 , · · · , xk ),

(9)

1

which can be proven to be C using a similar method as in Ref. 2. The derivative of the Lyapunov function Vk along (1) is V˙ k = V˙ k−1 +

k−1 X l=1

2µ−τ −rk ∂Wk x˙ l + ξk µ x˙ k ∂xl

k−1 2µ−τ −rk−1 X ∂Wk p ∗p 2 ≤ −(n + 2 − k) ξ12 + · · · + ξk−1 + ξk−1 µ (xkk−1 − xk k−1 ) + x˙ l ∂xl l=1

2µ−τ −rk µ

+ξk

2µ−τ −rk µ

k x∗p k+1 + φk (·) + ξk

k k (xpk+1 − x∗p k+1 )

(10)

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= 21−

rk pk−1 µ

|ξk |

τ +rk−1 µ

(11)

,

and by Lemma A.2 it can be seen that 2µ−τ −rk−1 µ

p

2µ−τ −rk−1

τ +rk−1

rk pk−1

) ≤ ξk−1 µ 21− µ |ξk | µ 1 2 ≤ ξk−1 + ck ξk2 , (12) 3 for a constant ck > 0. Using Lemma A.1, Assumption 2.1 can be rewritten as rk+1 pk rk+1 pk rk+1 pk |φk (d(t), x1 , . . . , xk )| ≤ c |ξ1 | µ + |ξ2 − β¯1 ξ1 | µ + · · · + |ξk − β¯k ξk | µ rk+1 pk rk+1 pk rk+1 pk ≤ c¯k |ξ1 | µ + |ξ2 | µ + · · · + |ξk | µ (13) ξk−1

∗pk−1

(xkk−1 − xk

for a constant c¯k > 0. By (13) and Lemma A.2 (with 2µ−τ −rk µ

ξk

φk (·) ≤ |ξk |

2µ−τ −rk µ

c¯k

k X

|ξi |

2µ−τ −rk µ

+

rk+1 pk µ

= 2)

rk+1 pk µ

i=1

1 2 1 2 2 ≤ ξ1 + ξ22 + · · · + ξk−2 (14) + ξk−1 + c˜k ξk2 2 3 Pk−1 ∂Wk x˙ l can be estifor a constant c˜k > 0. The third term in (10), namely l=1 ∂xl mated as the following proposition whose proof is included in the Appendix. Proposition 2.1. There is a constant cˆk > 0 such that k−1 1 2 X ∂Wk 1 2 2 x˙ l ≤ ξ + ξ22 + · · · + ξk−2 + ξk−1 + cˆk ξk2 . ∂xl 2 1 3 l=1

Substituting the estimates (12), (14), and the result of Proposition 2.1 into (10), we arrive at rk+1 pk 2µ−τ −rk µ 2 k V˙ k ≤ −(n + 1 − k) ξ12 + · · · + ξk−1 + ξk µ x∗p + (c + c ˜ + c ˆ )ξ k k k k k+1 2µ−τ −rk µ

+ξk

k k (xpk+1 − x∗p k+1 ).

Observe that a virtual controller of the form rk+1 pk µ

k x∗p k+1 = −ξk

rk+1 pk µ

βk = −ξk

[n + 1 − k + ck + c˜k + cˆk ] ,

(15)

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yields V˙ k ≤ −(n − k + 1)

k X

2µ−τ −rk µ

ξi2 + ξk

k k (xpk+1 − x∗p k+1 ).

i=1

This completes the inductive proof. The inductive argument shows that (8) holds for k = n + 1 with a set of virtual controllers (7). Hence, at the last step, choosing u = xn+1 = x∗n+1 = −ξn(rn +τ )/µ βn yields 2 V˙ n ≤ − ξ12 + · · · + ξn−1 + ξn2

(16) (17)

where V˙ n < 0, ∀x 6= 0 under (7), and Vn (x1 , · · · , xn ) is a positive definite and proper Lyapunov function of the form (9). Thus, (1)–(16) is globally asymptotically stable. Remark 2.1. In the case when τ is any nonnegative real number, we are still able to design a homogenous controller globally stabilizing the system (1) with necessary modification to preserve the sign of function [·]ri pi−1 /µ , where µ is defined as before, though may not be in IRodd . Specifically, for any real number ri pi−1 /µ > 0, we define [·]ri pi−1 /µ = sign(·)| · |ri pi−1 /µ .

(18)

Note that this function is differentiable, and for a constant σ ≥ 1, ∂ sign(g)|g|σ = σ|g|σ−1 . ∂g Using this function, we are able to design the controller without requiring ri pi−1 /µ to be odd. In this case, the controller can be constructed as u = −sign(ξn )|ξn |(rn +τ )/µ βn with ∗pk−1

where

x∗1

xk

= −sign(ξk−1 )|ξk−1 |

ξk

= xk − xk

µ rk

µ ∗ rk

,

rk pk−1 µ

βk−1 , and

k = 1, · · · , n,

= 0.

3. GLOBAL STABILIZATION BY HOMOGENEOUS OUTPUT FEEDBACK In this section, we show that under Assumption 2.1, the problem of global output feedback stabilization for system (1) is solvable. We first construct a homogeneous output feedback controller for the nominal nonlinear chain of power integrators: z˙1 = z2p1 , z˙2 = z3p2 , · · · , z˙n = v, y = z1 , pi ∈ IR≥1 odd , i = 1, · · · , n − 1.

(19)

Then, based on this output feedback controller, we will develop a scaled observer and controller to render the system (1) globally asymptotically stable under the polynomial growth condition (3).

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3.1. Homogeneous Output Feedback Control of Nominal System As a precursor to our ultimate goal of the global stabilization by output feedback of (1), we shall design an output feedback controller for a similar system without the perturbing nonlinear vector field. Once this feedback controller is established, we can then stabilize the augmented system (with φi (·)) by the appropriate choice of a scaling gain, as covered in the succeeding subsection. Theorem 3.1. Given a real number τ ≥ 0, there is a homogeneous output feedback controller of degree τ such that the nonlinear system (19) achieves global asymptotically stability. Proof. The construction of the homogeneous output feedback controller is accomplished in three steps. First, by Theorem 2.1, a homogeneous state-feedback stabilizer is explicitly constructed, we then design a homogeneous observer with a form unique to homogeneous stabilization techniques, and lastly, we replace the unmeasurable states with the estimates recovered from the observer. The closed-loop system can then be proven globally asymptotically stable by selecting an appropriate observer gain. For simplicity, we again assume that ri is odd and use the same µ as defined in the previous section. For other general ri , we adopt the technique stated in Remark 2.1 (i.e. (18)) and the same stabilization result can be achieved. State Feedback Controller: For nonlinear system (19), Assumption 2.1 is automatically satisfied since φi (·) is trivial. Hence, by Theorem 2.1, there is a homogeneous (with respect to the weight (4)) state feedback controller globally stabilizing (19). Specifically, there exists v ∗ (z) = −βn ξn(rn +τ )/µ

(20)

where µ

z1∗

= 0,

∗p zk k−1

= −ξk−1

rk pk−1 µ

ξ1 = z1r1 − z1∗ βk−1 ,

µ rk

µ r1

µ ∗ rk

ξk = z k − z k

,

(21)

,

for k = 2, · · · , n with appropriate constants β1 , · · · , βn > 0, such that 2µ−τ −rn V˙ n ≤ − ξ12 + · · · + ξn2 + ξn µ (v − v ∗ (z))

(22)

where Vn is a positive definite and proper Lyapunov function of the form n Z zi 2µ−τµ −ri X µ µ Vn (z1 , · · · , zn ) = s ri − zi∗ ri ds. i=1

zi∗

Homogeneous Observer Design: Next, a homogeneous observer is constructed in the vein of [1, 2]. η˙ 2 = −`1 zˆ2p1 , p

η˙ k = −`k−1 zˆk k−1 ,

zˆ2

r2

= [η2 + `1 z1 ] r1 ,

rk

zˆk = [ηk + `k−1 zˆk−1 ] rk−1 ,

k = 3, . . . , n

(23)

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where `i > 0, i = 1, · · · , n − 1 are the gains to be determined in later steps. Based on the estimated states zˆi , we design an output feedback controller of the form µ µ rnµ+τ µ µ rn−1 v(ˆ z ) = −βn zˆnrn + βn−1 zˆn−1 + · · · + β2 (ˆ z2r2 + β1 z1r1 ) · · · , (24) where the βi ’s may not be the same values as in the preceding section since there is no need to compensate for perturbing nonlinearities. For i = 2, · · · , n, we designate Ui =

Z

2µ−τ −ri−1 ri

zi

(ηi +`i−1 zi−1 )

2µ−τ −ri−1 ri−1

s

ri−1 2µ−τ −ri−1

− (ηi + `i−1 zi−1 ) ds.

By construction, it can be verified that Ui is C 1 . As a matter of fact, with a constant i−1 bi = 2µ−τr−r , we have the following i 2µ−τ −ri−1 −ri ri−1 ∂Ui ri ri = b i zi zi − (ηi + `i−1 zi−1 ) , ∂zi 2µ−τ −ri−1 2µ−τ −ri−1 ∂Ui ri ri−1 , = − zi − (ηi + `i−1 zi−1 ) ∂ηi 2µ−τ −ri−1 2µ−τ −ri−1 ∂Ui ri = −`i−1 zi − (ηi + `i−1 zi−1 ) ri−1 . ∂zi−1 Hence, the derivative of Ui along (19)-(23) is 2µ−τ −ri−1 −ri ri−1 ri ri pi ˙ zi − (ηi + `i−1 zi−1 ) Ui=zi+1 bi zi 2µ−τ −ri−1 2µ−τ −ri−1 p p ri −`i−1 (zi i−1 − zˆi i−1 ) zi − (ηi + `i−1 zi−1 ) ri−1 , p

µ

p

pn where zn+1 = v. Let ei = (zi i−1 − zˆi i−1 ) ri pi−1 . We now have 2µ−τ −ri−1 −ri ri−1 ri r pi U˙ i = zi+1 bi zi zi i − (ηi + `i−1 zi−1 ) 2µ−τ −ri−1 2µ−τ −ri−1 ri pi−1 ri ri −`i−1 ei µ zi − zˆi 2µ−τ −ri−1 ri pi−1 2µ−τ −ri−1 ri µ ri −`i−1 ei zˆi − (ηi + `i−1 zi−1 ) ,

(25)

i = 2, · · · , n. In what follows, we estimate the terms in (25). First, by Lemma A.1 1−

2µ−τ −ri−1

ri pi−1 and p = (|a − b|p ≤ 2p−1 |ap − bp |), with mi = 2 ri pi−1 2µ−τ −ri−1 2µ−τ −ri−1 p p (zi i−1 ) ri pi−1 − (ˆ zi i−1 ) ri pi−1 − `i−1 ei µ ri pi−1 µ

≤ −`i−1 mi ei

p

p

zi i−1 − zˆi i−1

2µ−τ −ri−1 ri pi−1

−ri−1 2µ−τ r p i i−1

≥ 1,

= −`i−1 mi e2i .

The remaining terms in (25) can be estimated using the following propositions whose proofs are included in the Appendix.

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Proposition 3.1. For i = 2, · · · , n − 1 i+1 2µ−τ −ri−1 −ri ri−1 1 X 2 ri ri pi zi+1 bi zi zi − (ηi + `i−1 zi−1 ) ≤ ξ + αi e2i + gi (`i−1 )e2i−1 12 j=i−1 j (26) where αi is a constant and gi is a continuous function of `i−1 with g2 (·) = 0. Proposition 3.2. For the controller v(ˆ z ), obtained by substituting the estimated states zˆ into (20), we have rn−1 n n 2µ−τ −rn−1 −rn X 1X 2 rn rn v(ˆ z )bn zi −(ηn +`n−1 zn−1 ) ≤ ξi + α ¯ e2i +gn (`n−1 )e2n−1 zn 8 i=1 i=2 (27) 0

where α ¯ is a constant and gn is a C function of `n−1 . Proposition 3.3. For i = 3, · · · , n 2µ−τ −ri−1 ri pi−1 2 2µ−τ −ri−1 ξ 2 +ξi−1 ri −`i−1 ei µ zˆi −(ηi +`i−1 zi−1 ) ri−1 ≤ e2i + i +hi (`i−1 )e2i−1 16 (28) where hi (`i−1 ) is a continuous function of `i−1 . With the help of the previous propositions, it can be shown that the derivative Pn of U = i=2 Ui is n

n−1

X 1X 2 U˙ ≤ ξi + (−`1 m2 + α2 + α ¯ + g3 (`2 ) + h3 (`2 ))e22 + (−`i−1 mi + αi + 1 + α ¯ 2 i=1 i=3 +gi+1 (`i ) + hi+1 (`i ))e2i + (−`n−1 + 1 + α ¯ )e2n ,

(29)

where mi , i = 2, . . . , n − 1, are known powers of 2. Determination of the Observer Gain `i : Due to the fact that states z2 , · · · , zn 2µ−τ −rn µ

are not measurable, the controller v = v(ˆ z ) results in a redundant term ξn ∗ v (z)) in (22). To deal with this, we have the following proposition.

(v−

Proposition 3.4. There is a constant α ˜ ≥ 0 such that 2µ−τ −rn µ

ξn

(v(ˆ z ) − v ∗ (z)) ≤

n n X 1X 2 ξi + α ˜ e2i . 4 i=1 i=2

(30)

Combining (29), (22) and (30) together yields 1 2 T˙ ≤ − ξ12 + · · · + ξn−1 + ξn2 − (`1 m1 − α2 − α ˜−α ¯ − g3 (`2 ) − h3 (`2 ))e22 4 n−1 X − (`i−1 mi − αi − 1 − α ˜−α ¯ − gi+1 (`i ) − hi+1 (`i ))e2i i=3

−(`n−1 − 1 − α ˜ − α)e ¯ 2n ,

(31)

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where the Lyapunov function T = Vn + U . Clearly, by choosing 1 +1+α ˜+α ¯, 4 1 `i−1 = m−1 + α + 1 + α ˜ + α ¯ + g (` ) + h (` ) , i i+1 i i+1 i i 4 1 + α + α ˜ + α ¯ + g (` ) + h (` ) , `1 = m−1 2 3 2 3 2 2 4

`n−1 =

i = n − 1, · · · , 3,

(31) becomes 

n X

n X



1 T˙ ≤ −  ξ2 + e2  . 4 i=1 i j=2 j

(32)

Note that from the construction of T , it is easily verified that T is positive definite and proper with respect to (z1 , · · · , zn , η2 , · · · , ηn )T =: Z.

(33)

Similarly, from the construction of the ξi and ei , the right hand side of (32) is negative definite. Therefore, the closed-loop system is globally asymptotically stable. Denoting fn+1 = η˙ 2 , fn = η˙ 3 , · · · , f2n−1 = η˙ n , it is straightforward to verify that the closed-loop system (19)-(23)-(24), which can be rewritten in the following compact form Z˙ = F (Z) = (z2p1 , · · · , znpn−1 , v(z1 , η2 , · · · , ηn ), fn+1 , · · · , f2n−1 )T ,

(34)

is homogeneous. In fact, by choosing the dilation weight ∆ = (R1 , R2 , · · · , R2n−1 ) = (r1 , r2 , · · · , rn , r1 , r2 , · · · , rn−1 ), | {z } | {z } for z1 , · · · , zn for η2 , · · · , ηn

(35)

with ri defined in (4), it can be shown that (34) is homogeneous of degree τ . In addition, T is homogeneous of degree 2µ − τ and the right hand side of (32) is homogeneous of degree 2µ. Remark 3.1. Note that the right hand side of (32) is negative definite and homogenous of degree 2µ. Hence, it can be shown that there is a constant c1 > 0 such that ∂T (Z) F (Z) ≤ −c1 kZk2µ ∆ ∂Z where kZk∆ =

qP

2n−1 i=1

kZi k2/ri .

(36)

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3.2. Global Output Feedback Stabilization of System (1) Utilization of the homogeneous controller and observer established in the preceding sections will enable us, in the following theorem, to globally stabilize the nonlinear system (1) with only its output feedback. Theorem 3.2. Under Assumption 2.1, the inherently nonlinear system (1) can be globally stabilized by output feedback. Proof: Under the new coordinates, denoting κ1 = 0 and κi = (κi−1 + 1)/pi−1 for i = 2, · · · , n, z1 = x1 , zi = xi /Lκi , i = 2, · · · , n, v = u/Lκn+1

(37)

with a constant L > 1, the system (1) can be rewritten as z˙1 = Lz2p1 + φ1 (·) φ2 (·) z˙2 = Lz3p2 + κ2 L .. . φn−1 (·) z˙n−1 = Lznpn−1 + κn−1 L φn (·) z˙n = Lv + κn . L Next, we construct an observer with the scaling gain L η˙ 2 = −L`1 zˆ2p1 , p

η˙ k = −L`k−1 zˆk k−1 ,

(38)

r2

zˆ2 = [η2 + `1 z1 ] r1 ,

rk

zˆk = [ηk + `k−1 zˆk−1 ] rk−1 ,

k = 3, . . . , n

(39)

where `i , i = 1, · · · , n−1 are the gains selected by (31) in Theorem 3.1. In addition, we design v using the same construction of (24). Specifically, v(ˆ z ) = −βn ξn(rn +τ )/µ (ˆ z) µ rnµ+τ µ µ µ rn−1 r2 r1 rn z 2 + β 1 z1 ) · · · . = −βn zˆn + βn−1 zˆn−1 + · · · + β2 (ˆ

(40)

Using the same notations (33) and (34), the closed-loop system (38)-(39)-(40) can be written as φ2 (·) φn (·) Z˙ = LF (Z) + (φ1 (·), κ2 , · · · , κn , 0, · · · , 0)T . (41) L L Note that the F (Z) in (41) has the exact same structure as (34) due to the use of same gains `i and βi . Hence, adopting the same Lyapunov function T (Z) used in preceding subsection, it can be concluded from Remark 3.1 that ∂T (Z) ∂T (Z) φ2 (·) φn (·) F (Z) + (φ1 (·), κ2 , · · · , κn , 0, · · · , 0)T T˙ = L ∂Z ∂Z L L ∂T (Z) φ2 (·) φn (·) 2µ ≤ −Lc1 kZk∆ + (φ1 (·), κ2 , · · · , κn , 0, · · · , 0)T . ∂Z L L

(42)

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Under the change of coordinates (37), we deduce from Assumption 2.1 and the nomenclature of L > 1 that r +τ r +τ r +τ φi (d(t), x1 , . . . , xi ) ≤ c |z1 | ir1 + |Lκ2 z2 | ir2 + · · · + |Lκi zi | iri L κi L κi ri +τ ri +τ ri +τ ≤ cL1−νi |z1 | r1 + |z2 | r2 + · · · + |zi | ri = cL1−νi

i X

|zj |

ri +τ rj

(43)

j=1

for a constant νi > 0, since it can be seen that by definition rj = τ κj + κj

1 p1 ···pj−1 ,

so

1 1 1 τ κj + κj p1 ···p − κi p1 ···p + τ) κj (τ κi + p1 ···p ri+1 pi i−1 j−1 i−1 − κ = − κi = i 1 1 rj τ κj + p1 ···p τ κ + j p1 ···pj−1 j−1 τ κj ≤ < 1, j = 2, · · · , i, i = 1, · · · , n. 1 τ κj + p1 ···p j−1

Recall that for i = 1, · · · , n − 1, then be seen that

∂T is homogeneous of degree 2µ − τ − ri . It can ∂Zi

ri +τ ri +τ ri +τ ∂T r r r ∂Zi |z1 | 1 + |z2 | 2 + · · · + |zi | i

(44)

is homogeneous of degree 2µ. With (43) and (44) in mind, we can find a constant ρi such that ∂T φi (·) ≤ ρi L1−νi kZk2µ ∆. ∂Zi Lκi

(45)

Substituting (45) into (42) yields T˙ |(38)−(39)−(40) ≤ −L(c1 −

n X

ρi L−νi )kZk2µ ∆.

(46)

i=1

Apparently, when L is large enough the right hand side of the (46) is negative definite. Consequently, the closed-loop system is globally asymptotically stable. 4. DISCUSSIONS In this section we first point out the generality of this method, specifically in Section 4.1, by incorporating the existing global output feedback stabilization results for high-order nonlinear systems as special cases of this framework. Section 4.2 then details the fact that this new scheme will yield a smoother control law than existing non-smooth solutions, as illustrated in Example 4.1, while also providing a way to stabilize systems of a form which made them ill-fitted for the current literature (see Examples 4.2 and 4.3).

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4.1. Unification of Existing Results This part of the section formally unifies the existing literature on the global output feedback stabilization of inherently nonlinear systems by incorporating the existing results as special cases of this design scheme. The following corollaries demonstrate the generalized framework of this methodology alluded to in the preceding sections, with Corollaries 4.1 and 4.2 incorporating the current results for output feedback stabilization of inherently nonlinear systems of Refs. 2 and 3, respectively, while Corollary 4.3 reveals that the problems covered by the homogeneous domination machinery of Ref. 1 are subsets of our realizable problem sets. It is apparent that when pi are any positive odd integers, by letting τ = 0 Assumption 2.1 reduces to the bound described in the following assumption of Ref. 2: Assumption 4.1. (Ref. 2) For (1) with pi ≥ 1 odd integers, i = 1, · · · , n, there is a constant c > 0 such that 1 1 1 |φi (d(t), x1 , . . . , xi )| ≤ c |x1 | p1 ···pi−1 + |x2 | p2 ···pi−1 + · · · + |xi−1 | pi−1 + |xi | , where r1 = 1, r2 = 1/p1 , . . . , rn = 1/(p1 p2 · · · pn−1 ). Corollary 4.1. (Ref. 2) Under Assumption 4.1 (Assumption 2.1 with τ = 0), there is an output feedback controller of the form (39)-(40) which will render the system (1) globally strong stable. In the case when pi = p, p ≥ 1 an odd integer, by selecting τ = p − 1, it is apparent that the growth condition of Ref. 3, which is assumed as: Assumption 4.2. (Ref. 3) p ≥ 1 and

Given (1), assume pi = p for a constant odd integer

|φi (d(t), x1 , . . . , xi )| ≤ c (|x1 |p + |x2 |p + · · · + |xi |p ) , i = 1, · · · , n, with c > 0 a constant, is contained in Assumption 2.1. As a consequence, Theorem 3.2 reduces to the following, which is the main result in Ref. 3. Corollary 4.2. (Ref. 3) When pi = p, p ≥ 1 an odd integer, under Assumption 4.2 (Assumption 2.1 with τ = p − 1), there is an output feedback controller of the form (39)-(40) which globally asymptotically stabilizes (1). In this case, choosing µ = p, it is apparent the controller is smooth in nature and recovers the result in Ref. 3. However, our approach avoids the issue of overcompensation of the nonlinearities (see Ref. 3). In our framework, all the perturbing nonlinearities will be suppressed by the selection of a single appropriate gain L (see Remark 4.1), while in Ref. 3 each of the state feedback controller and the observer

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gains atone for their respective nonlinear perturbations, resulting in an unnecessary doubling of the needed gains. For the case when pi = 1, Assumption 2.1 reduces to the assumption contained in Ref. 1: Assumption 4.3. (Ref. 1)

Given τ ≥ 0, for (1) with pi = 1, i = 1, · · · , n, iτ +1 iτ +1 |φi (d(t), x1 , . . . , xi )| ≤ c |x1 |iτ +1 + |x2 | 2τ +1 + · · · + |xi | (i−1)τ +1 ,

where c > 0 is a constant, and (1) becomes a linear chain of integrators perturbed by a nonlinear vector field, which is the system covered in the main result of Ref. 1. Corollary 4.3. (Ref. 1) When pi = 1, under Assumption 4.3 (Assumption 2.1 with pi = 1), there is an output feedback controller of the form (39)-(40) which will achieve global asymptotic stability of the system (1). 4.2. Improved Performance and Solutions to Previously Unsolved Problems The corollaries in previous subsection clearly show that the existing global output feedback stabilization results for inherently nonlinear systems are contained within this generalized framework. However, as illustrated in the following example, we should emphasize that our design methodology is not limited to using these corollaries and can be employed to offer stabilization under a more desirable performance criterion, such as achieving a smooth output feedback stabilizer when pi are any positive odd integers, whereas Corollary 4.1 (see Ref. 2) can only offer a non-smooth solution. Example 4.1. Consider the system x˙ 1 = x32 ,

x˙ 2 = x3 ,

x˙ 3 = u,

y = x1 .

(47)

For the output feedback stabilization of this example, we previously were limited to using Corollary 4.1, which encompasses the result of Ref. 2, leading to a non-smooth output feedback controller of the form η˙2 = −l1 x ˆ32 = −l1 (η2 + l1 x1 ),

η˙ 3 = −l2 x ˆ3 = −l2 (η3 + l2 x ˆ2 )

ˆ2 )3 + b2 (z2 + l1 x1 ) + b1 x1 u = −b3 (ˆ x33 + b2 x ˆ32 + b1 x1 )1/3 = −b3 (z3 + l2 x

1/3

.

Now, we have the freedom to choose τ , and thus µ, which will eventually lead to a smooth output feedback controller, such as one with the structure η˙ 2 = −`1 x ˆ32 = −`1 (η2 + `1 x1 )3 , 5/3

u = −β3 x ˆ3

η˙ 3 = −`2 xˆ3 = −`2 (η3 + `2 xˆ2 )3

− β2 x ˆ52 − β1 x51 = −β3 (η3 + `2 η2 + `2 `1 x1 )5 − β2 (η2 + `1 x1 )5 − β1 x51 .

Such freedoms epitomize the features of this design approach, enabling us to not only assimilate the existing literature into our framework, but also provides a scheme which can exhibit smoother performance than existing methods.

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In what follows, we show that Theorem 3.2 enables us to delve into output feedback stabilization problem sets which were previously unresolved. We first consider the system (2) from the Introduction with q = 2 . Example 4.2. Consider the inherently nonlinear system x˙ 1 = x32 ,

x˙ 2 = u + d(t)x22 ,

y = x1 ,

|d(t)| ≤ 1.

(48)

As discussed in the Introduction, neither Ref. 2 nor Ref. 18 can provide us a global output feedback controller for this system. However, we can verify that φ1 is trivial and φ2 = x22 , therefore Assumption 2.1 holds for p1 = 3, τ = 1/2, r1 = 1, r2 = 1/2, and µ = 3/2. By the form of τ the controller structure of Remark 2.1 is used and the output feedback controller is η˙ 2 = −L`1 sign(η2 + `1 y)|η2 + `1 y|3/2 , u = −L4/3 β2 sign sign(η2 + `1 y)|η2 + `1 y|3/2 + β1 sign(y)|y|3/2 2/3 × sign(η2 + `1 y)|η2 + `1 y|3/2 + β1 sign(y)|y|3/2 ,

where β1 , β2 , `1 , and L are appropriate gains. Figure 1 illustrates the computer simulation for (48) with β1 = 2, β2 = 10, and `1 = 12.

2 1 0 −1 −2

x −3 −4

x 0

5

10

15

20

1 2

25

30

4 2 0 −2 −4 −6 z hat

−8 −10

2

0

5

10

15 Time

20

25

30

Figure 1. State trajectories of the closed-loop system with (x1 , x2 , zˆ2 )(0) = (1, 2, 3) and L = 2 (chosen to accommodate φ2 = d(t)x22 ).

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In the following, the usefulness of this generalized methodology is validated by solving the global output feedback stabilization of a higher-order uncertain nonlinear system which cannot be stabilized using the existing schemes of Refs. 1, 2, and 3, particularly not by a smooth feedback controller. However, by the technique developed here a C 1 controller can be guaranteed for global output feedback stabilization of this system. Example 4.3. Consider the uncertain inherently nonlinear system 5/3

x˙ 1 = x2

x˙ 2 = x33 + d(t)ax22 x˙ 3 = u y = x1 ,

|d(t)| ≤ 1,

(49)

where a is a constant. It can easily be seen that the global output feedback stabilization of this system was previously impossible to guarantee due to the uncertain higher-order term added to a nonlinear chain of integrators with different non-integer powers. However, it is straightforward to see that Assumption 2.1 can be satisfied by choosing τ = 3/2. Therefore, by Theorem 3.2 there is an output feedback controller globally stabilizing system (49). By following the form of the homogeneous controller and observer as in our design scheme, 5/2

3/5

η˙ 2 = −L`1 sign (η2 + `1 y) |η2 + `1 y| = −L`1 zˆ2 (50) 3 3 2 3 η˙ 3 = −L`2 sign η3 + `2 (η2 + `1 y) η3 + `2 (η2 + `1 y) = −L`2 zˆ3 (51) 12 10 10 23 z2 )|ˆ z2 | 3 +β1 z15 ) zˆ35 +β2 (sign(ˆ z2 )|ˆ z2 | 3 +β1 z15 ) , u = −L 15 β3 sign zˆ35 +β2 (sign(ˆ (52)

it can be shown that (49) is globally asymptotically stable for any L > 0. Therefore, by this generalized framework, there is a large enough gain L such that output feedback controller (50)-(51)-(52) renders the system (49) globally asymptotically stable. Notice that the same output feedback controller structure and gains (β1 , β2 , β3 , `1 , `2 ) can be utilized when either a = 0 or a = 1, with the only difference residing in the scaling gain, L. Remark 4.1. To state what the previous example showed more formally, note that by the design methodology, once the specific gains (βi and `i ) are selected for the nominal nonlinear system (19), only the scaling gain L needs to be adjusted to accommodate various nonlinear terms, φi (·) under Assumption 2.1, exemplifying the universality of this approach in stabilizing uncertain nonlinear systems. 5. CONCLUSIONS This paper presented a generalized framework for the global output feedback stabilization of inherently nonlinear systems for which there exist two greatly different

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methods [2, 3] under distinctive conditions. However, it was shown that these two methods fall within the applicable system sets of this more general scheme, as does the existing homogeneous domination solution (see Ref. 1). A deciding factor in using this approach, compared to the previous methods, was seen in the ability to smoothly stabilize a system which previously had only a non-smooth stabilizer resolution. A unique feature of this formulation is the ability to design a homogeneous controller and observer for the stabilization of a nonlinear chain of integrators, and then use the scaling gain to compensate for a multitude of uncertain nonlinearities satisfying a particular growth condition. Under these qualities, we have developed a universal technique for the global output feedback stabilization of inherently nonlinear systems. Appendix A. Useful Inequalities The next three lemmas, given without proof, were used for the implicit tool of adding a power integrator, whose proofs can be found in Ref. 20, among other places. Lemma A.1. For x ∈ IR, y ∈ IR, p ≥ 1 is a constant, the following inequalities hold: |x + y|p ≤ 2p−1 |xp + y p |, 1 p

1 p

(A.1)

1 p

(|x| + |y|) ≤ |x| + |y| ≤ 2

p−1 p

1 p

(|x| + |y|) .

(A.2)

If p ∈ IR≥1 odd then |x − y|p ≤ 2p−1 |xp − y p |

and

1

1

|x p − y p | ≤ 2

p−1 p

1

|x − y| p .

(A.3)

Lemma A.2. Let c, d be positive constants. Given any positive number γ > 0, the following inequality holds: |x|c |y|d ≤

c d c γ|x|c+d + γ − d |y|c+d . c+d c+d

(A.4)

Lemma A.3. Let p ∈ IR≥1 odd and x, y be real-valued functions. Then, |xp − y p | ≤ p|x − y|(xp−1 + y p−1 ) ≤ c|x − y||(x − y)

p−1

+y

(A.5) p−1

|,

(A.6)

for a constant c > 0. Proof. (A.5) was proven in Ref. 20. The proof of (A.6) stems from (A.5) by noting xp−1 = (x − y + y)p−1 ≤ c (x − y)p−1 + y p−1 , where c = 1, if 1 < p ≤ 2 and c = 2p−2 , if p > 2. The resulting inequality follows.

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Appendix B. Proof of Propositions This part of the appendix contains the technical details of the proofs. Herein we use a generic constant c which represents any finite positive constant value and may be implicitly changed in various places. Nevertheless, the constant c is always independent of `i . Proof of Proposition 2.1: First, it can be seen that for l = 1, · · · , k − 1 ∂Wk 2µ−τ −rk −µ ∗ µ ∂xl x˙ l ≤ c|xk − xk ||ξk |

∗ rµk ∂xk 2µ−τ −µ µ ∂xl x˙ l ≤ c|ξk |

where the last inequality is from (A.3) with p = (A.2), µ r

µ rk

∗ rµk ∂xk ∂xl x˙ l (B.1)

> 1. By definition of x∗k and

µ r

µ−rl µ−rl ∂(xl l ) ∂xk∗ k ∂(β¯k−1 ξk−1 ) r r = =c = cxl l = c(ξl + x∗l l ) µ ∂xl ∂xl ∂xl l X µ−rl µ−rl = c(ξl + β¯l−1 ξl−1 ) µ ≤ c |ξi | µ . µ

(B.2)

i=l−1

This, together with (13) gives   ∗ rµk l l X X rl+1 pl ∂xk µ−rl |ξi | µ |xl+1 |pl + |ξj | µ  ∂xl x˙ l ≤ c j=1 i=l−1   rl+1 pl µ l l µ X X rl+1 pl µ−rl rl+1 ∗ ≤c |ξi | µ  ξl+1 + xl+1 + |ξj | µ  i=l−1

≤c

l X

i=l−1

|ξi |

µ−rl µ



|ξl+1 |

j=1

rl+1 pl µ

+ |ξl |

rl+1 pl µ

+

l X j=1

|ξj |

rl+1 pl µ



.

By Lemma A.2 and the fact that rl+1 pl = τ + rl , we have ∗ rµk l+1 l+1 X X µ−rl +rl+1 pl ∂xk µ+τ µ x ˙ ≤ c = c |ξi | µ |ξ | i ∂xl l i=1 i=1

(B.3)

for l = 1, · · · , k − 1. Clearly, Proposition 2.1 follows immediately from (B.1) and (B.3). Proof of Proposition 3.1: By the definition of zˆi , it can be shown that, with

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qi := 2µ − τ − ri−1 − ri ri−1 ri−1 ri−1 qi qi r r r r r pi pi zi+1 bi zi i zi i − zˆi i −`i−1 (zi−1 − zˆi−1 ) bi zi i zi i −(ηi +`i−1 zi−1 ) = zi+1 qi ri−1 ri−1 p p r pi zi i−1 ) ri pi−1 = zi+1 bi zi i (zi i−1 ) ri pi−1 − (ˆ pi−2 /pi−2 pi−2 /pi−2 −`i−1 (zi−1 − zˆi−1 ) .

(B.4)

Note that

ri−1 ri pi−1

and likewise,

ri−1 ri pi−1 ,

≤ 1. By (A.3) with p = r ri−1 ri−1 pi−1 rrpi−1 1− r pi−1 pi−1 ri pi−1 (z i i−1 i i−1 |e | µ , − (ˆ z ≤ 2 ) ) i i i p

i−2 (zi−1

/pi−2

p

i−2 − zˆi−1

By (21), we know that ri ri |zi | ≤ c |ξi | µ + |ξi−1 | µ ,

/pi−2

)≤2

1− p

1 i−2

|ei−1 |

ri−1 µ

.

(B.5)

(B.6)

ri+1 ri+1 |zi+1 | ≤ c |ξi+1 | µ + |ξi | µ .

With these facts in mind, by Young’s inequality, utilizing ri+1 pi = τ + ri and (A.1), the following holds i+1 qi h i X ri+1 pi ri+1 pi ih 2µ−ri−1 qi qi r pi zi+1 zi i ≤ c |ξi+1 | µ + |ξi | µ |ξi | µ + |ξi−1 | µ ≤ c |ξj | µ . (B.7) j=i−1

Applying (B.5), (B.6), and (B.7) to (B.4) yields, ri−1 i+1 qi X 2µ−ri−1 ri ri pi zi+1 bi zi zi − (ηi + `i−1 zi−1 ) ≤c |ξj | µ × j=i−1

r ri−1 ri−1 1− 1 1− i−1 2 ri pi−1 |ei | µ +`i−1 2 pi−2 |ei−1 | µ . (B.8)

Applying Young’s inequality to each term in (B.8) will lead to (26). In the case when i = 2, e1 = 0 since zˆ1 := z1 . Hence, we can simply set g2 = 0. Proof of Proposition 3.2: Similar to (B.4), (B.5), and (B.6), we have, with qn := 2µ − τ − rn−1 − rn , rn−1 qn rn rn v(ˆ z )bn zn zn − (ηn + `n−1 zn−1 ) ≤ c|v(ˆ z )|

n X

|ξj |

j=n−1

qn µ

rn−1 rn−1 rn−1 1− 1 21− rn |en | µ + `n−1 2 pn−2 |en−1 | µ .

(B.9)

By the homogeneity of v, r +τ

|v(ˆ z )| ≤ c kˆ z k∆nz

(B.10)

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where n X

kˆ z k∆z =

|ˆ zi |2/ri

i=1

!1/2

∆z = (r1 , · · · , rn ).

,

So, by the definition of the homogeneous norm, we have kˆ z k ∆z =

n X

|ˆ zi |

2 ri

! 12

=

i=1

≤c ≤c

n X i=1 n X

2 ! 21 n ri pi−1 r p X pi−1 i i−1 µ z − ei i i=1

|zi | |zi |

2 ri

1 ri

n X

+ !

|ei |

!1/2

2 µ

i=2

+c

i=1

n X

|ei |

1 µ

!

(B.11)

,

i=2

where zˆ1 = z1 . Likewise, using (21) to replace ξi for zi , together with (B.10) and (B.11), we have ! n n X X rn +τ rn +τ |v(ˆ z )| ≤ c |ξi | µ + |ei | µ . (B.12) i=1

i=2

Applying (B.12) to (B.9) yields, rn−1 qn rn rn v(ˆ z )bn zn zn − (ηn + `n−1 zn−1 ) ≤c

n X i=1

≤

|ξi |

rn +τ µ

+

n X i=2

|ei |

rn +τ µ

!

|ξn |

qn µ

+ |ξn+1 |

qn µ

rn−1 rn−1 rn−1 1− 1 × 21− rn |en | µ + `n−1 2 pn−2 |en−1 | µ

n n X 1X 2 ξi + α ¯ e2i + gn (`n−1 )e2n−1 , 8 i=1 i=2

(B.13)

for a constant α ¯ > 0. The last relation is obtained by applying Lemma A.2 to each individual term in the previous inequality. qi Proof of Proposition 3.3: By definition of zˆi and Lemma A.3 (p = ri−1 := 2µ−τ −ri−1 ri−1

> 1) we have, qi ri pi−1 qi ri µ ri−1 −`i−1 ei zˆi − (ηi + `i−1 zi−1 ) qi qi ri pi−1 ≤ `i−1 |ei | µ (ηi + `i−1 zˆi−1 ) ri−1 − (ηi + `i−1 zi−1 ) ri−1 ri pi−1

≤ c`i−1 |ei | µ |ηi + `i−1 zˆi−1 − (ηi + `i−1 zi−1 )| qi qi −1 −1 × (ηi + `i−1 zˆi−1 ) ri−1 − (`i−1 zˆi−1 − `i−1 zi−1 ) ri−1 ,

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where the last inequality invokes (A.6). Using (A.3) with p = pi−2 , qi ri pi−1 qi −`i−1 ei µ zˆiri − (ηi + `i−1 zi−1 ) ri−1 qi −ri−1 ri pi−1 ri−1 qi −ri−1 1+ p 1 ri i−2 . µ µ µ ≤ c`i−1 |ei | |ei−1 | + |`i−1 ei−1 | zˆi On the other hand, by (A.3) with p = pi−1 , r ri pi−1 i 1 ri p zˆi = (−ei µ + zi i−1 ) pi−1 ≤ c −eiµ + (|ξi | + β¯i−1 |ξi−1 |) µ ri ri ri ≤ c |ei | µ + |ξi | µ + β˜i−1 |ξi−1 | µ .

Thus, ri pi−1 µ

−`i−1 ei

qi ri

qi ri−1

1+ p

1

ri pi−1

ri−1

zˆi − (ηi + `i−1 zi−1 ) ≤ c`i−1 i−2 |ei | µ |ei−1 | µ q −r qi −ri−1 qi −ri−1 qi −ri−1 i i−1 × |ξi | µ + βˆi−1 |ξi−1 | µ +|ei | µ +|`i−1 ei−1 | µ .

By using Young’s inequality to each term in above relation, the desired result can be proven for a polynomial function hi (`i−1 ). µ Proof of Proposition 3.4: Firstly, let w(·) = v(·) rn +τ . By (A.3) of Lemma A.1, we have rn +τ rn +τ rn +τ z ) − w ∗ (z)| µ |v(ˆ z ) − v ∗ (z)| = w(ˆ z ) µ − w∗ (z) µ ≤ c |w(ˆ (B.14) Now, because w is at least C 1 , we can expand this function as Z 1 n X ri ∂w(X) ri pi−1 dλ. |w(ˆ z ) − w∗ (z)| ≤ c |ei | µ 1 ∂Xi Xi =(zpi−1 −λe µ ) pi−1 0 i=2 i

By the homogeneity of w ∗ (z) whose degree is µ, µ − ri . Hence,

i

∂w(X) is homogeneous of degree ∂Xi

ri pi−1 1 ∂w(X) p i ≤ ck(zi i−1 − λei µ ) pi−1 kµ−r ∆z ∂Xi X ! ! n n X X µ−ri µ−ri ≤c +c |zi | ri |ei | µ , i=1

i=2

for λ ∈ [0, 1]. Noting (B.12), we have ! ! n n X X µ−ri µ−ri ∂w(X) ri pi−1 ≤c |ξi | µ +c |ei | µ . 1 ∂Xi Xi =(zpi−1 −λe µ ) pi−1 i=1 i=2 i

i

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Therefore, 2µ−τ −rn µ

ξn

2µ−τ −rn µ

rn +τ

(v(ˆ z ) − v ∗ (z)) ≤ c|ξn

| |w(ˆ z ) − w∗ (z)| µ n rnµ+τ ri rnµ+τ 2µ−τ −rn X µ−ri µ−ri rµi ≤ c|ξn µ | + |ei | µ |ei | µ |ei | |ξi | µ ≤

n 1X

4

i=1

i=1 n X

ξi2 + α ˜

e2i

i=2

for a constant α ˜ ≥ 0. References 1. C. Qian. A homogeneous domination approach for global output feedback stabilization of a class of nonlinear systems. In: Proc. of 2005 American Control Conference, pages 4708–4715 (2005). 2. C. Qian and W. Lin. Recursive observer design and nonsmooth output feedback stabilization of inherently nonlinear systems. In: Proc. of 43rd IEEE Conference on Decision and Control, pages 4927–4932, Atlantis, Bahamas (2004). 3. B. Yang and W. Lin. Robust output feedback stabilization of uncertain nonlinear systems with uncontrollable and unobservable linearization. IEEE Trans. Automat. Control, 50(5):619–630 (2005). 4. F. Mazenc, L. Praly, and W. P. Dayawansa. Global stabilization by output feedback: examples and counterexamples. Systems Control Lett., 23(2):119–125 (1994). 5. D. Bestle and M. Zeitz. Canonical form observer design for non-linear time-variable systems. International Journal of Control, 38(2):419–431 (1983). 6. A. J. Krener and A. Isidori. Linearization by output injection and nonlinear observers. Systems Control Lett., 3(1):47–52 (1983). 7. A. J. Krener and W. Respondek. Nonlinear observers with linearizable error dynamics. SIAM J. Control Optim., 23(2):197–216 (1985). 8. A. J. Krener and M. Xiao. Nonlinear observer design in the Siegel domain. SIAM J. Control Optim., 41(3):932–953 (2002). 9. R. Marino and P. Tomei. Dynamic output feedback linearization and global stabilization. Systems Control Lett., 17(2):115–121 (1991). 10. G. Besancon. State affine systems and obsever based control. NOLCOS, 2:399–404 (1998). 11. J.-P. Gauthier, H. Hammouri, and S. Othman. A simple observer for nonlinear systems, applications to bioreactors. IEEE Trans. Automat. Control, 37(6):875–880 (1992). 12. H. K. Khalil and A. Saberi. Adaptive stabilization of a class of nonlinear systems using high-gain feedback. IEEE Trans. Automat. Control, 32(11):1031–1035 (1987). 13. L. Praly. Asymptotic stabilization via output feedback for lower triangular systems with output dependent incremental rate. In: Proc. of the 40th IEEE Conference on Decision and Control, pages 3808–3813 (2001). 14. J. Tsinias. A theorem on global stabilization of nonlinear systems by linear feedback. Systems Control Lett., 17(5):357–362 (1991). 15. C. Qian and W. Lin. Output feedback control of a class of nonlinear systems: a nonseparation principle paradigm. IEEE Trans. Automat. Control, 47(10):1710–1715, 2002.

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16. Z. Chen and J. Huang. Global output feedback stabilization for uncertain nonlinear systems with output dependent incremental rate. In: Proc. of 2004 American Control Conference, pages 3047–3052 (2004). 17. L. Praly and Z. Jiang. On global output feedback stabilization of uncertain nonlinear systems. In: Proc. of the 42nd IEEE Conference on Decision and Control, pages 1544–1549 (2003). 18. C. Qian and W. Lin. Smooth output feedback stabilization of planar systems without controllable/observable linearization. IEEE Trans. Automat. Control, 47(12):2068– 2073 (2002). 19. B. Yang and W. Lin. Further results on global stabilization of uncertain nonlinear systems by output feedback. Internat. J. Robust Nonlinear Control, 15(6):247–268 (2005). 20. C. Qian and W. Lin. A continuous feedback approach to global strong stabilization of nonlinear systems. IEEE Trans. Automat. Control, 46(7):1061–1079 (2001). 21. A. Bacciotti and L. Roiser. Liapunov Functions and Stability in Control Theory. Springer (2001). 22. W. P. Dayawansa, C. F. Martin, and G. Knowles. Asymptotic stabilization of a class of smooth two-dimensional systems. SIAM J. Control Optim., 28(6):1321–1349 (1990). 23. W. P. Dayawnasa. Recent advances in the stabilization problem for low dimensional systems. In: Proc. of 1992 IFAC NOLCOS, pages 1–8 (1992). 24. H. Hermes. Homogeneous coordinates and continuous asymptotically stabilizing feedback controls. In: Differential equations, pages 249–260. Dekker, New York (1991). 25. M. Kawski. Homogeneous stabilizing feedback laws. Control Theory Adv. Tech., 6(4):497–516 (1990).

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Chap27-XuYashan

A LINEAR QUADRATIC CONSTRAINED OPTIMAL FEEDBACK CONTROL PROBLEM

YASHAN XU∗ School of Mathematical Science, Fudan University, 220, Handan Road, Shanghai, 200433, China E-mail: [email protected]

A linear quadratic optimal control problem with constrained feedback control set is considered. The relationship between open-loop control problem and feedback closedloop control problem is presented. By a synthesis method, a sufficient condition is derived to ensure the existence and uniqueness of globally optimal nonlinear feedback control.

1. Introduction Classical theory of linear quadratic optimal control problems is pretty mature. 1 – 2 It has many applications in engineering and other various related fields.2 – 3 In a number of interesting physical problems, it is necessary to impose some additional constraints upon the control and/or the state. Examples are control with piecewise constraints,4 energy constraints,5 control with linear constraints,6 state constraints,7 – 8 path constraints,9– 10 state and control with linear inequality constraints11 – 12 and quadratic constraints,13 etc. In this paper, we would like to consider an important constraint which comes from the response of the controller with respect to his state. More precisely, consider the following constrained feedback closed-loop control set ϕ(·, ·) kϕ(t, y)k ≤ kkyk , where y is state variable and k is a given positive number. As a special case, the LQ control problem with constrained state feedback matrix ϕ(t, y) = K(t)y, kK(t)k ≤ k has been discussed.14 In that case, we found that globally optimal linear feedback does not necessarily exist. The purpose of this paper is to study the existence of globally optimal nonlinear feedback control in this expanded feedback control set. Let t ∈ [0, T ). Consider the following linear controlled system: y(s) ˙ = A(s)y(s) + B(s)ϕ(s, y(s)), s ∈ [t, T ], (1) y(t) = x ∈ lRn ,

∗ Work

partially supported by grant 10371024 of the China National Science Foundation. 215

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where A(·) ∈ C([0, T ]; lRn×n ), B(·) ∈ C([0, T ]; lRn×m ), and ϕ(·, ·) is a feedback control. The cost functional is defined to be Z T J(t, x; u(·, ·)) = [hQ(s)y(s), y(s)i + hR(s)ϕ(s, y(s)), ϕ(s, y(s))i]ds (2) t +hM y(T ), y(T )i with Q(·) ∈ C([0, T ]; lRn×n ), R(·) ∈ C([0, T ]; lRm×m ), and M ∈ lRn×n . For given k ∈ (0, +∞) and an initial condition (t, x) ∈ [0, T ) × lRn , feedback n m control set ΦF k (t, x) consists of all measurable functions ϕ(·, ·) : [0, T ] × lR → lR which satisfies kϕ(s, y)k ≤ kkyk, (s, y) ∈ [t, T ] × lRn ,

(3)

and state equation Eq(1) under the initial condition (t, x) admits a unique solution. Further define feedback control set ΦF k as follows \ ΦF ΦF (4) k = k (t, x). (t,x)∈[0,T )×lRn

Now we present the constrained feedback control LQ problem: Problem (LQ)F ¯ , ·) ∈ ΦF k . Find a ϕ(· k such that J(t, x; ϕ(· ¯ , ·)) =

inf

ϕ(· ,·)∈ΦF k

J(t, x; ϕ(· , ·)), ∀ (t, x) ∈ [0, T ] × lRn .

(5)

As a useful tool, the relax control theory has been used widely on the topic for the existence of an optimal open-loop control. But few works has been obtained in the case of the globally feedback control. Here the author would construct a globally feedback control via the relationship between the open-loop control and closed-loop control. The main difficulty of the paper is to obtain the uniqueness of the solution to the system equation under the constructing feedback control. 2. Primal Results In this section, we will pose an open-loop constrained LQ problem which is useful to deal with the original problem. In what follows, we fix k ∈ (0, +∞). Now we introduce a series of open-loop control sets: For any given (t, x) ∈ m 2 [0, T ) × lRn , let Φt,x k be the set of all open-loop controls u(·) ∈ L (t, T ; lR ) such that ku(s)k ≤ kky(s)k, a.e. s ∈ [t, T ], with y(·) being the state trajectory of system equation y(s) ˙ = A(s)y(s) + B(s)u(s), s ∈ [t, T ], y(t) = x. Then we present the following definitions.

(6)

(7)

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Definition 2.1. Given (t, x) ∈ [0, T ) × lRn and a feedback control ϕ ∈ ΦF k (t, x), (i) An open-loop control defined by u(·) = ϕ(·, y t,x (·)), s ∈ [t, T ]

(8)

is called an open-loop control associated with the feedback control ϕ(·, ·) ∈ ΦF k (t, x) at (t, x). (ii) A control set A ⊆ Φt,x k is called the open-loop control set associated with the feedback control set B ⊆ ΦF k (t, x) at (t, x) if A consists of all open-loop control associated with some feedback control in B at (t, x). Then, we have the following result. Lemma 2.1. Let (t, x) ∈ [0, T ) × lRn . Then the control set Φt,x k is the open-loop control set associated with the feedback control set ΦF (t, x). k Now we introduce an open-loop LQ problem with constraints as follows. Problem (LQ)k . For any (t, x) ∈ [0, T ) × lRn , find a u ¯(·) ∈ Φt,x k such that J(t, x; u ¯(·)) =

inf

def

u(·)∈Φt,x k

J(t, x; u(·)) = V∗k (t, x).

(9)

Further, the following two results will be used. Proposition 2.1. For any (t, x), if feedback control ϕ(·, ¯ ·) ∈ ΦF k (t, x) satisfies J(t, x; ϕ) ¯ =

inf

ϕ∈ΦF k (t,x)

J(t, x; ϕ),

then the open-loop control u ¯(·) associated with ϕ¯ solves Problem (LQ) k at (t, x). Proposition 2.2. Let (t, x) ∈ [0, T ) × lRn . If the open-loop control Problem (LQ)k admits a solution at (t, x), then there exists a ϕ(·, ¯ ·) ∈ ΦF k (t, x) such that J(t, x; ϕ(·, ¯ ·)) =

inf

ϕ(·,·)∈ΦF k (t,x)

J(t, x; ϕ(·, ·)).

(10)

3. Main Results In this section, we will construct a globally optimal nonlinear feedback control which uniquely solves Problem (LQ)F k by a synthesis method. First we pose the following hypothesis. Q(s) ≥ 0,

R(s) ≥ δI, ∀ s ∈ [0, T ],

M ≥ 0,

(11)

where δ > 0. By section 2, one can the discuss Problem (LQ)F k at the given (t, x) by solving Problem (LQ)k at (t, x). But it is still difficult to solve Problem (LQ)k at (t, x) directly. Our method is to introduce an open-loop control problem without constraints. Consider the following controlled system y(s) ˙ = A(s)y(s) + B(s)K(s)y(s)), s ∈ [t, T ], (12) y(t) = x ∈ lRn ,

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where K(·) is the control variable belonging to the admissible control set

def Kk = K(·) ∈ L2 (0, T ; lRm×n ) K(s)k ≤ k .

The corresponding cost functional is defined by Z T J(t, x; K(·)) = [hQ(s)y(s), y(s)i + hR(s)K(s)y(s), K(s)y(s)i]ds t

(13)

+hM y(T ), y(T )i. Thus we propose following problem. ¯ Problem (LQ)L k . For any given (t, x), find a K(·) ∈ Kk such that ¯ J(t, x; K(·)) = Denote def

ΦL k =

inf

K(·)∈Kk

(14)

J(t, x; K(·)).

ϕ(s, y) = K(s)y, ∀ (s, y) ∈ [0, T ] × lRn K(·) ∈ Kk

⊆ ΦF k.

It is clear that Problem(LQ)L k is essentially a linear feedback control problem. Set t,x Φk is also an open-loop control set associated with ΦL k at (t, x). Furthermore, we can obtain the following result as Proposition 2.1 and 2.2. Lemma 3.1. Problem (LQ)k is solvable at (t, x) if and only if Problem (LQ)L k admits a solution at (t, x). We emphasize that the existence and uniqueness of the solution to Problem L (LQ)t,x k depend on that of Problem (LQ)k at (t, x). First, so-called Cesari condiL tion ensures that Problem (LQ)k admits an optimal open-loop control at (t, x)15 . Second, we can obtain that there exists a δ > 0 such that the following two-point boundary problem admits at most one solution for any (k, t, x) ∈ [0, δ] × [0, T ] × lRn ,  BB > ψ   y˙ = Ay − min{ 21 kB > ψk, kkyk} > ,   kB ψk  > > + kB ψk kB ψk (15) > ˙  ψ = −A ψ − 2Qy + − 2k min , k y,   kyk 2kyk   y(t) = x, ψ(T ) = 2M y(T ). where [c]+ = max{c, 0} and it is conventional to assume that

BB > ψ = 0, for B > ψ = 0; min{ 21 kB > ψk, kkyk} > kB ψk > + > kB ψk kB ψk − 2k min , k y = 0, for y = 0. kyk 2kyk Thus, the uniqueness of the solution to Problem (LQ)L k at (t, x) is obtained. Further, we have Theorem 3.1. Assume (11) and Q(·) is uniformly positive definite. Then there exists a δ > 0 such that for any given (k, t, x) ∈ [0, δ] × [0, T ) × lR n , Problem (LQ)k admits at most one solution at (t, x).

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The main result of this paper can be stated as follows. Theorem 3.2. Assume (11) and Q(·) is uniformly positive definite. Then there exists a δ > 0 such that Problem (LQ)F k admits a unique globally optimal continuous feedback control for any k ≤ δ. The main idea behind the proof is as follows. First, present a feedback control ¯ ·) by the synthesis method from the optimal open-loop control set t,x ϕ(·, u ¯ (·), (t, x) ∈ [0, T ) × lRn whose element solves Problem (LQ)k at the given (t, x) respectively. Second, show that the system equation under feedback control ϕ(·, ¯ ·) admits a unique solution by virtue of the uniqueness of solution to Problem (LQ)k at (t, x) and the regularity of value function. Therefore, the feedback control ϕ(·, ¯ ·) solves the original problem. Acknowledgments The author would like to thank Professors J. Yong and H. Lou for many constructive suggestions. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

R. Kalman, Trans. ASME J. Basic Engr. 86D, 51 (1964). B. D. O. Anderson and J. Moore, Linear Optimal control, Prentice Hall (1971). J. Casti, SIAM Review, 22, 459 (1980). A. Matveev and V. Yakubovich, Dyn. Control, 7, 99 (1997). Y. Chen, Control Theory Appl., 16, 474 (1999). P. Brunovsky, J. Diff. Eqs., 38, 344 (1980). Z. Emirsajlow, Syst. Anal.Modelling Simulation, 4, 227 (1987). Z. Emirsajlow, IMA J. Math. Control Inform., 8, 179 (1991). Y. Liu, S. Ito, H. W. J. Lee, and K. L. Teo, J. Optim. Theory Appl., 108, 617 (2001). D. D. Thompson and R. A. Volz, SIAM J. Control, 13, 110 (1980). G. Chen, Appl. Anal., 41, 257 (1991). G. Stefani and P. Zezza, SIAM J. Control Optim., 35, 876 (1997). D. H. Martin and D. H. Jacobson, Automatica, 15, 431 (1979). Y. Xu, A linear quadratic control problem with constrained state feedback matrix, to appear. 15. L. Cesari, Lecture Notes in Math., Vol.979, 88 (1983).

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Chap28-YangBo

ON FINITE-TIME STABILIZATION OF A CLASS OF NONSMOOTHLY STABILIZABLE SYSTEMS∗

BO YANG Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, U.S.A. Email: [email protected] WEI LIN Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH 44106, U.S.A. Email: [email protected]

The problem of global finite-time stabilization is investigated in this paper. Using the finite-time Lyapunov stability theorem and the nonsmooth feedback design method developed recently, with a suitable twist, we prove that it is possible to achieve global finite-time stabilizability for a class of nonlinear systems with uncertainty, which has been known not to be smoothly stabilizable, even locally, but can be globally asymptotically stabilized by H¨ older continuous state feedback. Examples are presented to demonstrate the validity of the finite-time stabilization results obtained in the paper.

1. Introduction We consider a nonlinear system described by equations of the form x˙ 1 = xp21 + f1 (x1 ) x˙ 2 = xp32 + f2 (x1 , x2 ) .. . x˙ n = upn + fn (x1 , · · · , xn ),

(1)

where x = (x1 , · · · , xn )T ∈ IRn and u ∈ IR are the system state and input, respectively, pi ≥ 1, i = 1, · · · , n, are arbitrarily odd integers, and fi : IRi → IR, i = 1, · · · , n, are C 1 functions with fi (0, · · · , 0) = 0. It has been recognized that (1) represents a class of nonlinear systems which is not stabilizable by any smooth state feedback, even locally, for the reason that ∗ W.

Lin’s work was supported in part by the U.S. NSF under grants DMS-0203387 and ECS0400413, and in part by the AFRL Grant FA8651-05-C-0110. 220

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the linearized system of (1) may contain an uncontrollable mode associated with eigenvalues on the right-half plane. As a consequence, stabilization of (1) can usually be achieved by nonsmooth state feedback. Over the past fifteen years, the problem of feedback stabilization of nonsmoothly stabilizable nonlinear systems such as (1) has received considerable attention. For instance, the book 2 and the papers 10,11,16,17,22,23 studied the problem of local asymptotic stabilization via continuous but non-differentiable state feedback, for lower-dimensional (two or three-dimensional) systems with uncontrollable unstable linearization, while the works 7,9,35 investigated the local stabilization of the ndimensional nonlinear system (1), using the idea of homogeneous approximation and Hermes’ robust stability theorem for homogeneous systems.13,16,17,31 Recently, it has been proved in Refs. 28 and 29 that global strong stabilization of the nonlinear system (1) is indeed possible by using nonsmooth state feedback. In particular, a H¨ older continuous, globally stabilizing state feedback control law was explicitly constructed by the tool of adding a power integrator.28,29 The main purpose of the paper is to prove that using H¨ older continuous state feedback, global finite-time stabilization, instead of global asymptotic stabilization, can be achieved for the nonlinear system (1). The other objective of the paper is to demonstrate how the tool of adding a power integrator,28,29 with an appropriate twist, can be used to construct a H¨ older continuous controller which renders the trivial solution x = 0 of (1) not only global stable but also convergent in finite-time. The problem of finite-time stabilization arises naturally in various applications. A well-known example is the so-called dead-beat control system which has found wide applications in classical control engineering, for example, in process control and digital control, just to name a few. A more classical example is the time optimal control in which the concept of finite-time stability is automatically involved. To be precise, consider the problem of time-optimal control for a doubleintegrator system. By the well-known maximal principle, a time-optimal controller of bang-bang type can be derived, steering all the trajectories of the doubleintegrator system to the origin in a minimum time from any initial condition. The time-optimal control system thus obtained has a distinguished feature, namely, finite-time convergence rather than infinite settling time. Compared with the notion of asymptotic stability, finite-time stability requires essentially that a control system be stable in the sense of Lyapunov. Moreover, its trajectories converge to zero in finite time. Studying control systems that exhibit finite-time convergence is important for two reasons: 1) this class of systems usually has a faster convergent rate; 2) finite-time stable systems seem to perform better in the presence of uncertainties and disturbances.6 Finally, it is worth noticing that the notion of finite-time stability also plays a key role in the design of sliding mode controllers (see, for instance, Ref. 18), whose strategy is to steer all the trajectories to the sliding surface in finite time. The problem of finite-time stabilization has been studied, for instance, in Refs. 4,

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5, 6, 14, 19, 33, which demonstrated that finite-time stable systems enjoy not only faster convergence but also better robustness and disturbance rejection properties. Notably, a fundamental result on finite-time stability was obtained in Ref. 6, in which a Lyapunov theory for finite-time stability is presented. It provides a basic tool, within the finite-time framework, for analysis and synthesis of nonlinear control systems. The finite-time stability theory developed in Ref. 6 was then employed to derive C 0 finite-time stabilizing state controllers for the double integrator.5 Most of the aforementioned finite-time stabilization results are only applicable to lower dimensional control systems. Moreover, these results are local due to the use of homogeneous approximation. In the higher-dimensional case, Reference 19 considered primarily the local finite-time stabilization problem and proposed continuous finite-time stabilizers for a class of nonlinear systems such as triangular systems, using the homogeneous systems theory. Recently, we have addressed the problem of global finite-time stabilization for a family of uncertain nonlinear systems with controllable linearization. In particular, it was shown in Ref. 20 that for the nonlinear system (1) with pi = 1, i = 1, · · · , n, global finite-time stabilization is achievable by H¨ older continuous state feedback. In contrast to the standard backstepping design for global asymptotic stabilization, the feedback design method in Ref. 20 is more subtle and delicate for the two reasons: 1) nonsmooth state feedback control laws, rather than the smooth ones, must be constructed at every step of the recursive design procedure; 2) to guarantee global finite-time stability of the closed-loop system, the derivative of the control Lyapunov function V (x) along the trajectories of the closed-loop system must be not only negative definite but also less than −cV α (x), for suitable real numbers c > 0 and 0 < α < 1. In view of Ref. 20, an interesting theoretical question arises naturally: can global finite-time stabilization be achieved for the nonlinear system (1) with uncontrollable unstable linearization? In this work, we shall give an affirmative answer to this theoretical issue. In particular, we show how the global finite-time stabilization result obtained in Ref. 20 for feedback linearizable systems (i.e. system (1) with pi = 1, i = 1, · · · , n) can be extended to the nonlinear system (1). It should be emphasized that such an extension is by no means trivial. In fact, the finite-time feedback design method proposed in Ref. 20 cannot be applied to (1), due to the presence of uncontrollable unstable linearization of (1). Therefore, one of the main contributions of this paper is to show how to find, based on the theory of homogeneous systems (particularly, the idea of homogeneity with respect to a family of dilations (see Refs. 15, 16, 17, 22, 23, 24), a control Lyapunov function and a finite-time global stabilizer simultaneously for the nonlinear system (1), so that global finite-time stabilization of the closed-loop system can be concluded from the finite-time stability theorem.6 The rest of this paper is organized as follows: In Section 2, we review useful concepts and the Lyapunov theory that pertain to the notion of finite-time stability. The main results are shown in Section 3, where the existence of globally stabilizing

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finite-time controllers is proved for the non-smoothly stabilizable system (1). The controllers are explicitly constructed via an adding a power integrator method. In Section 4, we show how the finite-time stabilization result obtained in Section 3 can be extended to a broader class of uncertain nonlinear systems. Concluding remarks are given in Section 5. The appendix collects the proofs of four propositions that are used in the paper. A preliminary version of this work (see Ref. 36) was presented at the 16th IFAC World Congress, Prague, Czech, in July 2005. 2. Finite-Time Stability In this section, we review some concepts and terminologies related to the notion of finite-time stability and the corresponding Lyapunov stability theory. It is known that the classical Lyapunov stability theory (e.g., see Refs. 13 and 25) is only applicable to a differential equation whose solution from any initial condition is unique. A well-known sufficient condition for the existence of a unique solution of the autonomous system x˙ = f (x),

with n

f (0) = 0,

x ∈ IRn

(2)

n

is that the vector field f : IR → IR is locally Lipschitz continuous. The solution trajectories of the locally Lipschitz continuous system (2) can have at most asymptotic convergent rate. However, it is sometimes desirable to achieve finitetime convergence in practical applications, such as dead-beat control, sliding mode and time-optimal control. It should be pointed out that only the non-uniqueness of trajectories in backward time, or the non-Lipschitz continuous property of autonomous systems can render the finite-time convergence. A simple example is the scalar system 1

x˙ = −x 3 ,

x(0) = x0

whose solution trajectories are unique in forward time but not unique in backward time. They can be described by  3/2 2 2  sgn(x0 ) x03 − 32 t , 0 ≤ t < 23 x03 , x(t) = 2  0, t ≥ 23 x03 ,

which converges to x = 0 in finite time. This example suggests that in order to achieve finite-time stabilizability, non-smooth or at least non-Lipschitz continuous feedback must be employed, even if the controlled plant x˙ = f (x, u, t) is smooth. For the analysis and synthesis of non-Lipschitz continuous systems, new notions on stability and the corresponding Lyapunov stability theory must be introduced in the continuous framework. In Ref. 26, Kurzweil introduced the notion of global strong stability (GSS) for the continuous nonlinear system (2) and established the Lyapunov stability theory, without requiring uniqueness of the solution trajectories of (2).

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Definition 2.1. (p. 69 in Ref. 26) The trivial solution x = 0 of (2) is said to be globally strongly stable (GSS) if there are two functions B : (0, +∞) → (0, +∞) and T : (0, +∞) × (0, +∞) → (0, +∞) with B being increasing and lims→0 B(s) = 0, such that ∀α > 0 and ∀ε > 0, for every solution x(t) of (2) defined on [0, t1 ), 0 < t1 ≤ +∞ with kx(0)k ≤ α, there is a solution z(t) of (2) defined on [0, +∞) satisfying 1. z(t) = x(t), t ∈ [0, t1 ); 2. kz(t)k ≤ B(α), ∀t ≥ 0; 3. kz(t)k < ε, ∀t ≥ T (α, ε). This definition is a natural extension of global asymptotic stability introduced by Lyapunov for the autonomous system (2) when it has a unique solution. Using the concept of GSS, Kurzweil 26 proved that the Lyapunov’s second theorem remains true as long as the vector field f (x) is continuous (no local Lipschitz continuity is required). Suppose there exists a C 1 positive definitea and properb function ∂V f (x) is negative definite. Then, the trivial solution V : IRn → IR, such that ∂x x = 0 of system (2) is globally strongly stable. Theorem 2.1.

This theorem has been shown of paramount importance in establishing various global strong stabilization results by continuous state feedback for nonlinear control systems that are not smoothly stabilizable.28,29 Interestingly, Theorem 2.1 is analogous to Lyapunov’s second theorem and recovers the case of the so-called global asymptotic stability when the solution of system (2) is unique. Similar to the work of Bhat-Bernstein,6 the concept of global strong stability for the continuous system (2) can also be extended to the case of finite-time stability. The following definition is a slight generalization of Definition 2.2 in Ref. 6. Definition 2.2. The system (2) is said to be globally finite-time stable at x = 0 if the following statements hold: (1) Globally strong stability: The system (2) is globally strongly stable at x = 0. (2) Finite-time convergence: there exists a function T : IR n → [0, +∞), called the settling time, such that for every x ∈ IRn , every solution φ(t, x) of (2) starting from φ(0) = x is defined on the interval [0, T (x)) and satisfies limt→T (x) φ(t, x) = 0. Based on the definition above, it is not difficult to prove the following Lyapunov’s theorem on globally finite-time stability by Theorem 2.1 and the comparison principle (cf. Theorem 4.2 of Ref. 6). aA

function V : IRn → IR is called positive definite if V (x) > 0, ∀x 6= 0 and V (0) = 0. nonnegative function V : IRn → IR is called proper if for each a > 0, the set V −1 ([0, a]) is compact in IRn . bA

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Theorem 2.2. Suppose there exists a C 1 positive definite, proper function V : IRn → IR such that ∂V V˙ + cV α := f (x) + cV α ≤ 0 ∂x

(3)

for some real numbers c > 0 and α ∈ (0, 1). Then, the system (2) is globally finite1 time stable at x = 0. Moreover, if T is the settling time, then T (x) ≤ c(1−α) V (x)1−α n for all x ∈ IR . In the case of global strong stability, the Kurzweil’s stability theorem requires only Lf V (x) < 0, ∀x 6= 0. On the contrary, Theorem 2.2 on global finite-time stability requires a stronger condition, i.e. V˙ ≤ −cV α , ∀x 6= 0. c For this reason, global finite-time stabilization is a more difficult problem than global asymptotic stabilization. Indeed, according to Theorem 2.2, in order to achieve finite-time stabilization, one must construct not only a non-Lipschitz continuous state feedback control law (because finite-time convergence is not possible in the case of either smooth or Lipschitz-continuous dynamics), but also a control Lyapunov function V (x) satisfying the inequality (3). The latter is of course not easy to achieve and makes the construction of a control Lyapunov function much more subtle. In summary, although Theorem 2.2 has provided a basic tool for testing global finite-time stability of nonlinear systems, how to employ it to design globally finitetime stabilizing controllers for nonlinear systems with uncontrollable unstable linearization such as (1) is an interesting problem that will be investigated in this paper. We conclude this section with some useful lemmas to be used repeatedly in the sequel. Lemma 2.1. For x, y, p ∈ IR with p > 0, the following inequality holds: (|x| + |y|)p ≤ max(2p−1 , 1)(|x|p + |y|p ). As a consequence, when p =

c r

with c ≥ 1 and r ≥ 1 being odd integers,

|xp − y p | ≥ 21−p |x − y|p , p

p

|x − y | ≤ 2 Lemma 2.2. holds:

(4)

1−p

p

|x − y| ,

if p ≥ 1,

(5)

if p ≤ 1.

(6)

Given positive real numbers x, y, m, n, a, b, the following inequality axm y n ≤ bxm+n +

n m + n − m m+n − m m+n ( ) na n b ny . m+n m

(7)

The proofs are straightforward and can be found in Section 2 of Ref. 29. c In

Ref. 6, it has been shown that the sufficient condition (3) is also necessary for a continuous autonomous system to be finite-time stable, under the condition that the settling-time function T (x) is continuous at the origin.

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3. Global Finite-Time Stabilization by H¨ older Continuous Feedback Using the Lyapunov theory on finite-time stability introduced in the previous section, we can prove the following global finite-time stabilization result for the nonlinear system (1). Theorem 3.1. For a chain of odd power integrators perturbed by a C 1 lowertriangular vector field (1), there is a state feedback control law u = u(x) with u(0) = 0, which is H¨ older continuous, such that the trivial solution x = 0 of (1) is globally finite-time stable. Remark 3.1. Since the function fi (·) in (1) is C 1 and vanishes at the origin, by the Taylor theorem there is a smooth function γi (x1 , · · · , xi ) ≥ 0 satisfying |fi (x1 , · · · , xi )| ≤ (|x1 | + · · · + |xi |)γi (·),

i = 1, · · · , n.

(8)

This property will be used in the proof of Theorem 3.1 frequently.

Proof. The proof is based on the tool of adding a power integrator 28,29 with a suitable twist. In particular, by choosing an appropriate dilation and homogeneous degree,15,24 we simultaneously construct a C 1 control Lyapunov function satisfying the Lyapunov inequality V˙ + cV α ≤ 0, as well as a H¨ older continuous finite-time stabilizer. For the convenience of the reader, we break up the proof in three steps. Initial Step. Choose the C 1 Lyapunov function V1 (x1 ) = (8), we have

x21 2 .

Using (1) and

V˙ 1 (x1 ) = x1 (xp21 − x∗2 p1 ) + x1 x∗2 p1 + x1 f1 (x1 ) ≤ x1 (xp21 − x∗2 p1 ) + x1 x∗2 p1 + x21 γ1 (x1 ) ≤ x1 (xp21 − x∗2 p1 ) + x1 x∗2 p1 + x2−d γˆ1 (x1 ), 1 where d is a rational number whose denominator is an even integer while whose numerator is an odd integer, such that 1 1 d ∈ 0, := 0, ⊂ (0, 1) (9) Rn pn−1 · · · p1 + · · · + p2 p1 + p1 + 1 and γˆ1 (x1 ) ≥ xd1 γ1 (x1 ) with γˆ1 (x1 ) being a positive smooth function. It is of interest to note that such a rational number d always exists. In fact, one can simply pick d = 2Rn2 +1 . Then, the virtual controller x∗2 defined by x∗2 p1 = −x1−d (n + γˆ1 (x1 )), 1 or, equivalently, (1−d)/p1

x∗2 = −x1

(n + γˆ1 (x1 ))1/p1 := −xq12 β1 (x1 ),

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which is H¨ older continuous, yields V˙ 1 (x1 ) ≤ −nx2−d + x1 (xp21 − x∗2 p1 ), 1 where β1 (x1 ) = (n + γˆ1 (x1 ))1/p1 > 0 is a smooth function. Inductive Step. Suppose at step k − 1, there are a C 1 Lyapunov function Vk−1 (x1 , · · · , xk−1 ), which is positive definite and proper, a set of parameters q1 , · · · , qk , and H¨ older continuous virtual controllers x∗1 , · · · , x∗k , defined by q1 = 1, , q2 = q1p−d 1 .. . −d qk = qk−1 pk−1 ,

x∗1 = 0, x∗2 = −ξ1q2 β1 (x1 ), .. . qk ∗ xk = −ξk−1 βk−1 (x1 , · · · , xk−1 ),

1/q

ξ1 = x1 1 − x∗1 1/q1 = x1 , 1/q ξ2 = x2 2 − x∗2 1/q2 , (10) .. . 1/q ξk = xk k − x∗k 1/qk ,

with β1 (x1 ) > 0, · · · , βk−1 (x1 , · · · , xk−1 ) > 0, being smooth, such that 2 Vk−1 (x1 , · · · , xk−1 ) ≤ 2(ξ12 + · · · + ξk−1 ),

V˙ k−1 (x1 , · · · , xk−1 ) ≤ −(n − k + 2)(

k−1 X

(11) p

ξi2−d ) + ξk−1k−1 (xkk−1 − x∗k pk−1 ). (12) 2−q

i=1

From (9) and (10), it is easy to see that 1 = q1 > q2 > · · · > qk > 0. In addition, using (4) and (10) yields |xk | ≤ |ξk |qk + |ξk−1 |qk βk−1 (x1 , · · · , xk−1 ),

(13)

p |xkk−1 | ≤ (|ξk |qk pk−1 + |ξk−1 |qk pk−1 )β¯k−1 (x1 , · · · , xk−1 ),

(14)

or, equivalently, where β¯k−1 (x1 , · · · , xk−1 ) ≥ 0 is a smooth function. Now, we claim that (11) and (12) also hold at step k. To prove the claim, consider

with

Vk (x1 , · · · , xk ) = Vk−1 (x1 , · · · , xk−1 ) + Wk (x1 , · · · , xk ) Z xk Wk (· · · ) = (s1/qk − x∗k 1/qk )2−qk ds.

(15) (16)

x∗ k

The Lyapunov function Vk (x1 , · · · , xk ) thus defined has several important properties listed in the following two propositions, whose proofs are included in the appendix. Proposition 3.1. Wk (x1 , · · · , xk ) is C 1 . Moreover, ∂Wk = ξk2−qk , ∂xk Z ∂Wk ∂(x∗k 1/qk ) xk 1/qk = −(2 − qk ) (s − x∗k 1/qk )1−qk ds, ∂xl ∂xl x∗ k

l = 1, · · · , k − 1.

Proposition 3.2. Vk (x1 , · · · , xk ) is C 1 , positive definite and proper, and satisfies Vk (x1 , · · · , xk ) ≤ 2(ξ12 + · · · + ξk2 ).

(17)

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With the aid of Propositions 3.1 and 3.2, one can deduce from (12) that k−1

X ∂Wk ∂Wk V˙ k (·) = V˙ k−1 (·) + x˙ k + x˙ l ∂xk ∂xl l=1

≤ −(n − k + 2)(

k−1 X

p

k k ξi2−d ) + ξk−1k−1 (xkk−1 − x∗k pk−1 ) + ξk2−qk (xpk+1 − x∗p k+1 )

2−q

i=1

2−qk k +ξk2−qk x∗p fk (x1 , · · · , xk ) + k+1 + ξk

k−1 X l=1

∂Wk x˙ l . ∂xl

(18)

Next, we introduce additional two propositions which are useful when estimating the last two terms in inequality (18). Proposition 3.3. There are non-negative smooth functions γ˜k (x1 , · · · , xk ) and γ¯k (x1 , · · · , xk ) such that |fk (x1 , · · · , xk )| ≤ (|ξ1 |qk + · · · + |ξk |qk )˜ γk (x1 , · · · , xk ) |x˙ k | ≤ (|ξ1 |

qk −d

+ · · · + |ξk |

qk −d

)¯ γk (x1 , · · · , xk ).

(19) (20)

Proposition 3.4. There is a non-negative smooth function Ck,l (x1 , · · · , xk ) such that ∂(x∗ 1/qk ) k ≤ (|ξk−1 |1−ql +· · ·+|ξl−1 |1−ql )Ck,l (x1 , · · · , xk ), l = 1, · · · , k−1. (21) ∂xl The proofs of Propositions 3.3 and 3.4 are given in the appendix. Using Proposition 3.3 and Lemma 2.2, one has

|ξk2−qk fk (x1 , · · · , xk )| ≤ |ξk |2−qk (|ξ1 |qk −d + · · · + |ξk |qk −d )¯ γk (x1 , · · · , xk ) ≤

2−d ξ12−d + · · · + ξk−1 + ξk2−d ρ˜k (x1 , · · · , xk ), 3

(22)

where γ¯k (x1 , · · · , xk ) ≥ (ξ1d + · · · + ξkd )˜ γ (x1 , · · · , xk ) and ρ˜k (x1 , · · · , xk ) are nonnegative smooth functions. Using Lemmas 2.1 – 2.2 and observing that pk−1 = (qk−1 − d)/qk , we obtain p

|ξk−1k−1 (xkk−1 − x∗k pk−1 )| ≤ 2|ξk−1 |2−qk−1 |ξk |qk−1 −d ≤ 2−q

2−d ξk−1 + ck ξk2−d , 3

(23)

for a suitable ck > 0. To estimate the last term in (18), we observe from Proposition 3.4 that for l = 1, · · · , k − 1, ∂(x∗ 1/qk ) Z xk ∂Wk 1/q 1/qk ∗ k k 1−qk = (2 − qk ) (s − x ) ds k ∂xl ∂xl x∗ k

≤ (2 − qk )(|ξk−1 |1−ql + · · · + |ξl−1 |1−ql )Ck,l (·)|x∗k − xk ||ξk |1−qk ≤ 2(2 − qk )(|ξk−1 |1−ql + · · · + |ξl−1 |1−ql )Ck,l (·)|ξk |.

(24)

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Then, combining (24) and (20) results in k−1 k−1 l+1 X ∂W k−1 X X X k 1−ql x ˙ ≤ 2(2 − q )( |ξ | )C (·)|ξ |( |ξj |ql −d )¯ γk (·) l k j k,l k ∂xl j=1 l=1

l=1

j=l−1

ξ12−d

2−d + · · · + ξk−1 + ξk2−d ρ¯k (x1 , · · · , xk ), (25) 3 where ρ¯k (x1 , · · · , xk ) > 0 is a smooth function. The last inequality follows from Lemma 2.2. Substituting the estimates (22), (23) and (25) into (18), we arrive at

≤

V˙ k ≤ −(n − k + 1)(

k−1 X

∗pk 2−d k k ξi2−d ) + ξk2−qk [(xpk+1 − x∗p (ck + ρ˜k (·) + ρ¯k (·)). k+1 ) + xk+1 ] + ξk

i=1

From (9), it follows that qk − d > 0. Then, it is clear that the C 0 virtual controller qk −d k x∗p (n − k + 1 + ck + ρ˜k (·) + ρ¯k (·)) k+1 (x1 , · · · , xk ) = −ξk

(26)

or, equivalently, (qk −d)/pk

x∗k+1 = −ξk

q

(n−k +1+ck + ρ˜k (·)+ ρ¯k (·))1/pk := −ξkk+1 βk (x1 , · · · , xk ) (27)

with βk (·) := (n − k + 1 + ck + ρ˜k (·) + ρ¯k (·))1/pk > 0 being smooth and qk+1 := qk −d pk < qk being positive, is such that k k V˙ k ≤ −(n − k + 1)(ξ12−d + · · · + ξk2−d ) + ξk2−qk (xpk+1 − x∗p k+1 ).

This completes the inductive proof. Last Step. According to the inductive argument above, we conclude that at the n-th step, there is a H¨ older continuous state feedback control law u = x∗n+1 = −ξnqn+1 βn (x1 , · · · , xn )

(28)

with βn (·) > 0 being smooth and ξn defined by (10), and a C 1 positive definite and proper Lyapunov function Vn (x1 , · · · , xn ) of the form (15), such that Vn (x1 , · · · , xn ) ≤ 2(ξ12 + · · · + ξn2 ), V˙ n (x1 , · · · , xn ) ≤ −(ξ12−d + · · · + ξn2−d ). Finally, pick α =

2−d 2

(29) (30)

∈ (0, 1). Then, it is straightforward to show that

n n 1 1 X 2−d 1 X 1/q V˙ n + Vnα ≤ − ( ξi ) = − [ (xi i − x∗i 1/qi )2−d ], 4 2 i=1 2 i=1

(31)

which is negative definite. By Theorem 2.2, it is concluded from (31) that the trivial solution x = 0 of the closed-loop system (1)–(28) is globally finite-time stable. Remark 3.2. If we choose d = 0 in the proof of Theorem 3.1, it is deduced from (10) that q1 = 1, q2 = p11 , · · · , qn = p1 ···p1 n−1 . Hence, our proof is degenerated

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to the proof of Theorem 3.1 in Ref. 29 when d = 0. However, as indicated in Ref. 29, the choice of d = 0 results in only an asymptotic stabilizer rather than a finite-time stabilizer because only negative definiteness of V˙ n can be guaranteed, not V˙ n + 41 Vnα ≤ 0. Remark 3.3. Theorem 2 of Ref. 4 states that for a homogeneous system, finitetime stability is equivalent to asymptotic stability plus a negative homogeneous degree. For the non-homogeneous system (1), −d < 0 in the proof above can be viewed as a degree-like parameter and q1 , · · · , qn+1 > 0 are dilation-like parameters, which are determined by (10). Therefore, a crucial point in proving Theorem 3.1 is to determine the degree-like parameter −d, which must be negative in order to achieve finite-time stability. Once the assignments of d and q1 , · · · , qn+1 are completed, one can easily use the adding a power integrator technique to construct a non-Lipschitz continuous, finite-time stabilizer, as illustrated in the proof of Theorem 3.1. In the rest of this section, we use the two-dimensional example in Ref. 29 to demonstrate how a global finite-time stabilizer can be explicitly constructed, by suitably choosing the degree −d and dilation q1 , q2 and by employing the adding a power integrator technique. Example 3.1. Consider the planar system x˙ 1 = x32 + x1 ex1 x˙ 2 = u.

(32)

Clearly, the Jacobian linearization contains an uncontrollable mode associated with a positive eigenvalue. As shown in Ref. 29, the planar system (32) is not smoothly stabilizable but can be globally asymptotically stabilized by non-Lipschitz continuous state feedback. By Theorem 3.1, we now know that (32) is also globally finite-time stabilizable. To find a finite-time stabilizer, one can choose (as done in the proof of Theorem 3.1) 7 2 R2 = 1 + p1 = 4, d = > 0, q1 = 1, q2 = 9 27 and let V1 (x1 ) = that

x21 2 .

2

1

With the relation x19 ≤ (1 + x21 ) 9 in mind, it is easy to see 16

1 V˙ 1 (x1 ) ≤ x1 (x32 − x∗2 3 ) + x1 x∗2 3 + x21 ex1 ≤ x1 (x32 − x∗2 3 ) + x1 x∗2 3 + x19 ex1 (1 + x21 ) 9 .

h i 13 7 1 Hence, the virtual controller x∗2 := −x127 2 + ex1 (1 + x21 ) 9 renders 16

V˙ 1 (x1 ) ≤ −2x19 + x1 (x32 − x∗2 3 ).

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Denote ξ2 = x27 − x∗2 punov function

27 7

and construct a C 1 , positive definite and proper Lya-

V2 (x1 , x2 ) =

x21 + 2

Z

x2 x∗ 2

27

(s 7 − x∗2

27 7

47

) 27 ds.

Then, the time derivative of V2 (·) along the trajectories of (32) satisfies h i 16 47 7 1 7 7 V˙ 2 ≤ −2x19 + 1.2|x1 ||ξ2 | 9 + ξ227 u + 3C1 (x1 )|ξ2 |(|ξ2 | 9 + |x1 | 9 2 + 2ex1 (1 + x21 ) 9 ) 16 47 16 16 ≤ −x19 + ξ29 0.8 + 3C1 (x1 ) + 0.6C˜1 (x1 ) 9 + ξ227 u, where h

C1 (x1 ) = 2 + ex1 (1 + x21 )

1 9

i 97

h

+ 2 + ex1 (1 + x21 )

1 9

i 27

ex1 (1 + x21 )

10 9

and

27 ∂ x∗ 7 2 ≥ ∂x1

1 C˜1 (x1 ) = 6[1 + ex1 (1 + x21 ) 9 ]C1 (x1 ).

Clearly, the H¨ older continuous controller 1 16 u = −ξ227 1.8 + 3C1 (x1 ) + 0.6C˜1 (x1 ) 9

(33)

yields 16

16

V˙ 2 ≤ −x19 − ξ29 . Thus, 16 1 8 1 16 V˙ 2 + V29 ≤ − (x19 + ξ29 ), 4 2

(34)

which is negative definite. By Theorem 2.2, it follows from (34) that the trivial solution x = 0 of the closed-loop system (32)–(33) is globally finite-time stable. Remark 3.4. According to Theorem 3.1, it is concluded that the fourdimensional, underactuated unstable two degree of freedom mechanical system 32,28 x˙ 1 = x2 ,

x˙ 2 = x33 +

g sin x1 , l

x˙ 3 = x4 ,

x˙ 4 = u

is also globally finite-time stabilizable by nonsmooth state feedback. A globally finite-time stabilizing, nonsmooth state feedback controller can be explicitly designed, in a fashion similar to Example 3.1.

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4. Further Extension and Discussion In this section, we discuss how the global finite-time stabilization result derived so far can be extended to a more general class of nonlinear systems in the Hessenberg form 8 x˙ 1 = d1 (t)xp21 + f1 (t, x1 , x2 ) .. . x˙ n−1 = dn−1 (t)xpnn−1 + fn−1 (t, x1 , · · · , xn ) x˙ n = dn (t)upn + fn (t, x1 , · · · , xn , u)

(35)

where x = (x1 , · · · , xn )T ∈ IRn and u ∈ IR are the system state and input, respectively. For i = 1, · · · , n, pi is an odd positive integer, fi : IR × IRi+1 → IR, is a C 0 function with fi (t, 0, · · · , 0) = 0, ∀t ∈ IR, and di (t) is a C 0 function of time t, which represents an unknown time varying parameter. The following assumptions characterize a subclass of nonlinear systems (35). Assumption 4.1. For i = 1, · · · , n, there are positive real numbers λ and µ such that 0 < λ ≤ di (t) ≤ µ,

i = 1, · · · , n.

Assumption 4.2. For i = 1, · · · , n, fi (t, x1 , · · · , xi , xi+1 ) =

pX i −1

xji+1 ai,j (t, x1 , · · · , xi ),

(36)

j=0

where xn+1 = u. Moreover, there is a smooth function γi,j (x1 , · · · , xi ) ≥ 0 such that |ai,j (t, x1 , · · · , xi )| ≤ (|x1 | + · · · + |xi |)γi,j (x1 , · · · , xi ),

for j = 0, · · · , pi − 1. (37)

Remark 4.1. When ai,j (t, x1 , · · · , xi ) ≡ ai,j (x1 , · · · , xi ), if the function ai,j (x1 , · · · , xi ) is C 1 and ai,j (0, · · · , 0) = 0, there always exists a smooth function γi,j (x1 , · · · , xi ) ≥ 0 satisfying the property (37), as explained in Remark 3.1. The following result on global finite-time stabilizability is an extension of Theorem 3.1. Theorem 4.1. For a family of uncertain nonlinear systems (35) satisfying Assumption 4.1–4.2, there is a H¨ older continuous controller u = u(x) with u(0) = 0, which renders the trivial solution x = 0 of (35) globally finite-time stable. Proof. The proof can be done in the spirit similar to that of Theorem 3.1, with a more delicate estimation.

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To be precise, choose V1 (x1 ) = 21 . By Assumption 4.2, |f1 (t, x1 , x2 )| ≤ Pp1 −1 j j=0 |x2 | |x1 |γ1,j (x1 ). From Lemma 2.2, it follows that for any N > 0, p1 j p1 − j p j−j 1+ j p1 −j (x1 ) |x2 |p1 + N 1 |x1 | p1 −j γ1,j N p1 p1 j ≤ |x2 |p1 + |x1 |ˆ γ1,j (x1 ), j = 1, · · · , p1 − 1, N p1

|x2 |j |x1 |γ1,j (x1 ) ≤

j

j

p1

2 N p1 −j (1+x21 ) p1 −j (1+γ1,j (x1 )) p1 −j > 0 is a smooth function. where γˆ1,j (x1 ) := p1p−j 1 Then, picking N = 2j/λ yields

λ|x2 |p1 |f1 (t, x1 , x2 )| ≤ + |x1 |γ1 (x1 ), 2

γ1 (x1 ) :=

pX 1 −1

γˆ1,j (x1 ).

(38)

j=0

Therefore, it follows from (38) that V˙ 1 (x1 ) = x1 [d1 (t)xp21 + f1 (t, x1 , x2 )]

λ p1 ∗p1 2 1 |x1 |(|x∗p 2 | + |x2 | − |x2 | ) + x1 γ1 (x1 ) 2 λ λ 2−d 1 1 ρ¯1 (x1 ) + (µ + )|x1 ||xp21 − x∗p (39) + |x1 ||x∗p 2 | + x1 2 |, 2 2

1 1 ≤ x1 d1 (t)[x∗p + (xp21 − x∗p 2 2 )] + 1 ≤ d1 (t)x1 x∗p 2

where ρ¯1 (x1 ) ≥ xd1 γ1 (x1 ) is a non-negative smooth function, and d = 2Rn2 +1 — the same parameter used in the proof of Theorem 3.1. Then, the continuous virtual controller x∗2 defined by 1/p1 2(n + ρ¯1 (x1 )) ¯1 (x1 )) (1−d)/p1 2(n + ρ ∗ x∗2 p1 = −x1−d ⇐⇒ x := −xq12 β1 (x1 ) = −x 2 1 1 λ λ (40) 1 1 renders the term d1 (t)x1 x∗p ≤ −2x2−d (n + ρ¯1 (x1 )) < 0. In view of λ2 |x1 ||x∗p 2 | ≤ 2 1 x2−d (n + ρ¯1 (x1 )), it is not difficult to verify that 1 λ 1 V˙ 1 (x1 ) ≤ −nx2−d + (µ + )|x1 ||xp21 − x∗p 1 2 |. 2

(41)

Using an inductive argument, one can prove that a similar conclusion holds at the n-th step. That is, there are a C 1 positive definite and proper Lyapunov functions Vn : IRn → IR, a set of parameters 1 = q1 > q2 > · · · > qk > 0, and C 0 virtual controllers x∗1 , · · · , x∗n defined by q1 = 1, , q2 = q1p−d 1 .. . −d qn = qn−1 pn−1 , qn+1 = qnp−d , n

1/q

x∗1 = 0, ξ1 = x1 1 − x∗1 1/q1 = x1 , 1/q q x∗2 = −ξ12 β1 (x1 ), ξ2 = x2 2 − x∗2 1/q2 , .. .. . . 1/q qn ∗ xn = −ξn−1 βn−1 (x1 , · · · , xn−1 ), ξn = xn n − x∗n 1/qn , q u = x∗n+1 = −ξnn+1 βn (x1 , · · · , xn ) (42)

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with β1 (x1 ) > 0, · · · , βn−1 (x1 , · · · , xn−1 ) > 0, being smooth, such that Vn (x1 , · · · , xn ) ≤ 2(ξ12 + · · · + ξn2 ), V˙ n (x1 , · · · , xn ) ≤ −(ξ12−d + · · · + ξn2−d ).

(43) (44)

Similarly, let α := (2 − d)/2 ∈ (0, 1). A direct calculation shows that 1 1 (45) V˙ n + Vnα ≤ − (ξ12−d + · · · + ξn2−d ) ≤ 0. 4 2 By Theorem 2.2, the trivial solution x = 0 of the closed-loop system (35)–(42) is finite-time stable. It is clear from the proof above that Theorem 4.1 remains true when di (t) and ai,j (t, x1 , · · · , xi ) are replaced by di (t, x, u) and ai,j (t, x, u), respectively, for i = 1, · · · , n, as long as Assumptions 4.1–4.2 are fulfilled. This observation leads to the following extension. Corollary 4.1. Consider the uncertain nonlinear system i x˙ i = di (t, x, u)xpi+1 +

pX i −1

xji+1 ai,j (t, x, u),

i = 1, · · · , n

with

u = xn+1 . (46)

j=0

Suppose there is a smooth function γi,j (x1 , · · · , xi ) ≥ 0, such that |ai,j (t, x, u)| ≤ (|x1 | + · · · + |xi |)γi,j (x1 , · · · , xi ),

0 ≤ j ≤ pi − 1,

1 ≤ i ≤ n.

Moreover, there exist smooth functions λi (x1 , · · · , xi ) and µi (x1 , · · · , xi+1 ) satisfying 0 < λi (x1 , · · · , xi ) ≤ di (t, x, u) ≤ µi (x1 , · · · , xi , xi+1 ), i = 1, · · · , n. Then, system (46) is globally finite-time stabilizable by H¨ older continuous state feedback. The effectiveness of Theorem 4.1 can be illustrated by the following example. Example 4.1. Consider an affine system of the form x˙ 1 = xp2 + xp−2 ap−2 (x1 ) + · · · + x2 a1 (x1 ) + a0 (x1 ) 2 x˙ 2 = v,

(47)

where a0 (x1 ), a1 (x1 ), · · · , ap−2 (x1 ) are smooth functions, with a0 (0) = a1 (0) = · · · = ap−2 (0) = 0, and p ≥ 1 is an integer. It is of interest to note that (47) is representative of a class of two-dimensional affine systems. In fact, Jakubczyk and Respondek 21 proved that every smooth affine system in the plane, i.e., ξ˙ = f (ξ) + g(ξ)u, is feedback equivalent to (47) if g(0) and adpf g(0) are linearly independent. Some more general results were obtained in Refs. 8 and 30 later on, showing that (47) is indeed a special case of the so-called “p-normal form”.8,30 In other words, (47) is a normal form of two-dimensional affine systems when rank[g(0), adpf g(0)] = 2. In Ref. 28, it was proved that global asymptotic stabilization of (47) is possible by non-Lipschitz continuous state feedback, although there may not exist any smooth stabilizer for system (47).

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On the other hand, it is easy to verify that the planar system (47) satisfies Assumptions 4.1-4.2. By Theorem 4.1, global finite-time stabilization of (35) is achievable by H¨ older continuous state feedback as long as p is an odd integer. Moreover, a finite-time stabilizer can be designed in a manner similar to Example 3.1. For illustration, let us examine the planar system (47) with p = 3 and a1 (x1 ) = a0 (x1 ) = x1 , i.e., x˙ 1 = x32 + x1 x2 + x1 x˙ 2 = u,

(48)

which is not smoothly stabilizable. Moreover, the system (48) is neither in a triangular form nor homogeneous. Following the constructive proof of Theorem 4.1, one can design the continuous controller 1/27 u = −2ξ2 1365 + C(1 + x41 )(1 + ξ22 ) + c˜(x1 ) + 280˜ c(x1 )16/9 , (49) where C > 0 is a sufficiently large constant and 9 ∆ 27 ξ2 = x27 + x1 4 + 2(2 + x21 )2 7 n 9 2 o ∆ c˜(x1 ) = [84 + 45.5(2 + x21 )2 ] 4 + 2(2 + x21 )2 7 + 11x21 (2 + x21 ) 4 + 2(2 + x21 )2 7 . This controller renders the trivial solution (x1 , x2 ) = (0, 0) of the closed-loop system globally finite-time stable, which can be verified using the C 1 Lyapunov function Z x2 47 1 2 27 ∗ 27 27 V2 (x1 , x2 ) = x1 + ds ≤ 2(|x1 |2 + |ξ2 |2 ) s 7 − x2 7 2 x∗ 2 7 1 with x∗2 = −x127 4 + 2(2 + x21 )2 3 . Then, a direct calculation gives 16 16 1 8 V˙ 2 ≤ −x19 − ξ29 =⇒ V˙ 2 + V29 < 0, 4 which implies global finite-time stability.

∀(x1 , x2 ) 6= 0,

We conclude this section with an interesting generalization of the finite-time stabilization design method developed so far to cascade systems with zero dynamics. Consider a class of cascade systems described by equations of the form z˙ = F0 (t, z, x, u) x˙ 1 = d1 (t)xp21 + f1 (t, z, x1 , x2 ) .. . x˙ n = dn (t)upn + fn (t, z, x1 , · · · , xn , u), r

(50) n

where u ∈ IR is the system input, z ∈ IR and x = (x1 , · · · , xn )T ∈ IR are the system states, and pi , i = 1, · · · , n are positive odd integers. The functions F0 : IR × IRn+r+1 → IRr and fi : IR × IRi+1 → IR, i = 1, · · · , n are continuous functions

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satisfying F0 (t, 0, 0, 0) = 0 and fi (t, 0, · · · , 0) = 0, ∀t ∈ IR, and di (t) : IR → IR stand for time varying parameters and satisfy Assumption 4.1. To tackle the problem of finite-time global stabilization of (50) by continuous partial state feedback, we make the following assumptions. Assumption 4.3. There exist C 1 Lyapunov function U0 (z), which is positive definite and proper, and K∞ functions a ¯(·), a(·) such that ¯(||z||) a(||z||) ≤ U0 (z) ≤ a ∂U0 F0 (t, z, x, u) ≤ −c0 U0α0 (z) + β0 (x1 ), ∂z

(51)

where c0 > 0, α0 ∈ (0, 1) are constants, and β0 (·) is a C 2 positive definite function with β0 (0) = 0. Assumption 4.4. For i = 1, · · · , n, λ |fi (t, z, x1 , · · · , xi , xi+1 )| ≤ φ(z) + |xi+1 |pi + (|x1 | + · · · + |xi |)γi (x1 , · · · , xi ), (52) 2 where xn+1 = u, γi (·) ≥ 0 is a continuous function, and the function φ(z) > 0 satisfies φ2 (z) = O[aα0 (||z||)]

as

||z|| → 0.

(53)

Remark 4.2. Clearly, Assumption 4.3 can be regarded as a finite-time version of the input-to-state stability (ISS) condition. Moreover, when fi (·) =

pX i −1

xji+1 ai,j (z, x1 , · · · , xi )

j=0

1

with ai,j (·) being C and satisfying ai,j (0, 0, · · · , 0) = 0, the estimation (52), instead of (53), always holds by the variable separation lemma in Ref. 27. With the help of these two conditions, we can prove the following result. Theorem 4.2. Under Assumptions 4.1, 4.3 and 4.4, the cascade system (50) is globally finite-time stabilizable by continuous partial state feedback. Proof. The proof of this result is similar to that of Theorems 3.1 and 4.1. The only difference lies in the design of a partial-state other than full-state feedback controller for (50). In what follows, only a sketch of the proof is given to highlight the main difference. By changing the supply rate in Ref. 27, it follows from (51) and (53) that given any smooth function φ2 (z), there exists a smooth nondecreasing function η(·) with η(0) > 0 and a Lyapunov function Z U0 (z) V0 (z) = η(s)ds, 0

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which is C 1 , positive definite and proper, such that c0 c0 V˙ 0 ≤ − η(U0 (z))U0α0 (z)+βˆ0 (x1 ) ≤ − η(U0 (z))U0α0 (z)−rφ2 (||z||)+βˆ0 (x1 ), (54) 2 4 ∞ where βˆ0 (x1 ), with βˆ0 (0) = 0, is a C positive definite function. Then, by a similar estimation and inductive argument to the proof of Theorem 4.1, we can construct the Lyapunov functions Vi (z, x1 , · · · , xi ) = V0 (z) + W1 (x1 ) + · · · + Wi (x1 , · · · , xi ) where Wi (·) can be obtained from (16) and the virtual controllers x∗i are given in (42). Indeed, with an appropriate twist, this can be done from (52) and (54). Moreover, it is easy to verify that Vn satisfies Vn ≤ V0 (z) + 2(ξ12 + · · · + ξn2 ) c0 V˙ n ≤ − η(U0 (z))U0α0 (z) − (ξ12−d + · · · + ξn2−d ), 4 which guarantees the existence of c > 0 and α ∈ (0, 1) such that V˙ n ≤ −cVnα . 5. Conclusion We have studied the problem of global finite-time stabilization for a class of nonlinear systems. The systems under consideration usually involve an uncontrollable unstable Jacobian linearization (i.e., the uncontrollable modes have eigenvalues on the right half plane), and therefore are difficult to be controlled. Using the Lyapunov theory on finite-time stability,6 as well as the tool of adding a power integrator 28,29 with an appropriate modification (in particular, by subtly choosing a homogeneous degree −d and a family of dilations q1 , q2 , · · · , qn ), we have presented a systematic feedback design method for the explicit construction of globally finitetime stabilizing, H¨ older continuous state feedback control laws, as well as a C 1 control Lyapunov function that satisfies the Lyapunov finite-time stability inequality V˙ (x) ≤ −cV α (x), which is a stronger requirement than the traditional Lyapunov inequality V˙ < 0, ∀x 6= 0. While the results presented in this paper are theoretical in nature, it would be interesting to see some practical applications of the finite-time stabilization theory. This will certainly be an important subject for future investigations. We hope that this work would generate interest in the control community and make the control engineer be aware of the difficulty, subtlety and power of finite-time control strategies, and in turn stimulate research activities in the areas of finite-time nonlinear control theory and applications. Appendix A. Proofs of Key Propositions This appendix collects the proofs of Propositions 3.1—3.4 that are used in the proof of Theorem 3.1.

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Proof of Proposition 3.1: Note that 0 < qi < 1, i = 2, · · · , n. We can use an argument similar to the one in 28 to prove the smoothness of W (·) by calculating its partial derivatives. The reader is referred to the proof of Proposition B.1 in Ref. 28 for details. Proof of Proposition 3.2: In the case of x∗k ≤ xk , using (5) in Lemma 2.1 leads to Z xk 21−1/qk · qk Wk (·) ≥ (xk − x∗k )2/qk . (A.1) 21−1/qk (s − x∗k )(2−qk )/qk ds ≥ 2 x∗ k Similarly, it can be shown that (A.1) holds as well in the case of x∗k ≥ xk . Hence, Vk = Vk−1 + Wk ≥ Vk−1 + mk (xk − x∗k )2/qk ,

mk > 0.

This implies that Vk (·) is positive definite and proper. By (16) and Proposition 3.1, Vk (·) is a C 1 function. On the other hand, by Lemma 2.1 it is easy to obtain 1/qk

Wk (x1 , · · · , xk ) ≤ |xk − x∗k ||xk

− x∗k 1/qk |2−qk ≤ 21−qk |ξk |qk |ξk |2−qk ≤ 2|ξk |2 , (A.2)

which implies (17) immediately. Proof of Proposition 3.3: Keeping 0 < qk < qk−1 < · · · < q1 = 1 and (13) in mind, one deduces from (8) that |fk (x1 , · · · , xk )| ≤ (|x1 | + |x2 | + · · · + |xk |)γk (x1 , · · · , xk ) ≤ (|ξ1 |q1 + |ξ1 |q2 β1 (·) + |ξ2 |q2 + · · · + |ξk−1 |qk βk−1 (·) + |ξk |qk )γk (·) ≤ (|ξ1 |qk + · · · + |ξk |qk )˜ γk (x1 , · · · , xk ),

(A.3)

where γ˜k (x1 , · · · , xk ) ≥ 0 is a smooth function. Similarly, the inequalities (14) and (19) and the identity qk+1 pk = qk − d lead to k |x˙ k | ≤ |xpk+1 | + |fk (x1 , · · · , xk )|

≤ 2pk −1 (|ξk+1 |qk+1 pk + |ξk |qk+1 pk βk (·)pk ) + (|ξ1 |qk + · · · + |ξk |qk )˜ γk (x1 , · · · , xk ) ≤ (|ξ1 |qk −d + · · · + |ξk+1 |qk −d )¯ γk (x1 , · · · , xk ),

(A.4)

where γ¯k (x1 , · · · , xk ) ≥ 0 is a smooth function. Proof of Proposition 3.4: The proof can be done by using an inductive argument. Case k = 2: It is not difficult to verify that ∂(x∗ 1/q2 ) ∂[x β(x )] 1 1 2 ≤ ≤ C2,1 (x1 ). ∂x1 ∂x1

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Inductive Assumption: Assume at step k − 1, for l = 1, · · · , k − 2, there exist smooth functions Ck−1,l (·) ≥ 0 such that ∂(x∗1/qk−1 ) k−1 (A.5) ≤ (|ξk−2 |1−ql + · · · + |ξl−1 |1−ql )Ck−1,l (x1 , · · · , xk−1 ). ∂xl

Our goal is to prove that there are non-negative smooth functions Ck,l (·), l = 1, · · · , k − 1, such that ∂(x∗ 1/qk ) k ≤ (|ξk−1 |1−ql + · · · + |ξl−1 |1−ql )Ck,l (x1 , · · · , xk ). ∂xl

To begin with, consider the case when l = 1, · · · , k − 2. Let β˜k (·) := βk (·)1/qk . By (10), 1/qk−1 ∗1/qk−1 ∂(x∗ 1/qk ) ˜k−1 (·) ∂(x − x ) ∂ β k−1 k−1 k ≤ ξk−1 + β˜k−1 (·) ∂xl ∂xl ∂xl ∗1/q ∂(xk−1 k−1 ) ∂ β˜k−1 (·) ˜ = ξk−1 + β (·) k−1 , ∂xl ∂xl

which, together with (A.5), implies that ∂(x∗1/qk−1 ) ∂ β˜k−1 (·) ˜ k−1 ≤ ξk−1 + βk−1 (·)(|ξk−2 |1−ql + · · · + |ξl−1 |1−ql )Ck−1,l (·) ∂xl ∂xl

≤ (|ξk−1 |1−ql + · · · + |ξl−1 |1−ql )Ck,l (x1 , · · · , xk ), (A.6) ∂ β˜ (·) k−1 ˜ where Ck,l (x1 , · · · , xk ) ≥ + βk−1 (·)Ck−1,l (x1 , · · · , xk−1 ) is a non-negative ∂xl smooth function. Next, we shall prove that (A.6) also holds for l = k − 1. For, note that 1/q ∗1/q ∂(x∗ 1/qk ) ∂(xk−1k−1 − xk−1 k−1 ) ∂ β˜k−1 (·) ˜ k ≤ ξk−1 + βk−1 (·) ∂xk−1 ∂xk−1 ∂xk−1 ∗ q 1 −1 ∂ β˜k−1 (·) 1 ˜ k−1 = ξk−1 βk−1 (·)xk−1 + ∂xk−1 qk−1 1 ck ˜ ∂ β˜k−1 (·) qk−1 −1 βk−1 (·)(|ξk−1 |1−qk−1 + |ξk−2 |1−qk−1 β˜k−2 ≤ ξk−1 ) + ∂xk−1 qk−1 ≤ (|ξk−1 |1−ql + · · · + |ξl−1 |1−ql )Ck,k−1 (x1 , · · · , xk ),

(A.7)

where Ck,k−1 (x1 , · · · , xk ) ≥ 0 is a smooth function and ck > 0 is a constant. Putting (A.6) and (A.7) together completes the inductive proof. Thus, Proposition 3.4 is true.

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References 1. M. Athans and P. L. Falb, Optimal Control: An Introduction to Theory and Its Applications, New York: McGraw-Hill (1966). 2. A. Bacciotti and L. Rosier, Liapunov functions and stability in control theory, London: Springer (2001). 3. S. P. Bhat and D. S. Bernstein, Proc. American Control Conf., 1831 (1995). 4. S. P. Bhat and D. S. Bernstein, Proc. American Control Conf., 2513 (1997). 5. S. P. Bhat and D. S. Bernstein, IEEE Trans. Automat. Contr., 43, 678 (1998). 6. S. P. Bhat and D. S. Bernstein, SIAM J. Contr. Optim., 38, 751 (2000). 7. S. Celikovsky and E. Aranda-Bricaire, Syst. Contr. Lett., 36, 21 (1999). 8. D. Cheng and W. Lin, IEEE Trans. Aut. Contr., 48, 1242 (2003). 9. J. M. Coron and L. Praly, Syst. Contr. Lett., 17, 89 (1991). 10. W. P. Dayawansa, Proc. 2nd IFAC Symp. Nonlinear Control Systems Design, 1 (1992). 11. W. P. Dayawansa, C. F. Martin and G. Knowles, SIAM J. Control Optim., 28, 1321 (1990). 12. A. T. Fuller, Int. J. Contr., 3, 359 (1966). 13. W. Hahn, Stability of Motion, New York: Springer-Verlag (1967). 14. V. T. Haimo, SIAM J. Control Optim., 24, 760 1986. 15. H. Hermes, SIAM. Review, 33, 238 (1991). 16. H. Hermes, Syst. Contr. Lett., 12, 437 (1991). 17. H. Hermes, in: S. Elaydi (Ed.), Differential Equations Stability and Control, Lecture Notes in Applied Mathematics, Marcel Dekker, New York, 249 (1991). 18. R. M. Hirschorn, IEEE Trans. Automat. Contr., 46, 276 (2001). 19. Y. Hong, Syst. Contr. Lett., 46, 231 (2002). 20. X. Huang, W. Lin and B. Yang, Automatica, 41, 881 (2005). 21. B. Jakubczyk and W. Respondek, in M.A. Kaashoek et al., eds.: Robust Control of Linear Systems and Nonlinear Control, Boston, MA: Birkh¨ auser, 447 (1990). 22. M. Kawski, Systems Control Lett., 12, 169 (1989). 23. M. Kawski, Control Theory and Advanced Technology, 6, 497 (1990). 24. M. Kawski, Proc 3rd IFAC Symp. on Nonlinear Control Systems, 164 (1995). 25. H. Khalil, Nonlinear Systems, New York: Macmillan (1992). 26. J. Kurzweil, Amer. Math. Soc. Translations, 24, 19 (1956). 27. W. Lin and R. Pongvuthithum, IEEE Trans. Automat. Contr., 47, 1356 (2002). 28. C. Qian and W. Lin, IEEE Trans. Automat. Contr., 46, 1061 (2001). 29. C. Qian and W. Lin, Syst. Contr. Lett., 42, 185 (2001). 30. W. Respondek, Proc. 42nd IEEE CDC, 1574 (2003). 31. L. Rosier, Syst. Contr. Lett., 19, 467 (1992). 32. C. Rui, M. Reyhanoglu, I. Kolmanovsky, S. Cho and N.H. McClamroch, Proc. 36th IEEE Conf. Decision and Control, 3998, (1997). 33. E. P. Ryan, Int. J. Contr., 36, 549 (1979). 34. E. P. Ryan, Dynamics and Contr., 1, 83 (1991). 35. M. Tzamtzi and J. Tsinias, Syst. Contr. Lett., 38, 115 (1999). 36. B. Yang and W. Lin, Proc. 16th IFAC World Congress, paper code: Tu-M22-TO/4 (2005).

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THE ALGEBRAIC CRITERION FOR DELAY-INDEPENDENT STABILITY OF LINEAR SYSTEMS∗

XIN YU and CHAO XU Laboratory of Information and Optimization Technologies, Ningbo Institute of Technology, Zhejiang University, Ningbo, 315104, CHINA E-mail: [email protected] KANGSHENG LIU Department of Mathematics, Zhejiang University, Hangzhou, 310027, CHINA E-mail: [email protected]

This paper studies the stability of the time-delay systems. We first give a necessary and sufficient algebraic criterion for delay-independent stability of the time-delay systems. In the following, we will discuss the robust stability of time-delay systems. A sufficient condition for robust stability of time-delay systems will be given. Keywords: delay-independent stability, robust stability, algebraic criterion.

1. Introduction Consider the following linear time-delay system of retarded type with commensurate delays: n−1 m

XX dn dk + y(t − jh) = 0. a kj dtn dtk j=0 k=0

It is well known that the system is asymptotically stable if and only if its characteristic function P (s, e−hs ) satisfies the condition P (s, e−hs ) 6= 0,

<s ≥ 0,

(1)

where the characteristic function can be written as ∗ Supported

partially by the National Natural Science Foundations (no. 10501039,10271111) and Ningbo Doctorial Fundation (no. 2005a610005). 241

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P (s, e−hs ) = sn +

n−1 m XX

akj sk e−jhs .

(2)

k=0 j=0

The characteristic function is a quasi-polynomial which include both powers of the independent variable and exponential functions. There are extensive studies on the stability problems for time-delay systems(see Refs. 2, 4 and 8). A specific notion concerning stability of time-delay systems is asymptotical stability independent of delay. We say the quasi-polynomial (2) is asymptotically stable independent of delay if P (s, e−hs ) 6= 0,

<s ≥ 0, all rell numbers h ≥ 0.

This notion was formally introduced by Kamen in Refs. 11, 12, and has received considerable research interest in the past two decades. There are some paper discuss the asymptotical stability independent of delay (see Refs. 8, 11 and 16). But in contrast to polynomials, we have few coefficient criteria of stability for quasipolynomial. On the other hand the problem of robust stability of dynamical system has been intensively studied since the publication of Kharitonov’s celebrated theorem 13 in 1978. This theorem asserts that a interval polynomial family will be Hurwitz stable if and only if four special extreme polynomial will be Hurwitz stable. An important extension of Kharitonov’s theorem is the edge theorem discovered by Bartlett, Hollot and Huang. 1 The edge theorem states that the stability of a polytope of polynomials can be guaranteed by the stability of its one-dimensional exposed edge polynomials. Since the edge theorem was extended to time-delay systems by Fu et al., 5 many researchers focus on the problem of robust stability of time-delay systems. 3,6,14,15 But up to now, there are no coefficient criteria of robust stability for quasi-polynomial families. In this paper, we first give the necessary and sufficient algebraic criterion for delay-independent stability of a general time-delay system. According this criterion, to judge whether or not a quasi-polynomial is asymptotically stable independent of delay, we only need check the coefficients of (2). In the following, we will discuss the robust stability of (2). A sufficient condition for robust stability of time delay system will be obtained. 2. Algebraic Criterion for the Stability To the function P (s, e−hs ), we can associate a polynomial P (s, z) in two independent complex variables s and z given by

P (s, z) = sn +

n−1 m XX k=0 j=0

akj sk z j .

(3)

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Lemma 2.1. Given the polynomial P (s, z) defined by (3), the following conditions are equivalent. (i) (ii) (iii) (iv)

P (s, e−hs ) 6= 0, <s ≥ 0, all rell numbers h ≥ 0. P (s, z) 6= 0, <s ≥ 0, s 6= 0, |z| ≤ 1 and P (0, 1) 6= 0. P (s, z) 6= 0, <s = 0, s 6= 0, |z| = 1 and P (s, 1) 6= 0, <s ≥ 0; P (s, eiθ ) 6= 0, <s = 0, s 6= 0, θ ∈ (0, 2π) and P (s, 1) 6= 0, <s ≥ 0.

Proof. The implications (i) ⇔ (ii) ⇔ (iii) follow from Refs. 9 and 10. The implication (iii) ⇔ (iv) is obvious. Note that ix − 1 iθ |x ∈ lR . {e |θ ∈ (0, 2π)} = ix + 1 It is easy to see that P (s, eiθ ) 6= 0, <s = 0, s 6= 0, θ ∈ (0, 2π) ix − 1 ⇐⇒ P (iy, ) 6= 0, x ∈ lR, y ∈ lR, y 6= 0 ix + 1 ix − 1 ⇐⇒ (ix + 1)m P (iy, ) 6= 0, x ∈ lR, y ∈ lR, y = 6 0. ix + 1 Let R[x, y] and R[y] denote the space of all polynomials of two variables and one variable, respectively, and let (ix + 1)m P (iy,

ix − 1 ) = f (x, y) + ig(x, y), ix + 1

where f (x, y) = a0 (y)xl + a1 (y)xl−1 + · · · + al (y) ∈ R[x, y], g(x, y) = b0 (y)xp + b1 (y)xp−1 + · · · + bp (y) ∈ R[x, y], and ak (y), bj (y) ∈ R[y], k = 0, 1, · · · , l, j = 0, 1, · · · , p. Define a0 (y) a1 (y) a2 (y) · · · al (y) a0 (y) a1 (y) a2 (y) · · · ··· a0 (y) a1 (y) a2 (y) R(f, g)(y) = b0 (y) b1 (y) b2 (y) · · · bp (y) b0 (y) b1 (y) b2 (y) · · · ··· b0 (y) b1 (y) b2 (y)

al (y) ··· bp (y) ···

al (y) bp (y)

(4)

Let y0 ∈ lR. From the basic knowledge of algebra, we see R(f, g)(y0 ) = 0 if and only if either a0 (y0 ) = b0 (y0 ) = 0 or f (x, y0 ) and g(x, y0 ) have common divisor h(x) ∈ R[x] with ∂h(x) > 0.

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Theorem 2.1. The quasi-polynomial (2) is asymptotically stable independent of delay if and only if (i) P (s, 1) 6= 0, <s ≥ 0; (ii) R(f, g)(y) 6= 0, ∀y ∈ lR, y 6= 0. Proof. Sufficiency: According to Lemma 2.1 and above discussions, we only need prove that f (x, y) 6= 0, ∀x, y ∈ lR, y 6= 0. g(x, y) 6= 0, If not, there exist x0 , y0 ∈ lR, y0 6= 0 such that f (x0 , y0 ) = 0, g(x0 , y0 ) = 0. Thus, (x − x0 )|f (x, y0 ), (x − x0 )|g(x, y0 ), which implies that R(f, g)(y0 ) = 0. This will contradict (ii). Necessity: Obviously, (i) holds. In the following we will show (ii). If not, there exist y0 ∈ lR such that R(f, g)(y0 ) = 0. Thus we have either a0 (y0 ) = b0 (y0 ) = 0 or f (x, y0 ) and g(x, y0 ) have common divisor h(x) ∈ R[x] with ∂h(x) > 0. Noting that the condition (i) implies P (iy0 , 1) 6= 0, we have ix − 1 ) ix + 1 n−1 m XX = (ix + 1)m (iy0 )n + akj (iy)k (ix − 1)j (ix + 1)m−j (ix + 1)m P (iy0 ,



k=0 j=0

= im (iy0 )n + m

n−1 m XX k=0 j=0

m



akj (iy0 )k  xm + c1 (y0 )xm−1 + · · · + cm (y0 )

= i P (iy0 , 1)x + c1 (y0 )xm−1 + · · · + cm (y0 ) = im P (iy0 , 1) xm + i−m P (iy0 , 1)−1 [c1 (y0 )xm−1 + · · · + cm (y0 )] ,

where ci (y) ∈ R[y], i = 1, 2, · · · , m. Since

P (iy0 , 1) 6= 0 ⇐⇒

or b0 (y0 ) 6= 0.

Thus there exists h(x) ∈ R[x] with ∂h(x) > 0 such that h(x)|f (x, y0 ) and h(x)|g(x, y0 ) in R[x]. h(x) = 0 either has a real root x0 or a pair of complex roots x1 , x ¯1 . If h(x0 ) = 0, we have f (x0 , y0 ) = 0 and g(x0 , y0 ) = 0. Then we get a contradiction.

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If there exist a pair of complex x1 , x¯1 such that h(x x1 ) = 0, without the 1 ) = h(¯ ix1 −1 loss of generality, we assume 0. Thus ix < 1. Moreover, we have 1 +1 f (x1 , y0 ) = 0

g(x1 , y0 ) = 0,

which imply P (iy0 ,

ix1 − 1 ) = 0, ix1 + 1

we get a contradiction with (ii) in Lemma 2.1. Obviously, the above condition (i) can be easily checked by the Hurwitz principle. To judge (ii), we introduce the following lemma (Ref. 7, p. 201 or Ref. 17): Lemma 2.2. For a given real polynomial G(y) =

1 R(f, g)(y) = y n + c1 y n−1 + · · · + cn , α

let s1 , s2 , · · · , s2n−2 are the Newton’s sum of G(y), i.e., si = xi1 + xi2 + · · · + xin , where x1 , x2 , · · · , xn are the n roots of G(y) = 0. Let   n s1 s2 · · · sn−1  s1 s2 s3 · · · sn     S=  s2 s3 s4 · · · sn+1    ··· sn−1 sn sn+1 · · · s2n−2

and Dk be the k-th ordered main subdeterminant, k = 1, 2, · · · , n. Then the number of different real roots of G(y) = 0 is r0 − 2V (1, D1 , D2 , · · · , Dr0 ), where r0 is the rank of matrix S and V (1, D1 , D2 , · · · , Dr0 ) is the number of times 1, D1 , D2 , · · · , Dr0 change signs. And if Dk 6= 0, Dk+1 = Dk+2 = · · · = Dk+l = 0, Dk+l+1 6= 0, then define signDk+j = (−1)

j(j−1) 2

signDk .

Remark 2.1. The Newton sum can be calculated by the Newton formula sk + c1 sk−1 + c2 sk−2 + · · · + ck−1 s1 + kck = 0, 1 ≤ k ≤ n sk + c1 sk−1 + · · · + cn sk−n = 0, k > n, so all the verifications are just elementary algebraic calculations.

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3. Preliminary for Robustness Stability Let + [a− i , ai ] ⊂ lR,

i = 0, 1, 2, · · · , n.

Consider the following polynomial family ( ) n X − + n−i − + Π = P (s) = ai s | ai ∈ [ai , ai ], 0 ∈ / [an , an ] . i=0

Define V (ω) = {P (ω)|P (s) ∈ Π},

ω ∈ lR.

Lemma 3.1. For any P (s) ∈ Π, P (s) = 0 has no (positive or negative, respectively) real root if and only if 0 ∈ / V (ω) for any ω ∈ lR (ω > 0 or ω < 0, respectively). Proof. This lemma is obvious. We omit its proof. Theorem 3.1. For any P (s) ∈ Π, P (s) = 0 has no real root if and only if − − − 2 3 P1 (s) = a− n + an−1 s + an−2 s + an−3 s + · · · , + + + 2 3 P2 (s) = a+ n + an−1 s + an−2 s + an−3 s + · · ·

have no positive real root and + − + 2 3 P3 (s) = a− n + an−1 s + an−2 s + an−3 s + · · · , − + − 2 3 P4 (s) = a+ n + an−1 s + an−2 s + an−3 s + · · ·

have no negative real root. Proof. The necessity is obvious. We only need to prove the sufficiency. We will first show that for any P (s) ∈ Π, P (s) = 0 has no positive real root. Let ω > 0. It is not difficult to see that P1 (ω) ≤ P2 (ω) and V (ω) = [P1 (ω), P2 (ω)]. Suppose that there is a polynomial P (s) ∈ Π such that p(s) = 0 has a positive real root. From the Lemma 3.1, there is a ω 0 > 0 such that 0 ∈ V (ω 0 ). Noting + + V (0) = [a− / [a− / V (0). Since the extreme point P1 (ω) n , an ] and 0 ∈ n , an ], we have 0 ∈ and P2 (ω) of V (ω) continuously variate with ω, we can find a 0 < ω0 < ω 0 such that P1 (ω0 ) = 0 or P2 (ω0 ) = 0. Thus we get a contradiction. Using the similar method and the polynomial P3 and P4 , we can show that for any P (s) ∈ Π, P (s) = 0 has no negative real root. Obviously, 0 is not the root of any P (s) ∈ Π. Combining the above results, we conclude the proof of this theorem.

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4. Robust Stability Now, we will consider the robust stability of delay systems. + Let [a− kj , akj ] ⊂ lR, k = 1, 2, · · · , n − 1, j = 1, 2, · · · , m. Denote   n−1 m   XX F = P (s, e−hs ) = sn + akj sk e−jhs |akj ∈ [a− , a+ ] kj kj   k=0 j=0

and

n o + G = R(f, g)(y)|akj ∈ [a− , a ] , kj kj

where R(f, g)(y) is defined by (4). Obviously, according to Theorem 2.1, the following theorem holds. Theorem 4.1. The quasi-polynomial family F is stable independent of delay if and only if (i) The polynomial family   n−1 m   X X Π1 = P (s, 1) = sn + ( akj )sk |akj ∈ [a− , a+ ] kj kj   k=0 j=0

is stable; (ii) For any R(f, g)(y) ∈ G, R(f, g)(y) 6= 0, ∀y ∈ lR.

Remark 4.1. The condition (i) can be checked by Kharitonov’s theorem, we need only test the stability of four special extreme polynomial in Π1 .

13

i.e.,

Remark 4.2. We can also find the convenient method to judge the condition (ii). + In fact, since the coefficient akj continuously variate, there exist [α− r , αr ] ∈ lR, r = 0, 1, · · · , N such that the set G can be denoted by ( ) N X N −r − + G ⊂ R(y) = αr y |αr ∈ [αr , αr ] . r=0

Thus, by Theorem 3.1, we need only determine whether or not four polynomial in G have real roots. Using Lemma 2.2, we can easily complete this work. However, the coefficients αr (r = 0, 1, · · · , N ) are independent each other while the original polynomial family G has dependent coefficients. Therefore, this is a sufficient criterion but not necessary. 5. Example Case Study + − + − + − + Let [a− 1 , a1 ] = [4, 5], [a2 , a2 ] = [1, 3], [a3 , a3 ] = [4, 6], [a4 , a4 ] = [9, 11]. Consider the following time-delay differential equation:

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d2 d d u(t) + a2 u(t) + a2 u(t − h) + a3 u(t − h) + a4 = 0, dt2 dt dt

(5)

+ where ai ∈ [a− i , ai ], i = 1, 2, · · · , 4, are real numbers. Here its characteristic function is

P (s, e−hs ) = s2 + a1 s + a2 se−hs + a3 e−hs + a4 . Obviously,

Let

+ Π1 = P (s, 1) = s2 + (a1 + a2 )s + (a3 + a4 )|ai ∈ [a− i , ai ], i = 1, 2, 3, 4 . + + + 2 P1 (s) = s2 + (a+ 1 + a2 )s + (a3 + a4 ) = s + 8s + 17, + − − 2 P2 (s) = s2 + (a+ 1 + a2 )s + (a3 + a4 ) = s + 8s + 13, − + + 2 P3 (s) = s2 + (a− 1 + a2 )s + (a3 + a4 ) = s + 5s + 17, − − − 2 P4 (s) = s2 + (a− 1 + a2 )s + (a3 + a4 ) = s + 5s + 13.

It is easy to see that P1 (s), P2 (s), P3 (s), P4 (s) are stable. Thus by the well-known Kharitonov’s theorem 13 , the polynomial family Π1 is stable, i.e., the condition (i) in Theorem 4.1 is satisfied. Let ix − 1 (ix + 1)P (iy, ) ix + 1 = −(a1 + a2 )yx − a3 + a4 − y 2 + i{[−y 2 + (a3 + a4 )]x + (a1 − a2 )y]} = f (x, y) + ig(x, y), we have that

−(a1 + a2 )y −a3 + a4 − y 2 R(f, g)(y) = 2 −y + (a3 + a4 ) (a1 − a2 )y

= −y 4 + (2a4 − a21 − a22 )y 2 + a23 − a24 .

Thus + G = −y 4 + (2a4 − a21 − a22 )y 2 + a23 − a24 |ai ∈ [a− i , ai ], i = 1, 2, 3, 4 .

According to Theorem 3.1, to judge the condition (ii) in Theorem 4.1, we need only check the four polynomial P5 (y), P6 (y), P7 (y), P8 (y) have no real roots, where 2

2

2

2

2

2

2

2

+ + − + 2 4 2 P5 (y) = −y 4 + (2a− 4 − a1 − a2 )y + a3 − a4 = −y − 16y − 105, − − + − 2 4 2 P6 (y) = −y 4 + (2a+ 4 − a1 − a2 )y + a3 − a4 = −y + 5y − 45,

P7 (y) = P1 (y), P8 (y) = P2 (y). Obviously, P5 (y), P6 (y), P7 (y), P8 (y) have no real roots. Thus, the condition (ii) in Theorem 4.1 is satisfied. So the system (5) is stable independent of delay.

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References 1. A. C. Bartlett, C. V. Hollot, and L. Huang, Root locations of an entire polytope of polynomials: It ssuffices to check the edges, Mathematics of Control, Signals, and Systems, Vol. 1, 61-71 (1988). 2. R. E. Bellman and K. L. Cooke, Differential-Difference Equations, New York, Academic, 1963. 3. J. Chiasson and C. Abdallah, Robust stability of time delay systems: Theory, Proceedings of the 3rd IFAC Workshop on Time Delay Systems, pp. 125-130, Santa Fe, NM, Dec. 8-10, 2001. 4. L. E. El’sgol’ts, Intruduction to the Theory of Differential Equations with Deviating Arguments, San Francisco, CA, Holden-Day, 1966. 5. M. Fu, A. W. Olbrot, and M. P. Polis, Robust stability for time-delay systems: the edge theorem and graphical tests, IEEE Tran. Automat. Contr. 34, 813-820 (1989). 6. M. Fu, A. W. Olbrot, and M. P. Polis, Edge Theorem and Graphical Test for Robust Stability of Neutral Time-Delay Systems, Automatica 27, 739-742 (1991). 7. F. P. Gantmacher, Matrix Theory, Vol. 2, Chelsea. New York, 1959. 8. J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. 9. J. K. Hale, E. F. Infante, and F. S. P. Tsen, Stability in linear delay equations, J. Math. Anal. Appl. 115, 533-555 (1985). 10. D. Hertz, E. I. Jury, and E. Zeheb, Stability independent and dependent of delay for delay differential systems, J. Franklin Institute 318, 143-150 (1984). 11. E. W. Kamen, On the relationship between zero criteria for two-variable polynomials and asymptotic stability of delay differential equations, IEEE Tran. Automat. Contr. AC-25, 983-984 (1980). 12. E. W. Kamen, Linear systems with conmensurate time delays: Stability and stabilization independent of delay, IEEE Tran. Automat. Contr. AC-27, 367-375 (1982). 13. V. L. Kharitonov, Asymptotic stability of an equilibrium position of a family of systems of linear differential equations, Differentsialjnie Uravnenia 14, 2086-2088 (1978). 14. V. L. Kharitonov, A. P. Zhabko, Robust stability of time-delay systems, IEEE Tran. Automat. Contr. 39, 2388-2397 (1994). 15. V. L. Kharitonov, Interval stability of quasipolynomials, Robust Stability and Control, CRC Press, 439-446 (1991). 16. R. M. Lewis, B. D. O. Anderson, Necessary and sufficient conditions for delayindependent stability of linear autonomous systems, IEEE Tran. Automat. Contr. AC-25, 735-739 (1980). 17. Z. Zhang, An algebraic principle for the stability of difference operators, Journal of Differential Equations 136, 236-247 (1997).

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TRANSFORMATION METHOD OF SOLVING A TIME-OPTIMAL CONTROL PROBLEM WITH POINTWISE STATE CONSTRAINTS∗

SHANGWEI ZHU Department of Applied Mathematics, Shanxi Finance & Economics University, Taiyuan 030006, CHINA Email: [email protected] XUNJING LI Department of Mathematics, Fudan University, Shanghai 200433, CHINA

It is difficult to solve the optimal control problem with pointwise state constraints. The maximum principle and the dynamic programming method can hardly be used to solve this kind of problems. In this paper, a time optimal control problem for a fourth-order linear system with pointwise state constraints is investigated. By means of transformations, the concrete expressions for the optimal time and the optimal control are given.

1. Introduction The Pontryagin maximum principle plays an important role in optimal control theory. It is a known method that sinking out the optimal control by means of the maximum principle. However, in many applied problems, the trajectory of the controlled systems should often satisfy some constraint conditions (for example, see Refs. 1–6). It is very difficult to solve the optimal control problem with pointwise state constraints. The maximum principle and the dynamic programming method can hardly be used to solve this kind of problems. In the literature, various optimality conditions for the optimal control problems with pointwise state constraints are always based upon some prior assumptions which are difficult to be verified. To our best knowledge, there has been no general method in calculating the optimal control for a control problem with pointwise state constraints. The example which could be solved to the end has hardly been seen up to now. In this paper, we consider a time-optimal control problem for a forth-order linear system with pointwise state constraints and the explicit solution of the optimal ∗ This

work was partially supported by the Chinese NSF under grant 101310310, and the NSF under grant 10171059. 250

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control is given. Let [ U = [−1, 1], U(T ) = { u = u(·) u(·) : (0, T ) → U is measurable }, U = U(T ). T >0



  y1 01  y2  0 0   Y =  y3  , A =  0 0 y4 00

   00 0 0 1 0  , B =   and g(Y ) = y32 − 1. 0 0 1 00 1

(1.1)

For any u = u(·) ∈ U, set tu = sup{ T | u ∈ U(T ) } which depends on u. Consider the following linear system with pointwise state constraint: Y˙ (t) = AY (t) + Bu(t),

g(Y (t)) ≤ 0,

(0 ≤ t ≤ tu ).

(1.2)

For given h > 0, we study the optimal control problem, named problem (Ph ), that the system (1.2) steers a moving point Y (t) from the initial state Y0 = 0 to the target state Yh = (T, 0, 0, 0)τ in the shortest time. For the finite dimensional optimal control problem, the maximum principle with pointwise state constraints is given in Ref. 1 (see Theorem 6.25 of Ref. 1). The 6= 0 principle holds based upon the regularity condition, which says that ∂p(Y,u) ∂u holds on the boundary-arcs of the optimal trajectory. Nevertheless, any boundaryarc of the admitted trajectory of the problem (Ph ), if it exists, dose not satisfy the regularity condition. In fact, from p(Y, u) =<

∂g(Y ) , AY + Bu >= 2y3 y4 ∂Y

≡ 0. Therefore, the problem (Ph ) can not be solved by means we know that ∂p(Y,u) ∂u of the maximum principle (Theorem 6.25 of Ref. 1) if the optimal trajectory contains any boundary-arc. In order to overcome the difficulty above, we turn to study the other optimal ˆ T ) and (RT ), which will be stated in the next section. control problems (QT ), (Q By the relation among these problems, we finally obtain the explicit solution of the problem (Ph ). Our main result reads as follows. Theorem 1.1. For any h > 0, the problem (Ph ) has the following unique solution ¯ =u u ¯(·):

u ¯(t) =

  1 

−1

t ∈ (0, (1 − t ∈ ((1 −

√ 2 2 )t1 )

∪ (t1 , (1 +

√ 2 2 )t1 , t1 )

∪ ((1 +

√ 2 2 )t1 )

,

√ 2 2 )t1 , 2t1 )

(1.3)

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√ + 2 2, and  √ 1 t ∈ (0, 1) ∪ (t1 , t1 + 2) ∪ (2t1 − 2, 2t1 − 1) ,      √ √ u ¯(t) = 0 t ∈ (2, t1 − 2) ∪ (t1 + 2, 2t1 − 2) ,     √  −1 t ∈ (1, 2) ∪ (t1 − 2, t1 ) ∪ (2t1 − 1, 2t1 ) √ T¯ (h) when h > 17 and 6 + 2 2. Here, t1 = 2 √ √ 4 7  0 < h ≤ 17  2 ·3·h 6 +2 2 , T¯(h) = q √   2 + 2 + 4h h > 17 3 6 +2 2

when 0 < h ≤

17 6

(1.4)

(1.5)

is the optimal time of the problem (Ph ).

2. Transformation of the problem (Ph ) In this section, we firstly give some relative problems to (Ph ). Then we study the relation among these problems. Let x(t) = y1 (t), the system (1.2) which satisfy the initial conditions Y0 = 0 may be rewritten as   x(4) (t) = u(t), (0 ≤ t ≤ tu ), (2.6)  x(k) (0) = 0, k = 0, 1, 2, 3 |¨ x(t)| ≤ 1,

(0 ≤ t ≤ tu ).

(2.7)

For any u = u(·) ∈ U, the system (2.6) has a unique solution x(·) = xu (·). By Taylor formula, we easily get Z Z t 1 t 3 xu (t) = (t − s) u(s)ds, x ¨u (t) = (t − s)u(s)ds, 0 ≤ t ≤ tu . (2.8) 6 0 0 The terminal condition Y (tu ) = Yh may be stated as x(tu ) = h;

x(k) (tu ) = 0, k = 1 , 2, 3.

(2.9)

For any h > 0, set Uad (h) = { u ∈ U | tu < +∞, (2.7) and (2.9) hold for x(·) = xu (·) }.

(2.10)

Then, the optimal control problem mentioned above is stated as follows: ¯ ∈ Uad (h) such that Problem(Ph ): For given h > 0, find a u def

tu¯ = T¯(h) =

inf

u∈Uad (h)

tu .

(2.11)

Here, T¯(h)¯ u(if it exists) and xu¯ (·) is called an optimal time optimal control and optimal trajectory of the problem (Ph ), respectively. For any u ∈ Uad (h), u and

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xu (·) is called an admitted control and admitted trajectory of the problem (P h ), respectively. For any T > 0, set (k) U˜ad (T ) = { u ∈ U | tu = T, (2.7) holds for x(·) = xu (·) and xu (tu ) = 0, k = 1, 2, 3 },

Uˆad (T ) = { u ∈ U | tu = T, (2.7) holds for x(·) = xu (·) and x¨u (tu ) = 0 }. ˜ ∈ U˜ad (T ) such that Problem (QT ): For given T > 0, find a u def

xu˜ (T ) = s(T ) =

sup

xu (T ) .

(2.12)

˜ad (T ) u∈U

ˆ T ): For given T > 0, find a u ˆ ∈ Uˆad (T ) such that Problem (Q def

xuˆ (T ) = sˆ(T ) =

sup

xu (T ) .

ˆad (T ) u∈U

¯ ∈ U(T ) such that x¨v¯ (T ) = 0 and Problem (RT ): For given T > 0, find a v def

xv¯ (T ) = r(T ) = sup{ xv (T ) | v ∈ U(T ) and x ¨v (T ) = 0 }. We firstly give the following relation between problems (Ph ) and (QT ): Lemma 2.1. (i) If Uad (h) 6= ∅ for any h > 0, function s(·) defined by (2.12) is strict increasing and continuous, lim s(T ) = 0 ,

T →0+

lim s(T ) = +∞

T →+∞

(2.13)

then the following equalities hold: 0 < T¯(h) < +∞,

s(T¯h ) = h,

∀h > 0.

(2.14)

(ii) Furthermore, the problem (Ph ) has a solution for any h > 0 and the solutions of the problem (Ph ) and (QT¯(h) ) are the same if the problem (QT ) has a solution for any T > 0. Proof. (i) For h > 0, Uad (h) 6= ∅ implies 0 < T¯(h) < +∞. We can prove that s(T ) ≥ h,

∀T > T¯(h).

(2.15)

In fact, for T > T¯(h), there exists a u = u(·) ∈ Uad (h) such that tu ∈ [T¯(h), T ) Set v = v(·), where u(t), a.e. t ∈ (0, tu ); v(t) = (2.16) 0, a.e. t ∈ (tu , T ). Then, tv = T and Taylor formula implies that xu (t), t ∈ [0, tu ]; xv (t) = h, t ∈ (tu , T ].

(2.17)

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Consequently, v ∈ U˜ad (T ) and xu (T ) = h. Hence, (2.15) follows. we now prove that s(T ) ≤ h,

∀0 < T < T¯(h).

(2.18)

Otherwise, s(T ) > h implies that xu1 (T ) > h for certain u1 ∈ U˜ad (T ). By (2.13), s(T0 ) < h for certain T0 ∈ (0, T ). Therefore, there exists a u ∈ U˜ad (T0 ) such that tu = T0 and xu (tu ) < h. Set ˜ = λv + (1 − λ)u1 , u where, v = v(·) is defined by (2.16) and λ ∈ (0, 1) such that λxu (tu ) + (1 − λ)xu1 (T ) = h. ˜ ∈ U˜ad (T ) and xu˜ (T ) = h. consequently, u ˜ ∈ Uad (h) It is easy to confirm that u ¯ and T = tu˜ ≥ T (h). Hence, (2.18) follows. From (2.15), (2.18) and strict increasing character of s(·), we get (2.14). ˜ is a solution of the problem (QT¯(h) ), then u ˜ ∈ U˜ad (T¯(h)) and (ii) If u ¯ ¯ ¯ ˜ ∈ Uad (h) and tu˜ = T (h), namely, u ˜ is a xu˜ (T (h)) = s(T (h)) = h. Hence, u solution of the problem (Ph ). The converse can be proved analogously. ˆ T ). We now prove the relation between problems (QT ) and (Q Lemma 2.2. For any T > 0, the following equality holds: s(2T ) = 2ˆ s(T ).

(2.19)

ˆ T ) is solvFurthermore, the problem (Q2T ) is solvable if and only if the problem (Q able. Proof. For any u = u(·) ∈ U˜ad (2T ), define mapping G by 1 (Gu)(t) = u(t) − u(2T − t) , a.e. t ∈ (0, T ). (2.20) 2 It is easy to prove that G is a mapping from U˜ad (2T ) onto Uˆad (T ). Furthermore, for any v = v(·) ∈ Uˆad (T ), set   v(t), a.e. t ∈ (0, T ); u(t) = (2.21)  −v(2T − t), a.e. t ∈ (T, 2T ), then, (Gu)(·) = v(·). the equality (2.19) is deduced from the following relation: xu (2T ) = 2xGu (T ),

∀u ∈ U˜ad (2T ).

(2.22)

ˆ T ) are equivalent. Let us prove that the solvability of problem (Q2T ) and (Q ˆ ˜ be a solution of problem (Q2T ). By G˜ Suppose u u ∈ Uad (T ), (2.22) and (2.19), we have 1 1 xGu˜ (T ) = xu˜ (2T ) = s(2T ) = sˆ(T ), 2 2

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ˆ T ). The converse can be proved analo˜ )(·) is a solution of problem (Q hence, (Gu gously. ˆ T ) is the pointwise Finally, the only difference between the problem (RT ) and (Q constraint (2.7). By (2.8), any xu (·) satisfies (2.7) when T is small enough. Hence, ˆ T ) when T is small the solution of problem (RT ) must be a solution of problem (Q enough. 3. Solving the transformed problems Without any pointwise constranint, the problem (RT ) can be solved by means of the Pontryagin’s maximum principle (see Theorem 2.6 of Ref. 1 or Theorem 4.1.3 of Ref. 5, for example). We give the following result about the problem (RT ) directly, the proof will be omitted. 4

Lemma 3.1. For any T > 0, r(T ) = T48 , and the problem (RT ) has a unique solution √   1, 0 < t < (1 − 22 )T ; v¯(t) = (3.23) √  2 −1, (1 − 2 )T < t < T. A direct calculating indicates that ˆ Corollary 3.1. v¯(·) √ defined by (3.23) is also the unique solution of problem ( QT ) when 0 < T ≤ 2 + 2. √ When T ≥ 2 + 2, we suitably stretch the extremum point of the function x¨v 0 (·) √ into a segment x ¨v ≡ 1, where, v 0 = v¯(·) is corresponding to T = 2 + 2 in (3.23). ˆ T ) is obtained. Using the method, the solution of problem (Q ˆ T ) has a unique solution u ˆ =u Theorem 3.1. For any T > 0, the problem (Q ˆ(·), √ where, u ˆ(·) = v¯(·) is defined by (3.23) when 0 < T ≤ 2 + 2 and  1, t ∈ (0, 1),     √ √ u ˆ(t) = when T > 2 + 2. (3.24) 0, t ∈ (2, T − 2),    √  −1, t ∈ (1, 2) ∪ (T − 2, T )

The optimal value function is  4  T48 , sˆ(T ) =  T2 2 −T +

5 12 ,

√ 02+ 2 .

(3.25)

Proof. √According to Corollary 3.1, what need to be proved is the case when T > 2 + 2.

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Firstly, it is obvious that tuˆ = T . From (2.8) and (3.24), we get  4 t  t ∈ [0, 1] ,  24 ,       1 (t−1)2 (t−2)4  t ∈ (1, 2] ,  12 + 2 − 24 , xuˆ (t) =  (t−1)2  1  + , t ∈ (2, t1 ] ,  12 2      4  (t−1)2 1 1) − (t−t , t ∈ (t1 , T ] 12 + 2 24

and

x ¨uˆ (t) =

 2 t   2 ,      2    1 − (t−2) , 2    1,        1−

t ∈ [0, 1] , t ∈ (1, 2] , t ∈ (2, t1 ] ,

(t−t1 )2 2

, t ∈ (t1 , T ] . √ ˆ ∈ Uˆad (T ) and xuˆ (T ) = Here, t1 = T − 2. The expressions above imply that u T2 5 2 − T + 12 . Secondly, for any u = u(·) ∈ Uˆad (T ), setting y(t) = x ¨u (t) − x ¨uˆ (t), then we have y¨(t) = u(t)− u ˆ(t), a.e. t ∈ (0, T ); y(0) = y(0) ˙ = y(T ) = 0; y(2) = x ¨u (2)−1 ≤ 0 . We now prove that y(t) ≤ 0,

∀t ∈ [0, T ].

(3.26)

For 0 ≤ t ≤ 1, (3.26) comes from Z t y(t) = (t − s)[u(s) − 1]ds ≤ 0. 0

(3.26) holds obviously when 2 ≤ t ≤ t1 . Suppose that y(t0 ) > 0 hold for certain t0 ∈ (1, 2), thenthere exist some ξ ∈ (1, t0 ) and η ∈ (t0 , 2) such that y(ξ) ˙ =

y(t0 ) − y(1) > 0, t0 − 1

y(η) ˙ =

y(2) − y(t0 ) < 0, 2 − t0

y(η) ˙ − y(ξ) ˙ < 0.

On the other side, y¨(t) = u(t)−ˆ u(t) = u(t)+1 ≥ 0, a.e. t ∈ (1, 2), therefore y(η)− ˙ y(ξ) ˙ =

Z

η

y¨(t)dt ≥ 0, ξ

this contradiction indicates that (3.26) holds for any t ∈ (1, 2). For t ∈ (t1 , T ], (3.26) can be proved analogously. Finally, from (3.26), we get Z t xu (t) − xuˆ (t) = (t − s)y(s)ds ≤ 0, (∀u(·) ∈ Uˆad (T ) , ∀t ∈ [0, T ]). (3.27) 0

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√ ˆ T ) and (3.25) follows. When 0 < T ≤ 2 + 2, ˆ is a solution of problem (Q Hence, u ˆ the uniqueness of the solution comes from Corollary 3.1. √ Suppose u = u(·) ∈ Uad (T ) ˆ T ) for some T > 2 + 2. From is also a solution of problem (Q Z T (T − s)y(s)ds = xu (T ) − xuˆ (T ) = 0 0

ˆ . The uniqueness of and (3.26), we get y(s) = 0 (∀s ∈ [0, 1]), which implies u = u the solution is verified. Combining Theorem 3.1 and Lemma 2.2, we have the following conclusion. Theorem 3.2. For any T > 0, the problem (QT ) has the following unique solution ˜ =u u ˜(·): √ √   1, t ∈ (0, (2− 4 2)T ) ∪ ( T2 , (2+ 4 2)T ) , (3.28) u ˜(t) = √ √  (2− 2)T T (2+ 2)T −1, t∈( , 2)∪( ,T) 4 4 √ when 0 < T ≤ 4 + 2 2, and  √  t ∈ (0, 1) ∪ ( T2 , T +22 2 ) ∪ (T − 2, T − 1) ,  1,    √ √ u ˜(t) = (3.29) 0, t ∈ (2, T −22 2 ) ∪ ( T +22 2 , T − 2) ,    √   −1, t ∈ (1, 2) ∪ ( T −22 2 , T2 ) ∪ (T − 1, T ) √ when T > 4 + 2 2. The optimal value function is  4 √  2T7 ·3 , 04+2 2 . 4 − T + 6, ˆ T /2 ) by u ˆ=u Proof. Denoting the solution of problem (Q ˆ(·) and replacing T by T in (3.23) or (3.24), we have 2  u ˆ(t), 0 ≤ t ≤ T2 , u ˜(t) = (3.31)  −ˆ u(T − t), T2 < t ≤ T ;

˜ =u where, u ˜(·) is defined by (3.28) or (3.29). According to the proof of Lemma ˜ is a solution of the problem (QT ). From (2.19) and (3.25), we immediately 2.2, u get (3.30). We now prove the uniqueness. Let u = u(·) be also a solution of problem (QT ). ˆ T /2 ). Citing Theorem 3.1, we By (2.19) and (2.22), Gu is a solution of problem (Q ˆ which together with (2.20) implies get Gu = u u(t) − u(T − t) = 2ˆ u(t),

a.e. t ∈ (0,

T ). 2

(3.32)

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Set E+ = { t ∈ [0, T2 ] | u ˆ(t) = 1 }, E− = { t ∈ [0, T2 ] | u ˆ(t) = −1 }, ˆ(t) = 0 }. E0 = { t ∈ [0, T2 ] | u In view of |u(t)| ≤ 1 (a.e. t ∈ (0, T )) and (3.32), we have u(t) = −u(T − t) = ±1 = u ˆ(t), a.e. t ∈ E+ ∪ E− . (3.33) √ ˜. When 0 < T ≤ 4√+ 2 2 , E+ ∪ E− = [0, T2 ] , (3.33) and (3.31) imply u = u When T > 4 + 2 2 , E0 6= ∅ and E0 ∪ E+ ∪ E− = [0, T2 ]. By (2.8) and (3.32), we get x ¨u (t) − x ¨u (T − t) = 2¨ xuˆ (t),

∀t ∈ [0,

T ]. 2

In view of (2.7) and x ¨uˆ (t) = 1 (∀t ∈ E0 ) , the above equality results in x ¨u (t) = −¨ xu (T − t) = x ¨uˆ (t),

∀t ∈ E0

which implies u(t) = −u(T − t) = u ˆ(t),

a.e. t ∈ E0 .

(3.34)

˜ . The proof is completed. From (3.31), (3.33) and (3.34), we get u = u 4. The proof of the main result and some remarks ¯ =u Proof of Theorem 1.1. For any h > 0 and u ¯(·) defined by (1.3) or (1.4), a ¯ ∈ Uad (h). It is easy to verify that function s(·) defined direct calculating shows u by (3.30) is strict increasing, continuous and satisfies (2.13). According to Theorem ¯ =u 3.2 and Lemma 2.1, the problem (Ph ) has a unique solution u ¯(·) for any h > 0. Expression (1.5) may be obtained from (3.30) and (2.14), (1.3) and (1.4) is the refurbished version of (3.28) and (3.29), respectively. ¯ of the problems (Ph ) is not bang-bang when Remark 4.1. √ The optimal control u 17 h > 6 + 2 2. √ Remark 4.2. When h > 17 ¯(·) = xu¯ (·) of the 6 + 2 2 , the optimal trajectory x problem (Ph ) contains two boundary-arcs. Therefore, the problem (Ph ) can not be solved by means of the maximum principle with pointwise state constraints. Remark 4.3. Theorems 1.1, 3.1 and 3.2 still hold when U = {1, 0, −1}. However, ˆ T ) and (Ph ) have no solution when U = {1, −1} and T or h the problems (QT ), (Q is larger.

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References 1. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko, Mathematical Theory of Optimal Processes, Wiley, New York (1962). 2. N. Arada and J. P. Raymond, Optimal control problems with mixed control-state constraints, SIAM. J. Control Optim. 39, 1391-1407 (2000). 3. B. Hu and J. Yong, Pontryagin maximum principle for semilinear and quasilinear parabolic equations with pointwise state constraints, SIAM. J. Control Optim. 33, 1857-1880 (1995). 4. G. Wang and L. Wang, State-constrained optimal control governed by non-well-posed parabolic differential equations, SIAM. J. Control Optim. 40, 1517-1539 (2002). 5. X. Li and J. Yong, Optimal Control Theory For Infinite Dimensional Systems, Birkh¨ auser, Boston (1995). 6. R. F. Hartl, S. P. Sethi, and R. G. Vickson, A survey of the maximum principles for optimal control problems with state constraints, SIAM Review, 37, 181-218 (1995).

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Chap32-ChenQihong

OPTIMAL CONTROL OF QUASILINEAR ELLIPTIC OBSTACLE PROBLEMS∗

QIHONG CHEN and YUQUAN YE Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China [email protected] (Q. Chen), [email protected] (Y. Ye)

In this paper we consider an optimal control problem for a quasilinear elliptic obstacle variational inequality in which the obstacle is taken as the control. The cost functional of this optimal control problem is of Lagrange type in which the p-th power of Laplacian of the control appears. This feature leads that it is hard to derive the optimality system for the underlying problem. In this paper, the existence and optimality system for the optimal control are established. Keywords: Obstacle optimal control problem, existence, necessary condition. (AMS) Mathematics Subject Classification: 93C20, 35B37, 49J20, 49K20.

1. Introduction Suppose Ω ⊂ Rn is a bounded domain with a C 1,1 boundary ∂Ω. Let zd ∈ L2 (Ω) be a given target profile. For any ϕ ∈ W01,p (Ω), we define K(ϕ) = {v ∈ W01,p (Ω)|v ≥ ϕ

a.e.

x ∈ Ω}

and consider a quasilinear elliptic obstacle problem   yZ ∈ K(ϕ),  A(x, ∇y)∇(v − y)dx ≥ 0, ∀v ∈ K(ϕ)

(1)

(2)

Ω

where A(x, η) = (a1 (x, η), · · · , an (x, η)). Given an obstacle ϕ ∈ W01,p (Ω), under some further assumptions on A(x, η)(see (H1 ) and (H2 )), the variational inequality (2) is uniquely solvable (cf. [12]). We will denote by y = T (ϕ), the unique solution of (2) corresponding to ϕ. Let W = W01,p (Ω) ∩ W 2,p (Ω). We seek an obstacle ϕ¯ ∈ W so that the corresponding state y¯ = T (ϕ) ¯ is close to a desired target profile zd and the norm of ϕ¯ is ∗ This

work was supported partly by FANEDD grant 200218 and NSFC grant 10171059. 263

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not too large in W . For this purpose, we take as our objective functional: Z 1 1 2 p J(ϕ) = (T (ϕ) − zd ) + |∆ϕ| dx 2 p Ω

(3)

and pose the following optimal control problem: Problem (P) Find ϕ¯ ∈ W such that J(ϕ) ¯ = inf J(ϕ). ϕ∈W

(4)

One of the main feature of our problem is that the input control intervene is in the obstacle. Such a case is referred to as an optimal obstacle control problem. The first work on the optimal obstacle control problem was that of Adams, Lenhart and Yong [1] in 1998. For the homogeneous case, an optimal obstacle control problem for an elliptic variational inequality is considered. The key result of that paper is that the optimal control is equal to its corresponding state. By virtue of the properties of the homogeneous equation, the super-harmonic functions and the quadratic cost functional, the existence, uniqueness of the optimal control as well as characterizations of the optimal pair are established. Later, Chen studies indirect obstacle control problem in [6]. Lou considered the regularity of the obstacle control problem in [14] for the homogeneous case with the major term being p-Laplacian. Recently, Adams and Lenhart continue the work begun in [1]; a nonzero source term is added to the right hand side of the state equation. They soon find that even such a “minor” change is not a trivial alteration (cf. [2]). In our control Problem (P), the governing equation is a quasilinear elliptic obstacle variational inequality (1.2), which can be rewritten as the following (if y is smooth enough): min{−divA(x, ∇y(x)), y(x) − ϕ(x)} = 0 x∈Ω

where A(x, ·) is nonlinear. In practice, there are many real physical or geometrical problems related to the optimal control for obstacle variational inequalities (cf. [3, 4]) and, except some ideal cases, the governing equations are mostly quasilinear or nonlinear. A typical problem entering the framework is A(∇y) = |∇y|p−2 ∇y, which appears in the study 1 of non-Newtonian fluids (cf. [8]). Another example is A(∇y) = (1 + |∇y|2 )− 2 ∇y, in the study of minimal surface with obstacle (cf. [13, 15]). Similar cases occur in the evolutionary problems. For instance, in mathematical finance, the wellknown model of pricing American type contingent claims (option,e.g.) is a parabolic obstacle variational inequality, where the pay-off function serves as the obstacle. Controlling the obstacle amounts to designing the pay-off. When the volatility depends on the portfolio, the governing equation will be quasilinear. Thus, in some sense, nonlinearity (including quasilinearity) is more practical than linearity. It is clear that the techniques developed in [1, 2] do not work when quasilinearity or nonlinearity is involved in the governing system.

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The rest of this paper is organized as the following: In section 2, we give some assumptions with the state equation and some preliminaries needed in the sequel. In section 3, we prove the existence of an optimal control. We approximate the variational inequality (2) by a family of quasilinear elliptic equations in section 4. Finally, in section 5, utilizing the special structure of the approximate optimality system including the monotonicity of the leading differential operator, we get an auxiliary condition, and then establish the optimality system for the original problem. 2. Preliminaries We introduce the following assumptions: (H1 ). For any η = (η1 , · · · , ηn ) ∈ Rn , a( ·, η) is a measurable function on Ω with aj (·, 0) = 0 and for any x ∈ Ω, aj (x, ·) belongs to C 1 (Rn ),j = 1, · · · , n. (H2 ). For any p ≥ 2 and all ξ, η ∈ Rn n X ∂aj (x, η)ξi ξj ≥ Λ1 (k + |η|)p−2 |ξ|2 , ∂ηi i,j=1

n X ∂aj ≤ Λ2 |η|p−2 , (x, η) ∂ηi i,j=1

(5)

(6)

where k ∈ (0, 1], Λ1 and Λ2 are some positive constants. The following lemma is an immediate consequence of assumptions (H1 ) and (H2 ). Lemma 2.1. (cf. [5]) Under assumptions (H1 )-(H2 ), there are positive constants k1 and k2 depending only on n, p, Λ1 , Λ2 such that (a) n X

(aj (x, η) − aj (x, η 0 ))(ηj − ηj0 ) ≥ k1 |η − η 0 |p

(7)

j=1

(b) n X

|aj (x, η)| ≤ k2 |η|p−1 .

(8)

j=1

In this paper, we need some basic results in real analysis. Here, for reader’s convenience, we list the following lemmas without proof. Lemma 2.2. (cf. [7]) Then

Let 1 < p < ∞, {fn } is bounded in Lp , and fn → f ,a.e.. w

fn * f

in Lp (Ω).

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w

0

If fn → f in Lp (Ω) and gn * g in Lp (Ω), then Z Z fn gn dx → f gdx.

Lemma 2.3. (cf. [9])

Ω

Lemma 2.4. (cf. [10])

Ω

Let 1 < p < ∞, then for any A, B ∈ Rn we have

(|A|p−2 A − |B|p−2 B) · (A − B) ≥ 0.

(9)

The above inequality (9) is the so called “monotonicity inequality”, which will play an important role in deriving the necessary conditions. 3. Existence of an optimal control First, we prove the existence theorem for Problem (P). Theorem 3.1. Under assumptions (H1 ) and (H2 ), there exists at least an optimal control to Problem (P). Proof.

Let {ϕk } be a minimizing sequence i.e. lim J(ϕk ) = inf J(ϕ)

(10)

ϕ∈W

k→∞ 4

where yk = T (ϕk ) be the solution of (2) corresponding to ϕk i.e.  Z yk ∈ K(ϕk )  A(x, ∇yk )(v − yk ) ≥ 0, ∀v ∈ K(ϕk )

(11)

Ω

Due to the form of the functional (3), bounds on {J(ϕk )} imply the existence of an obstacle ϕ¯ such that on a subsequence (still denoted by {ϕk }), w

ϕk * ϕ, ¯ in s ϕk → ϕ, ¯ in

W 2,p (Ω), W01,p (Ω).

By taking v = ϕk in (11), we see that Z Z Z k1 |∇yk |p dx ≤ A(x, ∇yk ) · ∇yk dx ≤ Ω

≤

Ω

k2 k∇yk kp−1 Lp k∇ϕk kLp ,

(12)

A(x, ∇yk )∇ϕk dx Ω

then, k∇yk kLp ≤ C

(13)

where C is independent of k. Thus, for some subsequence, we have the convergence w

yk * y¯, in W01,p (Ω), s yk → y¯, in Lp (Ω), yk → y¯, a.e. in Ω.

(14)

In the following we prove that y¯ is the solution of (2) corresponding to ϕ, ¯ i.e. y¯ = T (ϕ). ¯

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Step 1. We first prove that Z (A(x, ∇yk ) − A(x, ∇¯ y ))(∇yk − ∇¯ y )dx → 0 (k → ∞).

(15)

Ω

By Lemma 2.1 and (13), we get kA(x, ∇yk )kLp0 ≤ C.

(16)

By (11), (12) and (14), we have y¯ ≥ ϕ, ¯ a.e. in Ω.

(17)

Let zk = y¯ ∨ ϕk . We can easily get (note (12) and (17)) s

zk → y¯ in W01,p (Ω). Taking v = zk in (11), we have Z A(x, ∇yk )∇(zk − yk )dx ≥ 0,

(18)

Ω

so,

Z

A(x, ∇yk )∇yk dx ≤ Ω

Z

A(x, ∇yk )∇zk dx.

(19)

Ω

By Lemma 2.1, (19), (16) and (14), we have Z 0≤ (A(x, ∇yk ) − A(x, ∇¯ y ))(∇yk − ∇¯ y )dx Z Ω Z ≤ A(x, ∇yk ) · ∇(zk − y¯)dx − A(x, ∇¯ y )(∇yk − ∇¯ y )dx Ω ZΩ ≤ kA(x, ∇yk )kLp0 k∇(zk − y¯)kLp − A(x, ∇¯ y )(∇yk − ∇¯ y )dx Ω

→ 0 (k → ∞) i.e.

Z

(A(x, ∇yk ) − A(x, ∇¯ y ))(∇yk − ∇¯ y )dx → 0 (k → ∞)

(20)

Ω s

w

Step 2. We prove that ∇yk → ∇¯ y in Lp (Ω) and A(x, ∇yk ) * A(x, ∇¯ y ) in Lp (Ω). By Lemma 2.1, we have Z Z k1 |∇(yk − y¯)|p dx ≤ (A(x, ∇yk ) − A(x, ∇¯ y ))(∇yk − ∇¯ y )dx. 0

Ω

Ω

So, s

∇yk → ∇¯ y , in Lp (Ω)

(k → ∞).

(21)

Thus, as k → ∞, s

yk → y¯, ∇yk → ∇¯ y, yk → y¯,

in W01,p (Ω), a.e. in Ω, a.e. in Ω.

(22)

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By (H1 ) we get A(x, ·) ∈ C 1 (Rn ). Thus A(x, ∇yk ) → A(x, ∇¯ y ),

a.e. in Ω.

(23)

0

(24)

So, by (16) and Lemma 2.3, we get w

A(x, ∇yk ) * A(x, ∇¯ y ), in Lp (Ω). Step 3. We prove that y¯ = T (ϕ). ¯ For any v ∈ K(ϕ), ¯ we have v ∨ ϕk ∈ K(ϕk ) and by (12), s

v ∨ ϕk → v ∨ ϕ¯ = v,

in W01,p (Ω).

Thus by (11), (22), (24) and Lemma 2.3, we get Z Z 0≤ A(x, ∇yk ) · ∇(v ∨ ϕk − yk )dx → A(x, ∇¯ y ) · ∇(v − y¯)dx. Ω

(25)

Ω

This, together with (17), gives our conclusion. By (22), (14), (10) and the weak lower semi-continuity of Lp -norm Z 1 1 (T (ϕ) ¯ − zd )2 + |∆ϕ| ¯ p dx J(ϕ) ¯ = 2Z p Ω 1 1 2 p (yk − zd ) + |∆ϕk | dx ≤ lim inf k→∞ 2 p Ω = lim J(ϕk ) = inf J(ϕ) k→∞

(26)

ϕ∈W

Hence, ϕ¯ is an optimal control for Problem (P).

4. Approximate problems In this section we introduce a family of approximate problems. Let ε > 0 and ϕ ∈ W , we consider the following quasilinear equation ( 1 −divA(x, ∇yε ) + β(yε − ϕ) = 0, ε yε |∂Ω = 0 where

  0 0 ≤ r < +∞,    2 1 − ≤ r < 0, β(r) = −r 2   1 1   r + −∞ < r < − . 4 2

(27)

(28)

Clearly, β(·) ∈ C 1 (Ω) and β 0 (·) ≥ 0. Under (H1 )and (H2 ), (4.1) admits a unique solution (denoted by Tε (ϕ)) for any fixed ϕ ∈ W (cf. [11]). Let (¯ y , ϕ) ¯ be an optimal pair of Problem (P), we introduce the cost functional Z 1 1 1 Jε (ϕ) = (Tε (ϕ) − zd )2 + |∆ϕ|p + |ϕ − ϕ| ¯ p dx (29) 2 p p Ω and the approximate control problems

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Problem (Pε ) Find a ϕε ∈ W such that Jε (ϕε ) = inf Jε (ϕ)

(30)

ϕ∈W

Theorem 4.1. Problem (Pε ) admits an optimal pair (yε , ϕε ). The proof of Theorem 4.1 is similar to Theorem 3.1 with some suitable modifications. Now, we establish the following convergence theorem. This result is crucial in the next section. Theorem 4.2. Let (yε , ϕε ) be optimal pair for Problem (Pε ). Then s

W01,p (Ω) W 2,p (Ω) W01,p (Ω)

yε → y¯, in w ¯ in ϕε * ϕ, s ϕε → ϕ, ¯ in

(31)

where (¯ y , ϕ) ¯ is the given optimal pair for Problem (P). Proof. First,we note that Z Z 1 1 |∆ϕε |p dx ≤ Jε (ϕε ) ≤ Jε (0) = p Ω 2

|zd |2 dx + Ω

1 p

Z

|ϕ| ¯ p dx Ω

and thus k∆ϕε kLp ≤ C

(32)

where C is independent of ε. Then we may assume, extracting some subsequence if necessary, w

ϕε * ϕ∗ , in W 2,p (Ω), s ϕε → ϕ∗ , in W01,p (Ω). We see that Z Z 1 A(x, ∇yε )∇(ϕε − yε )dx = − ε Ω By (H2 ) and Lemma 2.1, we get Z Z p k1 ≤ |∇yε | dx ≤ Ω Z ≤

β(yε − ϕε )(ϕε − yε )dx ≥ 0.

(33)

(34)

Ω

A(x, ∇yε ) · ∇yε dx Ω

A(x, ∇yε ) · ∇ϕε dx

(35)

Ω

≤ k2 k∇yε kp−1 Lp k∇ϕε kLP .

Then, k∇yε kLp ≤

k2 k∇ϕε kLp k1

(36)

and thus k∇yε kLp ≤ C

(37)

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where C is independent of ε. Hence, for some subsequence, we have the convergence (as ε → 0) w

yε * y ∗ , in W01,p (Ω), s yε → y ∗ , in Lp (Ω).

(38)

By (27), we know that, for any ψ ∈ W01,p (Ω), ψ ≥ 0, Z Z 0≤ (−β(yε − ϕε ))ψdx = ε A(x, ∇yε ) · ∇ψdx ΩZ Ω ≤ εk2 |∇yε |p−1 |∇ψ|dx ≤ εk2 k∇yε kp−1 Lp k∇ψkLp → 0 (ε → 0).

(39)

Ω

Then, by Fatou’s lemma, Z Z 0≤ (−β(y ∗ − ϕ∗ ))ψdx ≤ lim inf ε→0

Ω

(−β(yε − ϕε ))ψdx = 0

(40)

Ω

This implies that β(y ∗ − ϕ∗ ) = 0. By the definition of β(·), we have y ∗ ≥ ϕ∗ a.e. in Ω.

(41)

By Lemma 2.1 and (27), we have Z k1 |∇y ∗ − ∇yε |p dx Z Ω ≤ (A(x, ∇y ∗ ) − A(x, ∇yε )) · ∇(y ∗ − yε )dx Z Z Ω 1 β(yε − ϕε )(y ∗ ∨ ϕε − yε )dx = A(x, ∇y ∗ ) · ∇(y ∗ − yε )dx + ε {y <ϕ } Ω ε ε Z − A(x, ∇yε ) · ∇(y ∗ − y ∗ ∨ ϕε )dx Z Ω ∗ ∗ ≤ A(x, ∇y ∗ ) · ∇(y ∗ − yε )dx + k2 k∇yε kp−1 Lp k∇(y − y ∨ ϕε )kLp .

(42)

Ω

s

From (39) and y ∗ ∨ ϕε → y ∗ in W01,p (Ω), we have Z |∇y ∗ − ∇yε |p dx → 0 (ε → 0)

(43)

Ω

and thus, s

yε → y ∗ , in W01,p (Ω), ∇yε → ∇y ∗ , a.e. in Ω.

(44)

By (H1 ), we know A(x, ·) ∈ C 1 (Rn ) and then A(x, ∇yε ) → A(x, ∇y ∗ ), a.e. in Ω.

(45)

From (H2 ) and (38), we have kA(x, ∇yε )kLp0 ≤ C.

(46)

Combining (46) and (47), we have w

0

A(x, ∇yε ) * A(x, ∇y ∗ ), in Lp (Ω).

(47)

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Now, we prove that y ∗ = T (ϕ∗ ). First, (4.5) shows that y ∗ ∈ K(ϕ∗ ). For any v ∈ K(ϕ∗ ), we have s

v ∨ ϕε → v, and

in W01,p (Ω)

Z 1 β(yε − ϕε )(v ∨ ϕε − yε )dx 0≤− Zε Ω = A(x, ∇yε ) · ∇(v ∨ ϕε − yε )dx ZΩ → A(x, ∇y ∗ ) · ∇(v − y ∗ )dx (ε → 0).

(48)

Ω

Thus, y = T (ϕ ), i.e. y is the solution of (1.2) corresponding to ϕ∗ . Finally, we prove that ϕ∗ = ϕ, ¯ y ∗ = y¯. From (34), (39) and the weak lower semicontinuity of Lp -norm,we have Z 1 1 J(ϕ∗ ) = (T (ϕ∗ ) − zd )2 + |∆ϕ∗ |p dx p Z Ω 2 1 1 1 ∗ 2 ≤ (T (ϕ ) − zd ) + |∆ϕ∗ |p + |ϕ∗ − ϕ| ¯ p dx 2Z p p Ω 1 1 1 2 (Tε (ϕε ) − zd ) + |∆ϕε |p + |ϕε − ϕ| ¯ p dx ≤ lim inf ε 2 p p Ω = lim inf Jε (ϕε ) ≤ lim sup Jε (ϕε ) ≤ lim sup Jε (ϕ) ¯ ε ε ε Z 1 1 (Tε (ϕ) ¯ − zd )2 + |∆ϕ| ¯ p dx = lim sup 2 p ε Ω Z 1 1 = (T (ϕ) ¯ − zd )2 + |∆ϕ| ¯ p dx 2 p Ω = J(ϕ). ¯ ∗

∗

∗

Then, J(ϕ∗ ) ≤ J(ϕ∗ ) +

1 p

Z

|ϕ∗ − ϕ| ¯ p dx ≤ J(ϕ) ¯

(49)

(50)

Ω

On ¯ ≤ J(ϕ∗ ). Thus Z the other hand, ϕ¯ is optimal to Problem (P), and then J(ϕ) |ϕ∗ − ϕ| ¯ p dx = 0, i.e. ϕ∗ = ϕ¯ a.e. in Ω. By the uniqueness of the solution for Ω

(1.2), we get y ∗ = y¯. The proof is completed.

(51)

5. Optimality conditions We first prove the following optimality conditions for approximate Problem (Pε ). Lemma 5.1. Assume that (yε , ϕε ) is an optimal pair for Problem (Pε ). Then there exists a unique pε ∈ H01 (Ω), such that the triple (yε , ϕε , pε ) satisfies the following

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equations: Z

A(x, ∇yε ) · ∇ωdx + Ω

Z =

Z +

Z Ω

ZΩ

∇φT

Z

1 β(yε − ϕε )ωdx = 0, Ωε

∂A (x, ∇yε )T ∇pε dx + ∂η

Z

∀ω ∈ W01,p (Ω).

1 0 β (yε − ϕε )pε φdx Ωε

∀φ ∈ H01 (Ω).

(yε − zd )φdx, Ω

|ϕε − ϕ| ¯ p−2 (ϕε − ϕ)ψdx ¯ +

Z

1 0 β (yε − ϕε )pε ψdx Ωε

|∆ϕε |p−2 (∆ϕε )∆ψdx = 0,

(52)

(53)

(54)

∀ψ ∈ W.

Ω

Proof. Let (yε , ϕε ) be an optimal pair for Problem (Pε ). Take ψ ∈ W . Similar to [5], one can show that Tε (ϕε + δψ) − Tε (ϕε ) → ξε δ

(δ → 0),

(55)

where ξε is the solution of the following equation:   −div ∂A (x, ∇y )∇ξ + 1 β 0 (y − ϕ )ξ = 1 β 0 (y − ϕ )ψ, ε ε ε ε ε ε ε ∂η ε ε  ξε |∂Ω = 0.

(56)

As ϕε is optimal to Problem (Pε ), we get Jε (ϕε + δψ) − Jε (ϕε ) 0 ≤ lim inf δ→0 δ Z = {(yε − zd )ξε + |ϕε − ϕ| ¯ p−2 (ϕε − ϕ) ¯ ·ψ Ω +|∆ϕε |p−2 (∆ϕε )

(57)

· ∆ψ}dx.

Let pε ∈ H01 (Ω) be the solution of the equation:   −div( ∂A (x, ∇y )T ∇p ) + 1 β 0 (y − ϕ )p = y − z ε ε ε ε ε ε d ∂η ε  p | =0 ε ∂Ω

in Ω

(58)

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Then we have Z 0≤ {(yε − zd )ξε + |∆ϕε |p−2 (∆ϕε ) · ∆ψ + |ϕε − ϕ| ¯ p−2 (ϕε − ϕ) ¯ · ψ}dx Z Ω ∂A 1 = {[−div( (x, ∇yε )T ∇pε ) + β 0 (yε − ϕε )pε ]ξε + |ϕε − ϕ| ¯ p−2 (ϕε − ϕ) ¯ ·ψ ∂η ε Ω p−2 +|∆ϕ Z ε | (∆ϕε ) · ∆ψ}dx ∂A 1 = {∇ξεT (x, ∇yε )T ∇pε + β 0 (yε − ϕε )pε ξε + |ϕε − ϕ| ¯ p−2 (ϕε − ϕ) ¯ ·ψ ∂η ε Ω p−2 +|∆ϕ Z ε | (∆ϕε ) · ∆ψ}dx 1 ∂A (x, ∇yε )∇ξε )T ∇pε + β 0 (yε − ϕε )pε ξε + |ϕε − ϕ| ¯ p−2 (ϕε − ϕ) ¯ ·ψ = {( ∂η ε Ω p−2 +|∆ϕ Z ε | (∆ϕε ) · ∆ψ}dx 1 = {pε · β 0 (yε − ϕε )ψ + |ϕε − ϕ| ¯ p−2 (ϕε − ϕ) ¯ ·ψ ε Ω p−2 +|∆ϕε | (∆ϕε ) · ∆ψ}dx. (59) Since ψ ∈ W is arbitrary, we obtain that Z 1 {(|∆ϕε |p−2 ∆ϕε )∆ψ + |ϕε − ϕ| ¯ p−2 (ϕε − ϕ)ψ ¯ + β 0 (yε − ϕε )pε ψ}dx = 0. (60) ε Ω The proof is completed.

Lemma 5.2. Under the same assumptions in Lemma 5.1, we have, as ε → 0, w

pε * p¯,

in H01 (Ω),

1 ∗ − β 0 (yε − ϕε )pε * µ, ε

¯ in H −1 (Ω) ∩ M(Ω)

(61)

(62)

¯ is the set of all regular signed measures on Ω. ¯ where pε solves (53) and M(Ω) Proof. Taking φ = pε in (53), we have Z Z Z 1 0 2 T ∂A β (yε − ϕε )(pε ) dx + ∇pε ∇pε dx = (yε − zd )pε dx. ε Ω ∂η Ω Ω From (38) and (H2 ), we get Z Z 1 Λ1 (k + |∇yε |)p−2 |∇pε |2 dx + ε Ω

(63)

β 0 (yε − ϕε )(pε )2 dx ≤ Ckpε kL2 (Ω) . Ω

Then, kpε kH01 (Ω) ≤ C. Thus, (61) holds. Now, we prove (62).

(64)

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From (53), we get Z 1 0 β (yε − ϕε )pε φdx| | Ω Zε Z ≤| (yε − zd )φdx| + |

∂A(x, ∇yε )T ∇pε dx| ∂η Ω Ω ≤ C(kφkL2 (Ω) + kpε kH01 (Ω) k∇φkL2 (Ω) ) ∇φT

(65)

and thus (by (64)) 1 k β 0 (yε − ϕε )pε kH −1 (Ω) ≤ C. ε

(66)

Let Sδ (r) ∈ C 1 (R) be a family of smooth approximation to sign r, satisfying the following: Sδ0 (r) ≥ 0, and

 1 Sδ (r) = 0  −1

∀r ∈ R,

if if if

r > δ, r = 0, r < −δ.

Multiplying the equation of (58) by εSδ (pε ) and integrating it over Ω, then by (H2 ) and (38) we can get Z β 0 (yε − ϕε )pε Sδ (pε )dx ≤ Cε. Ω

Letting δ → 0, we have kβ 0 (yε − ϕε )pε kL1 (Ω) ≤ Cε.

(67)

Obviously, (62) holds. To obtain the optimality conditions for Problem (P), we first prove an auxiliary lemma, which is in fact equivalent to the necessary condition (stated in Theorem 5.4). Lemma 5.3. Let ϕ¯ be the optimal control to Problem (P). Then Z Z − (ϕ¯ − v)dµ + |∆v|p−2 ∆v∆(ϕ¯ − v)dx ≤ 0, ∀v ∈ W Ω

(68)

Ω

Proof. By (54),for any ψ ∈ W , we get Z Z 1 |ϕε − ϕ| ¯ p−2 (ϕε − ϕ) ¯ + β 0 (yε − ϕε )pε ψdx + ε Ω

|∆ϕε |p−2 ∆ϕε ∆ψdx = 0. Ω

From Lemma 2.4, for any v ∈ W , we have Z (|∆ϕε |p−2 ∆ϕε − |∆v|p−2 ∆v)(∆ϕε − ∆v)dx ≥ 0. Ω

(69)

(70)

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Taking ψ = ϕε − v in (69) yields R ¯ p−2 (ϕε − ϕ) ¯ + 1ε β 0 (yε − ϕε )pε (ϕε − v)dx ΩZ |ϕε − ϕ| + |∆ϕε |p−2 ∆ϕε ∆(ϕε − v)dx = 0.

(71)

Ω

Then, by (70), Z

1 |ϕε − ϕ| ¯ p−2 (ϕε − ϕ) ¯ + β 0 (yε − ϕε )pε (ϕε − v)dx ε ZΩ p−2 + |∆v| ∆v∆(ϕε − v)dx ≤ 0.

(72)

Ω

As p ≥ 2 and ϕε − v ∈ W , using (66) and (32) we get Z 1 0 | β (yε − ϕε )pε (ϕε − ϕ)dx| ¯ Ωε ≤ Ckϕε − ϕk ¯ H01 (Ω) → 0 (ε → 0). Then, by (62) we get Z 1 0 β (yε − ϕε )pε (ϕε − v)dx Z Ωε Z 1 0 = β (yε − ϕε )pε (ϕε − ϕ)dx ¯ + ZΩ ε →−

1 Ωε

β 0 (yε − ϕε )pε (ϕ¯ − v)dx

(73)

(74)

(ϕ¯ − v)dµ (ε → 0).

Ω

In the meantime, by Theorem 4.2 we have Z |ϕε − ϕ| ¯ p−2 (ϕε − ϕ)(ϕ ¯ ε − v)dx → 0 (ε → 0)

(75)

Ω

and

Z

|∆v|

p−2

∆v∆(ϕε − v)dx →

Ω

Z

|∆v|p−2 ∆v∆(ϕ¯ − v)dx (ε → 0).

Consequently, from (72), (74), (75) and (76) we get Z Z − (ϕ¯ − v)dµ + |∆v|p−2 ∆v∆(ϕ¯ − v)dx ≤ 0. Ω

(76)

Ω

(77)

Ω

This proves our Lemma. Now, we are in a position to derive the necessary conditions for Problem (P). Theorem 5.1. Let y¯, ϕ¯ be an optimal pair to Problem (P). Then there exists p¯ ∈ ¯ such that H01 (Ω) and µ ∈ H −1 (Ω) ∩ M(Ω) Z A(x, ∇¯ y )∇(v − y¯)dx ≥ 0, ∀v ∈ K(ϕ). ¯ (78) Ω

Z

∇φT Ω

Z Z ∂A (x, ∇¯ y )T ∇¯ pdx = φdµ + (¯ y − zd )φdx, ∀φ ∈ H01 (Ω). ∂η Ω Ω Z Z |∆ϕ| ¯ p−2 (∆ϕ)∆ψdx ¯ = ψdµ, ∀ψ ∈ W. Ω

Ω

(79) (80)

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Z

Proof. We first prove (79). By (53), we get Z Z ∂A 1 0 ∇φT (x, ∇yε )T ∇pε dx+ β (yε −ϕε )pε φdx = ∂η Ω Ωε

(yε −zd )φdx, Ω

∀φ ∈ H01 (Ω). (81)

From (5.11) and Theorem 4.2, for any φ ∈ H01 (Ω), we have Z Z 1 0 β (yε (ϕε ) − ϕε )pε φdx → − φdµ. Ωε Ω Z

(yε − zd )φdx → Ω

Z

(82)

(¯ y − zd )φdx.

(83)

Ω

Obviously,

∂A ∂A T T ∇φ (x, ∇yε ) ∇pε − (x, ∇¯ y ) ∇¯ p dx ∂η ∂η Ω Z ∂A ∂A = ∇φT (x, ∇yε )T − (x, ∇¯ y )T ∇pε dx ∂η ∂η ZΩ ∂A (x, ∇¯ y )T (∇pε − ∇¯ p)dx. + ∇φT ∂η Ω Z

T

(84)

From Lemma 5.2, we get Z ∂A ∇φT (x, ∇¯ y )T (∇pε − ∇¯ p)dx → 0 (ε → 0). ∂η Ω By (H2 ), we know that

(85)

n ∂a P j (x, η) is bounded. Then by Lebesgue dominated i,j=1 ∂ηi

convergence theorem 2 Z T ∂A ∂A T T ∇φ (x, ∇yε ) − (x, ∇¯ y ) dx → 0 ∂η ∂η Ω Thus, using H¨ older’s inequality, we have Z ∂A ∂A T T T ∇φ (x, ∇y ) − (x, ∇¯ y ) ∇p dx ε ε ∂η ∂η Ω 2 !1/2 Z Z T ∂A ∂A ∇φ ≤ (x, ∇yε )T − (x, ∇¯ y )T dx ∂η ∂η Ω → 0.

(ε → 0).

2

|∇pε | dx Ω

(86)

1/2

(87)

Hence, Z

∇φT Ω

and (79) holds.

∂A (x, ∇yε )T ∇pε → ∂η

Z

∇φT Ω

∂A (x, ∇¯ y )T ∇¯ pdx ∂η

(ε → 0)

(88)

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To get (80), we need only to prove that for any ψ ∈ W , as ε → 0, Z Z |∆ϕε |p−2 ∆ϕε ∆ψdx → |∆ϕ| ¯ p−2 ∆ϕ∆ψdx, ¯ Z Ω Ω |ϕε − ϕ| ¯ p−2 (ϕε − ϕ)ψdx ¯ → 0, Z Ω Z 1 0 β (yε − ϕε )pε ψdx → − ψdµ. Ωε Ω Then, passing to the limit, we can obtain (80) from (54). Applying H¨ older’s inequality, we get Z p | |ϕε − ϕ| ¯ p−2 (ϕε − ϕ)ψdx| ¯ ≤ kϕε − ϕk ¯ p−1 Lp (Ω) kψkL (Ω) .

(89)

(90)

Ω

Hence, by (32), we get

and by (62) we have

Z Z

|ϕε − ϕ| ¯ p−2 (ϕε − ϕ)ψdx ¯ → 0,

(91)

Ω

1 0 β (yε − ϕε )pε ψdx → − Ωε

Z

(92)

ψdµ. Ω

0

0

Since |∆ϕε |p−2 ∆ϕε is bounded in Lp (Ω), there exists a F ∈ Lp (Ω) such that w

0

|∆ϕε |p−2 ∆ϕε * F, in Lp (Ω). Thus, letting ε → 0, we get from (54) that Z Z F ∆ψdx − Ω

ψdµ = 0.

(94)

Ω

In particular, by taking ψ = ϕ¯ − v, we see that Z Z F ∆(ϕ¯ − v)dx − (ϕ¯ − v)dµ = 0. Ω

(93)

(95)

Ω

Then, applying the auxiliary condition (Lemma 5.3), we have Z (F − |∆v|p−2 ∆v)∆(ϕ¯ − v)dx ≥ 0.

(96)

Ω

Given ξ ∈ W . Let v = ϕ¯ + δξ, δ > 0, then Z − (F − |∆(ϕ¯ + δξ)|p−2 ∆(ϕ¯ + δξ))∆ξdx ≥ 0.

(97)

Ω

As δ → 0+ , we have −

Z

(F − |∆ϕ| ¯ p−2 ∆ϕ)∆ξdx ¯ ≥ 0.

(98)

Ω

On the other hand, let v = ϕ¯ − δξ, δ > 0, in the same way above, we get Z − (F − |∆ϕ| ¯ p−2 ∆ϕ)∆ξdx ¯ ≤ 0. Ω

(99)

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Combining (98) and (99), we have Z (F − |∆ϕ| ¯ p−2 ∆ϕ)∆ξdx ¯ = 0.

(100)

Ω

Since ξ ∈ W is arbitrary, we can assert that F = |∆ϕ| ¯ p−2 ∆ϕ, ¯ a.e. in Ω. This combined with (91)-(93) proves (89).

(101)

Remark 5.1. We can further prove that suppµ ⊂ {x ∈ Ω|¯ y (x) = ϕ(x)}, ¯

(102)

¯ with suppη ⊂ Ω+ , which is understood as the following: for any η ∈ C(Ω) < µ, η >M(Ω),C( ¯ ¯ = 0, Ω)

(103)

Ω+ = {x ∈ Ω|¯ y (x) > ϕ(x)}. ¯

(104)

where

References 1. D. R. Adams, S. M. Lenhart and J. Yong, Optimal control of obstacle for elliptic variational inequality, Appl. Math. Optim. 38, 121-140 (1998). 2. D. R. Adams and S. M. Lenhart, An obstacle control problem with a source term, Appl. Math. Optim. 47, 59-78 (2002). 3. V. Barbu, Necessary conditions for nonconvex distributed control problems governed by elliptic variational inequalities, J. Math. Anal. Appl. 80, 566-597 (1981). 4. V. Barbu, Optimal control of variational inequalities, Pitman, London (1984). 5. E. Casas and L. A. Fernandez, Distributed control of systems governed by a general class of quasilinear elliptic equations, J. Diff. Equations 104(1), 20-47 (1993). 6. Q. Chen, Indirect obstacle control problem for semilinear elliptic variational inequalities, SIAM J. Control Optim. 38, 138-158 (1999). 7. M. Chen, D. Deng and R. Long, Real Analyse, High Education Press (1993). 8. J. I. Diaz, Nonlinear partial differential equations and free boundaries, Vol. I. Elliptic Equations, research Notes in Math., Vol. 106, Pitman, London (1985). 9. L. C. Evans, A new proof of local C 1,α regularity for solutions of certain degenerate elliptic PDES, J. Diff. Equations 45, 356-373 (1982). 10. M. Fuchs and N. Fusco, Partial regularity results for vector valued functions which minimize certain functions having nonquadratic growth under smooth side conditions, J. Reine Angew. Math. 390, 67-78 (1988). 11. D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, 2nd Ed., Springer-Verlag, Berlin (1983). 12. J. Henonen, T. Kilel¨ ainen, and O. Martio, Nonlinear potential theory of second order degenerate elliptic partial differential equations, Oxford University Press (1993). 13. D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press (1980). 14. H. Lou H, On the regularity of an obstacle control problem, J. Math. Anal. Appl. 258, 32-51 (2001). 15. J. Rodrigues, Obstacle problem in mathematical physics, North Holland, 134 (1987).

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Chap33-GaoHang

CONTROLLABILITY OF A NONLINEAR DEGENERATE PARABOLIC SYSTEM WITH BILINEAR CONTROL∗

PING LIN Corresponding author, Graduate Student, School of Mathematics & Statistics, Northeast Normal University, Changchun 130024, CHINA E-mail: [email protected]. HANG GAO School of Mathematics & Statistics, Northeast Normal University, Changchun 130024, CHINA Email: [email protected] XU LIU Department of Mathematics, College of Science, Zhejiang University, Hangzhou 310000 and School of Mathematics & Statistics, Northeast Normal University, Changchun 130024, CHINA

In this paper, we discuss the controllability of a nonlinear degenerate parabolic system with bilinear control. Based on the shrinking property of the solutions, we prove that the system is not globally approximate controllable. Furthermore, we give an approximate controllability result for a special target. We also prove that the system is not globally exact null controllable by comparison principle. Keywords: Degenerate parabolic system, bilinear control, non-controllability, approximate controllability, exact null controllability.

1. Introduction and main results Let Ω be a bounded smooth domain of Rn with smooth boundary ∂Ω, we consider the controllability of the following degenerate bilinear control system not in ∗ This

work was partially supported by the NSF of China under grant 10471021. 279

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divergence form,   ut − u∆u = vu, in QT , u = 0, on ∂Ω × (0, T ),  u(x, 0) = u0 (x), in Ω,

(1)

where QT = Ω × (0, T ), 0 ≤ u0 ∈ C(Ω) ∩ H01 (Ω), v ∈ C α,α/2 (QT ), 0 < α < 1 is the control. Let us remind the reader that it is said that the system at hand is globally approximately controllable in Lp (Ω) (1 ≤ p < ∞) at time T > 0, if for every target u1 ∈ Lp (Ω) and for every ε > 0, there exists a control v in some given space such that the corresponding solution u satisfies ku(T ) − u1 kLp (Ω) < ε. In turn, it is said that the system at hand is exact null controllable at time T > 0, if, by selecting a suitable available control, it can be steered from any initial state within the given time-interval [0, T ] from any initial state to zero exactly. It is well-known that [2-11] a rather general semilinear parabolic equation, governed in a bounded domain by the classical either boundary or addictive locally distributed control is globally approximately controllable. The methods of these works make use of the fixed-point argument and the fact that such semilinear equations can be viewed as “linear equations” with the coefficients uniformly bounded in some sense. As to the works on controllability of the bilinear parabolic systems, in the pioneering work [12] by Ball, Mardsen and Slemrod, the global approximate controllability of the rod equation utt + uxxx + k(t)uxx = 0 with hinged ends and of the wave equation utt − uxx + k(t)u = 0 with Dirichlet boundary conditions, where k is control (the axial load), was shown by making use of the nonharmonic Fourier series approach under the additional (nontraditional) assumption that all the modes in the initial data are active. We also refer to [13] exploring the ideas of [12] in the context of simultaneous control of the rod equation and Schr¨ odinger equation. In [14], A.Y. Khapalov discussed the non-negative approximate controllability of the parabolic system with superlinear term ut = 4u + vu − f (x, t, u, ∇u)(∗), governed by a bilinear control v. It shows that the system (∗) is nonnegative globally approximately controllable in L2 (Ω), that is, for every ε > 0 and nonnegative u0 , ud ∈ L2 (Ω), u0 6= 0, there exists a T = T (ε, u0, ud ) and a bilinear control v ∈ L∞ (QT ) such that for all solutions of the system corresponding to the latter satisfy ku(T ) − u1 kL2 (Ω) < ε. In [15], the global approximate controllability of a semilinear heat equation with superlinear term ut = 4u + k(t)u + χω (x)v(x, t) − f (x, t, u, ∇u) was established at any positive time T > 0 in the case when a pair of controls govern the system at hand: (a) the traditional internal either locally distributed or lumped control v and (b) a piecewise constant bilinear control k. In one dimension space the method of [15] was further extended in [16] to the case dealing with bilinear controls only.

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Non-controllability of a control system at hand is of great interest to many people in recent years. It is pointed out in [14] in general, the system (∗) is not approximately controllable in any reasonable linear space. This can be illustrated by a quick analysis of the linear truncated version of it with f = 0. Indeed, due to the maximum principle, if, e.g., the initial state u0 (x) is nonnegative, then the maximum principle implies that the corresponding solution u(x, t) to the truncated linear version must remain nonnegative for all t > 0, regardless of the choice of v. Hence, one is unable to reach any of the “nonnegative” target states from a nonnegative initial state. In [10], it is proved that, for each β > 2, there exist functions f = f (s) with f (0) = 0 and |f (s)| ∼ |s|log β (1 + |s|) as |s| → ∞ such that the semilinear parabolic system yt − ∆y + f (y) = χω v is not null controllable for all T > 0. The proof of it is based on the fact that there are initial data which lead to blow-up before time T , whatever the control is. Arguments of this kind are well-known. For instance, see J. Henry [17] for the proof of the lack of approximate controllability of the heat equation with nonlinear absorption terms; see also O.Yu. Imanuvilov [18] and A. Fursikov and O.Yu. Imanuvilov [2] for examples of systems that fail to be null-controllable with power-like nonlinearities, i.e. in the more restrictive class of nonlinear terms growing at infinity, like |s|p with p > 1. Recall that, in the context of the semilinear wave equation, due to the finite speed propagation property, if blow-up occurs, exact-controllability cannot hold (see [19]). As we know, up to now, there are only a few works on the approximate controllability governed by degenerate parabolic equations. In [23], it is proved that the P-Laplace equation with traditionally locally distributive control is not controllable for the property of the finite propagation of the solution when time T is sufficiently small; However, as the control is acted on the entire Ω, the P-Laplace control system is globally approximately controllable. But a little is known for the degenerate equation with bilinear control like (1). The main interests in this paper center on the non-controllability of (1). We also prove that (1) is approximately null controllable by constant bilinear control. The main result are as follows. Theorem 1.1. Assume that 0 ≤ u0 ∈ C(Ω) ∩ H01 (Ω), supp u0 ⊂⊂ Ω, then the system (1) is not globally approximate controllable in Lp (Ω) for a.e. T > 0 and any 1 ≤ p < ∞. But for the special target 0, we have Theorem 1.2. System (1) is approximately null controllable in L 2 (Ω), that is, for any T > 0, 0 ≤ u0 ∈ C(Ω) ∩ H01 (Ω) and ε > 0, there exists a bilinear control v ∈ C α,α/2 (QT ) such that any solution of (1) with this control satisfies ku(T )kL2 (Ω) ≤ ε.

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Theorem 1.3. System (1) is not globally exact null controllable. The rest of this paper is organized as follows. In Section 2, we shall prove the existence of the solution of (1). In Section 3, we shall give the prove of Theorem 1.1. In Section 4, we shall prove Theorem 1.2. In Section 5, we shall give the counter example to prove that (1) is not globally exact null controllable. 2. Existence Let QT = Ω × (0, T ), U = L∞ (QT ) ∩ L2 (0, T ; H01 (Ω)). We give the definition of the weak solution to (1) as follows, Definition 2.1. 0 ≤ u ∈ U is a weak solution of (1) on [0, T ] if Z T Z TZ Z TZ < ut , ϕ > dt + ∇u∇(uϕ)dxdt = vuϕdxdt, 0

0

Ω

0

(2)

Ω

for every ϕ ∈ U , ut ∈ L2 (QT ), lim

t→0

Z

|u(x, t) − u0 (x)|dx = 0.

(3)

Ω

Let L = kvkL∞(QT ) , for any ε > 0, consider the sequence of problems   uεt − uε ∆uε = vuε + Lε, in QT , u = ε, on ∂Ω × (0, T ),  ε uε (x, 0) = u0 (x) + ε, in Ω.

(4)

The following regularity property gives us the main tool for proving the existence of a weak solution. Lemma 2.1. If uε is a classical solution of (4), then for all α ∈ (0, 1), Z TZ |∇uε |2 dxdt ≤ Cα . uα 0 Ω ε Z

T 0

Z

(uεt )2 dxdt ≤ C.

(5)

(6)

Ω

Proof. For any ε > 0, equation (4) admits a unique classical solution (see [26]). By maximum principle ([26]), we have ε ≤ uε ≤ C1 ,

(7)

where C1 depends on ku0 kL∞ (Ω) , kvkL∞(QT ) and T and independent of ε. Moreover, we have the following comparison theorem, uε1 ≤ uε2 , if 0 < ε1 < ε2 .

(8)

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Multiplying (4) by ϕ := u1α and integrating over QT , we have ε Z T Z TZ 1 uε < uεt , α > dt + ∇uε ∇( α )dxdt u u 0 0 Ω ε ε −

Z

Σ

∂uε 1−α (uε ) dsdt = ∂ν

Z

Z

T 0

v(uε )

1−α

dxdt +

Ω

Z

Z

T 0

L Ω

ε dxdt, uε α

∂ ∂ν

where Σ = ∂Ω × (0, T ), is the outward normal derivative to the boundary ∂Ω. ε Noticing that ∂u ≤ 0 and (7), we have ∂ν Z Z TZ |∇uε |2 1 (uε )(1−α) (T )dx + (1 − α) dxdt 1−α Ω uα 0 Ω ε ≤

1 1−α

Z

(uε )(1−α) (0)dx + L Ω

Z

T 0

Z

(uε )

1−α

dxdt +

Ω

Z

T 0

Z

Lε1−α dxdt. Ω

The integral on the right is bounded independent of ε, thus Z TZ |∇uε |2 ≤ Cα . uα 0 Ω ε Multiplying (4) by ϕ := uuεtε and integrating over QT , we have Z TZ Z (uεt )2 dxdt + |∇uε |2 (t)dx uε 0 Ω Ω =

Z

2

|∇uε | (0)dx + Ω

Z

T 0

Z

vuεt dxdt + Ω

Z

T 0

Z

Lε Ω

uεt dxdt. uε

Using (7) and (8) and ε-Young inequality, we have Z TZ Z Z TZ 1 2 2 (uεt ) dxdt ≤ |∇uε | (0)dx + δ(uεt )2 dxdt + C(δ), C1 0 Ω Ω 0 Ω where C(δ) is independent of ε. Select δ > 0 sufficiently small, we have Z TZ (uεt )2 dxdt ≤ C. 0

(9)

Ω

This proves Lemma 2.1. From Lemma 2.1, we have Theorem 2.1. (Existence)For every 0 ≤ u0 ∈ C(Ω) ∩ H01 (Ω) and every v ∈ C α,α/2 (QT ), problem (1) admits a weak solution on [0, T ]. Proof. In fact, From (5) and (6), we can assume that without loss of generality that uε → u

weakly in L2 (0, T ; H 1 (Ω)),

uε → u

strongly in L2 (QT ),

uεt → ut

weakly in L2 (QT ).

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Now we have to show that, indeed u satisfies Z T Z TZ Z < ut , ϕ > dt + ∇u∇(uϕ)dxdt − 0

0

Ω

T 0

Z

vuϕdxdt = 0

(10)

Ω

for every ϕ ∈ U . Then we need “strong” convergence of (∇uε )ε>0 in order to go to the limit in Z T Z TZ < uεt , ϕ > dt + (|∇uε |2 ϕ + uε ∇uε ∇ϕ)dxdt 0

−

Z

T 0

Z

0

Ω

v(x, t)uε ϕdxdt = 0.

(11)

Ω

For this purpose, we take as a test function in (11) ϕ := u2ε − (ε2 + u2 ), this yields Z T Z TZ < uεt , u2ε − (ε2 + u2 ) > dt + |∇uε |2 (u2ε − (ε2 + u2 ))dxdt 0

Z

1 + 2

T 0

1 =− 2

Z

Z

0

0

Ω

|∇(u2ε − u2 )|2 dxdt

Ω T Z

∇(u

2

)∇(u2ε

2

− u )dxdt +

Ω

Z

T 0

Z

v(x, t)uε (u2ε − (ε2 + u2 ))dxdt. (12) Ω

In view of (8), we have uε ≥ u a.e. in QT . Furthermore, due to the estimations previously obtained the first term on the left and the expression on the right convergence to zero as ε → 0, then Z TZ lim |∇(u2ε − u2 )|2 dxdt = 0. ε→0

The strong convergence of

0

Ω

∇(u2ε )

to ∇(u2 ) in L2 (QT ) furthermore implies

∇uε → ∇u in L2 (K)

(13)

strongly, where K denotes any set of the form {(x, t) ∈ QT | δ ∈ R, δ > 0, u(x, t) ≥ δ}. Combining (5) and (13), we obtain |∇uε |2 → |∇u|2

strongly in L1 (QT ).

In fact, we have

+

Z

0

T

Z

Z

T 0

Z

2

2

||∇uε | − |∇u| |dxdt ≤ δ Ω

|χ{uε ≥δ} |∇uε |2 − χ{u≥δ} |∇u|2 |dxdt + Ω

Z

T 0

Z

α

Z

T 0

Z

Ω

|∇uε |2 dxdt uα ε

(1 − χ{u≥δ} )|∇u|2 dxdt Ω

(14)

1, x ∈ A, When δ > 0 is such that {(x, t) ∈ QT |u(x, t) ≥ δ} is 0, otherwise. not empty. (otherwise it is trivial)

where χ(A) =

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Concluding, we can go to the limit in (11) as ε → 0 obtaining u satisfies (10). On the other hand, by (6), we have Z Z Z t |uε (x, t) − u0 (x) − ε|dx ≤ | uεt dτ |dx Ω

Ω

≤ t1/2 |Ω|1/2 (

Z Z Ω

0

t

(uεt )2 dτ dx))1/2 ≤ Ct1/2 , 0

where C is independent of ε. Let ε → 0, we have Z |u(x, t) − u0 (x)|dx ≤ Ct1/2 , Ω

Z

|u(x, t) − u0 (x)|dx → 0, as t → 0. Ω

Hence, problem (1) admits a weak solution, this proves theorem 2.1. Remark 2.1. We would like to mention that independently M. Ughi [19] has investigated (1) (v = 0) in one space dimension. In her article there is an explicit counterexample to uniqueness of (1). 3. Proof of Theorem 1.1 In order to prove Theorem 1.1, we need the following lemma. Lemma 3.1. (Shrinking of Support)For every T > 0 and every v ∈ C α,α/2 (QT ), let u be a weak solution of the problem (1) on [0, T ], then supp u(·, t) ⊂ supp u0 a.e. in (0, T ). Proof. Let ye = e−rt u, e c(x, t) = r − v(x, t), then ye satisfies  rt y − ce(x, t)y, in QT ,  yet = e ye∆e ye = 0, on ∂Ω × (0, T ),  ye(x, 0) = u0 (x), in Ω.

(15)

Let 0 < σ1 < 1 and set

ψσ1 (x) = inf {1;

1 dist(x, supp u0 ∪ ∂Ω)}. σ1

Obviously ψσ1 ∈ C(Ω) ∩ H01 (Ω) and ψσ1 (x)u0 (x) = 0 a.e. in Ω, define φ(x, t) = where ye is the solution of the problem (3.1). Multiplying (3.1) by φ and integrating the resulting relation over Qt = Ω × (0, t), we have Z tZ Z tZ yet φdxds + ers |∇e y |2 φdxds 0 Ω 0 Ω Z tZ Z tZ rs + e ye∇e y ∇φdxds + ce(x, s)e y φdxds = 0. (16) ψσ1 (x) y e(x,t)+ε ,

0

Ω

0

Ω

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Noticing that ce(x, s) and ye are nonnegative, we have Z tZ Z tZ Z tZ rs 2 yet φdxds + e |∇e y | φdxds + ers ye∇e y ∇φdxds ≤ 0 0

Ω

0

Ω

∇φ =

0

1 ((e y + ε)∇ψσ1 − ψσ1 ∇y). (e y + ε)2

Substituting (18) in (17), we obtain Z tZ Z tZ ψσ 1 ψσ 1 dxds + ers |∇e y |2 dxds yet ye + ε ye + ε 0 Ω 0 Ω Z tZ Z tZ ∇ψσ1 ψσ 1 + ers ye∇e y dxds − ers ye|∇e y |2 dxds ≤ 0. y e + ε (e y + ε)2 0 Ω 0 Ω Since

Z tZ 0

ers |∇e y |2 Ω

=

Z tZ

ψσ 1 dxds − ye + ε

ers ψσ1 |∇e y |2 {

0

Ω

=

Z tZ 0

it follows that

Z tZ 0

≤ ert ( that is,

Therefore,

Z Z

Ω

0

Ω

ers ye|∇e y |2

0

(19)

ye 1 − }dxds ye + ε (e y + ε)2

ers ψσ1 |∇e ye|2

ε dxds ≥ 0 (e y + ε)2

Z tZ ψσ 1 ∇ψσ1 dxds ≤ − ers ye∇e y dxds ye + ε ye + ε 0 Ω Z Z TZ |∇e y |2 dxdt)1/2 ( |∇ψσ1 |2 dxdt)1/2 Ω

(18)

ψσ 1 dxds (e y + ε)2

yet

Ω T

Z

Z tZ

(17)

Ω

0

(20)

Ω

√ ψσ1 [ln(e y(x, t) + ε) − ln(u0 + ε)]dx ≤ Cert t. Ω

√ χ(supp ψσ1 )ψσ1 [ln(e y(x, t) + ε) − lnε]dx ≤ Cert t. Ω

From this inequality, we see that for every σ1 sufficiently small, Z √ χ({(x, t)|e y (x, t) > δ} ∩ {x|ψσ1 (x) = 1})(ln(e y(x, t) + ε) − lnε)dx ≤ Cert t. Ω

√ Since Cert t is independent of ε, we may conclude that

mes({(x, t)|e y (x, t) > δ} ∩ {x|ψσ1 (x) = 1}) = 0. Due to the arbitrariness of σ1 ∈ (0, 1), we obtain supp ye(·, t) ⊂ supp u0 a.e. in (0, T ).

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Noticing that

we have

supp ye(·, t) = supp u(·, t), ∀t ∈ (0, T ), supp u(·, t) ⊂ supp u0 a.e. in (0, T ).

This proves the Lemma 3.1. Proof of Theorem 1.1. From Lemma 3.1, it is easily seen that if supp u0 ⊂⊂ Ω, then for every T > 0 and for every v ∈ C α,α/2 (QT ), we have any of the corresponding solutions u of (1) on [0, T ] satisfies supp u(·, t) ⊂⊂ suppu0 ⊂ Ω a.e. in (0, T ). On the other hand, for every 1 ≤ p < ∞, there exist ε0 > 0 and u1 ≥ 0 such that ku1 kLp (Ω\supp u0 ) ≥ ε0 , this implies that the system (1) could not be globally approximately controllable by any bilinear control v in C α,α/2 (QT ) for a.e. T > 0, this proves the theorem 1.1. Remark 3.1. From the proof of theorem 1.1, we can see that there exists the nonzero target such that the system (1) could not be approximately controllable by bilinear control. But for the zero target, you will find that (1) is controllable in the following proof of Theorem 1.2. 4. Proof of Theorem 1.2 Proof. For every T > 0, every 0 ≤ u0 ∈ C(Ω) ∩ H01 (Ω) and every v ∈ C α,α/2 (QT ), by theorem 2.1, the following system   ut = u∆u + vu, in QT , (21) u = 0, on ∂Ω × (0, T )  u(x, 0) = u0 (x), in Ω,

admits at least one weak solution. Hence, if we choose v = −v0 , where v0 is a positive constant, then for any solution u of (21) with 0 ≤ u ∈ U and ut ∈ L2 (QT ), multiplying (21) by ev0 t u and integrating over QT , we have Z Z Z TZ Z Z 1 T 1 T v0 t 2 v0 t 2 (e u )t dx + 2 e u|∇u| dxdt + v0 ev0 t u2 dx = 0. (22) 2 0 Ω 2 0 Ω 0 Ω Noticing that v0 > 0 and u ≥ 0, we have Z Z 2 −v0 T u (T )dx ≤ e u20 (x)dx. Ω

Ω

(23)

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Then as v0 → +∞, (23) implies for every ε > 0, there exists a positive constant v0 > 0 such that for every solution of (21), we have ku(·, T )kL2(Ω) ≤ ε. This completes the proof of Theorem 1.2. 5. Proof of Theorem 1.3 In order to prove Theorem 1.3, we need the following comparison principle, Lemma 5.1. (Comparison theorem) Let u1 be a supersolution of (1) such that u1 ≥ ε > 0 in QT , and u2 a subsolution of (1). If u2 (·, 0) ≤ u1 (·, 0), then u2 ≤ u1 a.e. in QT . Proof. Take as a test function in (2.1) ϕ1 =

signδ ((u2 − u1 )+ ) , u1

ϕ2 =

signδ ((u2 − u1 )+ ) , u2

where signδ (z) = sign(z)inf(|z|/δ, 1), then we get Z TZ (log u2 − log u1 )t signδ ((u2 − u1 )+ )dxdt 0

+

Z

Ω

T 0

Z

|∇(u2 − u1 )|2 sign0δ ((u2 − u1 )+ )dxdt = 0. Ω

Hence, Z

T 0

Z

(log u2 − log u1 )t signδ ((u2 − u1 )+ )dxdt ≤ 0. Ω

Let δ → 0, we have Z TZ 0

(log u2 − log u1 )t sign((u2 − u1 )+ )dxdt ≤ 0. Ω

Noticing that sign((u2 − u1 )+ ) = sign((logu2 − logu1 )+ ), we have

Z

T 0

Z

(log u2 − log u1 )t sign((logu2 − logu1 )+ )dxdt ≤ 0. Ω

By u2 (·, 0) ≤ u1 (·, 0), we have Z (log u2 − log u1 )(x, t)dx ≤ 0, Ω

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which implies (log u2 − log u1 )(x, t) = 0 a.e. in QT . Hence, u2 ≤ u1 a.e. in QT . The following theorem gives the main tool for proving Theorem 1.3, Theorem 5.1. Let 0 ≤ u0 ∈ C(Ω) ∩ H01 (Ω) satisfy: u0 > 0 in Int(suppu0 ), then for every v ∈ C α,α/2 (QT ) and every δ > 0, there exists βδ > 0 such that u(x, t) ≥ βδ a.e.in Ωδ × (0, T ), where Ωδ = {x ∈ supp u0 |dist(x, ∂supp u0 ) ≥ δ} and u is the solution which obtained in section 2, (i.e. u = lim uε , where uε is the solution of(4)). ε→0

Proof. For every δ > 0, it is easily seen that there exists βeδ ∈ (0, 1) such that u0 (x) ≥ βeδ in Ωδ/2 , for every v ∈ C α,α/2 (QT ), let ( βeδ φδ/2 e−(λδ/2 +L)t , in Ωδ/2 × (0, T ) (24) uδ = 0, otherwise where L = kvkL∞ (QT ) , ∆φδ/2 + λδ/2 φδ/2 = 0 in Ωδ/2 , φδ/2 > 0 in Ωδ/2 , φδ/2 = 0 on ∂Ωδ/2 . It is easy to verify that uδ is subsolution of (1) and for every ε > 0, uε is supersolution of (1), by comparison theorem uε ≥ uδ a.e. in QT , and in particular uε ≥ βeδ inf φδ/2 e−(λδ/2 +L)t ≥ inf φδ/2 e−(λδ/2 +L)T = βδ > 0 Ωδ

Ωδ

from which as ε → 0, u ≥ βδ > 0 a.e. in Ωδ × (0, T ). Theorem 5.1 implies Theorem 1.3. References

1. R. Dal Passo and S. Luckhaus, A degenerate diffusion problem not in divergence form, J. Diff. Eqs., 39(3), 378-412 (1981). 2. A.V. Fursikov, O. Yu. Imanuvilov, Controllability of Evolution Equations, Seoul University (1996). 3. V. Barbu, The Carleman inequality for linear parabolic equations in Lq norm, Differential and Integral Equations, 15, 513-525 (2002). 4. V. Barbu, A. Rascanu, G. Tessitore, Carleman estimates and controllability of stochastic heat equation with multiplicative noise, Applied Math. Optim., 47, 97-120 (2003). 5. C. Fabre, J.P. Puel, E. Zuazua, Approximate controllability of the semilinear heat equation, Proceeding of the Royal Society of Edinburgh, 125A, 31-61 (1995). 6. D.L. Russal, A unified boundary controllability theory for hyperbolic and parabolic differential equations, Studies in Applied Math., 52, 189-212 (1973). 7. E. Zuaua, Finite dimensional null controllability for the semilinear heat equation, Math. Pure Appl., 76, 237-264 (1997).

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8. Lue de Teresa, Approximate controllability of a semilinear heat equation in R N , Siam. J. Control Optim., 36, 2128-2147 (1998). 9. E. Zuazua, Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities, Control Cybernet., 28, 665-683 (1999). 10. E. Fern´ anez-Cara, E. Zuaua, Null and approximate controllability for weakly blowingup semilinear heat equations, Non Lin´ eaire., 17, 583-616 (2000). 11. E. Fern´ anez-Cara, Null controllability of the semilinear heat equation, Esiam: Control Optim. Calc. Var., 2, 87-107 (1997). 12. J.M. Ball, J.E. Mardsen, M. Slemrod, Controllability for distributed bilinear systems, Siam J. Control Optim. 20 (1982) 575-597. 13. K. Kime, Simultaneous control of a rod equation and a simple Schr¨ odinger equation, Systems Control Lett., 24, 301-306 (1995). 14. A.Y. Khapalov, Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: a qualitative approach, Siam J. Control Optim., 41, 1886-1990 (2003). 15. A.Y. Khapalov, Bilinear control for global controllability of the semilinear parabolic equations with superlinear terms, in Control of Nonlinear Distributed Prameter Systems, G. Chen, I. Lasiecka, and J. Zhou, eds., Marcel Dekker, New York, 139-155 (2001). 16. A.Y. Khapalov, Global non-negative controllability of the semilinear parabolic equation governed by bilinear control, Esaim. Control Optim. Calc. Var., 7, 269-283 (2002). 17. J. Henry, Contrˆ ole d’un r´eacteur enzymatique a ` l’aide de mod`eles a ` param`etres distribu´es: Quelques probl`emes de contrˆ olabilit´e de syst`emes paraboliques. Ph. D. Thesis, Universit´e Paris VI (1978). 18. O.Yu. Imanuvilov, Exact boundary controllability of the parabolic equation, Russian Math. Surveys, 48, 211-212 (1993). 19. E. Zuazua, Exact controllability for the semilinear wave equation in one space dimension, Ann. IHP, Analyse non Lin´ eaire, 10, 109-129 (1993). 20. M. Ughi, A degenerate parabolic equation modelling the spread of an epidemic, Ann. Mat. Pura. Appl., 143, 385-400 (1986). 21. A.Y. Khapalov, On bilinear controllability of the parabolic equation with the reactiondiffusion term satisfying Newton’s Law. J. Comput. Appl. Math, 21, 1-23 (2002). 22. Yin Jinxue, Gao Hang, Behavior of solutions to a degenerate diffusion problem, Acta Mathematicae Applicatae Sinica, 13(2), 188-195 (1997). 23. Hang Gao, Peidong Lei, Bo Zhang, A class of nonlinear degenerate integrodifferential control systems, Siam J. Control Optim., 43(3), 986-1010 (2004). 24. S. M¨ uller, Strong convergence and arbitrarily slow decay of energy for a class of bilinear control problems, J. Differential Equations, 81, 50-67 (1989). 25. M.E. Bradley, S. Lenhart, J. Yong, Bilinear optimal control of the velocity term in a Kirchhoff plate equation, J. Math. Appl. 238, 451-467 (1999). 26. O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Mono. 23, AMS, Providence RI (1968).

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Chap34-PanLiping

NEAR-OPTIMAL CONTROLS TO INFINITE DIMENSIONAL LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEM∗

LIPING PAN School of Mathematical Science, Fudan University, 220 Handan Road, Shanghai 200433, China E-mail: [email protected] QIHONG CHEN School of Mathematical Science, Fudan University, 777 Guoding Road, Shanghai 200433, China E-mail: [email protected]

This paper discusses how to find the near-optimal controls to (a class of) infinite dimensional linear-quadratic optimal control problems.

1. Introduction Linear-quadratic optimal control problems for infinite dimensional systems are a class of important objects of study in optimal control theory. The standard method to solve them is: solve the corresponding integral Riccati equations and then give the optimal (state) feedback controls via the solutions of the Riccati equations. However, this standard method is much difficult for use because the integral Riccati equation is a nonlinear operator(mapping a infinite dimensional space into itself)valued function equation, which is very hard to solve. In this paper we put forward an algorithm to find ε near-optimal controls to linear-quadratic optimal control problems for a class of infinite dimensional systems whose dynamic operators generate compact C0 -semigroups (on the state space). Notice that in applications the use effect of ε near-optimal controls are satisfying so long as the positive number ε is small enough. The key step of the algorithm we suggest here is to solve some largescale systems of linear algebraic equations with fine numerical properties, which is one of those work that large computers are good at. The near-optimal controls one can obtain by the above algorithm are continuous piecewise linear functions, which are easily performed in engineering by the holding device of order 1. ∗ The

work is supported by the Chinese NSF under grants 10571030 and 10171059. 291

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2. Formulation of our problem Suppose the following hypotheses hold: (H1) t0 , t1 ∈ R1 , t0 < t1 ; (H2) U and X are (real) Hilbert spaces, B ∈ L(U, X), A : D(A) → X generates a compact C0 -semigroup on X and f, x0 ∈ X; ∞ (H3) {Xn }n=1 is a sequence of finite dimensional subspaces of X, dim(Xn ) ≥ 1, n = 1, 2, · · · , lim dim(Xn ) = ∞, and there exists a sequence of projection operators in n→∞

∞

L(X) which we denoted them by {Πn }n=1 such that  Πn X = X n ,    Πn etA = etA Πn , ∀t ∈ [0, +∞),    ∀x ∈ X, lim | Πn x − x |= 0;

n = 1, 2, · · · ,

n→∞

(H4) Q(·) and W (·) are convex functions in C2 (X, R1 ); (H5) ∀G ∈ (0, +∞), supx∈X,|x|≤G | Q(x) |< +∞, supx∈X,|x|≤G | W (x) |< +∞, supx∈X,|x|≤G | (∇x Q)(x) |< +∞, supx∈X,|x|≤G | (∇x W )(x) |< +∞; 2

2

(H6) ∀G ∈ (0, +∞), supx∈X,|x|≤G | ∂∂xQ2 (x) |< +∞, supx∈X,|x|≤G | ∂∂xW2 (x) |< +∞; (H7) Q0 and W0 are semi-positive definite (self-adjoint) operators in L(X); (H8) R is a self-adjoint operator in L(U ) for which there exists a positive number δ0 > 0 such that ∀u ∈ U,

hRu, ui ≥ δ0 | u|2 .

For the infinite dimensional linear controlled system

x(t, u(·)) := e(t−t0 )A x0 +

Z

t

e(t−τ )A [Bu(τ ) + f ]dτ, t0

(1)

∀u(·) ∈ U := L2 (t0 , t1 ; U ), t ∈ [t0 , t1 ], the (nonquadratic) objective functional Z t1 1 J(u(·)) := [hQ(x(t, u(·)) + hRu(t), u(t)i]dt + W (x(t1 , u(·))), 2 t0

∀u(·) ∈ U (2)

and arbitrarily given positive number ε, if u ˆε (·) ∈ U such that J(ˆ uε (·)) ≤ ε + inf u(·)∈U J(u(·)),

(3)

then we call u ˆε (·) a ε near-optimal control to the following optimal control problem (LNQP) Find u ˆ(·) ∈ U such that J(ˆ u(·)) = min J(u(·)). u(·)∈U

(4)

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(When ∀x ∈ X,

Q(x) =

1 1 hQ0 x, xi, W (x) = hW0 x, xi, 2 2

(5)

we especially denote (LNQP) by (LQP) which is called linear-quadratic optimal control problem as usual.) The main purpose of this paper is to derive an algorithm to find ε near-optimal controls to (LQP).

3. A convergence result ∗

By (H2)-(H3) and (e·A )∗ = e·A , it is easy to see that Xn ⊆ D(A) ∩ D(A∗ ), AXn ∪ ∗ A∗ Xn ⊆ Xn and the restriction of e·A (resp. e·A ) on Xn is a C0 -semigroup (on finite dimensional space Xn ) whose generator An (resp. A∗n ) is the restriction of ∗ A(resp. A∗ ) on Xn , and (e·An )∗ = e·An , n = 1, 2, · · · . Consider the following optimal control problem (LNQP)n Find u ˆn (·) ∈ U such that

Jn (ˆ un (·)) = min Jn (u(·)),

(6)

u(·)∈U

where xn (t, u(·)) = e(t−t0 )An xn0 +

Z

t

et−τ )An [Bn u(τ )+fn ]dτ,

∀u(·) ∈ U, t ∈ [t0 , t1 ], (7)

t0

xn0 := Πn x0 ∈ Xn , Bn := Πn B ∈ L(U, Xn ), fn := Πn f ∈ Xn and

Jn (u(·)) :=

Z

t1 t0

1 [Q(xn (t, u(·))) + hRu(t), u(t)i]dt + W (xn (t1 , u(·))), 2

∀u(·) ∈ U (8)

n = 1, 2, · · · (When (5) holds we especially denote (LNQP)n by (LQP)n .), and (LNQP). From the discusses of [2], we can know that (if (H1)∼(H4) and (H6) hold then) there exists an unique solution (i. e. optimal control) to (LNQP)n , n = 1, 2, · · · and (LNQP) also has an unique solution. Let us denote the optimal control of (LNQP) by u ˆ(·), xˆ(·) := x(·, u ˆ), and the one of (LNQP)n by u ˆn (·), x ˆn (·) := n n x (·, u ˆ ), n = 1, 2, · · · ; then by Pontryagin’s maximum principle we have ∀t ∈ [t0 , t1 ],

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 n u ˆ (t) = R−1 B ∗ Πn ψˆn (t),         Z t1  ∗ ψˆn (t) := −[ e(σ−t)A Πn (∇x Q)(ˆ xn (σ))dσ   t       ∗  + e(t1 −t)A Πn (∇x W )(ˆ xn (t1 ))], n = 1, 2, · · · ,

 ˆ u ˆ(t) = R−1 B ∗ ψ(t),         Z t1  ∗ ˆ ψ(t) := −[ e(σ−t)A (∇x Q)(ˆ x(σ))dσ   t       ∗  + e(t1 −t)A (∇x W )(ˆ x(t1 ))].

(9)

Theorem 3.1. Let (H1)∼(H5) and (H8) hold. Then lim

max | u ˆn (t) − u ˆ(t) |= 0

n→∞ t0 ≤t≤t1

(10)

and lim min Jn (u(·)) = min J(u(·)) (i. e. lim Jn (ˆ un (·)) = J(ˆ u(·))).

n→∞ u(·)∈U

u(·)∈U

n→∞

(11)

Proof. First, from (9) we immediately know  ˆ(·) ∈ C([t0 , t1 ], X), u 

(12)

u ˆn (·) ∈ C([t0 , t1 ], Xn ), n = 1, 2, · · · ;

Second, by (H4)

Q(z) ≥ Q(y) + h(∇x Q)(y), z − yi, ∀y, z ∈ X. W (z) ≥ W (y) + h(∇x W )(y), z − yi, Besides (12)–(13), noting (H8), we have ∀u(·) ∈ U,

(13)

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Jn (u(·)) ≥

Z

t1

Q(e

(t−t0 )A

Π n x0 +

t0

W (e(t1 −t0 )A Πn x0 + Z

Z

t1

Z

t1

t

e(t−σ)A Πn f dσ)dt+ t0

e(t1 −σ)A Πn f dσ)+

t0

h(∇x Q)(e(t−t0 )A Πn x0 +

t0

h(∇x W )(e(t1 −t0 )A Πn x0 +

Z

Z

t1

t

e(t−σ)A Πn f dσ), t0

e(t1 −σ)A Πn f dσ),

t0

Z

Z

t

e(t−τ )A Πn Bu(τ )dτ idt+ t0 t1

e(t1 −τ )A Πn Bu(τ )dτ i+

t0

δ0 k u(·)k2U 2 =

δ0 2 k u(·)kU + hB ∗ Πn [ 2

Z

t1

∗

e(t−·)A (∇x Q)(e(t−t0 )A Πn x0 +

·

∗

e(t1 −·)A (∇x W )(e(t1 −t0 )A Πn x0 + Z

t1 t0

Q(e(t−t0 )A Πn x0 +

W (e(t1 −t0 )A Πn x0 +

Z

Z

t1

Z

t1

Z

t

e(t−σ)A Πn f dσ)dt+ t0

e(t1 −σ)A Πn f dσ)], u(·)iU +

t0

t

e(t−σ)A Πn f dσ)dt+ t0

e(t1 −σ)A Πn f dσ).

t0

(14)

Therefore, define

αn :=k B ∗ Πn [

Z

t1

∗

e(t−·)A (∇x Q)(e(t−t0 )A Πn x0 +

·

∗

e(t1 −·)A (∇x W )(e(t1 −t0 )A Πn x0 +

βn :=

Z

t1

Z

t1

Z

t

e(t−σ)A Πn f dσ)dt+ t0

e(t1 −t0 )A Πn f dσ)]kU ,

(15)

t0

Q(e(t1 −t0 )A Πn x0 )dt + W (e(t1 −t0 )A Πn x0 )

t0

then we easily deduce

∀u(·) ∈ U, Jn (u(·)) ≥

δ0 2 k u(·)kU − αn k u(·)kU + βn , 2

(16)

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n = 1, 2, · · · from (14). Noting that e·A is a C0 -semigroup on X ⇒ there exist two positive numbers M and ω such that

∀t ∈ [0, +∞), | etA |≤ M eωt

(17)

and by (15), for n = 1, 2, · · · we have: α2n =

Z

t1

| B ∗ Πn [ t0

+

Z

+

Z

Z

t1 τ

∗

e(t−τ )A (∇x Q)(e(t−t0 )A Πn x0

t

∗

e(t−σ)A Πn f dσ)dt + e(t1 −τ )A (∇x W )(e(t1 −t0 )A Πn x0 t0 t1

2

e(t1 −σ)A Πn f dσ)]| dτ

t0 2

2

≤ 2 | B ∗ | | Πn | [

+

Z

+

Z

Z

t1

| t0

Z

t1

(18)

τ

t

2

e(t−σ)A Πn f dσ)dt| dτ + t0 t1

∗

e(t−τ )A (∇x Q)(e(t−t0 )A Πn x0 Z

t1 t0

∗

| e(t1 −τ )A (∇x W )(e(t1 −t0 )A Πn x0

e(t1 −σ)A Πn f dσ)|2 dτ ]

t0

≤ 2 | B|2 (t1 − t0 )[M eω(t1 −t0 ) ]2 × {(t1 − t0 )2 [

| (∇x Q)(x) |]2 + [

sup x∈X,|x|≤N

| (∇x W )(x) |]2 }

sup x∈X,|x|≤N

(where

N := M eω(t1 −t0 ) [| x0 | +(t1 − t0 ) | f |]),

(19)

which implies αn ≤ α :=M | B | eω(t1 −t0 ) × {2{(t1 − t0 )2 [

(20) sup

x∈X,|x|≤N

| (∇x Q)(x) |]2 + [

sup x∈X,|x|≤N

1

| (∇x W )(x) |]2 }} 2 ;

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| βn |≤

Z

t1

| Q(e

(t−t0 )A

Π n x0 +

t0

+ | W (e(t1 −t0 )A Πn x0 + ≤ β := (t1 − t0 )

sup

Z

Z

t

e(t−σ)A Πn f dσ) | dt t0 t1

e(t1 −σ)A Πn f dσ) |

(21)

t0

| Q(x) | +

x∈X,|x|≤N

sup

| W (x) | .

x∈X,|x|≤N

Thus, from (16), (20) and (21) we immediately know that Jn (u(·)) ≥

δ0 k u(·)k2U − α k u(·)k2U − β 2

δ0 α α2 α2 = (k u(·)kU − )2 − (β + ) ≥ −(β + ) 2 δ0 2δ0 2δ0

(22)

∀u(·) ∈ U. Taking u(·) = u ˆn (·) in the above we get −(β +

α2 δ0 α α2 ) ≤ (k u ˆn (·)kU − )2 − (β + ) ≤ Jn (ˆ un (·)) 2δ0 2 δ0 2δ0

(23)

= min Jn (u(·)) ≤ Jn (0) = βn ≤ β. u(·)∈U

Hence, α |k u ˆ (·)kU − |≤ δ0 n

s

2β +

α2 , 2δ0

with n = 1, 2, · · · . Thereby we successively have s α2 α n + 2β + , sup k u ˆ (·)kU ≤ γ := δ0 2δ0 n≥1

(24)

(25)

sup max | x ˆn (t) | n≥1 t0 ≤t≤t1

≤M eω(t1 −t0 ) [| x0 | + | B |

√ t1 − t 0 k u ˆn (·)kU + (t1 − t0 ) | f |]

(26)

√ √ =M1 := M eω(t1 −t0 ) [| x0 | + t1 − t0 ( t1 − t0 | f | + | B | γ)], sup max | ψˆn (t) | n≥1 t0 ≤t≤t1

≤M2 := M eω(t1 −t0 ) [(t1 − t0 ) +

sup

| (∇x Q)(x) |

x∈X,|x|≤M1

sup

| (∇x W )(x) |]

x∈X,|x|≤M1

and (by (9), the above and (H8) we immediately know)

(27)

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sup max | u ˆn (t) |≤

n≥1 t0 ≤t≤t1

| B | M2 . δ0

(28)

Next, take a (number) η ∈ (0, t1 − t0 ) arbitrarily. ∀t0 , t” ∈ [t0 , t1 − η], t0 ≤ t”, according to (9) and (26), we have | ψˆn (t”) − ψˆn (t0 ) | =|

Z

+

t”

0

∗

e(σ−t )A Πn (∇x Q)(ˆ xn (σ))dσ

t0

Z

t1

0

∗

∗

[e(σ−t )A − e(σ−t”)A ]Πn (∇x Q)(ˆ xn (σ))dσ

t” 0

+ [e(t1 −t )A − e(t1 −t”)A ]Πn (∇x W )(ˆ xn (t1 )) | 0

∗

≤ | [e(t”−t )A − IX ] Z t1 ∗ e(σ−t”)A Πn (∇x Q)(ˆ xn (σ))dσ ×[ t”

(t1 −t”)A∗

+e Z +|

t”

0

∗

e(σ−t )A Πn (∇x Q)(ˆ xn (σ))dσ |

t1

− IX ]e

+

Z

ηA∗

∗

e(σ−t”−η)A Πn (∇x Q)(ˆ xn (σ))dσ

t”+η

(t1 −η−t”)A∗

+e Z +

(29)

t0

(t”−t0 )A∗

≤ | [e Z ×[

Πn (∇x W )(ˆ xn (t1 ))] |

t”+η

Πn (∇x W )(ˆ xn (t1 ))] | 0

∗

∗

| e(σ−t )A − e(σ−t”)A || (∇x Q)(ˆ xn (σ)) | dσ t” t”

0

∗

| e(σ−t )A || (∇x Q)(ˆ xn (σ)) | dσ

t0 0

∗

≤ max | [e(t”−t )A − IX ]x | x∈eηA∗ S

+ M eω(t1 −t0 ) (2η + t” − t0 )

sup

| (∇x Q)(x) |

x∈X,|x|≤M1

where

(

S := x ∈ X | x |≤ M eω(t1 −t0 ) (t1 − t0 ) +

sup x∈X,|x|≤M1

sup

| (∇x Q)(x) |

x∈X,|x|≤M1

) | (∇x W )(x) | .

(30)

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Therefore lim

sup | ψˆn (t0 ) − ψˆn (t”) |

lim

{ max | [e|t −t”|A − IX ]x |

t0 ,t”∈[t0 ,t1 −η], |t0 −t”|↓0 n≥1

≤

0

∗

t0 ,t”∈[t0 ,t1 −η], |t0 −t”|↓0 x∈eηA∗ S

+ M eω(t1 −t0 ) (2η+ | t0 − t” |)

sup

| (∇x Q)(x) |}

(31)

x∈X,|x|≤M1

=2ηM eω(t1 −t0 )

sup

| (∇x Q)(x) | .

x∈X,|x|≤M1

Consequently (note the arbitrariness of η) lim

sup | ψˆn (t0 ) − ψˆn (t”) |

lim

sup | ψˆn (t0 ) − ψˆn (t”) |

t0 ,t”∈[t0 ,t1 −η], |t0 −t”|↓0 n≥1

≤

(32)

t0 ,t”∈[t0 ,t1 −˜ η ], |t0 −t”|↓0 n≥1

≤ 2˜ η M eω(t1 −t0 )

sup

| (∇x Q)(x) |,

∀˜ η ∈ (0, η],

x∈X,|x|≤M1

which means sup | ψˆn (t0 ) − ψˆn (t”) |= 0.

lim

t0 ,t”∈[t0 ,t1 −η], |t0 −t”|↓0 n≥1

(33)

Now let us testify ∀t ∈ [t0 , t1 − η], {ψˆn (t)|n = 1, 2, · · · } is totally bounded. Take (a) t ∈ [t0 , t1 − η] arbitrarily. ∀ε > 0, take a δε ∈ (0, η) small enough such that Z t+δε ∗ e(σ−t)A Πn (∇x Q)(ˆ xn (σ))dσ | sup | n≥1 t (34) ε | (∇x Q)(x) |< . ≤M eω(t1 −t0 ) δε sup 3 x∈X,|x|≤M1 Define φˆnδε (t) := −[

Z

t1 t+δε

∗

∗

e(σ−t−δε )A Πn (∇x Q)(ˆ xn (σ))dσ + e(t1 −t−δε )A Πn (∇x W )(ˆ xn (t1 ))], n = 1, 2, · · · ,

Sδε := {φˆnδε (t)|n = 1, 2, · · · }. (35) Obviously Sδε ⊆ S,

(36)

so ∗

∗

eδε A Sδε ⊆ eδε A S.

(37)

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∗

The above inclusion together with the compactness of eδε A S imply eδε A Sδε is totally bounded. Therefore a finite subset of natural numbers set {n1 (ε), · · · , nk() ()} such that ∗

n (ε)

∃j ∈ {1, · · · , k(ε)}, | eδε A [φnδε (t) − φδεj

∀n ∈ {1, 2, · · · },

(t)] |<

ε . 3

(38)

By the definition of ψˆn (·) in (9), (34) and above, ∀n ∈ {1, 2, · · · },

∃j ∈ {1, · · · , k(ε)},

∗ n (ε) | ψˆn (t) − ψˆnj (ε) (t) |=| eδε A [φˆnδε (t) − φˆδεj (t) Z t+δε ∗ e(σ−t)A Πn (∇x Q)(ˆ xn (σ))dσ −

t

+

Z

t+δε t

∗

e(σ−t)A Πnj (ε) (∇x Q)(ˆ xnj (ε) (σ))dσ |

∗ n (ε) ≤ | eδε A [φˆnδε (t) − φˆδεj (t)] | Z t+δε ∗ e(σ−t)A Πn (∇x Q)(ˆ xn (σ))dσ | +|

(39)

t

+|

Z

t+δε t

∗

e(σ−t)A Πnj (ε) (∇x Q)(ˆ xnj (ε) (σ))dσ |

ε 2ε < + = ε, 3 3 hence {ψˆn (t)|n = 1, 2, · · · } is a totally bounded subset of Hilbert space X. So far, we have had (27), (33) and the just proved relative compactness of {ψˆn (t)|n = 1, 2, · · · } (∀t ∈ [t0 , t1 − η]), ∀η ∈ (0, t1 − t0 ). According to Arzela-Ascoli theorem (infinite ∞ dimensional case) and using diagonal method, for {ψˆnm (·)}m=1 (an arbitrary sub∞ ∞ sequence of {ψˆn (·)}n=1 ) we can abstract a subsequence {ψˆnml (·)}l=1 (from among ∞ {ψˆnm (·)}m=1 ) to which there exists a ψˆ∞ (·) ∈ C([t0 , t1 ), X) such that ∀η ∈ (0, t1 − t0 ), lim

max

l→∞ t0 ≤t≤t1 −η

| ψˆnml (t) − ψˆ∞ (t) |= 0.

(40)

Letting u ˆ∞ (t) := R−1 B ∗ ψˆ∞ (t), ∀t ∈ [t0 , t1 )

(41)

and reviewing the first expression of (9), (H3) and (H8), we immediately have from (40) that

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∀t ∈ [t0 , t1 ), | u ˆnml (t) − u ˆ∞ (t) |=| R−1 B ∗ [Πnml ψˆnml (t) − ψˆ∞ (t)] | ≤| R−1 || B | {| Πnml [ψˆnml (t) − ψˆ∞ (t)] | + | (Πnml − IX )ψˆ∞ (t) |} ≤

(42)

1 | B | [| ψˆnml (t) − ψˆ∞ (t) | + | (Πnml − IX )ψˆ∞ (t) |] → 0(l → ∞). δ0

The above together with (28) implies (by Lebesgue dominated convergence theorem) sup

| B | M2 , δ0

|u ˆ∞ (t) |≤

t0 ≤t
lim k u ˆ

l→∞

nml

(43)

(·) − u ˆ (·)kU = 0. ∞

Letting xˆ∞ (t) := x(t, u ˆ∞ (·)), ∀t ∈ [t0 , t1 ],

(44)

then by x ˆn (·) := x(·, u ˆn ), n = 1, 2, · · · , the above, (H3), (43) and Lebesgue dominated convergence theorem we have lim max | xˆnml (t) − xˆ∞ (t) |

l→∞ t0 ≤t≤t1

= lim max | e(t−t0 )A (Πnml − IX )x0 + l→∞ t0 ≤t≤t1

+

Z

+

Z

t t0

Z

t t0

e(t−τ )A (Πnml − IX )f dτ

e(t−τ )A Πnml B[ˆ unml (τ ) − u ˆ∞ (τ )]dτ

t t0

e(t−τ )A (Πnml − IX )B u ˆ∞ (τ )dτ |

(45)

≤ M eω(t1 −t0 ) lim | (Πnml − IX )x0 | +(t1 − t0 ) | (Πnml − IX )f | l→∞

p ˆnml (τ ) − u ˆ∞ (τ )kU + | B | (t1 − t0 ) k u Z t1 1 + [(t1 − t0 ) | (Πnml − IX )B u ˆ∞ (τ )|2 dτ ] 2 t0

= 0. By the first expression of (9), (40), (42), the above and (H3) we know  ∞ ˆ (t) = R−1 B ∗ ψˆ∞ (t),  u Z t1 ∗ ∗   ψˆ∞ (t) = −[ e(σ−t)A (∇x Q)(ˆ x∞ (σ))dσ + e(t1 −t)A (∇x W )(ˆ x∞ (t1 ))] t

for any t ∈ [t0 , t1 ] right away

(46)

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(additionally define ψˆ∞ (t1 ) := −(∇x W )(ˆ x∞ (t1 )), u ˆ∞ (t1 ) := R−1 B ∗ ψˆ∞ (t1 )).

(47)

Relations (46)–(47), together with Hypotheses (H4) and (H8) etc. imply u ˆ∞ (·) = u ˆ(·);

(48)

ˆ xˆ∞ (·) = xˆ(·), ψˆ∞ (·) = ψ(·).

(49)

consequently

Thus, from (9), (45), the above and (H3) we immediately know ˆ |= 0. lim max | ψˆnml (t) − ψ(t)

l→∞ t0 ≤t≤t1

(50)

Consequently lim max | u ˆ nml − u ˆ(t) |

l→∞ t0 ≤t≤t1

ˆ = lim max | R−1 B ∗ [Πnml ψˆnml (t) − ψ(t)] | l→∞ t0 ≤t≤t1

≤

|B| ˆ ˆ |} lim max {| Πnml [ψˆnml (t) − ψ(t)] | + | (Πnml − IX )ψ(t) δ0 l→∞ t0 ≤t≤t1

≤

|B| ˆ | + lim [ lim max | ψˆnml (t) − ψ(t) max | (Πnml − IX )x |] ˆ 0 ,t1 ]) l→∞ x∈ψ([t δ0 l→∞ t0 ≤t≤t1

(51)

=0 ˆ ˆ 0 , t1 ]) is a compact subset of X), by which (note ψ(·) ∈ C([t0 , t1 ], X) ⇒ ψ([t ∞ and {nm }m=1 is an arbitrarily taken subsequence of natural numbers sequence we immediately get (10). Finally, we prove (11): On one hand from min Jn (u(·)) ≤ Jn (ˆ u(·)) → J(ˆ u(·))(n → ∞)

u(·)∈U

(52)

we immediately know lim min Jn (u(·)) ≤ J(ˆ u(·)) = min J(u(·)).

n→∞ u(·)∈U

u(·)∈U

(53)

On the other hand by min J(u(·))

u(·)∈U

≤J(ˆ un (·)) = J(ˆ un (·)) − Jn (ˆ un (·)) + Jn (ˆ un (·)) n

n

=J(ˆ u (·)) − Jn (ˆ u (·)) + min Jn (u(·)) u(·)∈U

(54)

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and J(ˆ un (·)) − Jn (ˆ un (·)) Z t1 = [Q(x(t, u ˆn (·))) − Q(xn (t, u ˆn (·)))]dt + W (x(t1 , u ˆn (·))) − W (xn (t1 , u ˆn (·))) =−

Z

t0

t1

h t0

+ + −

Z

Z Z

e(t−τ )A (Πn − IX )[B u ˆ(τ ) + f ]dτ t0 t

Z

t t0

−

Z

+ h{

Z

e(t−τ )A (Πn − IX )[B u ˆ(τ ) + f ]dτ + t0 t

Z

t

e(t−τ )A Πn B[ˆ un (τ ) − u ˆ(τ )]dτ t0

e(t−τ )A B[ˆ un (τ ) − u ˆ(τ )]dτ idt

t0 1

(∇x W )(x(t1 , u ˆn (·)) + θ{e(t1 −t0 )A (Πn − IX )x0

0

Z

+

Z

−

e(t−τ )A B[ˆ un (τ ) − u ˆ(τ )]dτ })dθ, e(t−t0 )A (Πn − IX )x0

t

+

+

e(t−τ )A Πn B[ˆ un (τ ) − u ˆ(τ )]dτ

t0

Z

+

(∇x Q)(x(t, xˆn (·)) + θ{e(t−t0 )A (Πn − IX )x0

0 t

+

−

1

t1

e(t1 −τ )A (Πn − IX )[B u ˆ(τ ) + f ]dτ

t0 t1

e(t1 −τ )A Πn B[ˆ un (τ ) − u ˆ(τ )]dτ

t0

Z

t1 t0

Z

Z

t1

e(t1 −τ )A B[ˆ un (τ ) − u ˆ(τ )]dτ )dθ}, e(t1 −t0 )A (Πn − IX )x0 e(t1 −τ )A (Πn − IX )[B u ˆ(τ ) + f ]dτ

t0 t1

e(t1 −τ )A Πn B[ˆ un (τ ) − u ˆ(τ )]dτ

t0

Z

t1

e(t1 −τ )A B[ˆ un (τ ) − u ˆ(τ )]dτ i

t0

→ 0 (n → ∞), (55) with n = 1, 2, · · · , we immediately have min J(u(·)) ≤ lim min Jn (u(·)).

u(·)∈U

(53)–(56) means (11) is true.

n→∞ u(·)∈U

(56)

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4. Necessary and sufficient conditions of near-optimal controls This section is devoted to deriving necessary and sufficient conditions of nearoptimal controls to (LNQP) and (LQP) respectively. Simply denote L2 (t0 , t1 ; X) by X . Define the two operators Γ : X → X and Γ1 : X → X by Z t y(·) 7→ (Γy)(t) := e(t−τ )A y(τ )dτ, ∀t ∈ [t0 , t1 ], t0 (57) Z t1 (t1 −τ )A y(·) 7→ Γ1 y(·) := e y(τ )dτ, t0

respectively. Clearly, Γ ∈ L(X ) and Γ1 ∈ L(X , X). Moreover, it is easy to verify that  ∗ ( ∗  Γ : X → X, Γ1 : X → X , Z t1 (58) ∗ (σ−t)A∗ ∗  z 7→ e(t1 −t)A z. e z(σ)dσ, ∀t ∈ [t0 , t1 ],  z(·) 7→ (Γ z)(t) := t

Let ε0 be a arbitrarily fixed positive number and r (t1 − t0 )[e2ω(t1 −t0 ) − 1] CΓ :=k ΓkL(X ) (≤ M ), 2ω r e2ω(t1 −t0 ) − 1 CΓ1 :=k Γ1 kL(X ) (≤ M ), 2ω Ψ0 := Γ∗ Q0 Γ + Γ∗1 W0 Γ1 , Φ0 := RIU + B ∗ Ψ0 B ≥ δ0 IU , G := BR−1 B ∗ ,

g0 (·) := x(·, 0),  r  e2ω(t1 −t0 ) − 1   α0 := M | B |   2ω        × (t1 − t0 ) max | (∇x Q)(x) |  | |f |   x∈X,|x|≤M [eω(t1 −t0 ) (|x0 |+ |f ω )− ω ]        | (∇ W )(x) | + max x   | |f |  x∈X,|x|≤M [eω(t1 −t0 ) (|x0 |+ |f  ω )− ω ]     ≥| B | [CΓ k (∇x Q)(g0 (·))kX + CΓ1 | (∇x W )(g0 (t1 )) |],  β0 := (t1 − t0 ) max | Q(x) |   | |f |  x∈X,|x|≤M [eω(t1 −t0 ) (|x0 |+ |f  ω )− ω ]     + max | W (x) |   |f | |f |  x∈X,|x|≤M [eω(t1 −t0 ) (|x0 |+ ω )− ω ]    Z t1      ≥ Q(g0 (t))dt + W (g0 (t1 )) = J(0),    t0   s     α0 2 α20   + ( + 2β0 + ε0 ),  γ0 := δ0 δ0 2δ0

(59)

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    C1             C2                    C3             C4                  C5          C6          

:= M | B |

r

e2ω(t1 −t0 ) − 1 , 2ω

|f | |f | := M [e (| x0 | + )− ] + γ0 ω ω r e2ω(t1 −t0 ) − 1 , ≥ max | g0 (t) | +γ0 t0 ≤t≤t1 2ω ω(t1 −t0 )

:=

r

e2ω(t1 −t0 ) − 1 2ω

M 2 [e2ω(t1 −t0 ) − 1] , ω

1 ∂2Q := C1 {C3 { [(t1 − t0 ) max √ | ( 2 )(x) |]2 2 x∈X,|x|≤C3 +C2 ε0 ∂x +[

max

√ x∈X,|x|≤C3 +C2 ε0

|(

1 ∂2W )(x) |]2 }} 2 , ∂x2

:=| R | + | B | C4 + 1. :=

M 2 | B | [e2ω(t1 −t0 ) − 1] [| Q0 | (t1 − t0 )+ | W0 |] 2ωδ0 2

× (1 +

M 2 | B| [e2ω(t1 −t0 ) − 1] [| Q0 | (t1 − t0 )+ | W0 |]) 2ωδ0

  2   | B |k Ψ0 kL(X ) | B| k Ψ0 kL(X )    ≥ (1 + ),   δ0 δ0       | B | C6 + 1   , C7 :=   δ0    r    |f | |f | β0  ω(t1 −t0 )  C8 := M [e (| x0 | + )− ] + C1    ω ω δ0    s     J(0)   ≥ max | g0 (t) | +C1 ,   t ≤t≤t δ0 0 1     s  r    2β 2J(0)  0  C9 := + C 7 ε0 ≥ + C 7 ε0 ,    δ0 δ0       C10 := C8 + C1 C7 ε0 ,     r    1 2β0   C11 := {[| Q0 | (t1 − t0 )+ | W0 |]C1 C7 (C8 + C10 )+ | R | C7 [C9 + ]}    2 δ0    s     1 2J(0)   ≥ {[| Q0 | (t1 − t0 )+ | W0 |]C1 C7 (C8 + C10 )+ | R | C7 [C9 + ]}.  2 δ0

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Theorem 4.1. (Necessary Conditions of Near-Optimal Controls to (LNQP)) Let (H1)∼(H6) and (H8) hold; Suppose that ε ∈ (0, ε0 ] and u ˆε (·) is a ε near-optimal control to (LNQP). Then √ k Rˆ uε (·) − B ∗ ψˆε (·)kU ≤ C5 ε,

(60)

where

ψˆε (t) := −[

Z

t1 t

∗

∗

e(σ−t)A (∇x Q)(ˆ xε (σ))dσ + e(t1 −t)A (∇x W )(ˆ xε (t1 ))]

(61)

while x ˆε (·) := x(·, u ˆε ) (i.e. the state function of system (1) corresponding to (control function) u ˆε (·)). Proof. According to Ekeland’s variational principle, u ˆ ε (·) is a ε near-optimal control to (LNQP) implies  J(¯ uε (·)) ≤ J(ˆ uε (·)),    √ ku ¯ε (·) − u ˆε (·))kU ≤ ε,   √  J(¯ uε (·)) ≤ J(v(·)) + ε k v(·) − u ¯ε (·)kU , ∀v(·) ∈ U.

∃¯ uε (·) ∈ U,

(62)

Taking v(·) = u ¯ε (·) + λ[u(·) − u ¯ε (·)],

λ ∈ (0, 1), u(·) ∈ U

(63)

in the third inequality in (62), we get for ∀ ∈ (0, 1) and u(·) ∈ U, √ J(¯ uε (·) + λ[u(·) − u ¯ε (·)]) − J(¯ uε (·)) ≥ − ε k u(·) − u ¯ε (·)kU . λ

(64)

Define x ¯ε (·) :=x(·, u ¯ε ), Z t1 ∗ ∗ ψ¯ε (·) = − [ e(σ−·)A (∇x Q)(¯ xε (σ))dσ + e(t1 −·)A (∇x W )(¯ xε (t1 ))]. ·

(65)

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Consequently, we have √ J(¯ uε (·) + λ[u(·) − u ¯ε (·)]) − J(¯ uε (·)) ε k u(·) − u ¯ε (·)kU ≤ lim λ↓0 λ Z t1 Z t = {h(∇x Q)(¯ xε (t)), e(t−τ )A B[u(τ ) − u ¯ε (τ )]dτ i −

t0

t0

+ hR¯ uε (t), u(t) − u ¯ε (t)i}dt Z t1 + h(∇x W )(¯ xε (t1 )), e(t1 −τ )A B[u(τ ) − u ¯ε (τ )]dτ i =

Z

t1

hB [ ∗

t0

+e =

Z

t1

Z

(66)

t0

t1

∗

e(τ −t)A (∇x Q)(¯ xε (τ ))dτ

t

(t1 −t)A∗

(∇x W )(¯ xε (t1 ))] + R¯ uε (t), u(t) − u ¯ε (t)idt

hR¯ uε (t) − B ∗ ψ¯ε (t), u(t) − u ¯ε (t)idt, ∀u(·) ∈ U,

t0

which obviously is equivalent to Z

t1

hR¯ uε (t) − B ∗ ψ¯ε (t), w(t)idt +

√ ε k w(·)kU ≥ 0, ∀w(·) ∈ U.

(67)

t0

Hence − k R¯ uε (·) − B ∗ ψ¯ε (·)kU + =

inf

w(·)∈U ,kw(·)kU =1

√ ε

[hR¯ uε (·) − B ∗ ψ¯ε (·), w(·)iU +

√ ε k w(·)kU ] ≥ 0.

(68)

Therefore k R¯ uε (·) − B ∗ ψ¯ε (·)kU ≤

√ ε.

(69)

By the second inequality of (62), max | x ¯ε (t) − x ˆε (t) |= max |

t0 ≤t≤t1

t0 ≤t≤t1

≤M | B | max [ t0 ≤t≤t1

≤M | B | eωt

sZ

Z t

t0

Z

t

e(t−τ )A B[¯ uε (τ ) − u ˆε (τ )]dτ | t0

t

eω(t−τ ) | u ¯ε (τ ) − u ˆε (τ ) | dτ t0

√ e−2ωτ dτ k u ¯ε (·) − u ˆε (·)kU ≤ C1 ε.

(70)

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From (H4) and (H8), we have ε0 + β0 = ε0 + J(0) ≥ε + inf u(·)∈U J(u(·)) ≥ J(ˆ uε (·)) ≥

Z

t1

[Q(g0 (t)) + h(∇x Q)(g0 (t)), (ΓB u ˆε )(t)i]dt t0

+

δ0 2 ku ˆε kU + W (g0 (t1 )) + h(∇x W )(g0 (t1 )), Γ1 B u ˆε (·)i 2

(71)

δ0 2 ≥ ku ˆε (·)kU + h(∇x Q)(g0 (·)), (ΓB u ˆε )(·)iX 2 + h(∇x W )(g0 (t1 )), Γ1 B u ˆ ε i − β0 =

δ0 ku ˆε (·)k2U + hB ∗ [Γ∗ (∇x Q)(g0 ) + Γ∗1 (∇x W )(g0 (t1 ))](·), uˆε (·)iU − β0 2

≥

δ0 2 ku ˆε kU − α0 k u ˆε (·)kU − β0 2

Furthermore, ε0 ≥

δ0 δ0 α0 2 α2 2 ku ˆε (·)kU − α0 k u ˆε kU − 2β0 = [k u ˆε (·)kU − ] − 0 − 2β0 , 2 2 δ0 2δ0

α0 |k u ˆε (·)kU − |≤ δ0

s

2 α20 ( + 2β0 + ε0 ), δ0 2δ0

ku ˆε (·)kU ≤ γ0 .

(72)

(73)

(74)

Consequently, max | xˆε (t) |= max | g0 (t) +

t0 ≤t≤t1

t0 ≤t≤t1

≤ max | g0 (t) | +M | B | max [ t0 ≤t≤t1

≤ max | g0 (t) | + t0 ≤t≤t1

t0 ≤t≤t1

r

Z

Z

t

e(t−τ )A B u ˆε (τ )dτ | t0 t

eω(t−τ ) | u ˆε (τ )dτ ]

t0

e2ω(t1 −t0 ) − 1 ku ˆε (·)kU ≤ C2 . 2ω

(75)

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From (H6), (70) and the above, we have k ψ¯ε (·) − ψˆε (·)kX Z t1 Z t1 ∗ = | e(τ −t)A [(∇x Q)(ˆ xε (τ )) − (∇x Q)(¯ xε Q)(¯ xε (τ ))]dτ 2

t−0

t

(t1 −t)A∗

+e Z t1 Z ≤M 2 [ t0

t1

2

[(∇x W )(ˆ xε (t1 )) − (∇x W )(¯ xε (t1 ))]| dt

eω(τ −t) | (∇x Q)(∇x Q)(ˆ xε (τ )) − (∇x Q)(¯ xε (τ )) | dτ

t

+ eω(t1 −t) | (∇x W )(ˆ xε (t1 )) − (∇x W )(¯ xε (t1 )) |]2 dt Z t1 Z t1 2 ≤2M { [ eω(τ −t) | (∇x Q)(ˆ xε (τ )) − (∇x Q)(¯ xε (τ )) | dτ ]2 dt t0

+[

Z

2

Z

≤2M {

t1

t

2

e2ω(t1 −t) dt] | (∇x W )(ˆ xε (t1 )) − (∇x W )(¯ xε (t1 ))| }

t0 t1

[ t0

Z

t1

e

2ω(τ −t)

dτ ][

t

Z

t1

2

| (∇x Q)(ˆ xε (τ )) − (∇x Q)(¯ xε (τ ))| dτ ]dt t

e2ω(t1 −t0 ) − 1 | (∇x W )(ˆ xε (t1 )) − (∇x W )(¯ xε (t1 ))|2 } 2ω Z t1 Z t1 M2 | (∇x Q)(ˆ xε (τ )) − (∇x Q)(¯ xε (τ ))|2 dτ ]dt = [e2ω(t1 −t) − 1][ { ω t t0 +

2

+ [e2ω(t1 −t0 ) − 1] | (∇x W )(ˆ xε (t1 )) − (∇x W )(¯ xε (t1 ))| } Z t1 2 (τ − t0 ) | (∇x Q)(ˆ xε (τ )) − (∇x Q)(¯ xε (τ ))| dτ ≤C3 [ t0

+ | (∇x W )(ˆ xε (t1 )) − (∇x W )(¯ xε (t1 ))|2 ] Z t1 Z 1 2 ∂ Q =C3 [ (τ − t0 ) | ( 2 )(ˆ xε (τ ) + θ(¯ xε (τ ) − x ˆε (τ )))dθ|2 | x ¯ε (τ ) − xˆε (τ )|2 dτ ∂x t0 0 Z 1 2 ∂ W 2 2 +| ( )(ˆ xε (t1 ) + θ(¯ xε (t1 ) − x ˆε (t1 )))dθ| | x ¯ε (t1 ) − x ˆε (t1 )| ] 2 ∂x 0 ≤C3 C12 ε{ +[

(t1 − t0 )2 ∂2Q ε[ max √ | ( 2 )(x) |]2 2 ∂x x∈X,|x|≤C2 +C1 ε0 max

√ ( x∈X,|x|≤C2 +C1 ε0

∂2W )(x)]2 } = C42 ε, ∂x2 (76)

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namely √ k ψ¯ε (·) − ψˆε (·)kX ≤ C4 ε.

(77)

Finally, by (69), the above and the second inequality of (62), we have k Rˆ uε (·) − B ∗ ψˆε (·)kU =k R[ˆ uε (·) − u ¯ε (·)] + R¯ uε (·) − B ∗ ψ¯ε (·) + B ∗ [ψ¯ε (·) − ψˆε (·)]kU √ ≤| R |k u ˆε (·) − u ¯ε (·)kU + ε+ | B |k ψ¯ε (·) − ψˆε (·)kX √ √ √ ≤ (| R | +1) ε+ | B | C4 ε = C5 ε

(78)

at once. Theorem 4.2. (Sufficient Conditions of Near-Optimal Controls to ((LQP)) Let (H1)∼(H3), (H7)-(H8) and (5) hold. Let ε ∈ (0, ε 0 ] and u ˆε (·) ∈ U such that k Rˆ uε (·) − B ∗ ψˆε (·)kU ≤ ε where ψˆε (·) := −[

Z

t1

∗

(79)

∗

e(σ−·)A Q0 xˆε (σ)dσ + e(t1 −·)A x ˆε (t1 )]

(80)

·

and x ˆε (·) := x(·, u ˆε ).

(81)

Then u ˆε (·) is a C11 ε near-optimal control to (LQP). Proof. Define rˆε (·) := Rˆ uε (·) − B ∗ ψˆε (·),

(82)

u ˆε (·) = R−1 [B ∗ ψˆε (·) + rˆε (·)].

(83)

then

ˆε (t1 )] ψˆε = −[Γ∗ Q0 x ˆε + Γ∗1 W0 x = − {Γ∗ Q0 (g0 + ΓB u ˆε ) + Γ∗1 W0 [g0 (t1 ) + Γ1 B u ˆε ]} = − [Γ∗ Q0 g0 + Γ∗1 W0 g0 (t1 ) + Ψ0 B u ˆε ]

(84)

= − [Γ∗ Q0 g0 + Γ∗1 W0 g0 (t1 ) + Ψ0 BR−1 (B ∗ ψˆε + rˆε )] = − [Ψ0 Gψˆεˆ + Γ∗ Q0 g0 + Γ∗1 W0 g0 (t1 ) + Ψ0 BR−1 rˆε ], which implies ˆε (IX + Ψ0 G)ψˆε = h

(85)

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where ˆ ε := − [Γ∗ Q0 g0 + Γ∗ W0 g0 (t1 ) + Ψ0 BR−1 rˆε ] h 1 = − {Γ∗ Q0 (g0 + ΓBR−1 rˆε ) + Γ∗1 W0 [g0 (t1 ) + Γ1 BR−1 rˆε ]}.

(86)

ˆ (see the second expression of (9)) and (58), we In addition, by the definition of ψ(·) have ψˆ = −[Γ∗ Q0 xˆ + Γ∗1 W0 x ˆ(t1 )] = − {Γ∗ Q0 (g0 + ΓB u ˆ) + Γ∗1 W0 [g0 (t1 ) + Γ1 B u ˆ]} = − [Ψ0 B u ˆ + Γ∗ Q0 g0 + Γ∗1 W0 g0 (t1 )]

(87)

ˆ = − Ψ0 Gψˆ + h where ˆ h := −[Γ∗ Q0 g0 + Γ∗1 W0 g0 (t1 )].

(4.32)

From (4.31) we immediately know ˆ (IX + Ψ0 G)ψˆ = h.

(88)

From (85) together with the above, we have ˆ ε − h) ˆ = (IX − Ψ0 BΦ−1 B ∗ )Ψ0 BR−1 rˆε . ψˆε − ψˆ = (IX + Ψ0 G)−1 (h 0

(89)

Furthermore, −1 ˆ ≤k IX − Φ0 BΦ−1 B ∗ k k ψˆε − ψk rˆε kX 0 X L(X ) k Ψ0 BR −1 ≤ (1+ | B| k Ψ0 kL(X ) | B ∗ |k Φ−1 |k rˆε kU 0 kL(U ) ) k Ψ0 kL(X ) | B || R

≤

| B |k Ψ0 kL(X ) δ0

(90)

2

(1 +

| B| k Ψ0 kL(X ) δ0

)ε ≤ C6 ε.

Consequently, ku ˆε (·) − u ˆ(·)kU =k R−1 R[ˆ uε (·) − u ˆ(·)]kU ≤| R−1 |k Rˆ uε (·) − Rˆ u(·)kU ≤

1 1 ˆ ˆ k B ∗ ψˆε (·) + rˆε (·) − B ∗ ψ(·)k {k B ∗ [ψˆε (·) − ψ(·)]k ˆε kU } U ≤ U+ k r δ0 δ0

≤

1 | B | C6 + 1 ˆ [| B |k ψˆε (·) − ψ(·)k ε = C7 ε X + ε] ≤ δ0 δ0

(91)

which yields ku ˆε (·)kU ≤k u ˆ(·)kU + k u ˆε (·) − u ˆ(·)kU ≤k u ˆ(·)kU + C7 ε ≤k u ˆ(·)kU + C7 ε0

(92)

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and max | xˆε (t) − xˆ(t) |

t0 ≤t≤t1

= max | t0 ≤t≤t1

Z

t

e(t−τ )A B[ˆ uε (τ ) − u ˆ(τ )]dτ | t0

Z

≤M | B | max

t0 ≤t≤t1

≤M | B |

r

(93)

t

e

ω(t−τ )

|u ˆε (τ ) − u ˆ(τ ) | dτ

t0

e2ω(t1 −t0 ) − 1 ku ˆε (·) − u ˆ(·)kU ≤ C1 C7 ε. 2ω

By (H7)-(H8) and the optimality of u ˆ(·) (It is the optimal control to (LQP).) and (H7)-(H8) we can easily get

ku ˆ(·)kU ≤

s

2J(0) , δ0

(94)

which implies max | x ˆ(t) |= max | g0 (t) +

t0 ≤t≤t1

t0 ≤t≤t1

≤ max | g0 (t) | +M | B | t0 ≤t≤t1

≤ max | g0 (t) | +C1 t0 ≤t≤t1

s

Z

t

Z

t

e(t−τ )A B u ˆ(τ )dτ | t0

eω(t−τ ) | u ˆ(τ ) | dτ ≤ max | g0 (t) | +C1 k u ˆ(·)kU t0 ≤t≤t1

t0

2J(0) ≤ C8 . δ0 (95)

(92) together with (94) imply

ku ˆε (·)kU ≤

s

2J(0) + C 7 ε0 = C 9 δ0

(96)

and (95) together with (93) imply

max | x ˆε (t) |≤ max | x ˆ(t) | + max | x ˆε (t) − xˆ(t) |

t0 ≤t≤t1

t0 ≤t≤t1

t0 ≤t≤t1

≤C8 + C1 C7 ε ≤ C8 + C1 C7 ε0 = C10 .

(97)

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Finally, by (97), (95), (93), (96), (94) and (91), we have | J(ˆ uε (·)) − J(ˆ u(·)) | Z t1 1 = | [hQ0 x ˆε (t), x ˆε (t)i − hQ0 xˆ(t), xˆ(t)i 2 t0 + hRˆ uε (t), u ˆε (t)i − hRˆ u(t), u ˆ(t)i]dt + hW0 xˆε (t1 ), x ˆε (t1 )i − hW0 x ˆ(t1 ), x ˆ(t1 )i | Z t1 1 [| hQ0 x ˆε (t), xˆε (t)i − hQ0 x ˆ(t), xˆ(t)i | ≤ { 2 t0 + | hRˆ uε (t), u ˆε (t)i − hRˆ u(t), u ˆ(t)i |]dt + | hW0 x ˆε (t1 ), xˆε (t1 )i − hW0 x ˆ(t1 ), x ˆ(t1 )i |} Z t1 1 {| hQ0 [ˆ xε (t) − x ˆ(t)], xˆε (t)i | + | hQ0 xˆ(t), xˆε (t) − xˆ(t)i | ≤ { 2 t0 + | R[ˆ uε (t) − u ˆ(t)], u ˆε (t)i+ | hRˆ u(t), u ˆε (t) − u ˆ(t)i |}dt + | hW0 [ˆ xε (t1 ) − x ˆ(t1 )], xˆε (t1 )i | + | hW0 x ˆ(t1 ), xˆε (t1 ) − x ˆ(t1 )i |} Z t1 1 ≤ {| Q0 | [| x ˆε (t) | + | xˆ(t) |] | x ˆε (t) − x ˆ(t) | dt 2 t0 Z t1 [| u ˆε (t) | + | u ˆ(t) |] | u ˆε (t) − u ˆ(t) | dt +|R| t0

+ | W0 | [| x ˆε (t1 ) | + | x ˆ(t1 ) |] | x ˆε (t1 ) − xˆ(t1 ) |} 1 ≤ {[| Q0 | (t1 − t0 )+ | W0 |](C10 + C8 )C1 C7 ε 2 + | R | [k u ˆε (·)kU + k u ˆ(·)kU ] k u ˆε (·) − u ˆ(·)kU } 1 {[| Q0 | (t1 − t0 )+ | W0 |]C1 C7 (C8 + C10 )ε 2 s 2J(0) + | R | [C9 + ]C7 ε} ≤ C11 ε. δ0

≤

Hence u ˆε (·) is a C11 ε near-optimal control to (LQP). 5. Algorithm to find near-optimal controls to (LQP) By a completely analogous argument, we can deduce the following result:

(98)

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Theorem 5.1. Let (H1)∼(H6) and (H8) hold. Let n be an arbitrary positive integer. Let xn0 := Πn x0 ,

fn := Πn f,

Gn := Bn R−1 Bn∗ (= Πn GΠn ),

(99)

and hN :=

t1 − t 0 , N

tk,N := t0 + khN , k = 0, 1, · · · , N,

N = 1, 2, · · · .

(100)

1 −t0 ) Then for any positive integer N ∈ [3 max( (|An |+1)(t , 1), +∞) we have the fol2 lowing two assertions. (1) There exists a unique solution to the following algebraic system of equations

 hN hN hN   (IXn − An )x1 − Gn (3ψ1 + ψ2 ) = (IXn + An )xn0 + hN fn ,   2 4 2       h h   − (IXn + N An )xk−1 + (IXn − N An )xk   2 2      hN    − Gn (ψk−1 + 2ψk + ψk+1 )   4       = hN fn , k = 2, · · · , N − 1,        hN hN   − (IXn + An )xN −1 + (IXn − An )xN   2 2       hN   − Gn (ψN −1 + 3ψN ) = hN fn ,   4     1 hN 1 Πn [(∇x Q)( (x1 + xn0 )) + (∇x Q)( (x1 + x2 ))]  2 2 2      hN ∗ hN ∗    + (IXn − An )ψ1 − (IXn + A )ψ2 = 0,   2 2 n      hN 1 1   Πn [(∇x Q)( (xk−1 + xk )) + (∇x Q)( (xk + xk+1 ))]    2 2 2      hN ∗ hN ∗   + (IXn − An )ψk − (IXn + A )ψk+1 = 0,    2 2 n       k = 2, · · · , N − 1,       hN 1   Πn [ (∇x Q)( (xN −1 + xN ))    2 2      hN ∗   + (∇x W )(xN )] + (IXn − A )ψN = 0, 2 n

(101)

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in Xn2N , which we denote it by 

x ˆn1,N

  ..  .   n x  ˆN,N  n  ψˆ  1,N   ..  . 

n ψˆN,N



       .      

(2) Define  n  u ˆn0,N := R−1 Bn∗ ψˆ1,N ,      1 n n + ψˆk+1,N ), k = 1, · · · , N − 1, u ˆnk,N := Bn∗ (ψˆk,N  2     u n ˆnN,N := R−1 Bn∗ ψˆN,N .

(102)

There exists a positive constant Cn which can be determined by

(An , Bn , fn , xn0 , Q(·), R, W (·), t0 , t1 )

such that for ∀t ∈ [tk−1,N , tk,N ] and k = 0, 1, · · · , N ,

u ˆnN (t) :=

t − tk−1,N n tk,N − t n u ˆk−1,N + u ˆk,N hN hN

(103)

makes a Cn h2N near-optimal control (fuction) to (LNQP)n . Corollary 5.1. Let (H1)∼(H3), (H7)-(H8) and (5) hold, and n be an arbitrary positive integer. We still adopt those related notations in the statement of Theorem 1 −t0 ) , 1), +∞), we have the 5.1. Then for any positive integer N ∈ [3 max( (|An |+1)(t 2 following two assertions.

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(1) There exists a unique solution to the following linear algebraic system of equations  hN hN hN   (IXn − An )x1 − Gn (3ψ1 + ψ2 ) = (IXn + An )xn0 + hN fn ,   2 4 2      hN hN hN   − (IXn + An )xk−1 + (IXn − An )xk − Gn (ψk−1 + 2ψk + ψk+1 ) = hN fn ,   2 2 4      k = 2, · · · , N − 1,       hN hN hN   An )xN −1 + (IXn − An )xN − Gn (ψN −1 + 3ψN ) = hN fn ,  − (IXn + 2 2 4  hN hN ∗ hN ∗   Πn Q0 (2x1 + x2 + xn0 ) + (IXn − An )ψ1 − (IXn + A )ψ2 = 0,   4 2 2 n      hN ∗ hN ∗  hN Π Q (x  An )ψk − (IXn + A )ψk+1 = 0, n 0 k−1 + 2xk + xk+1 ) + (IXn −   4 2 2 n      k = 2, · · · , N − 1,         Πn [ hN Q0 xN −1 + (W0 + hN Q0 )xN ] + (IXn − hN A∗n )ψN = 0 4 4 2 (104) in Xn2N . (2) There exists a positive constant Cn which can be determined by (An , Bn , fn , xn0 , Q(·), R, W (·), t0 , t1 ) such that u ˆnN (·) makes a Cn h2N near-optimal control (fuction) to (LQP)n . Besides C1 ∼ C11 in (59), we additionally let  | f | [eω(t−t0 ) − 1]  ω(t−t0 )  }, C := M max {e | x | +  12 0   t0 ≤t≤t1 ω     1 2    C13 := C12 [| Q0 | (t1 − t0 )+ | W0 |],   2 p 1  C14 := C5 C7 {C1 [| Q0 | (t1 − t0 )+ | W0 |](C1 C5 C7 ∆(ε0 ) + 2M1 )    2   p    + | R | (C5 C7 ∆(ε0 ) + 2γ)},     p  C15 := C5 C14 + 1.

(105)

Now we give an algorithm. Algorithm 5.1. Step 1

n := 1.

(106)

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Step 2 Take p C11 C15 2 (| An | +1)(t1 − t0 ) Nn := [max((t1 − t0 ) Cn ( ) , 3 max( , 1))] + 1 ε 2

(107)

√ 1 −t0 ) (where [max((t1 − t0 ) Cn ( C11εC15 )2 , 3 max( (|An |+1)(t , 1))] denotes 2 √ the maximum integer which is not more than max((t1 − t0 ) Cn ( C11εC15 )2 , 1 −t0 ) 3 max( (|An |+1)(t , 1))), and 2 hNn :=

t1 − t 0 ; Nn

(108)

solve the following system of linear algebraic equations  hN n hN n  An )x1 − Gn (3ψ1 + ψ2 ) (IXn −    2 4     hN n    = (IXn + An )xn0 + hNn fn ,   2      hN n hN n   An )xk−1 + (IXn − An )xk − (IXn +   2 2      hN n   − Gn (ψk−1 + 2ψk + ψk+1 ) = hNn fn ,   4       k = 2, · · · , Nn − 1,       hN n hN n   − (IXn + An )xNn −1 + (IXn − An )xNn   2 2      hN n   − Gn (ψNn −1 + 3ψNn ) = hNn fn ,  4  hN n hN n ∗   Πn Q0 (2x1 + x2 + xn0 ) + (IXn − A )ψ1    4 2 n     hN n ∗   A )ψ2 = 0, − (IXn +    2 n     hN n ∗ hN n    Πn Q0 (xk−1 + 2xk + xk+1 ) + (IXn − A )ψk   4 2 n      hN n ∗   − (IXn + A )ψk+1 = 0,   2 n      k = 2, · · · , Nn − 1,       hN hN n    Πn [ n Q0 xNn −1 + (W0 + Q0 )xNn ]   4 4      hN n ∗  + (IXn − A )ψNn = 0; 2 n

(109)

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let the solution of the above system of equations be denoted by  n  x ˆ1,Nn   ..   .     n x   ˆNn ,Nn   ˆn   ψ1,N  n     ..   .   n ˆ ψ Nn ,Nn

and tk,Nn := t0 + khNn , k = 0, 1, · · · , Nn ,  n n  u ˆ := R−1 Bn∗ ψˆ1,N ,  n  0,Nn   1 n n + ψˆk+1,N ), k = 1, · · · , Nn − 1, u ˆnk,Nn := Bn∗ (ψˆk,N n n  2     n u ˆ := R−1 B ∗ ψˆn , Nn ,Nn

u ˆnNn (t) :=

n

(110)

Nn ,Nn

t − tk−1,Nn n tk,Nn − t n u ˆk−1,Nn + u ˆk,Nn , hN n hN n

∀t ∈ [tk−1,Nn , tk,Nn ], k = 0, 1, · · · , Nn . Step 3 x ˆnNn (t) := e(t−t0 )A x0 + n ψˆN (t) n

:= −[

Check whether

Z

t1

e

Z

t t0

(σ−t)A∗

t u ˆnNn (·)

Z

e(t−τ )A [B u ˆnNn (τ ) + f ]dτ, ∀t ∈ [t0 , t1 ], (111)

Q0 xˆnNn (σ)dσ

+e

(t1 −t)A∗

W0 x ˆnNn (t1 )],

∀t ∈ [t0 , t1 ].

satisfies t1 t0

n | Rˆ unNn (t) − B ∗ ψˆN (t)| dt ≤ ( n 2

ε 2 ) . C11

(112)

If not, let n := n + 1;

(113)

then goto step 2. Otherwise, let nε := n, u ˆε (·) := u ˆnNεnε (·).

(114)

In virtue of algorithm 5.1 one can obtain a varepsilon near-optimal control for arbitrarily given ε ∈ (0, ε0 ) (For further details, please see the following Theorem 5.2 together with its proof.). Theorem 5.2. Let (H1)∼(H3), (H7)-(H8) and (5) hold. Then nε is a finite positive integer and u ˆε (·) is a ε near-optimal control to (LQP).

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Proof. From (107) we immediately know Cn h2Nn =

ε Cn (t1 − t0 )2 ≤ ∆(ε) := ( )4 ; Nn2 C11 C15

(115)

hence according to Corollary 5.1 u ˆnNn (·) is a ∆(ε) near-optimal control to (LQP)n , namely Jn (ˆ uNn (·)) ≤ ∆(ε) + min Jn (u(·)). u(·)∈U

Consequently by Theorem 4.1 and  tAn |e |≤| etA |≤ M eωt , ∀t ∈ [0, +∞),       | Bn |=| Πn B |≤| Πn || B |≤| B |,   | fn |=| Πn f |≤| Πn || f |≤| f |,     | xn0 |=| Πn x0 |≤| Πn || x0 |≤| x0 |

we know

k Rˆ unNn (·) − B ∗ ψˆn (·, u ˆnNn )kU ≤ C5 where ψˆn (·, u ˆnNn ) := − [

Z

t1

(117)

(118)

∗

e(σ−·)An Πn (∇x Q)(xn (σ, u ˆnNn ))dσ

·

(t1 −·)A∗ n

+e Z =−[

p ∆(ε)

(116)

t1

e

Πn (∇x W )(xn (t1 , u ˆnNn ))]

(σ−·)A∗

Πn (∇x Q)(x

n

(119)

(σ, u ˆnNn ))dσ

· ∗

+ e(t1 −·)A Πn (∇x W )(xn (t1 , u ˆnNn ))]. n

max | x (t, 0) |= max |

t0 ≤t≤t1

t0 ≤t≤t1

e(t−t0 )A xn0

≤ M max {e t0 ≤t≤t1

ω(t−t0 )

+

Z

t

e(t−τ )A fn dτ | t0

| Πn x0 | +[

≤ M max {eω(t−t0 ) | x0 | + t0 ≤t≤t1

| Jn (0) |=

1 | 2

Z

t1

Z

t

eω(t−τ ) dτ ] | Πn f |}

t0

| f | [eω(t−t0 ) − 1] } = C12 , ω (120)

hQ0 xn (t, 0), xn (t, 0)idt + hW0 xn (t1 , 0), xn (t1 , 0)i |

t0

≤

1 [| Q0 | (t1 − t0 )+ | W0 |][ max | xn (t, 0) |]2 t0 ≤t≤t1 2

≤

1 [| Q0 | (t1 − t0 )+ | W0 |]N 2 = C13 . 2

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Then by the proof of Theorem 4.2 and (117) we know p ku ˆnNn (·) − u ˆn (·)kU ≤ C5 C7 ∆(ε).

(121)

From the above,

| J(ˆ unNn (·)) − J(ˆ un (·)) | Z t1 1 ≤ { [| hQ0 x(t, u ˆnNn (·)), x(t, u ˆnNn (·))i − hQ0 x(t, u ˆn (·)), x(t, uˆn (·))i | 2 t0 + | hRˆ unNn (t), u ˆnNn (t)i − dhRˆ un (t), u ˆn (t)i |]dt + | hW0 x(t1 , u ˆnNn (·)), x(t1 , u ˆnNn (·))i − hW0 x(t1 , u ˆn (·)), x(t1 , u ˆn (·))i |} Z t1 1 {| hQ0 [x(t, u ˆnNn (·)) − x(t, u ˆn (·))], x(t, uˆnNn (·)) − x(t, u ˆn (·))i = { 2 t0 + 2hQ0 x(t, u ˆn (·)), x(t, u ˆnNn (·)) − x(t, u ˆn (·))i | + | R[ˆ unNn (t) − u ˆn (t)], u ˆnNn (t) − u ˆn (t)i + 2hRˆ un (t), u ˆnNn (t) − u ˆn (t)i |}dt + | hW0 [x(t1 , u ˆnNn (·)) − x(t1 , u ˆn (·))], x(t1 , u ˆnNn (·)) − x(t1 , u ˆn (·))i

(122)

+ 2hW0 x(t1 , u ˆn (·)), x(t1 , u ˆnNn (·)) − x(t1 , u ˆn (·))i |} Z t1 1 2 ≤ {| Q0 | [ | x(t, u ˆnNn (·)) − x(t, u ˆn (·))| dt 2 t0 Z t1 | x(t, u ˆn (·)) || x(t, u ˆnNn (·)) − x(t, u ˆn (·)) | dt] +2 t0

+|R|[

Z

t1 t0

2

|u ˆnNn (t) − u ˆn (t)| dt + 2

Z

t1

t0

+ | W0 | [| x(t1 , u ˆnNn (·)) − x(t1 , u ˆn (·))|

|u ˆn (t) || u ˆnNn (t) − u ˆn (t) | dt]

2

+ 2 | x(t1 , u ˆn (·)) || x(t1 , u ˆnNn (·)) − x(t1 , u ˆn (·)) |]}, max | x(t, u ˆnNn (·)) − x(t, u ˆn (·)) |

t0 ≤t≤t1

= max | t0 ≤t≤t1

Z

t t0

e(t−τ )A B[ˆ unNn (τ ) − u ˆn (τ )]dτ |

≤M | B | max

t0 ≤t≤t1

≤M | B | (121) & (59)

≤

C 1 C5 C7

r

Z

t t0

eω(t−τ ) | u ˆnNn (τ ) − u ˆn (τ ) | dτ

e2ω(t1 −t0 ) − 1 ku ˆnNn (·) − u ˆn (·)kU 2ω

p ∆(ε)

(123)

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and (28)–(29), we know that | J(ˆ unNn (·)) − J(ˆ un (·)) | p 1 ≤ {[| Q0 | (t1 − t0 )+ | W0 |][(C1 C5 C7 )2 ∆(ε) + 2M1 C1 C5 C7 ∆(ε)] 2 p + | R | [(C5 C7 )2 ∆(ε) + 2γC5 C7 ∆(ε)]}

p 1 ≤ C5 C7 {C1 [| Q0 | (t1 − t0 )+ | W0 |](C1 C5 C7 ∆(ε0 ) + 2M1 ) 2 p p + | R | (C5 C7 ∆(ε0 ) + 2γ)} ∆(ε) p =C14 ∆(ε).

(124)

Review (11), which means for any ε ∈ (0, ε0 ] there exists a (natural number) mε such that | J(ˆ un (·)) − min J(u(·)) |≤

p

n = mε , mε + 1, · · · .

(125)

≤ | J(ˆ unNn (·)) − J(ˆ un (·)) | + | J(ˆ un (·)) − min J(u(·)) |

(126)

u(·)∈U

∆(ε),

So | J(ˆ unNn (·)) − min J(u(·)) | u(·)∈U

u(·)∈U

(124)−(125)

≤

(C14 + 1)

p ∆(ε),

n = mε , mε + 1, · · · .

Hence, according to Theorem 4.1 (and the above) we (immediately) have for n = mε , mε + 1, · · · , n (·)kU ≤ C5 k Rˆ unNn (·) − B ∗ ψˆN n

p 1 1 ε C14 + 1∆(ε) 4 = C15 ∆(ε) 4 = . C11

(127)

The above implies 1 ≤ n ε ≤ mε ,

(128)

and in accordance with Theorem 4.2 (and (126)) we (immediately) know u ˆ nNεnε (·) is a ε near-optimal to (LQP).

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References 1. Y. C. You, Optimal control for linear systems with quadratic indefinite criterion on Hilbert spaces, Chin. Ann. of Math., 4B (1), pp. 21–32 (1983). 2. Y. C. You, A nonquadratic Bolza problem and a quasi-Riccati equation for distributed parameter systems, SIAM J. Control & Optim., 25 (4), pp. 905–920 (1987). 3. L. P. Pan and X. H. Yu, Near-Optimal controls to the linear-nonquadratic optimal control problem, submitted to Systems Science & Mathematics (in Chinese).

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Chap35-WangGengsheng

SHAPE OPTIMIZATION OF STATIONARY NAVIER-STOKES EQUATION∗

GENGSHENG WANG, LIJUAN WANG and DONGHUI YANG Department of Mathematics, Wuhan University, Wuhan, Hubei, 430072, CHINA Emails: [email protected] (G. Wang), [email protected] (L. Wang), [email protected] (D. Yang)

In this work, we obtain some properties for the family of open subsets being starlike with respect to a ball. Based on these, we prove the existence of solutions of some shape optimization for stationary Navier-Stokes equations. Keywords: Shape optimization, stationary Navier-Stokes equation. AMS Subject Classifications: 49Q10, 65K10.

1. Introduction Let D and B be open bounded subsets with Lipschitz boundaries in R N , N = 2, 3, such that D ⊂⊂ B, and O be a family of open subsets included in D which will be precised later. Consider the following stationary Navier-Stokes equation in domain B \ Ω where Ω ∈ O:   −ν4u + (u · ∇)u + ∇p = f in B \ Ω,   (1) div u = 0 in B \ Ω,   u = 0 on ∂B ∪ ∂Ω,

where ν > 0 is the viscosity constant of the fluid and f ∈ (L2 (B))N is a given function, u denotes the velocity while p denotes the pressure. ∞ For each open subset ω ⊂ RN , we denote C0,σ (ω) = {u ∈ C0∞ (ω)N ; divu = 0} k·k

1

H 1 ∞ ∞ (ω) and H0,σ (ω) = C0,σ , the completion of C0,σ (ω) in the norm of H 1 (ω)N . 1 We say that u is a weak solution of (1) if u ∈ H0,σ (B \ Ω) and Z Z Z ∇u · ∇ϕ dx + (u · ∇)u · ϕ dx = f · ϕ dx, (2)

B\Ω

B\Ω

B\Ω

∗ This

work was supported by National Natural Science Foundation of China, Grant No. 10401041 and 10471053 and by the Grant of Key Laboratory—Optimal Control and Discrete Mathematics of Hubei Province. 323

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324 ∞ for all ϕ ∈ C0,σ (B \ Ω). It is well known that (see Ref. 6) for each Ω ∈ O, equation (1) has at least one weak solutions, moreover, there exists a positive constant C(γ, d) depending only on the viscosity constant γ and the width d of set B such that if

kf kL2 (B)n < C(γ, d)

(3)

then the weak solution of (1) corresponding to each Ω is unique. We shall assume in this paper that (3) holds. In this paper, we shall study the following shape optimization problem

(P )

MinΩ∈O

Z

F (x, uΩ , ∇uΩ ) dx B\Ω

where uΩ is a weak solution of (1) corresponding to Ω ∈ O, F (x, ξ, η) : B × R N × RN ×N → R+ is continuous and satisfies that there exists a positive constant c such that |F (x, ξ, η)| ≤ c(1 + |ξ|2RN + |η|2RN ×N )

(4)

for all (x, ξ, η) ∈ B × RN × RN ×N . Now we are going to explicate the family O. Let Ω ⊂ R N be an open and bounded subset, we say that Ω is starlike with respect to a point x0 ∈ Ω if the line {x0 + te; t ∈ R} intersects the boundary ∂Ω in exact two points for each vector e ∈ RN . We say that Ω is starlike with respect to an open ball U (x, r) with center x and radius r if U (x, r) ⊂ Ω and Ω is starlike with respect to every point of U (x, r). We define O = {Ω ⊂ D; Ω is open and starlike with respect to some open ball U (x, r) with r ≥ r0 }, where r0 is a given positive constant. The topology on O is induced from the Hausdorff-Pompeiu distance between the complementary sets, i.e., ρ(Ω1 , Ω2 ) = max{supx∈D\Ω1 d(x, D \ Ω2 ), supy∈D\Ω2 d(D \ Ω1 , y)}

(5)

for ∀ Ω1 , Ω2 ∈ O, where d(·, ·) denotes the Euclidean metric in RN . We denote by Hlim, the limit in the sense of (5). In this work, we obtain the following fundamental results on the family O: ∞ ∞ (i) If {Ωm }∞ m=1 ⊂ O, then there exists a subsequence {Ωmk }k=1 of {Ωm }m=1 such that HlimΩmk = Ω and Ω ∈ O; ∧

(ii) (Γ-property for O) If {Ωm }∞ m=1 ⊂ O and Ω ∈ O be such that HlimΩm = Ω, then for any open subset K with K ⊂ B \ Ω, there is an integer m(K) > 0 such that for all m ≥ m(K), K ⊂ B \ Ωm .

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Based on these, we obtain the existence of the optimal solutions for problem (P ). 2. Some properties related to the family O In this section, we shall use the following notations: δ(K1 , K2 ) = max{ sup d(x, K2 ), sup (K1 , y)}, x∈K1

y∈K2

N

where K1 and K2 are compact subsets in R ; U (x, r) denotes the open ball in RN with center x and radius r; L(x, y), x, y ∈ RN , x 6= y, denotes the line passing through x and y in RN ; C(α, y, z), α ∈ R, y, z ∈ RN , denotes the cone in RN with vertex z, direction y and angle α; −→ yz , y, z ∈ RN , z 6= 0, denotes the ray in RN with starting point y and direction z; d(A, B), A, B ⊂ RN , denotes the distance between sets A and B in RN . The following three results were given and proved in Ref. 5 and will be used in this paper. Lemma 2.1. Let A, An , n = 1, 2, · · · , be compact subsets in RN such that δ(An , A) → 0, then A is the set of all accumulation points of the sequences {x n } such that xn ∈ An for each n. ˜ An , A˜n , n = 1, 2, · · · , be compact subsets in Rn such that Lemma 2.2. Let A, A, ˜ ˜ → 0. Suppose that An ⊂ A˜n for each n, Then A ⊂ A. ˜ δ(An , A) → 0 and δ(An , A) Lemma 2.3. (Γ−property for O) Assume that {Ωn }∞ n=1 ⊂ O, Ω0 ∈ O and Ω0 = HlimΩn . Then for each open subset K satisfying K ⊂ Ω0 , there exists a positive integer nK (depending on K) such that K ⊂ Ωn for all n ≥ nK . One of the main results in this paper is as follows. ∞ Theorem 2.1. For any sequence {Ωn }∞ n=1 ⊂ O, there is a subsequence {Ωnk }k=1 ∞ ∗ ∗ of {Ωn }n=1 and Ω ∈ O such that HlimΩnk = Ω .

In order to prove this theorems, we shall first prove the following lemmas. Lemma 2.4. Let {xn } ⊂ RN and {rn } ⊂ R be such that rn ≥ r0 > 0 and HlimU (xn , rn ) = U ⊂ RN . Then there exists x0 ∈ RN such that U (x0 , r0 ) ⊂ U and on a subsequence, xn → x0 . Proof. Since U (xn , rn ) ⊂ D and D is a bounded subset in RN , there exists a subsequence of {xn }, still denoted by itself, such that xn → x 0

(6)

U (x0 , r0 ) ⊂ U.

(7)

for some x0 ∈ D. We claim that

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By contradiction, we assume that there did exist y0 ∈ U (x0 , r0 ) and y0 6∈ U . Since U = HlimU (xn , rn ), i.e., δ(D \ U (xn , rn ), D \ U ) → 0. It follows from Lemma 2.1 that there exists a sequence {yn } satisfying yn ∈ D \ U (xn , rn ) and yn → y0 .

(8)

Hence d(yn , xn ) ≥ rn ≥ r0 . By (6) and (8) we may pass to the limit for n → ∞ to get d(y0 , x0 ) ≥ r0 , which leads to a contradiction and implies (7) as desired. This completes the proof. ∼

Lemma 2.5. Let U (x0 , r0 ) be any open ball in D and {Ωn }∞ n=1 ⊂O be such that HlimΩn = Ω, where ∼

O ≡ {Ω ⊂ D; Ω is starlike with respect to every point of U (x0 , r0 )}. ∼

Then Ω ∈O . Proof. Since HlimΩn = Ω, we have Ω ⊂ D and δ(D \ Ωn , D \ Ω) → 0. It is clear that D \ Ωn ⊂ D \ U (x0 , r0 ), then it follows from Lemma 2.2 that D \ Ω ⊂ D \ U (x0 , r0 ), i.e., U (x0 , r0 ) ⊂ Ω. Thus it suffices to show that Ω is starlike with respect to U (x0 , r0 ). To this end, we assume by contradiction that there existed y0 ∈ U (x0 , r0 ) and e0 ∈ RN with |e0 | = 1 such that the line {y0 + te0 : t ∈ R} intersects ∂Ω at three points, denoted by zi = y0 + ti e0 , i = 1, 2, 3. since y0 ∈ U (x0 , r0 ) ⊂ Ω, and zi ∈ ∂Ω, i = 1, 2, 3, then we have ti 6= 0, i = 1, 2, 3 and at least two of them are both positive or negative. Without loss of generality, we may assume that t1 > t2 > 0 . Since y0 ∈ U (x0 , r0 ), there exists ry0 > 0 such that U (y0 , ry0 ) ⊂⊂ U (x0 , r0 ). First we claim that there is r > 0 such that for any z10 ∈ U (z1 , r) and z20 ∈ U (z2 , r), r y L(z10 , z20 ) ∩ U y0 , 0 6= ∅. (9) 2 Indeed, we may assume by rotating and relabelling the coordinate axes that ∼ y0 = 0 and e0 = (0, · · · , 0, 1). Then zi = (0, · · · , 0, ti ), i = 1, 2. Let r 0 > 0 be such ∼ 2 | |t2 |−ry0 , }, it is clear that that r 0 < min{ |t1 −t 2 2 r ∼ ∼ ∼ y U (z1 , r 0 ) ∩ U (z2 , r 0 ) = ∅ and U (z2 , r 0 ) ∩ U y0 , 0 = ∅. (10) 2 ∼

∼

For any x ∈ U (z1 , r 0 ), y ∈ U (z2 , r 0 ), we have L(x, y) ∩ {z ∈ Rn ; z = (z1 , · · · , zN ), zN = 0} = ((t0 x + (1 − t0 )y)0 , 0)

(11)

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for some t0 ∈ R, where (t0 x + (1 − t0 )y)0 denotes the first (N − 1) components of t0 + (1 − t0 )y in RN . By (10) we get yN > xN > 0, and by (11) we have t0 xN + (1 − t0 )yN = 0, i.e., yN |t0 | = t0 = > 0, (12) yN − xN ∼

∼

∼

∼

Since t2 − r 0 < yN < t2 + r 0 , t1 − r 0 < xN < t1 + r 0 , we have ∼

∼

(13)

(t2 − t1 ) − 2 r 0 < yN − xN < (t2 − t1 ) + 2 r 0 . It follows from (12) and (13) that ∼

|t0 | = t0 =

yN t2 + r 0 ≤ ∼ ≡ c(z1 , z2 ). yN − xN (t2 − t1 ) − 2 r 0

and |((t0 x + (1 − t0 )y)0 , 0)|RN = |(t0 x)0 + ((1 − t0 )y)0 |RN −1 ≤ |t0 ||x0 |RN −1 + (1 + |t0 |)|y 0 |RN −1 ≤ |t0 ||x − z1 |RN + (1 + |t0 |)|y − z2 |RN

Let r = min

(

≤ c(z1 , z2 )|x − z1 |RN + (1 + c(z1 , z2 ))|y − z2 |RN . ) r y0 ∼ r 0, , then for any z10 ∈ U (z1 , r), z20 ∈ U (z2 , r), we 3(1 + 2c(z1 , z2 ))

have L(z10 , z20 ) ∩ {z ∈ Rn ; z = (z1 , · · · , zN ), zN = 0} = ((t00 z10 + (1 − t00 )z20 )0 , 0) ∼

(14)

∼

for some t00 ∈ R, since U (z1 , r) ⊂ U (z1 , r 0 ) and U (z2 , r) ⊂ U (z2 , r 0 ). Then |t00 | < c(z1 , z2 ),

(15)

By (14) and (15), we have |((t00 z10 + (1 − t00 )z20 )0 , 0)|RN ≤ c(z1 , z2 )|z10 − z1 |RN + (1 + c(z1 , z2 ))|z20 − z2 |RN r

r

y0 y0 + (1 + c(z1 + z2 )) 3(1+2c(z ≤ c(z1 , z2 ) 3(1+2c(z 1 +z2 )) 1 ,z2 ))

=

ry 0 3

<

ry 0 2

.

This implies ((t00 z10 +(1−t00 )z20 )0 , 0) ∈ U (0, ry0 ), from which (9) follows immediately. Next we claim that there is a positive integer n0 such that for each n ≥ n0 , there exist yn ∈ Ωn and en ∈ RN with |en | = 1 satisfying Card(Ln ∩ ∂Ωn ) ≥ 3.

(16)

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Here Ln ≡ {yn + ten , t ∈ R} and Card(Ln ∩ ∂Ωn ) denotes the cardinality of the set Ln ∩ ∂Ωn . Indeed, since z1 ∈ ∂Ω, there are w1 ∈ Ω and rw1 ∈ (0, r) such that U (w1 , rw1 ) ⊂⊂ Ω ∩ U (z1 , r),

(17)

which, together with (9), implies that r y L(w1 , z2 ) ∩ U y0 , 0 6= ∅. 2 ry z2 − y 1 Fix y1 ∈ L(w1 , z2 ) ∩ U (y0 , 0 ) and let ey1 = , then 2 |z2 − y1 | L(w1 , z2 ) = {y1 + sey1 ; s ∈ R}. By (10), we have r ry y U (z2 , r)∩U (w1 , rw1 ) = ∅, U (z2 , r)∩U y0 , 0 = ∅ and U (w1 , rw1 )∩U y0 , 0 = ∅, 2 2 so there exist s1 6= 0 and s2 6= 0 with the same sign such that w1 = y1 + s1 ey1 and z2 = y1 + s2 ey1 , it is clear that s1 > s2 . On the other hand, Since z2 ∈ ∂Ω and w1 ∈ Ω, by Lemma A in the Appendix, −→ we see that the ray − y− 1 ey1 reaches the boundary ∂Ω at z4 after passing through w1 . Let z4 = y1 + s4 ey1 , then s4 > s1 > s2 since z4 6= z2 . Notice that L(yz1 , yz2 ) ∩ U (w1 , rw1 ) 6= ∅

(18)

for any yz1 ∈ U (z1 , rw1 ) and yz2 ∈ U (z2 , rw1 ). Then it follows from (8) and Lemma 2.3 that there is a positive integer n(w1 ) such that U (w1 , rw1 ) ⊂⊂ Ωn as n ≥ n(w1 ). Moreover, since z2 , z4 ∈ D \ Ω, it follows from Lemma 2.1, we get that there exist z2n ∈ D \ Ωn and z4n ∈ D \ Ωn satisfying z2n → z2 and z4n → z4 , so for rw1 > 0, there exists an integer n0 > n(w1 ), such that as n ≥ n0 , z2n ∈ U (z2 , rw1 ),

z4n ∈ U (z4 , rw1 ) and U (w1 , rw1 ) ⊂⊂ Ωn .

(19)

For each n ≥ n0 , it follows from (9) and (18) that there exists y1n ∈ L(z2n , z4n ) ∩ U (w1 , rw1 ) while it follows from (9) and (19) that there exists yn ∈ L(z2n , z4n ) ∩ y n − yn r U (y0 , y20 ). Let en = 1n , then points z2n , z4n and y1n are in the ray − yn−e→ n and |y1 − yn | z2n = yn + sn2 en , z4n = yn + sn4 en , y1n = yn + sn1 en with 0 < sn2 < sn1 < sn4 .

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By (19), we have y1n ∈ U (w1 , rw1 ) ⊂⊂ Ωn . Since z2n ∈ D \ Ωn , there exists a unique point wn ∈ ∂Ωn ∩ − yn−e→ n such that wn = yn + l1n en with sn2 ≤ l1n < sn1 . By the same arguments as above, we obtain that there exists a unique point ∼ w n ∈ ∂Ωn ∩ − yn−e→ n such that ∼

wn = yn + l2n en with sn1 < l2n ≤ sn4 . ∼ ∼ yn−→ en . On the other hand, the Notice that wn 6=wn and wn , w n are in the ray − −−−−−→ ray yn (−en ) must reach the boundary ∂Ωn at at least one point which is different ∼ from wn and w n . So (16) follows. ∼

However, since Ωn ∈O , Card({yn + ten } ∩ ∂Ωn ) = 2, which contradicts to (16). So the line {y0 + te0 ; t ∈ R} can not intersects ∂Ω at three or more than three points. This completes the proof. Proof of Theorem 2.1. For any sequence {Ωn }∞ n=1 ⊂ O, there exist a ∞ ∞ subsequence {Ωnk }k=1 of {Ωn }n=1 and an open subset Ω such that Ω = HlimΩnk . It suffices to show that Ω ∈ O. By the definition of O, Lemma 2.2 and Lemma 2.4, there exist U (xnk , rnk ) ⊂ Ωnk , rnk ≥ r0 and an open subset U such that U = HlimU (xnk , rnk ) and U (x0 , r0 ) ⊂ U ⊂ Ω, where x0 = limk→∞ xnk . For any 0 < < r0 , since U (x0 , r0 −) ⊂⊂ U (x0 , r0 ) ⊂ U , it follows from Lemma 2.3 that U (x0 , r0 − ) ⊂⊂ U (xnk , rnk ) as k is large enough. So Ωnk is starlike with respect to U (x0 , r0 − ) as k is large enough. This combined with Lemma 2.5 implies that Ω is starlike with respect to U (x0 , r0 − ), Hence Ω is starlike with respect to U (x0 , r0 ). This completes the proof. ∧

Theorem 2.2. (Γ −property for O) Assume that {Ωn }∞ n=1 ⊂ O and Ω0 = HlimΩn . Then for any open subset K satisfying K ⊂ B \ Ω0 , there exists a positive integer nK such that K ⊂ B \ Ωn for n ≥ nK . In order to prove Theorem 2.2, we need the following Lemmas. Lemma 2.6. If Ω is starlike with respect to U (0, r0 ) and r > 1, then d(∂Ω, ∂Ωr ) = d(Ω, ∂Ωr ) ≥ r0 (r − 1), where Ωr = {ry : y ∈ Ω}.

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Proof. Since Ω ⊂ Ωr (cf. Ref. 2), we have d(Ω, ∂Ωr ) = d(∂Ω, ∂Ωr ). Notice that d(∂Ω, ∂Ωr ) = inf x∈∂Ω,z∈∂Ωr |x − z|, f (x, z) ≡ |x − z| is continuous in Rn × Rn and ∂Ω × ∂Ωr is a compact in Rn × Rn , then there are x0 ∈ ∂Ω, z0 ∈ ∂Ωr satisfying d(∂Ω, ∂Ωr ) = d(x0 , z0 ).

(20) −→

Since z0 ∈ ∂Ωr and Ω is starlike with respect to the point of 0, the ray 0z0 −→

intersect ∂Ω at a unique point and denote it by y0 . Since 0, y0 , z0 ∈0z0 , there is s > 0 such that sy0 = z0 . First we claim that s = r.

(21)

If (21) was not true, then there are two cases: (i) 0 < s < r and (ii) 0 < r < s. In case (i), we have |sr −1 y0 | < |y0 |, from which we get sr −1 y0 ∈ Ω because y0 ∈ ∂Ω and Ω is starlike with respect to the point 0. Then by the definition of Ωr , we get z0 = sy0 ∈ Ωr . This leads to a contradiction. For case (ii), we have |sr −1 y0 | > |y0 |, from which we get sr −1 y0 6∈ Ω because y0 ∈ ∂Ω and Ω is starlike with respect to the point 0. Thus z0 = sy0 6∈ rΩ = Ωr . This contradict to z0 ∈ ∂Ωr . Next we claim that

r0 C arcsin , y0 , y0 ∩ ∂Ω = ∅. (22) |y0 | ∼ r0 We assume by contradiction that there did exist y 0 ∈ C arcsin , y0 , y0 ∩ ∂Ω. |y0 | We note that U (0, r ) is the maximal ball with the center 0 in the cone 0 r0 r0 C arcsin , −y0 , y0 , which means that U (0, r0 ) ⊂ C arcsin , −y0 , y0 and |y0 | |y0 | r0 for any z ∈ C arcsin , −y0 , y0 , |y0 | L(z, y0 ) ∩ U (0, r0 ) 6= ∅. (23) r0 r0 Since C arcsin , y0 , y0 and C arcsin , −y0 , y0 are symmetric about y0 , |y0 | |y 0| ∧ ∼ r0 then there exists a point y 0 ∈ C arcsin , −y0 , y0 , which is the symmetric to y 0 |y0 | ∧ ∧ about y0 , hence y 0 ∈ L(y0 , y˜0 ), and by (23) we have L(y 0 , y0 ) ∩ U (0, r0 ) 6= ∅.

∧

∼

Thus the lines L(y0 , y0 ) and L( y 0 , y0 ) are the same and intersect with ∂Ω at y0 −−− −−−−→ ∼ ∼ ∧ and y 0 , which are in the ray y1 ( y 0 −y0 ) for any y1 ∈ (y , y0 ) ∩ U (0, r0 ). However Ω −−−−−−−→ is starlike with respect to the point y1 , i.e., y1 (˜ y0 − y0 ) intersect ∂Ω only one point, which leads to a contradiction. r0 Because z0 = ry0 ∈ L(0, y0 ) ∩ C arcsin , y0 , y0 ∩ ∂Ωr and |y0 | r0 r0 r0 d(z0 , ∂C(arcsin , y0 , y0 )) ≥ |ry0 −y0 | sin(arcsin ) = (r −1)|y0 | = r0 (r −1), |y0 | |y0 | |y0 |

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there exists a ball U (z0 , r0 (r − 1)) with center z0 and radius r0 (r − 1), such that r0 , y0 , y0 . (24) U (z0 , r0 (r − 1)) ⊂ C arcsin |y0 | other hand, since x0 ∈ ∂Ω, it follows from (2.17) that x0 ∈ 6 On the r0 C arcsin , y0 , y0 ∩ ∂Ω, which together with (21) and (24) implies that x0 ∈ 6 |y0 | U (ry0 , r0 (r − 1)). Then by (20), we get d(Ω, ∂Ωr ) = d(∂Ω, ∂Ωr ) = |x0 − z0 | = |x0 − ry0 | ≥ r0 (r − 1). This completes the proof.

Since D ⊂⊂ B, there exists a constant ρ0 > 1 such that D ⊂⊂ ρ−1 0 B, where = {ρ−1 x : x ∈ B}. 0

ρ−1 0 B

∧

Lemma 2.7. Let O = {Ω ⊂ D; Ω is starlike with respect to U (0, r0 )}. Suppose ∧

that Ω, Ωn ∈O, n = 1, 2, · · · , are such that HlimΩn = Ω. Then for any ρ0 satisfying −1 ρ0 ρ0 ρ0 > 1 and D ⊂ ρ−1 0 B = {ρ0 x; x ∈ B}, δ(B \ Ωn , B \ Ω ) → 0. Proof. It is clear that Ωρ0 is open and there exists a subsequence {B \ Ωρn0k } of {B \ Ωρn0 } such that δ(B \ Ωρn0k , A) → 0

(25)

for some compact subset A in RN . We claim that A = B \ Ω ρ0 .

(26)

−1 −1 ρ0 To this end, we let x ∈ B \ Ωρ0 , then ρ−1 0 x ∈ ρ0 (B \ Ω ) = ρ0 B \ Ω. Since −1 HlimΩn = Ω and D ⊂⊂ ρ0 B, it follows that −1 δ(ρ−1 0 B \ Ωn , ρ0 B \ Ω) → 0.

Then by Lemma 2.1, there exists a sequence {xn } such that xn ∈ ρ−1 0 B \ Ωn and −1 −1 xn → ρ0 x, which implies ρ0 xn → x. Since ρ0 xn ∈ ρ0 (ρ0 B \ Ωn ) = B \ Ωρn0 , it follows from Lemma 2.1 again that x ∈ A and so B \ Ωρ0 ⊂ A. Now let x ∈ A. By (25) and by Lemma 2.1, there exists a sequence {xnk }∞ k=1 such that xnk ∈ B \ Ωρn0k and x = limk→∞ xnk . One can check easily that −1 ρ−1 0 xn k ∈ ρ 0 B \ Ω n k

−1 and ρ−1 0 xnk → ρ0 x,

−1 −1 −1 moreover, δ(ρ−1 0 B\Ωnk , ρ0 B\Ω) → 0, Then by Lemma 2.1 again, ρ0 x ∈ ρ0 B\Ω, i.e., x ∈ B \ Ωρ0 . Thus A ⊂ B \ Ωρ0 and (26) follows. Finally, we claim that

δ(B \ Ωρn0 , B \ Ωρm0 ) → 0 as n, m → ∞.

(27)

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Indeed, as n, m → ∞, δ(D \ Ωn , D \ Ωm ) → 0 and −1 δ(ρ−1 0 B \ Ωn , ρ0 B \ Ωm ) → 0,

so δ(B \ Ωρn0 , B \ Ωρm0 ) = max{supx∈B\Ωρm0 infy∈B\Ωρn0 |x − y|, supx∈B\Ωρn0 infy∈B\Ωρm0 |x − y|} = max{supρ−1 x∈ρ−1 B\Ω infρ−1 y∈ρ−1 B\Ω |x − y|, 0

m

0

0

n

0

supρ−1 x∈ρ−1 B\Ω infρ−1 y∈ρ−1 B\Ω |x − y|} 0

=

0

n

0

0

m

ρ0 max{supρ−1 x∈ρ−1 B\Ω infρ−1 y∈ρ−1 B\Ω |ρ−1 0 x m n 0

0

0

0

− ρ−1 0 y|,

−1 supρ−1 x∈ρ−1 B\Ω infρ−1 y∈ρ−1 B\Ω |ρ−1 0 x − ρ0 y|} 0

=

ρ0 δ(ρ−1 0 B

\

0

Ωn , ρ−1 0 B

n

0

0

m

\ Ωm ) → 0,

from which (27) follows. Now by (25), (26) and (27), δ(B \ Ωρn0 , B \ Ωρ0 ) → 0. This completes the proof. Now we are going to prove Theorem 2.2. ∼

Proof of Theorem 2.2. Since Ω0 = HlimΩn and {Ωn }∞ n=1 ⊂O , it follows ∼

from Theorem 2.1 that Ω0 ∈O . After carefully checking the proof of Theorem 2.1, we find that there exists x0 ∈ D such that Ω0 and Ωn , for n large enough, are starlike with respect to U (x0 , r20 ). We assume without loss of generality that x0 = 0. Let K ⊂ RN be open and such that K ⊂ B \ Ω0 . Then there exists ρ1 with ρ1 1 < ρ1 < ρ0 such that K ⊂ B \ Ω0 . In order to prove the theorem, it suffices to ρ1 show that Ωn ⊂ Ω0 for n large enough. By contradiction, we assume that there existed a subsequence of {xn }, still denoted by itself, such that xn ∈ Ωn and xn 6∈ Ωρ01 . Then by Lemma 2.6, we get r0 U (xn , (ρ1 − 1)) ⊂ Ωρn1 for all n. (28) 2 Notice that there is a subsequence of {xn }, still denoted by itself, such that HlimU (xn , r20 (ρ1 − 1)) = U for a certain open set U . Then by Lemma 2.4, we get U (xn , r20 (ρ1 − 1)) ⊂ U and on a subsequence of {xn }, denoted by itself again, xn → x0 . Thus by Lemma 2.7 we obtain r0 δ(B \ U (xn , (ρ1 − 1)), B \ U ) → 0. 2 Now it follows from Lemma 2.2 and (28) that U ⊂ Ωρ01 . Thus r0 U (x0 , (ρ − 1)) ⊂ Ωρ01 . (29) 2

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On the other hand, since xn ∈ Ωn \ Ωρ01 ⊂ D \ Ωρ01 , it follows that x0 = limn→∞ xn ∈ D \ Ωρ01 which contradicts to (29). This completes the proof. 3. Existence of shape optimization In this section, we shall prove the existence of problems (P ). Theorem 3.1. The shape optimization problem (P ) has at least one solution. Proof. Throughout the proof of Theorem 3.1, we shall use sptu to denote the support of u. Z Let d = MinΩ∈O

exists a sequence

F (x, uΩ , ∇uΩ ) dx. It is obvious that d > −∞. Then there

B\Ω {Ωn }∞ n=1

⊂ O such that Z F (x, un , ∇un ) dx, d = lim n→∞

(30)

B\Ωn

where un ≡ uΩn is the weak solution of (1.1). By Theorem 2.1, there exist a ∗ ∗ subsequence of {Ωn }∞ n=1 , still denoted by itself, and Ω ∈ O such that Ω = HlimΩn . By taking u = un , Ω = Ωn in (2), we get Z Z Z ∞ ν ∇un · ∇ϕ dx + (un · ∇)un · ϕ dx = f · ϕ dx, ∀ ϕ ∈ C0,σ (B \ Ωn ). B\Ωn

B\Ωn

B\Ω

1 Since un ∈ H0,σ (B \ Ωn ), by the latter equality, we have Z Z 2 ν |∇un | dx = f · un dx, B\Ωn

which implies that

Z

(31)

B\Ωn

|∇un |2 dx ≤ C, B\Ωn

here and throughout the proof of Theorem 3.1, C denotes several positive constants independent of n. Let   un (x) in B \ Ωn , ∧ un (x) = (32) 0 in Ωn ∪ (RN \ B), ∧

∧

1 N N then {un }∞ n=1 is bounded in (H (R )) . Hence there exists subsequence of {un }, still denoted by itself, such that ∧

∧

un →u weakly in (H 1 (RN ))N and strongly in (L2 (B))N ∧

∧

(33)

∧

for some u∈ (H01 (B))N with u (x) = 0 in RN \ B and div u (x) = 0 in RN . Now we claim that ∧

1 u (x) ∈ H0,σ (B \ Ω∗ ).

(34)

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Since Ω∗ is Lipschitz due to the starlikeness of Ω∗ with respect to a ball, it suffices ∧ to show that u∈ H01 (B \ Ω∗ )N (cf. Ref. 2 or 6). To this end, we only need to show that ∧

u (x) = 0 a.e. in Ω∗ .

(35)

Indeed, for any open subset K satisfying K ⊂ Ω∗ , by Lemma 2.3, there is a positive integer nK , such that K ⊂ Ωn ⊂ D ⊂⊂ B for all n ≥ nK . Thus Z Z Z ∧ ∧ ∧ 2 2 u u | (x)| dx = lim | n (x)| dx ≤ lim |un (x)|2 dx = 0, n→∞

K

n→∞

K

Ωn

∧

which implies that u (x) = 0 a.e. in K. Since K ⊂ K ⊂ Ω∗ is arbitrary, (35) and consequently (34) follow. Then we claim that Z Z Z ∧ ∧ ∧ ∗ ∞ ∇ u ·∇ϕ dx+ (u ·∇) u ·ϕ dx = f ·ϕ dx ∀ ϕ(x) ∈ C0,σ (B \ Ω ), ν B\Ω

∗

B\Ω

∗

B\Ω

∗

(36) i.e., ν

Z

∧

sptϕ

∇ u ·∇ϕ dx +

Z

∧

sptϕ

∧

(u ·∇) u ·ϕ dx =

Z

sptϕ

f · ϕ dx

∗

∞ for each ϕ(x) ∈ C0,σ (B \ Ω ). Let ∧

ϕ=

 ∗  ϕ in B \ Ω ,

(37)

 0 in (RN \ B) ∪ Ω∗ ,

by Theorem 2.2, there exists a positive integer n0 (ϕ), such that ∧

spt ϕ = sptϕ ⊂ B \ Ωn , for all n ≥ n0 (ϕ). ∧ ∞ Then for each n ≥ n0 (ϕ), ϕ∈ C0,σ (B \ Ωn ). So by (2), we have Z Z Z ∧ ∧ ∧ ∧ ∧ ν ∇un · ∇ ϕ dx + (un · ∇)un · ϕ dx = B\Ωn

B\Ωn

∧

f · ϕ dx,

B\Ωn

which, together with (37), implies Z Z Z ∧ ∧ ∧ ν ∇un · ∇ϕ dx + (un · ∇)un · ϕ dx = f · ϕ dx, sptϕ sptϕ sptϕ passing to the limit for n → ∞ and using (33) we get (36). Finally, we claim that Z ∧ ∧ F (x, u, ∇ u) dx. d≥ B\Ω

∗

(38)

We notice that ∧

∧

un →u strongly in (H 1 (RN ))N .

(39)

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Indeed, since ν

Z

∧

B\Ω

∗

|∇ u |2 dx =

Z

∧

B\Ω

∗

f · u dx,

it follow from (31), (32), (33) and (35) that Z Z Z Z ∧ ∧ ∧ ν |∇un |2 dx − ν |∇ u |2 dx = ν |∇un |2 dx − ν |∇ u |2 dx ∗ B B\Ω B B\Ω Z Z Z Z n ∧ ∧ ∧ = f · un dx − f · u dx = f · un dx − f · u dx → 0. B\Ωn

B\Ω

∗

B

B

This together with (33) implies (39). ∗ Let B \ Ω = ∪∞ j=1 Gj , where Gj , j = 1, · · · , be open and bounded subsets in ∗ B \ Ω such that Gj ⊂ Gj+1 . By Theorem 2.2, we obtain that for each j, there exists a positive integer nj such that Gj ⊂ B \ Ωn as n ≥ nj . So limn→∞

Z

∧

B\Ωn

∧

F (x, un , ∇un ) dx ≥ limn→∞

Z

∧

∧

F (x, un , ∇un ) dx.

(40)

Gj

By (30), (39), (40) and the assumption (4), using Fatou’s Lemma, we get Z ∧ ∧ d≥ F (x, u, ∇ u) dx.

(41)

Gj

It is clear that lim χGj (x) = χB\Ω∗ (x),

j→∞

∗

where χGj and χB\Ω∗ are the characteristic functions of Gj and B \ Ω respectively. Thus by (41) and Fatou’s Lemma again, we get Z Z ∧ ∧ ∧ ∧ d ≥ limj→∞ χGj F (x, u, ∇ u) dx F (x, u, ∇ u) dx dx = limj→∞ ∗ Gj B\Ω Z ∧ ∧ ≥ F (x, u, ∇ u) dx. B\Ω

∗

from which (38) follows. By (34), (36) and (38), we obtain that Ω∗ is a solution of problem (P ). This completes the proof. Appendix A. Lemma A.1. Let U ⊂ RN , N ≥ 1, be an open subset. For any x ∈ U and y ∈ RN \ U, there exists a positive constant 0 < t < 1 such that x + t(y − x) ∈ ∂U .

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Proof. Since x ∈ U and U is an open subset of RN , there exists a positive constant 0 < t0 < 1 such that x + t0 (y − x) ∈ U. Let t∗ = sup{0 ≤ t ≤ 1 : x + t(y − x) ∈ U }. Then t0 ≤ t∗ ≤ 1. Now we claim that t∗ 6= 1. By contradiction, there exists a sequence {tn }∞ n=1 satisfying 0 ≤ tn ≤ 1, x + tn (y − x) ∈ U and tn → 1 as n → ∞. Hence x + tn (y − x) → y ∈ U, which leads to a contradiction. Next we claim that x + t∗ (y − x) ∈ ∂U. By contradiction, we suppose that x + t∗ (y − x) ∈ U or x + t∗ (y − x) ∈ RN \ U . If x + t∗ (y − x) ∈ U , since U is an open subset, there exists a positive constant 0 < δ < 1 − t∗ , such that x + (t∗ + δ)(y − x) ∈ U, which contradicts with the fact that t∗ = sup{0 ≤ t ≤ 1 : x + t(y − x) ∈ U }. If x + t∗ (y − x) ∈ RN \ U, since RN \ U is an open subset, there exists a positive ∼

constant 0 < δ < t∗ , such that ∼

x + t(y − x) ∈ RN \ U for all δ ≤ t ≤ t∗ , which contradicts with the fact that t∗ = sup{0 ≤ t ≤ 1 : x + t(y − x) ∈ U }. This completes the proof.

References 1. C. Dellacherie, Analytical Sets, Capacities and Hausdorff Measures, Lectures Notes in Applied Mathematics 295, Springer-Verlag, Berlin, 1972. 2. Giovanni P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equation, Springer-Verlag, New York, 1994. 3. K. Kuratowski, Introduction to Set Theory and Topology, Pergamon Press, Oxford, 1961. 4. W. B. Liu, J. E. Rubio, Local convergence and optimal shape design, SIAM Journal on Control and Optimization 30, 49-62 (1992).

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5. O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, Berlin, 1984. 6. Hermann Sohr: The Navier-Stokes Equations: An Elementary Functional Analytic Approach, Birkh¨ auser, Berlin, 2001. 7. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Press Syndicate of the University of Cambridge, 1993. 8. D. Tiba, A property of Sobolev spaces and existence in optimal design, Applied Mathematics and Optimization 47, 45-58 (2003). 9. G. Wang, L. Wang and D. Yang, Shape Optimization of Elliptic Equation in Exterior Domain, Submitted .

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Chap36-XiangXiaoLing

SOLUTION MAP OF STRONGLY NONLINEAR IMPULSIVE EVOLUTION INCLUSIONS∗

X. XIANG Department of Mathematics, Guizhou University, Guiyang 550025, Guizhou, China E-mail: [email protected] Y. PENG Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China and Department of Mathematics, Zunyi Normal College, Zunyi 563002, Guizhou, China E-mail: [email protected] W. WEI Department of Mathematics, Guizhou University, Guiyang 550025, Guizhou, China E-mail: [email protected]

In this paper strongly nonlinear impulsive differential inclusions are considered. It is proved that the Solution Set is not empty. The compactness and upper semicontinuity of solution map are also obtained. Keywords: Pseudomonotone operator, L−generalized pseudomotonicity, Evolution triple, Compact embedding, Impulse. Subjclass: 34G20, 34K30, 35A05, 93C25.

1. Introduction Differential inclusions x(t) ˙ + A(t, x(t)) ∈ F (t, x(t)) a.e. t ∈ I

(1)

feature prominently in modern treatments of Optimal Control Theory. The condition (1), summarizing constraints on allowable velocities, provides a convenient framework for optimal control problem which may have solutions and derive the optimality conditions. Particularly when the data are nonsmooth, often the very ∗ This

work is supported by the National Science Foundation of China under Grand No. 10361002. 338

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statement of optimality conditions make reference to differential inclusion. A great deal of problems related to controlled systems governed by deterministic differential equations can be successfully solved by using methods of set-valued analysis. Of course the existence of solutions and properties of solution maps, such as compactness and continuity, for a differential inclusion are very important and basic, when we study optimal control problem. In recent years differential inclusions have been investigated by many authors.3,4,7,13,15 On the other hand, in order to study the phenomena that can not be modelled by traditional initial value problems such as the dynamics of population subjected to abrupt change, the impulsive conditions were introduced. The impulsive conditions are the combination of the traditional initial value and the short-term perturbations whose duration can be negligible in comparison with duration of the process. They have advantage over the traditional initial value problems because they can be used to model such a class of phenomena. There are a lot of papers discussing impulsive differential equation on finite dimensional spaces and on infinite dimensional spaces (see Refs. 1, 5, 10, and 19). Recently, we have done some works on semilinear (strongly nonlinear) impulsive differential equations and optimal controls (see Refs. 14 and 16). Of course, when we discuss impulsive control system, we should study impulsive differential inclusion. To our knowledge, few paper discussed the impulsive differential inclusions except one paper discussing semilinear impulsive differential inclusions given by us(see Ref. 18). In this paper we consider a class of strongly nonlinear impulsive differential inclusion. Existence of solutions (i.e., solution map is not empty) and some properties of solution map are obtained. Our approach is based on techniques and results from the theory of multi-valued analysis and nonlinear operator of monotone type. At first, by using concept of L−generalized pseudo-monotonicity given by N.S. papageorgiou 12 and modifying his idea, we obtain an existence result for strongly nonlinear differential inclusions under weaker assumptions. At the same time, compactness and upper semicontinuity of solution map in C(I, H) are also proved. Based the results obtained, we discuss the strongly nonlinear impulsive differential inclusions. We not only prove that solution map is not empty, but also give some interesting properties of solution map, such as compactness and continuity. In next section, for the convenience of the reader, we recall the basic definitions and preliminary results. In section 3, we discuss strongly nonlinear differential inclusions. The last section is devoted to strongly nonlinear impulse differential inclusions. 2. Mathematical Preliminaries Let (Ω, Σ) be a measurable space and X be a separable Banach space. Throughout this paper, we will be using the following notations given by Ref. 9:

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Pf (c) (X) = A⊆ X|A is nonempty, closed (and convex) ; P(w)k(c) (X) = A ⊆ X|A is nonempty, (weakly) compact and convex . A multifunction F : Ω −→ Pf (X) is said to be measurable if for all x ∈ X, the function ω 7−→ d(x, F (ω)) = inf kx − zk z ∈ F (ω)

is measurable. A multifunction F : Ω −→ 2X \ {∅} is said to be graph measurable if GrF = (ω, x) ∈ Ω × X x ∈ F (ω)

belongs to Σ × B(X) with B(X) being the Borel σ−field of X. For Pf (X)−valued multifunctions measurability implies graph measurability. For 1 ≤ p ≤ ∞, we define SFp to be the set of all Lp (Ω, X)−selectors of F (·), i.e. SFp = f ∈ Lp (Ω, X) f (ω) ∈ F (ω) a.e. . p X Note that for a graph measurable multifunction F : Ω −→ 2 \ ∅, the set SpF is nonempty if and only if the function ω −→ inf kxk x ∈ F (ω) belongs to L (Ω) (See, Lemma 3.2 , Page 175 of Ref. 9). Let Z be another Hausdorff topological space. A multifunction F : X −→ Z 2 \ {∅} is said to be lower semicontinuous (l.s.c) (upper semicontinuous (u.s.c)) if for C ⊆ Z closed, the set

F + (C) = {x ∈ X|F (x) ⊆ C} T respectively, F − (C) = {x ∈ X|F (x) C 6= ∅} is closed in X (see Ref. 9 ). Let H be a separable Hilbert space and V be a dense subspace carrying the structure of a separable reflexive Banach space, which is embedded into H continuously. Identifying H with its dual, we have V ⊆ H ⊆ V ∗ , where V ∗ is the topological dual of V , with all embedding being continuous and dense. Such a triple of spaces is known in the literature as “evolution triple”. Throughout this work we will assume that the embedding of X into H is compact. Let hx, yi denote the pairing of an element x ∈ V and an element y ∈ V ∗ , if x, y ∈ H, then hx, yi = (y, x), where (·, ·) is the scalar product on H. The norm of any Banach space X will be denoted by k · kX . Let 0 < t ≤ T < +∞, It = [0, t], I = [0, T ]. Let p, q ≥ 1 such that 2 ≤ p < +∞ and p1 + 1q = 1. For economy of notation, we write Ltp (V ) = Lp (It , V ), Ltq (V ∗ ) = Lq (It , V ∗ ). It follows from reflexivity of V both Ltp (V ), Ltq (V ∗ ) are reflexive Banach space. The pairing of Ltp (V ) and Ltq (V ∗ ) is demoted hh·, ·iit . In particular for t = T , we use << ·, · >> = << ·, · >>T . Clearly for x, y ∈ L2 (H), have << x, y >> = ((x, y)), where ((·, ·)) is the scalar product in Hilbert space L2 (H).

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Define

Wpq = Wpq (I) = x x ∈ Lp (V ), x˙ ∈ Lq (V ∗ ) ,

kxk2Wpq = kxk2Lp(V ) + kxk ˙ 2Lq (V ∗ ) ,

Wpq , k·kWpq is a Banach space and the embedding Wpq ,→ C(I, H) is continuous. If the embedding V ,→ H is compact, the embedding Wpq ,→ Lp (H) is also compact. Similarly, we can define Wpq ([t1 , t2 ]) (see Ref. 20). An operator A : X −→ X ∗ is said to be demicontinuous if xn −→ x in X implies w that A(xn ) −→ A(x) in X ∗ as n −→ ∞. We said that A(·) is pseudomonotone if w xn −→ x in X and limhA(xn ), xn − xi ≤ 0 imply that hA(x), x − yi ≤ limhA(xn ), xn − yi for all y ∈ X.

A demicontinuous monotone operator is pseudomonotone. The operator A(·) is said w to be of “type(S)+ ” if xn −→ x in X, and if limhA(xn ), xn − xi ≤ 0, then xn −→ x in X. If A is demicontinuous and of type(S)+ , then A is pseudomonotone (See Ref. 20). Now, let Y be a reflexive Banach space. Let L : D ⊆ Y → Y ∗ and be a linear densely defined maximal monotone operator, K : Y → Y ∗ , K(·) is “L− pseudomonotone” if it is demicontinuous, bounded (i.e., maps bounded sets to bounded ones) and if {yn }n≥1 ⊆ D is such that w

w

yn −→ y in Y , L(yn ) −→ L(y) in Y ∗ and

then we have

lim K(yn ), yn − y Y ∗ Y ≤ 0, w

K(yn ) −→ K(y) in Y ∗ as n → ∞, (K(yn ), yn )Y ∗ Y −→ (K(y), y)Y ∗ Y as n → ∞. ∗

Also, we say that the multi-valued operator G : Y −→ 2Y \ {∅} is “L−generalized pseudomonotone” if a) for every y ∈ Y , G(y) ∈ Pwkc (Y ∗ ); b) G(·) is u.s.c from every finite-dimensional subspace Z of Y into Yw∗ ; w w w c) {yn } ⊆ D, yn −→ y in Y , L(yn ) −→ L(y) in Y ∗ , yn∗ ∈ V (yn ), yn∗ −→ y ∗ in Y ∗ , lim (yn∗ , yn )Y ∗ Y ≤ (y ∗ , y)Y ∗ Y ⇒ (y ∗ , y) ∈ GrG, (yn∗ , yn )Y ∗ Y −→ (y ∗ , y)Y ∗ Y . Finally, we say that K(·) ( respectively G(·)) is coercive if hK(y), yi −→ ∞ respectively, kyk→∞ kykY inf [hz, yiY ∗ Y ; z ∈ G(y)] lim −→ ∞. kykY kyk→∞ lim

The following surjectivity result for L−generalized pseudomonotone operator will be used in the sequel (See Proposition 1 of Ref. 11 ).

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Proposition 2.1. If Y is a reflexive and strictly convex Banach space, L : D ⊆ Y −→ Y ∗ is a linear densely defined maximal monotone operator and G : Y −→ ∗ 2Y \ ∅ is an L−generalized pseudomonotone operator which is coercive then R(L + V ) = Y ∗. 3. Existence of Solutions of Differential Inclusion In this section, we consider the following evolution inclusion x(t) ˙ + A(t, x(t)) ∈ F (t, x(t)), x(0) = x0 ∈ H.

(2)

We impose the following hypotheses on the date of (2) [A] A : I × V −→ V ∗ , t 7−→ A(t, x) is measurable, x 7−→ A(t, x) is demicontinuous and pseudomonotone, q kA(t, x)kV ∗ ≤ a1 (t) + c1 kxkp−1 V , a1 (·) ∈ L (I);

hA(t, x), xi ≥ ckxkpV − c0 kxkp−1 − a(t) a.e. c > 0, c0 ≥ 0, a ∈ L1 (I). V [F] F : I × H −→ Pf c (H) is a set-valued mapping, t 7−→ F (t, x) is measurable, x 7−→ F (t, x) is u.s.c., kF (t, x)kH ≤ sup kykH y ∈ F (t, x) ≤ a2 (t) + ckxkk−1 H

where a2 ∈ Lq (I), c2 ≥ 0, 2 ≤ k < p. By a solution of inclusion (2), we mean a function x ∈ Wpq such that x satisfies following evolution equation in weak sense x(t) ˙ + A(t, x(t)) = f (t) a.e. on I, x(0) = x0 , with f ∈ SFq (·,x(·)) 20 . The set of solutions of inclusion (2) corresponding to x0 is denoted by S(x0 ) ⊆ C(I, H). At first, by similar arguments of the proof of Theorem 2.A of Ref. 17, one can obtain a priori estimation on solution of (2). Lemma 3.1. Under assumptions [A] and [F], there exists a positive constant M > 0 such that for all x ∈ S(x0 ), we have kxkWpg ≤ M , kxkC(I,H) ≤ M Proof. Assume that x ∈ S(x0 ), then there exists g(·) ∈ SFq (·,x(·)) = Lq (I, H) g(t) ∈ F (t, x(t)) a.e. , such that x satisfies x(t) ˙ + A(t, x(t)) = g(t), x(0) = 0. By assumption [F], we have kg(t, x(t))kH ≤ a2 (t) + c2 kxkk−1 V .

g ∈

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Using assumptions, integration by parts and Cauchy inequality, one can verify that for all t ∈ [0, T ], we have kx(t)k2H + rkxkpLt (V ) ≤ M1 + N1 kxkLt (V )

(k−1)q

p

p

where r > 0, M1 , N1 ≥ 0, (k − 1)q < p, hence there exists a constant M > 0 such that kxkWpq ≤ M , kxkC(I,H) ≤ M. Theorem 3.1. If assumptions [A] and [F] hold, and x0 = 0, then S(x0 ) 6= ∅. Proof. We consider the following evolution inclusion x(t) ˙ + A(t, x(t)) ∈ F (t, x(t)) x(0) = 0

(3)

which is a special case of (2). ∗ Let L : D ⊆ Lp (I, V ) −→ L q (I, V ) be the linear operator defined by Lx = x˙ for all x ∈ D = x ∈ Wpq (I) x(0) = 0 . L is a densely defined linear maximal monotone operator (see page 855 of Ref. 20). b be the Nemitsky operator corresponding to A, A b : Lp (I, V ) −→ Lq (I, V ∗ ) Let A is the demicontinuous and pseudomonotone and

b ≤e a1 + c1 kxkp−1

A(x) Lp (I,V ) , Lq (I,V ∗ )

DD

b A(x), x

EE

≥ ckxkpLp (I,V ) − d1 kxkp−1 Lp (I,V ) − d2

where c > 0, d1 , d2 are constants (see Ref. 20 ). Let G be the multi-valued Nemitsky operator corresponding to (−F ) then G : Lp (I, H) −→ Pwkc (Lq (I, H)). In fact, at first, by assumptions of [F], q G(x) = S−F (·,x(·)) = g ∈ Lq (I, H) g(t) ∈ −F (t, x(t)) a.e. 6= ∅

(See, Proposition 1.7, page 142 of Ref. 9) and G(x) is convex and bounded in w Lq (I, H). Due to reflexivity of Lq (I, H) we may assume {gn } ⊆ G(x) and gen −→ g s in Lq (I, H). By convexity of G(x) we can find {e gn } ⊆ G(x) such that ge −→ g in Lq (I, H) and assume gen (t) −→ g(t) a.e. on I in H.

Again By virtue of assumption [F], we have

g(t) ∈ F (t, x(t)) a.e. on I. This implies g ∈ G(x).

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G(x) is weakly compact. Define V : Lp (I, V ) → Pwkc (Lq (I, V ∗ )) given by b V (x) = A(x) + G(x), ∀x ∈ Lp (I, V ).

It can be seen from proposition 2.1 that in order to prove the existence of solutions of (3) we have to show V is L−generalized pseudomonotone and coercive. b is demicontinuous and L−generalized (see Ref. 17). We show that At first, A V : Lp (I, V ) → Pwkc (Lq (I, V ∗ )) is u.s.c.. Define n o \ V − (C) = x ∈ Lp (I, V ) V (x) C 6= ∅

where C is an nonempty and weakly closed set of Lq (I, H). We show that V −1 (C) s is closed. In fact, let {xn } ⊆ V −1 (C) and xn −→ x in Lp (I, V ), we may assume T xn (t) −→ x(t) a.e. on I in V . Let un ∈ V (xn ) C, then {un } ⊆ Lq (I, V ∗ ) and {un } is bounded (See Lemma 3.1). There exists a subsequence, relabelled as {un }, such that w

un −→ u in Lq (I, V ∗ ). b n )+gn (gn ∈ G(xn )) Obviously, u ∈ C. Since un ∈ V (xn ), we may assume un = A(x w ∗ and gn −→ g in Lq (I, V ). By convergence Theorem (See Theorem 7.2.2 of Ref. 6), we have g(x) ∈ G(x(t)) a.e. on I b is demicontinuous, hence A

w b n ) −→ A(x A(x) in Lq (I, V ∗ ) w b n ) + gn −→ b un = A(x u = A(x) + g in Lq (I, V ∗ ). T where u ∈ V (x) C, x ∈ V −1 (C). This implies that V : Lp (I, V ) −→ Pwkc (Lq (I, V ∗ )) is u.s.c.. Now we demonstrate that V satisfies last condition of L–generalized pseudow w monotonicity. Suppose {xn } ⊆ D, xn −→ x in Lp (I, V ) and L(xn ) = x˙ n −→ w ∗ ∗ L(x) = x˙ in Lq (I, V ), un ∈ V (xn ), un −→ u in Lq (I, V ) and

lim << un , xn >> ≤ << u, x >> . We will show (x, u) ∈ GrV and << un , xn >>−→<< u, x >>. b n ) + gn , gn ∈ G(xn ), {gn } is bounded in Lq (I, V ∗ ), we may assume Let un = A(x w w gn −→ g in Lq (I, V ∗ ). Since xn −→ x in Wpq and the embedding Wpq ,→ Lp (I, V ) is compact, we have s

xn −→ x in Lp (I, H). Hence << gn , xn − x >>−→ 0.

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On the other hand, lim << A(xn ), xn − x >> = lim << A(xn ) + gn , xn − x >> − << gn , xn − x >> = lim << A(xn ) + gn , xn >> − << A(xn ) + gn , x >> = lim << un , xn >> − << u, x >>

≤ 0. b is L−pseudomonotone, we have Since A and

i.e.,

w b b n ) −→ A(x A(x)

in Lq (I, V ∗ )

DD EE DD EE b n ), xn b A(x −→ A(x), x , << un , xn >>−→<< u, x >> .

b At last we show that V is coercive. Let x ∈ Lp (I, V ), u ∈ V (x) i.e., u = A(x)+g, where g ∈ G(x), we have EE DD b A(x), x + << g, x >> . << u, x >> = By virtue of assumptions [A] and [F] we have

k ckxkpLp (V ) − d1 kxkp−1 Lp (V ) − d2 kxkLp (V ) − d3 ≤ << u, x >>

where c > 0, d1 , d2 , d3 are constants. It is easy to see inf

[<< u, x >>| u ∈ V (x)] −→ +∞. kxkLp (V )

It can be seen from proposition 2.1, that R(L + V ) = Lq (I, V ∗ ). This implies that S(x0 ) 6= Ø for x0 = 0. Now we remove the assumption x0 = 0. Theorem 3.2. For any x0 ∈ H, under assumptions [A] and [F] the solution map S(x0 ) of evolution inclusion (2) is not empty. Proof. At first we assume x0 ∈ V . Define A1 (t, x) = A(t, x + x0 ), F1 (t, x) = f (t, x + x0 ) One can verify that A1 and F1 satisfy the assumption [A] and [F]. By Theorem 3.A, we obtain S(x0 ) 6= ∅. If x0 ∈ H, there exists a sequence {x0n } ⊆ V such that x0n −→ x0

in H

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there exists xn ∈ Wpq such that x˙ n (t) + A(t, xn (t)) = fn (t) xn (0) = x0n where fn (t) ∈ F (t, xn (t)) a.e. on I. By Lemma 3.1, we may assume w

in Lp (I, V ),

w

in Lq (I, V ∗ ),

w

in Lq (I, H).

xn −→ x x˙ n −→ x˙ fn −→ f Further, we have w

xn −→ x s

xn −→ x

in Wpq , in Lp (I, H),

s

xn (t) −→ x(t)

a.e. on I in H,

xn (0) = x0n −→ x0

in H.

Repeating the procedure of proof of Theorem 3.1, we have f ∈ −G(x). By integration by parts in Wpq , we have hhx˙ n − x, ˙ xn − xii =

1 1 k(xn − x)(T )k2H − k(xn − x)(0)k2H 2 2

and hhx˙ n , xn − xii =

1 1 k(xn − x)(T )k2H − k(xn − x)(0)k2H + hhx, ˙ xn − xii , 2 2

hence hhx˙ n , x − xn ii ≤

1 kxn (0) − x(0)kH + hhx, ˙ x − xn ii , 2

lim hhx˙ n , x − xn ii ≤ 0.

n→∞

Since

we have lim

n→∞

= lim

n→∞

DD DD

b n ) + fn , x˙ n = −A(x b n ), xn − x A(x

b n ) − fn , xn − x A(x

= lim hhx˙ n , x − xn ii n→∞

EE

EE

+ lim hhfn , xn − xii n→∞

≤ 0. b is L−pseudomonotone, we have A

w b b n ) −→ A(x A(x) in Lp (I, V ),

DD EE DD EE b n ), xn b A(x −→ A(x), x .

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This implies that DD EE b n ), y x˙ n + A(x = ((fn , y)) , as n → ∞, we have

This implies that

∀y ∈ Lq (I, V ∗ )

DD EE b x˙ + A(x), y = ((f, y)). x ∈ S(x0 ),

that is S(x0 ) 6= ∅. We have solution map S(·) : H → 2C(I,H) \ ∅. Now we will establish the basic properties of solution map. Theorem 3.3. Solution map S(·) : H → Pf k (C(I, H)) is u.s.c.. Proof. According to Theorem 3.B, solution map S(·) : H → 2C(I,H) \ ∅. At first, we show ∀x0 ∈ H, S(x0 ) is compact in C(I, H). Let {xn } ⊆ S(x0 ), by a standard priori estimate (see Lemma 3.1), there exists M > 0 such that kxn kWpq ≤ M, kxn kC(I,H) ≤ M. T We may assume that there exists x ∈ Wpq C(I, H) such that as n → ∞ w

in Wpq ,

s

in Lp (I, H),

xn −→ x xn −→ x

xn (t) −→ x(t)

a.e. on I in H.

Suppose xn ∈ S(x0 ), fn (·) ∈ F (·, xn (·)), xn satisfies the following equation. x˙ n (t) + A(t, xn (t)) = fn (t), xn (0) = x0 , {fn } is bounded in Lq (I, H), we may assume w

fn −→ f

in Lq (H).

By virtue of weak compactness of G (See the proof of Theorem 3.A), we have f ∈ G(x). Using integration by pars in Wpq one can verify DD EE DD EE b n ), xn − x b n ) − fn , xn − x + ((fn , xn − x)) A(x = A(x = hhx˙ n , x − xn ii + ((fn , xn − x)),

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and 1 hhx˙ n , x − xn ii = − kxn (T ) − x(T )k2 − hhx, ˙ xn − xii , 2EE DD b n ), xn − x lim A(x ≤ 0.

b is L−pseudomonotone, we have A

w b b n ) −→ A(x A(x)

and

Hence

DD EE DD EE b n ), xn b A(x −→ A(x), x . x ∈ S(x0 ).

Since 1 kxn (t) − x(t)kH = hhx˙ n − x, ˙ xn − xiit 2 = hhfn − f, xn − xiit +

DD

we have xn → x

b b n ), xn − x A(x) − A(x

EE

t

,

in C(I, H)

This implies that S(x0 ) is compact in C(I, H). Now we turn to show that the solution map is u.s.c. in C(I, H). Suppose C is a closed set of C(I, H). Set n o \ S −1 (C) = x ∈ H S(x) C 6= Ø

If {xn0 } ⊆ S −1 (C) and

xn0 → x0

in H,

T we have to prove x0 ∈ S −1 (C), i.e. there exists x ∈ S(x0 ) C. T Since xn0 ∈ S −1 (C), there exists xn ∈ S(xn0 ) C. Similar to the proof of Theorem 3.B, one can verify that there exists x ∈ Wpq (I) such that (passing to a subsequence if necessary) w

in Wpq ,

xn −→ x

s

in Lp (I, H),

xn −→ x

in C(I, H),

xn −→ x

and x ∈ S(x0 ). It follows that x ∈ C from closedness of C.

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4. Impulsive Evolution Inclusion In this section, we study the impulsive evolution inclusion. Suppose 0 = t0 < t1 < · · · , tm+1 = T , D = {t1 , · · · , tm }. Define I = [0, T ], Ik = [tk , tk+1 ] (k = 1, 2, · · · , m). Consider the following impulsive evolution inclusion x(t) ˙ + A(t, x(t)) ∈ F (t, x(t)), I \ D, (4) − − x(0) = x0 , x(t+ k ) = x(tk ) + Jk (x(tk )), where Jk is a map from H to H present the jump of state at tk . Define P CWpq (I) ≡ x : I \ D → V x|Ik ∈ Wpq (Ik ), k = 0, 1, 2, · · · , m , endowed the norm

x

P CWpq

=

m X

x

P CWpq (Ik )

k=0

P CWpq (I) is Banach space. P C(I, H) ≡ x : I \ D −→ H x|Ik ∈ C(Ik , H), k = 0, 1 · · · , m , endowed the norm

x

P C(I,H)

=

m X

x

C(Ik ,H)

k=0

P C(I, H) is a Banach space. Since Wpq (Ik ) ,→ C(Ik , H)(k = 0, 1, 2, · · · , m), we have P CWpq (I) ,→ P C(I, H). Definition 4.1. x ∈ P CWpq (I) is called P CWpq −solution of (4) if there exists f ∈ SFq (·,x(·)) and x satisfies x(t) ˙ + A(t, x(t)) = f (t),

t ∈ Ik (k = 1, 2, · · · , m)

in the weak sense and − − x(0) = x0 , x t+ . k = x tk + H k x tk

e 0 ). The set of all solutions of (4) corresponding to x0 is denoted by S(x By virtue of Theorem 3.B we can obtain the following existence of solution. Suppose: [J]: Jk : H −→ H is a bounded map (k = 1, 2, · · · , m). e 0 ), set of solutions of Theorem 4.1. Under assumption [A], [F] and [J], the S(x (4), is not empty.

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Proof. (1) for t ∈ [0, t1 ] = I0 , consider the following evolution inclusion without impulsive x(t) ˙ + A(t, x(t)) ∈ F (t, x(t)), (5) x(0) = x0 . If follows from Theorem 3.B that

Let x(0) (·) ∈ S(x0 )(I0 ), x(0)

S(x0 ) 6= ∅. T ∈ Wpq (I0 ) C(I0 , H). Define

(0) − x1 = x(0) (t− (t1 )) ∈ H. 1 ) + J1 (x

(2) For t ∈ [t1 , t2 ] = I1 , consider the following evolution inclusion without impulsive x(t) ˙ + A(t, x(t)) ∈ F (t, x(t)), (6) x(t1 ) = x1 . Again by Theorem 3.B that

Let x(1)

S(x1 ) 6= ∅. T ∈ S(x1 )(I1 ), x(1) ∈ Wpq (I1 ) C([t1 , t2 ], H) satisfies (4). Define (1) − x2 = x(1) t+ t2 + J2 x(0) t− . 2 −x 2

For t ∈ I2 , we consider the following evolution inclusion x(t) ˙ + A(t, x(t)) ∈ F (t, x(t)), x(t2 ) = x2 ,

(7)

we have x(2) ∈ S(x2 )(I2 ). Step by step, we have x(k) ∈ S(xk )(Ik ), k = 0, 1, 2, · · · , m. Define x(t) = x(k) (t),

t ∈ Ik .

It is easy to see that x(·) ∈ P CWpq and is a solution of (4). This implies that e 0 ) 6= ∅. S(x

e Now, we discuss the properties of solution map S(·). Suppose [J1 ] Jk : H −→ H (k = 1, 2, · · · , m) are locally Lip-continuous, i.e., for every ρ > 0, there exists L(ρ) > 0 such that kJk (x1 ) − Jk (x2 )kH ≤ L(ρ)kx1 − x2 kH provided kx1 kH , kx2 kH ≤ ρ.

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We have the following result similar to Theorem 3.C. e : H 7→ 2P C(I,H) \{∅} is u.s.c. with compact Theorem 4.2. Solution map of (4) S(·) values. e 0 ) is compact in C(I, H). Suppose Proof. At first we show that ∀x0 ∈ H, S(x e 0 ), {xn } ⊆ S(x

There exists a constant M > 0 such that kxn kP CWpq ≤ M, Define

n

x(0) n

o

kxn kP C(I,H) ≤ M.

= {xn |I0 } ⊆ S(x0 )(I0 )

By Theorem 3.C there exists a subsequence {xn |I0 }, relabelled {xn }, and x0 ∈ Wpq (I0 ) ,→ C(I0 , H) such that x(0) ∈ S(x0 )(I0 ) and xn |I0 −→ x(0) in C(I0 , H). 0 Of course, xn t− 1 −→ x (t1 ) in H. Define − xn1 = xn t− , x1 = x0 (t1 ) + J1 x0 (t1 ) 1 + J 1 x t1 By assumption [J1 ], we have

xn1 → x1

in H

Consider solution map S(·)(I1 ) given by the following evolution inclusion without impulse x(t) ˙ + A(t, x(t)) ∈ F (t, x(t)) x(t1 ) = x By Theorem 3.C, S(·)(I1 ) : H → Pf k (C(I1 , H)) is u.s.c.. Define

C ≡ x ∈ C(I1 , H) kxkC(I1 ,H) ≤ M ,

{xn } ⊆ S −1 (C)(I1 ),

x1 ∈ S −1 (C)(I1 )

It follows from proof n ofo Theorem 3.C there exists a subsequence of {xn |I1 }, (1) relabelled {xn |I1 } = xn , and x(1) ∈ Wpq (I1 ) ⊂ C(I1 , H) such that (1) x(1) n −→ x

in C(I, H)

and x(1) ∈ S(x1 )(I1 ). Step by step, we obtain family of functions x(i) , i = 0, 1, 2, · · · , m, x(i) ∈ Wpq (Ii ) ,→ C(Ii , H). Define x∗ (t) = x(i) (t),

t ∈ Ii (i = 0, 1, 2, · · · , m)

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e 0 ) and It is easy to see that x∗ ∈ S(x xn −→ x∗

in P C(I, H).

e 0 ) is compact. This implies that S(x e : H → Pf k (P C(I, H)) is u.s.c.. Next we show that solution map S(·) C is a closed set of P C(I, H). We demote Ck = C(Ik ) = x |Ik x ∈ C (i = 0, 1, 2, · · · , m), n o \ e Se−1 (C) = x ∈ H S(x) C 6= Ø . e Suppose xn0 ⊆ Se−1 (C) and xn0 −→ x0 . There exists xn ∈ S(x) · · · ). Define

T

C (n = 1, 2,

x(i) n = xn |Ii (i = 1, 2, · · · , m). It can be seen from the proof of Theorem 3.C that we can assume that (0) x(0) in C(I0 , H) n −→ x

T where x(0) ∈ S x(0) (I0 ) C0 , x0 ∈ S −1 (C0 )(I0 ). Similarly, let (0) − xn1 = xn t+ t1 + J1 x(0) t− , 1 , x1 = x 1 T then xn1 −→ x1 in H. xn1 ∈ S −1 (C1 ), x1n ∈ S(xn )(I1 ) C1 (n = 1, 2, · · · ). We T may assume that there exists x(1) ∈ S(x1 )(I1 ) C1 such that (1) x(1) in C(I1 , H) n −→ x

Step by step, we have family x(i) (i = 0, 1, · · · , m) of functions. Define x∗∗ (t) = x(i) (t), t ∈ Ii . e 0 ) T C, x0 ∈ Se−1 (C). The proof is completed. then x∗∗ ∈ S(x

Our results can be used to discuss some classes of evolution inclusions. Due to pages limited, we give a simple examples of distributed parameter feedback control system without proof. Let Ω ⊂ Rn be a bounded domain with smooth boundary and I = [0, T ], D = {t1 , t2 , · · · , t9 } ⊂ [0, T ], ti = iT 10 , i = 1, 2, · · · , 9. Consider  ∂x(t,y) P  − 4x(t, y) + λ ni=1 cosx(t, y) ∂x(t,y)  ∂t ∂yi    = f (t, y, x(t, y))u(t, y), (t, y) ∈ (0, T ] × Ω,  (8) x(t, y)|I×∂Ω = 0, x(0, y) = x0 (y), y ∈ Ω,   1  x(ti + 0, y) − x(ti − 0, y) = 2 x(ti − 0, y),    u(t, y) ∈ U (t, y; x(t, y)), a.e. y ∈ Ω,

where U : I × Ω × R −→ Pf (R) is a graph measurable multifunction and |λ| < λ1 , (λ1 is the first eigenvalue of (−∆, H01 (Ω)).

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Under reasonable assumptions we can discuss (8) in evolution triple H01 (Ω) ,→ L2 (Ω) ,→ H −1 (Ω) and prove that solution map S(x0 )(Im ) ⊂ T L2 Im , H01 (Ω) C(Im , L2 (Ω)) (m = 1, · · · , 10) are nonempty for x0 ∈ L2 (Ω) and has some kinds of compactness and continuity. References 1. N. U. Ahmed, K. L. Teo and S. H. Hou, Nonlinear impulsive systems on infinite dimensional space, Nonlinear Anal. 54, 907-925 (2003). 2. N. U. Ahmed and X. Xiang, Existence of solutions for a class of nonlinear evolution equations with nonmonotone pertubations, Nonlin. Anal. TMA, 22, 81-89 (1994). 3. N. U. Ahmed and X. Xiang, Differential inclusions on Banach spaces and their optimal control, Nonlin. Func. Anal. Appl., 8, 461-488 (2003). 4. N. U. Ahmed and X. Xiang, Optimal real relaxed controls for differential inclusion, Stock. Anal. Appl., 12, 1-10 (2003). 5. N. U. Ahmed and X. Xiang, Nonlinear uncertain systems and Necessary condition of optimality, SIAM J. Control. Optim., 35, 1755-1772 (1997). 6. J. Aubin and H. Frankowska, Set-Valued Analysis, Birkh¨ auser, 1990. 7. F. H. Clarke, Optimization and Nonsmooth Analysis, reprinted as Vol. 5 of Classics in Applied Mathematics, Wiley-Interscience, New York, 1983. 8. N. Hirano, Nonlinear evolution equations with nonmonotonic pertubations, Nonlin. Anal. TMA, 13, 599-609 (1989). 9. S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, (Volume 1. Theory), Kluwer ACAdemic Publishers, London, 1997. 10. V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore-London, 1989. 11. N. S. Papageorgiou, F. Papalini, and N. Yannakakis, Nonmonotone, Nonlinear Evolution Inclusions, Mathematical and Computer Modelling, 32, 1345-1365 (2000). 12. N. S. Papageorgiou and N. Shahazd, Properties of solution set of Nonlinear Evolution Inclusion. Appl. Math. Optim., 36, 1-20 (1997). 13. R. Vinter, Optimal control. Birkh¨ auser Boston, Berlin, 2002. 14. W. Wei and X. Xiang, Strongly nonlinear impulsive systems and Optimal Control, to appear. 15. X. Xiang and N. U. Ahmed, Necessary conditions of optimality for differential inclusion on Banach spaces, Nonlinear Analysis, TMA 30, 5434-5445 (1997). 16. X. Xiang, Y. Peng, and W. Wei, A general class of nonlinear impulsive integral differential equations and optimal controls on Banach spaces (Proceeding of AIMS 5th meeting on dynamical system and differential equations). 17. X. Xiang, P.Sattagathem, and W. Wei, Relaxed Control for a Class of Strongly Nonlinear Delay Evolution Equations. Nonlinear Analysis, TMA 52, 703-723 (2003). 18. X. Xiang and W. Wei, Mild solution for a class of nonlinear impulsive evolution inclusion on Banach space, to appear in Asian Bulletin of Mathematics. 19. T. Yang, Impulsive control theory, Springer Verlag, Berlin, 2001. 20. E. Zeidler, Nonlinear Functional Analysis and its Application, Springer-Verlag, New York, 1990.

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STUDY ON REPAIRABLE SERIES SYSTEM WITH TWO COMPONENTS A SEMIGROUP APPROACH

HONG XU and DINGHUA SHI Department of Mathematics, Shanghai University, Shanghai 200436, China E-mail: [email protected] In this paper, we will consider a series repairable system with two components, suppose that the lifetime for one of which is an Erlang-n distribution, but another lifetime and the repairing time are general continuous distributions. Using the density evolution method based on vector Markov process, the system can be described as a group of differential equations with integral terms. We will consider the problem as an Abstract Cauchy Problem in some Banach space and prove that the corresponding operator is a generator of a positive contraction semi-group. The main result is: if the initial distribution is in the domain of the operator, the equations exists uniquely a solution with probability meaning.

1. Introduction In [1], the author successfully set up many models for random systems, in which some partial differential and integral equations were obtained. Especially for a series system with two components, a group of differential equations with integral terms were obtained in the case that the lifetime for one of which is an Erlang-n distribution, but another lifetime and the repairing time are general distributions. The equations can be transferred into an abstract Cauchy problem as done in [3], where Fattorini dealt with some kinds of partial differential equations in a semi-group method after transforming them into a corresponding abstract Cauchy problem (ACP). In recent years some probability models have been treated in a similar way [2]. The outline of the paper is as following. In the section 2, the model discussed will be presented and the corresponding ACP will be given. In the section 3, the properties of the operator appeared in the ACP will be proved. The operator is a generator of a positive contract semi-group. Finally, in the section 4 the main result will be obtained. We will prove that in the case that the initial distribution is in the domain of the operator, the original equations exists a probability solution and the solution is also unique. 2. Description of the model and the equations The repairable series system discussed here is consisted with two components and will stop working if one of the components is out of the work. If the first component 354

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goes wrong, the second component stops working and after the system restart to work it will continue to work just as the system has not been wrong. But if the second component is wrong, the first component will continue to work for the repairing of the second component. There is only one repairer in the system. The first component has superior repairing order than the second has. Suppose that random variables of the lifetimes and repairing times are mutually independent. And in order to general, we assume that the two components are old at the beginning. The lifetimes of first component is of Erlang-n distribution and the parameters corresponding to work or failure are respectively λ1 and λ3 . The lifetime distribution of second component is denoted as F2 and the distributions of repairing time are denoted as Gi , i = 1, 2 for the first and second components respectively. Define the sates of the system S(t) as following. If the system is working normally then S(t) = 0; if the first component is out of working then S(t) = 1, in this case the second component will stop working; if the second component is failure the first component will continue to work for the repairing of the component in trouble, then S(t) = 2; if both components are in malfunction then S(t) = 3. Because the distribution is general the random variable S(t) is not a Markov process in the state space E = {0, 1, 2, 3}. Denote the age at time t of the component i as X i (t) and the repairing time elapsed at time t as Yi (t) (i = 1, 2). Then the stochastic process {S(t), Xi (t), Yi (t)} is a vector Markov process. Because the distribution of age for the first component is an n-order Erland distribution, the complementary variable X1 (t) is discrete. The complementary variables X2 (t), Y1 (t) and Y2 (t) are nonnegative. Denote p1 (t, x2 , y1 )dx2 dy1 = P {S(t) = 1, x2 < X2 (t) ≤ x2 + dx2 , y1 < Y1 (t) ≤ y1 + dy1 }, p3 (t, y1 , y2 )dy1 dy2 = P {S(t) = 3, y1 < Y1 (t) ≤ y1 + dy1 , y2 < Y2 (t) ≤ y2 + dy}, p0i (t, x2 , y1 ) = P {S(t) = 0, X1 (t) = i, x2 < X2 (t) ≤ x2 + dx2 , y1 < Y1 ≤ y1 + dy1 }, p2i (t, x2 , y1 ) = P {S(t) = 2, X1 (t) = i, x2 < X2 (t) ≤ x2 + dx2 , y1 < Y1 ≤ y1 + dy1 } here i = 1, 2, · · · , n. The following equations can be derived: [

∂ ∂ + + µ1 (y1 )]p1 (t, x2 , y1 ) = 0, ∂t ∂y1

(1)

[

∂ ∂ + + µ1 (y1 )]p3 (t, y1 , y2 ) = 0, ∂t ∂y1

(2)

∂ ∂ [ + + λ1 + λ2 (x2 )]p01 (t, x2 ) = ∂t ∂x2 [

Z

∞

p1 (t, x2 , y1 )µ(y1 )dy1 ,

(3)

0

∂ ∂ + + λ1 + λ2 (x2 )]p0j (t, x2 ) = λ1 p0,j−1 (t, x2 ), j = 2, · · · , n, ∂t ∂x2

(4)

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[

[

∂ ∂ + + λ3 + µ2 (y2 )]p21 (t, y2 ) = ∂t ∂y2

Z

∞

p3 (t, y1 , y2 )µ(y1 )dy1 ,

(5)

0

∂ ∂ + + λ3 + µ2 (y2 )]p2j (t, y2 ) = λ31 p2,j−1 (t, y2 ), j = 2, · · · , n, ∂t ∂y2

(6)

The boundary conditions:

p0j (t, 0) =

p2j (t, 0) =

Z Z

∞

p2,j (t, y2 )µ2 (y2 )dy2 , j = 1, · · · , n,

(7)

p0,j (t, x2 )λ2 (x2 )dx2 , j = 1, · · · , n,

(8)

0 ∞

0

p1 (t, x2 , 0) = λ1 p0n (t, x2 ),

(9)

p3 (t, 0, y2 ) = λ3 p2n (t, y2 ).

(10)

Initial conditions: p0j (0, x2 ) = ϕ(x2 ), j = {1, · · · , n}; 2) , µ1 (y1 ) = Here λ2 (x2 ) = Ff¯22(x (x2 ) Define states space

g1 (y1 ) ¯ 1 (y1 ) G

and the others are zeros.

and µ2 (y2 ) =

g2 (y2 ) ¯ 2 (y2 ) G

(11)

are the hazard functions.

  p = (p1 (x2 , y1 ), p3 (y1 , y2 ), p01 (x2 ), · · · , p0n (x2 ), p21 (y2 ), · · · , p2n (y2 ))          ∈ L1 ([0, ∞) × [0, ∞)) × L1 ([0, ∞) × [0, ∞)) × L1 [0, ∞) × · · · × L1 [0, ∞) . X= n n X X       k p0j kL1 + k p2j kL1    k p k=k p1 kL1 + k p3 kL1 +  j=1

j=1

It is obvious that X is a Banach space. Define the operator



 0  R ∞ p (x , y )µ (y )dy   0 1 2 1 1 1 1     0   A0   ..   p +  Ap =  A1 . , R∞   0 p3 (y1 , y2 )µ1 (y1 )dy1  A2   ..     . 0

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in which − ∂y∂ 1 − µ1 (y1 )

A0 = 

− ∂y∂ 1 − µ1 (y1 )

!

,

∂ − λ1 − λ2 (x2 ) − ∂x 2  ∂ λ1 − ∂x − λ1 − λ2 (x2 )  2  A1 =  .. .. . . 



  A2 =   

− ∂y∂ 2 − λ3 − µ2 (y2 ) λ3 − ∂y∂ 2 − λ3 − µ2 (y2 ) .. .. . .



  ,  

∂ λ1 − ∂x − λ1 − λ2 (x2 ) 2 

λ3 − ∂y∂ 2 − λ3 − µ2 (y2 )

  .  

The domain of the operator is   ∂pi   1      ∂y1 ∈ L ([0, ∞) × [0, ∞)),             p is absolutely continuous, i = 1, 3;  i           p (x , 0) = λ p (x ), p (0, y ) = λ p (y ); 1 2 1 0n 2 3 2 3 2n 2       Z  ∞  p2,j (y2 )µ2 (y2 )dy2 , j = 1, · · · , n, D(A) = p ∈ X p0j (0) =       Z0 ∞         p (0) = p (y )λ (x )dx , j = 1, · · · , n;   2j 0,j 2 2 2 2     0       0 1    pij ∈ L [0, ∞), pij is absolutely continuous          for i = 0, 2and j = 1, · · · , n.

Obviously, A is a linear operator. Now, the original problem can be considered as an abstract Cauchy problem in the Banach space X: dp = Ap(t), dt

t ∈ [0, ∞),

p(0) = (0, 0, · · · , 0, ϕ(x2 ), 0, · · · , 0).

(12)

(13)

In [1], the equations (1)–(11) were solved using Laplace transforms under the assumption that the solution exists really. Here we will prove the existence and the uniqueness of solutions for the system (12)–(13). 3. Properties of the operator A Definition 3.1. Let E be a Banach space. For u ∈ E, the set 2

2

Θ(u) = {u∗ ∈ E ∗ |k u∗ k =k uk = hu∗ , ui}

(14)

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is called the duality set of u. A map θ : E → E ∗ is called a duality map if θ(u) ∈ Θ(u) (u ∈ E).

(15)

In our case, the map defined as follows is a duality map: ϑ(p) =k p k (χp1 (x2 , y1 ), χp3 (y1 , y2 ), χp01 (x2 ), · · · , χp0n (x2 ), χp21 (y2 ), · · · , χp2n (y2 )), here    p(x, y) χp = | p(x, y)|  0

(p 6= 0) . (p = 0)

In fact Z Z

Z Z

hϑ(p), pi =k p k [ χp1 (x2 , y1 )p1 (x2 , y1 )dx2 dy1 + χp3 (y1 , y2 )p1 (y1 , y2 )dy1 dy2 + Z Z 2 χp01 (x2 )p01 (x2 )dx2 + · · · + χp2n (y2 )dy2 ] =k pk . This implies that it is a duality map. Definition 3.2. Let E be a Banach space, an operator G : E → E is said to be dissipative if hθ(u), Gui ≤ 0, (u ∈ D(G)) where θ is a duality map. Lemma 3.1. A is a dissipative operator. Proof. It is not too difficulty to verify hϑ(p), Api ≤ 0. In fact hϑ(p), Api =k p k (term1+term2+term3+term4+term5+term6+term7+term8),

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here term1 =

Z Z

term5 =

n−1 XZ

χp0,j+1 (x2 )λ1 p0j (x2 )dx2 ,

n−1 XZ

χp2,j+1 (y2 )λ3 p2j (y2 )dy2 ,

∂ p1 (x2 , y1 ) − µ1 (y1 )p1 (x2 , y1 )]dx2 dy1 , ∂y1 Z Z ∂ term2 = χp3 (y1 , y2 )[− p3 (y1 , y2 ) − µ1 (y1 )p3 (y1 , y2 )]dy2 dy1 , ∂y1 n Z X ∂ χp0j (x2 )[− term3 = p0j (x2 ) − λ1 p0j (x2 ) − λ2 (x2 )p0j (x2 )]dx2 , ∂x2 j=1 n Z X ∂ χp2j (y2 )[− term4 = p2j (y2 ) − λ3 p2j (y2 ) − µ2 (y2 )p2j (y2 )]dy2 , ∂y 2 j=1 χp1 (x2 , y1 )[−

j=1

term6 =

j=1

term7 = term8 =

Z Z Z Z

χp01 (x2 )p1 (x2 , y1 )µ1 (y1 )dy1 dx2 , χp21 (y2 )p3 (y1 , y2 )µ1 (y1 )dy1 dx2 .

Pay attention to the boundary conditions (7)–(10): term1 ≤ term2 ≤ term3 ≤ term4 ≤

Z

Z

λ1 p0n (x2 ) | dx2 − λ3 p2n (y2 ) | dy2 −

n Z X

j=1 n Z X

Z Z

Z Z

| p1 (x2 , y1 ) | µ1 (y1 )dy1 dx2 , | p3 (y1 , y2 ) | µ1 (y1 )dy1 dy2 ,

[−λ1 | p0j (x2 ) − λ2 (x2 ) | p0j (x2 ) |]dx2 + [−λ3 | p2j (y2 ) − µ2 (y2 ) | p2j (y2 ) |]dy2 +

j=1

term5 ≤

n−1 XZ

λ1 | p0j (x2 ) | dx2 ,

n−1 XZ

λ3 | p2j (y2 ) | dy2 ,

j=1

term7 ≤ term8 ≤

Z Z

Z Z

j=1 n Z X j=1

j=1

term6 ≤

n Z X

| p1 (x2 , y1 ) | µ − 1(y1 )dy1 dx2 , | p3 (y1 , y2 ) | µ − 1(y1 )dy1 dy2 .

µ2 (y2 ) | p2j (y2 ) | dy2 , λ2 (x2 ) | p0j (x2 ) | dx2 ,

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Hence, term1 + term2 + term7 + term8 ≤ term3 + term4 + term5 + term6 ≤

Z

Z

λ1 | p0n (x2 ) | dx2 +

Z

λ3 | p2n (y2 ) | dy2 , Z (−λ1 | p0n (x2 ) |)dx2 + (−λ3 | p2n (y2 ) |)dy2 .

That means hϑ(p), Api ≤ 0. Lemma 3.2. A is densely defined. Proof. The set  0  pij ∈ L1 [0, ∞), p0ij is absolutely continuous with compact support,         i = 0, 2, j = 1, · · · , n; ˜ = p∈X X  ∂pi    1 1     ∂y ∈ L [0, ∞) × L [0, ∞), pi is absolutely continuous i = 1, 3 1

is dense in X. This conclusion is not too difficult to understand for the reason that all infinitely differential functions with compact support sets is dense in the space L1 [0, ∞). For any p in X, suppose the compact support set is [α, β). Define p˜ as follows: p˜i = pi , i = 1, 3, ( p0j , x2 ∈ [α, ∞], p˜0j (x2 ) = k0j (x2 − α), x2 ∈ [0, α), ( p2j , y2 ∈ [α, ∞), p˜2j (x2 ) = k2j (y2 − α), y2 ∈ [0, α),

j = 1, · · · , n j = 1, · · · , n

where k0j k2j

Z Z 1 1 ∞ 1 ∞ =− [ p2j (y2 )µ2 (y2 )dy2 + p0j (x2 )λ2 (x2 )dx2 ], 1 − ( α2 )2 α α 2 α j = 1, · · · , n. Z ∞ Z ∞ 1 1 1 =− [ p2j (y2 )µ2 (y2 )dy2 + p0j (x2 )λ2 (x2 )dx2 ], 1 − ( α2 )2 2 α α α

It is easy to verify that p˜ belongs to D(A); and k p˜ − p k=

n α2 X α→0 [k0j + k2j ] → 0. 2 j=0

˜ also dense in X. It implies that D(A) is dense in X, From Lemmas 3.1 and 3.2, we obtain: Theorem 3.3. A is a densely defined and dissipative operator. But we are not sure whether A is closed or not in general situations. Lemma 3.4. Let A0 be densely defined and dissipative with respect to a duality map θ0 : D(A0 ) → E ∗ . Then: (a) A0 is closable; (b) Let A = A¯0 , thus there exists

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a duality map θ : D(A) → E ∗ whose restriction to D(A0 ) coincides with θ0 such that A is dissipative with respect to θ. The proof can be found in [3]. ¯ It is easy to obtain Let us denote the closed extension of the operator A as A. by Lemma 3.4 the following conclusion. Lemma 3.5. A¯ is a densely defined, closed and dissipative operator. Lemma 3.6. The disjoint operators for the A and A¯ are same, i.e. A∗ = A¯∗ . The result is obviously true considering the domains of both operators are dense. ¯ ¯ = X for any γ > 0. Lemma 3.7. We have that (γI − A)D( A) Proof. It is sufficient to prove that (γI − A¯∗ )q = 0 only exists zero solution. By hAp, qi = hp, A∗ qi, we have: ∂ qp (x2 , y1 ) − µ1 (y1 )qp1 (x2 , y1 ) + µ1 (y1 )qp01 (x2 ), ∂y1 1 ∂ qp (y1 , y2 ) − µ1 (y1 )qp3 (y1 , y2 ) + µ1 (y1 )qp21 (x2 ), A∗ qp3 (y1 , y1 ) = ∂y1 3 ∂ A∗ qp0j (x2 ) = qp (x2 ) − [λ1 + λ2 (x2 )]qp0j (x2 ) + λ2 (x2 )qp2j (0) + λ1 qp0,j+1 (x2 ), ∂x2 0j 1 ≤ j ≤ n − 1, ∂ qp (x2 ) − [λ1 + λ2 (x2 )]qp0n (x2 ) + λ2 (x2 )qp2n (0) + λ1 qp1 (x2 , 0), A∗ qp0n (x2 ) = ∂x2 0n ∂ A∗ qp2j (y2 ) = qp (y2 ) − [λ3 + µ2 (y2 )]qp2j (y2 ) + µ2 (y2 )qp0j (0) + λ3 qp2,j+1 (y2 ), ∂y2 2j

A∗ qp1 (x2 , y1 ) =

1 ≤ j ≤ n − 1, ∂ A∗ qp2n (y2 ) = qp (y2 ) − [λ3 + µ2 (y2 )]qp2n (y2 ) + µ2 (y2 )qp0n (0) + λ3 qp3 (0, y2 ). ∂y2 2n The equation (γI − A∗ )q = 0, γ ≥ 0 is equivalent to the following groups of differential equations: ∂ qp (x2 , y1 ) = [γ + µ1 (y1 )]qp1 (x2 , y1 ) − µ1 (y1 )qp01 (x2 ), (16) ∂y1 1 ∂ qp (y1 , y2 ) = [γ + µ1 (y1 )]qp3 (y1 , y2 ) − µ1 (y1 )qp21 (x2 ), ∂y1 3 ∂ qp (x2 ) = [γ+λ1 +λ2 (x2 )]qp0j (x2 )−λ2 (x2 )qp2j (0)−λ1 qp0,j+1 (x2 ), ∂x2 0j

(17) 1 ≤ j ≤ n−1, (18)

∂ qp (x2 ) = [γ + λ1 + λ2 (x2 )]qp0n (x2 ) − λ2 (x2 )qp2n (0) − λ1 qp1 (x2 , 0), ∂x2 0n ∂ qp (y2 ) = [γ+λ3 +µ2 (y2 )]qp2j (y2 )−µ(y2 )qp0j (0)−λ3 qp2,j+1 (y2 ), ∂y2 2j

(19)

1 ≤ j ≤ n−1, (20)

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∂ qp (y2 ) = [γ + λ3 + µ2 (y2 )]qp2n (y2 ) − µ(y2 )qp0n (0) − λ3 qp3 (0, y2 ). ∂y2 2n

(21)

Define ¯ 1 (y1 )qp1 (x2 , y1 ), Q3 (y1 , y2 ) = G ¯ 1 (y1 )qp3 (y1 , y2 ), Q1 (x2 , y1 ) = G ¯ 2 (x2 )qp2j (y2 ), 1 ≤ j ≤ n. Q0j (x2 ) = F¯2 (x2 )qp0j (x2 ), Q2j (y2 ) = G Now the equations (2)–(8) transfer to: Q01 (x2 ) ∂ , Q1 (x2 , y1 ) = γQ1 (x2 , y1 ) − g1 (y1 ) ¯ ∂y1 F2 (x2 )

(22)

∂ Q21 (y2 ) Q3 (y1 , y2 ) = γQ3 (y1 , y2 ) − g1 (y1 ) ¯ , ∂y1 G2 (y2 )

(23)

∂ Q0j (x2 ) = (γ + λ1 )Q0j (x2 ) − f2(x2 )Q2j (0) − λ1 Q0,j+1 (x2 ), 1 ≤ j ≤ n − 1, (24) ∂x2 ∂ Q0n (x2 ) = (γ + λ1 )Q0n (x2 ) − f2 (x2 )Q2n (0) − λ1 F¯2 (x2 )Qp1 (x2 , 0), ∂x2

(25)

∂ Q2j (y2 ) = (γ + λ3 )Q2j (y2 ) − g2(y2 )Q2j (0) − λ3 Q2,j+1 (y2 ), 1 ≤ j ≤ n − 1, (26) ∂y2 ∂ ¯ 2 (y2 )Qp3 (0, y2 ). Q2n (y2 ) = (γ + λ13 )Q2n (y2 ) − g2 (y2 )Q0n (0) − λ3 G ∂y2

(27)

Solve the equations (22) and (23): Q1 (x2 , y1 ) = eγy1 Q1 (x2 , 0) − [

Z

y1 0

Q01 (x2 ) eγ(y1 −τ ) g1 (τ )dτ ] ¯ . F2 (x2 )

(28)

Hence Q1 (x2 , 0) = [

Z

∞ 0

Q01 (x2 ) ˜ 1 (γ) Q01 (x2 ) . e−γτ g1 (τ )dτ ] ¯ =G F2 (x2 ) F¯2 (x2 )

(29)

And Q3 (y1 , y2 ) = eγy1 Q3 (0, y2 ) − [

Z

y1 0

Q21 (y2 ) eγ(y1 −τ ) g1 (τ )dτ ] ¯ . G2 (y2 )

(30)

Hence Q3 (0, y2 ) = [

Z

∞ 0

Q21 (y2 ) ˜ 1 (γ) Q21 (y2 ) . =G e−γτ g1 (τ )dτ ] ¯ ¯ 2 (y2 ) G2 (y2 ) G

(31)

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Using (29), (31), the equations (24)–(27) can be rewritten as: 

  Q01 (x2 ) (γ + λ1 ) −λ1 ..    0 (γ + λ1 )    .   .. .. ∂     .. . .  = .   ∂x2     ..    0 ··· . ˜ −λ G (γ) 0 1 1 Q0n (x2 )

  Q01 (x2 ) 0 ··· 0 ..   −λ1 · · · 0   .    .. ..    . . . .. 0      .. ..   ..  . . −λ1   . · · · · · · (γ + λ1 ) Q0n (x2 )



 Q21 (0) ..     .     . .. − f2 (x2 )  ,     ..   . Q2n (0)

  Q21 (y2 ) (γ + λ3 ) −λ3 ..    0 (γ + λ3 )    .   .. .. ∂     .. . . =  .   ∂y2     ..    0 ··· . ˜ −λ G (γ) 0 3 1 Q2n (y2 ) 

0 −λ3 .. . .. . ···

  Q21 (y2 ) ··· 0 ..   ··· 0   .    ..    . .. . 0      ..   ..  . −λ3   . · · · (γ + λ3 ) Q2n (y2 )

 Q01 (0) ..     .     . .. − g2 (y2 )  .     ..   . 

Q0n (0)

Denote 

 Q21  ..   .      Q2 =  ...  ,    .   ..  Q2n



 Q01  ..   .      Q0 =  ...  .    .   ..  Q0n

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So



 −(γ + λ1 ) λ1 0 ··· 0   0 −(γ + λ1 ) λ1   Z ∞   .. .. .. ..   . . Q0 (0) = [ exp{ . .  τ }f2 (τ )dτ ]Q2 (0)   0 ..     0 ··· ··· . λ1 ˜ λ1 G1 (γ) 0 · · · · · · −(γ + λ1 )   0 1 0 ··· 0  0  0 1   Z ∞  . .. . . . .   . . .  =[ exp{ . .  λ1 τ }f2 (τ )e−(γ+λ1 )τ dτ ]Q2 (0),   0 ..   . 1  0 ··· ˜ 1 (γ) 0 · · · · · · 0 G   −(γ + λ3 ) λ3 0 ··· 0   0 −(γ + λ3 ) λ3   Z ∞   .. .. .. ..   . . Q2 (0) = [ exp{ . .  τ }g2 (τ )dτ ]Q0 (0)   0 ..     0 ··· ··· . λ3 ˜ λ3 G1 (γ) 0 · · · · · · −(γ + λ3 )   0 1 0 ··· 0   0 0 1   Z ∞  . .. . . . .   . . .  =[ exp{ . .  λ3 τ }g2 (τ )e−(γ+λ3 )τ dτ ]Q0 (0).   0 ..   . 1  0 ··· ˜ 1 (γ) 0 · · · · · · 0 G

Because

 1 0 ··· 0   0 1     .. . . . .   . . .     ..    0 · · · · · · . −1  ˜ 1 (γ) 0 · · · · · · 0 −G 

0 0 .. .

has different eigenvalues, so it must have n independent eigenvectors. Characteristic equation   s −1 0 · · · 0   0 s −1     .. .. . . . .   ˜ 1 (γ) = 0 . . det  . .  = sn − G   .    0 · · · · · · .. −1  ˜ 1 (γ) 0 · · · · · · s −G

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has n different roots ν1 , · · ·  0 1  0 0   . ..  K  .. .    0 ··· ˜ 1 (γ) 0 G Then Q0 (0) = K[

Z

, νn , so there exists nonsingular matrix K, such that    0 ··· 0 ν1 0 · · · · · · 0   0 ν2 · · · · · · 0  1     .. .. . . .. ..  .. ..   −1  . . K =  . . . . .  .    . . ..   .. .. · · · νn−1 0  . 1 ··· ··· 0 0 0 · · · 0 νn

∞

diag[eλ1 ν1 τ , · · · , eλ1 νn τ ]e−(γ+λ1 )τ f2 (τ )dτ ]K −1 Q2 (0)

0

(32)

= Kdiag[f˜2 (γ + λ1 − λ1 ν1 ), · · · , f˜2 (γ + λ1 − λ1 νn )]K −1 Q2 (0), Q2 (0) = K[

Z

∞

diag[eλ3 ν1 τ , · · · , eλ3 νn τ ]e−(γ+λ3 )τ g2 (τ )dτ ]K −1 Q0 (0) 0

= Kdiag[˜ g2 (γ + λ3 − λ3 ν1 ), · · · , g˜2 (γ + λ3 − λ3 νn )]K

−1

(33)

Q0 (0).

It can be proved that f˜2 (γ + λ1 − λ1 νj )˜ g2 (γ + λ3 − λ3 νj ) 6= 1,

j = 1, · · · , n.

(34)

˜ 1 (γ) |≤ 1, j = 1, · · · , n, hence for any γ > 0 In fact, we have | νj |=| G | f˜2 (γ + λ1 − λ1 νj )˜ g2 (γ + λ3 − λ3 νj ) |< 1,

j = 1, · · · , n.

It is implied from (32), (33) and (34) that Q0 (0) = Q2 (0) = 0. This is equivalent to that q = 0. We obtain the required result. We have proved the following theorem considering Lemmas 3.5 and 3.7. Theorem 3.8. A¯ is a dissipative densely defined and closed operator, furthermore ¯ ¯ = X. (γI − A)D( A) 4. The existence and uniqueness of the solution Theorem 4.1. Let A ∈ C+ (1, 0) then A is dissipative and (λI − A)D(A) = E (λ > 0).

(35)

Conversely, let A be densely defined, dissipative with respect to some duality map, and let (35) be satisfied for some λ0 > 0 then A ∈ C+ (1, 0) Here C+ (1, 0) denote the set of contract positive semi-groups. The proof can be found in [3]. Using this theorem we can obtain the corresponding result for our operator. Theorem 4.2. The operator A¯ generates a positive contract semi-group T (t).

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Consider the following ACP dp ¯ = Ap(t), dt

t ∈ [0, ∞),

p(0) = (0, 0, · · · , 0, ϕ(x2 ), 0, · · · , 0).

(36) (37)

Theorem 4.3. There exists a unique probability solution for the ACP (36), (37) if ¯ the initial distribution of (37) belongs to the domain of A. ¯ It is easy to Proof. Denote T (t) the semi-group generated by the operator A. know by the semi-group theory that T (t)p(0) is a solution of the ACP (36),(37), and if p(0) ≥ 0, by the Theorem 4.1, then T (t)p(0) ≥ 0. Now we will prove that for any p(0) ≥ 0 k T (t)p(0) k=k p(0) k . Let Φ(t) = (1, · · · , 1), then k T (t)p(0) k= hT (t)p(0), Φ(t)i is differentiable. And d d k T (t)p(0) k= h T (t)p(0), (1, · · · 1)i = hT (t)p(0), A¯∗ (1, · · · , 1)i = 0. dt dt The last equality is implied by the result A¯∗ (1, · · · , 1) = 0. Hence k T (t)p(0) k=k p(0) k= 1. Corollary 4.4. If p(0) ∈ D(A) then p(t) = T (t)p(0) is a probability solution of the ACP (2.13),(2.14). Proof. In this case, d ¯ T (t)p(0) = T (t)Ap(0) = T (t)Ap(0) = AT (t)p(0). dt The probability condition holds for the same reason as in Theorem 4.3. References 1. D. H. Shi, Density Evolution Method in Stochastic Models, Science Press, Beijin, 1999. 2. G. Gupur, X. Z. Li and G. T. Zhu, Functional Analysis Method in Queuing Theory, Research Information Ltd., UK, 2001. 3. H. O. Fattorini, The Cauchy Problems, Addision-Wesles, Massachusetts, 1983. 4. A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, New York, 1983.

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Chap38-YouYuncheng

FINITE DIMENSIONAL REDUCTION OF GLOBAL DYNAMICS AND LATTICE DYNAMICS OF A DAMPED NONLINEAR WAVE EQUATION

YUNCHENG YOU Department of Mathematics University of South Florida Tampa, FL 33620-5700, USA E-mail: [email protected]

Global dynamics and related lattice dynamics of a damped Boussinesq nonlinear wave equation are studied. The existence of a global attractor for the solution semiflow in the energy space is proved by the dual exchange technique in showing precompact pseudometric. It is shown that the global attractor has a finite Hausdorff dimension. For the related lattice system, the existence of a global attractor is proved, which is upper semicontinuous with respect to finite-dimensional approximations.

1. Introduction In the period 1872–1877, J. Boussinesq published a series of articles, cf. Boussinesq, 2 in which he derived a nonlinear evolutionary equation to model the propagation of long waves of small amplitude over the surface of shallow water. There is an extensive literature of publications on the Boussinesq-type equations, most with spatial dimension one, cf. Chen et al., 3 Clarkson, 5 Hirota, 6 Li and Chen, 7 Miles, 8 Nakamura, 9 Straughan, 11 Varlamov, 13 and You. 14 The research results therein range from construction of exact solutions, solitary wave solutions, rational solutions, to the generated or coupled Boussinesq equations, and to the applications. A typical one-dimensional Boussinesq equation is in the form utt + uxxxx + βuxx + κux uxx + γ u2 xx = f, where β, κ and γ are real constants, and f can be an external input or a control function. Li and Chen7 studied the dissipative Schr¨ odinger-Boussinesq equations and proved the existence of a global attractor with finite fractal dimension, but the Boussinesq-type equation in their couple systems features a strong damping −∆ut which makes the dissipativity much stronger. In You14 a nonlinear feedback which involves (u, ut ) is found to guarantee the global existence of solutions of the Boussinesq equations and to achieve a global stabilization for solutions initially of finite energy. 367

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From the mathematical viewpoint, the Boussinesq equations form an important branch in the area of nonlinear wave equations. The Boussinesq equations can be regarded as equally challenging and significant in the hyperbolic evolutionary equation arena as the Kuramoto-Sivashinsky equations in the parabolic evolutionary equation arena. In this paper we shall investigate the following generalized, weakly damped Boussinesq equation, 2 utt + δut + uxxxx + βuxx − κ (ux ) uxx + γ u2 xx = f, u ∈ Ω = (0, 1), t > 0, u(0, t) = u(1, t) = ux (0, t) = ux (1, t) = 0, t ≥ 0, ¯ = [0, 1], u(x, 0) = u0 (x), ut (x, 0) = u1 (x), x ∈ Ω

(1) where δ and κ are positive constants, β and γ are real constants, and f = f (x) ∈ L2 (Ω), which is time-invariant. Note that the domain Ω can be any bounded interval and the clamped boundary condition can be changed to the hinged homogeneous boundary condition. Let H = L2 (0, 1), whose inner product and norm will be denoted by h·, ·i and k · k respectively. Define a linear operator A : D(A) → H by d4 ϕ , ϕ ∈ D(A) = H 4 (0, 1) ∩ H02 (0, 1). dx4 It is shown that the operator A is coercively positive, self-adjoint, and with compact resolvent. Hence −A generates a compact, analytic, C0 -semigroup e−At , t ≥ 0. Moreover, it can be shown that the space

V = D A1/2 with the norm kvkV = A1/2 v = kvxx k Aϕ =

is a Hilbert space. The IBVP (1) can be formulated into an initial value problem of the following abstract Boussinesq equation on the product space E = V × H, dw + GW = F (w), t > 0, dt T w(0) = w0 = (u0 , u1 ) ∈ E,

(2)

where w(t) = (u, ut )T , the linear operator G and the nonlinear operator F are given by 0 −I G= : D(G) = D(A) × V → E, A δI and F (w) =

0 −βuxx + κ (ux )2 uxx − γ u2 xx + f

: E → E.

One can check that F is a locally Lipschitz continuous mapping and that −G generates a C0 -group e−Gt on the space E, which is differentiable for t > 0, cf. Corollary 38.8 in Sell and You 10 . Thus, the local existence theory of solutions to

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semilinear evolutionary equations, cf. Sections 4.6 and 4.7 of Sell and You10 , can be applied to the IVP (2). For any w0 ∈ E, the IVP (2) admits a unique, maximally defined, mild solution w(t) on I = [0, T ). Now we show the global existence and the boundedness of each mild solution w(t) = w (t; w0 ), which is actually a strong solution in E due to the fact that the C0 -semigroup e−Gt , t > 0, is differentiable and E is a Hilbert space, by Theorem 46.2 in Sell and You10 . Due to the space limit, all proofs will be sketched only. Theorem 1.1. For any initial status w0 ∈ E and any given f ∈ H, the unique strong solution w(t) of the IVP (2) exists globally on I = [0, ∞), provided that the condition √ δ ≥ 2 5κ−1 |γ|

(3)

is satisfied. For the solution semiflow {S(t)}t≥0 generated by the abstract Boussinesq equation (2) on E, there exists a bounded absorbing ball Br0 ⊂ E centered at the origin and of radius r0 in the sense that, for every bounded set Z ⊂ E, there is a finite time tZ ≥ 0 such that S(t) Z ⊂ Br0 ,

for all t ≥ tZ .

(4)

Proof. Since w (t; w0 ) is a strong solution of the equation (2) in E, its first component u(t) = u (t; u0 , u1 ), t > 0, satisfies the original Boussinesq equation (1) almost everywhere. Let ρ > 0 and 0 < ≤ 1 be undetermined constants. Taking the inner product of (1) with 2ρut + u in H, we get d P (t) + N (t) = 0, dt

(5)

1 P (t) = ρ kut k2 + ρ kuxx k2 − ρβ kux k2 + hu, ut i + δ kuk2 2 Z Z 1 4 2 + ρκ |ux | dx − 2ργ u |ux | dx − 2ρhf, ui, 6 Ω Ω

(6)

Z 2 2 2 2 N (t) = (2ρδ − ) kut k + kuxx k − β kux k + 2ργ ut |ux | dx Ω Z Z 1 4 + κ |ux | dx − 2γ u |ux |2 dx − hf, ui. 3 Ω Ω

(7)

where

and

By using Sobolev imbedding properties, Poincar´e inequality, and integration by

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parts, we can deduce that Z 1 2 4 ρN (t) − P (t) = 2 ρ2 δ − ρ kut k + ρκ |ux | dx − 2 hu, ut i 6 Ω Z 1 2 2 2 2 − δkuk + 2ρ γ ut |ux | dx + ρhf, ui 2 Ω 5 8ρ3 γ 2 1 2 2 ≥ 2ρ δ − ρ − kut k + ρκλ1 kuk4 2 κ 24 1 1 − 2 δ + 2ρ−1 kuk2 − ρ3 kf k2 2 2

(8)

where λ1 > 0 is the smallest eigenvalue of the operator A. By the condition (3) and taking ρ = ρ0 =

δκ , 8γ 2

we have 2ρδ −

5 8p2 γ 2 8γ 2 − =− 2 κ κ 8γ 2 ≥− κ

ρ−

δκ 8γ 2

δκ ρ− 8γ

+

2

= 0,

δ2 κ 5 − 8γ 2 2

for any 0 < ≤ 1.

Besides we have 1 1 2 2 ρ0 κλ1 kuk4 − δ + 2ρ−1 0 kuk 24 2 " # 2 6 36 1 −1 −1 2 2 2 2 ρ0 κλ1 kuk − δ + 2ρ0 − δ + 2ρ0 = 24 ρ0 κλ1 (ρ0 κλ1 )2 2 2 3 3 ≥− 2 δ + 2ρ−1 ≥− δ + 2ρ−1 , 0 0 2ρ0 κλ1 2ρ0 κλ1 since 0 < ≤ 1. Simply taking = 1 and letting 0 = 1/ρ0 , then we obtain 2 30 1 N (t) ≥ 0 P (t) − (δ + 20 )2 + ρ20 kf k2 , 2κλ1 2

(9)

from (8). Denote the uniform constant in the square bracket by C1 . From (5) and (9) it follows that d P (t) + 0 P (t) ≤ C1 , dt

t > 0,

and consequently P (t) ≤ P (0) e−0 t + C1 /0 ,

t ≥ 0.

(10)

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On the other hand, from (6) with = 1 and ρ = ρ0 , we get Z 1 1 2 2 2 |ux |4 dx P (t) ≥ ρ0 kut k + ρ0 kuxx k − ρ0 β kux k + ρ0 κ 2 42 Ω 1 1 7γ 2 ρ0 + δ− − ρ0 kuk2 − ρ0 kf k2 − 2 2ρ0 κ 1 2 2 ≥ ρ0 kut k + ρ0 kuxx k − C2 , 2

(11)

where C2 =

21 2ρ0 κ

ρ0 |β| +

2 p 1 1 7γ 2 ρ0 λ1 δ − − ρ0 + ρ0 kf k2 . − 2 2ρ0 κ

Substituting (11) into (10), we find that kw(t)k2E ≤

2 2 (P (t) + C2 ) ≤ P (0) e−0 t + C1 /0 + C2 , ρ0 ρ0

t ≥ 0,

(12)

which implies that lim sup kw(t)k2E ≤ 2 (C1 + C2 0 ) .

(13)

t→∞

Finally, (12) implies that the solution w(t) = w (t; w0 ) exists globally on [0, ∞) and (13) in further implies that any bounded ball Br0 with r0 > 2 (C1 + C2 0 ) is an absorbing ball for this solution semigroup S(t), t ≥ 0. 2. The Existence of a Global Attractor We take the precompact pseudometric approach to prove the asymptotical compactness of the solution semigroup S(t) of the abstract Boussinesq equation (2), which together with the shown absorbing property implies the existence of a global attractor for S(t). Let us introduce a functional 2

Π(u, v) = kuxx k +

1 δθkuk2 + θhu, vi + kvk2 , 2

where θ > 0 is a small parameter. For ϕ = (u, v)T ∈ E, Π(u, v) is equivalent to T kϕk2E . For any two solutions wi = (ui (t), ∂t ui ) of the equation (2), i = 1, 2, we shall denote the difference by W (t) = w1 (t) − w2 (t) and accordingly U (t) = u1 (t) − u2 (t). Lemma 2.1. For any given bounded set M ⊂ E, there exists a constant θ > 0 and a constant L0 = L0 (M, θ) > 0 such that for any two solutions as above of the equation (2), with wi (0) ∈ M , i = 1, 2, it holds that Π (U (t), Ut (t)) ≤ e−θt Π (U (0), Ut (0)) + L0 sup kU (s)k + sup kUx (s)kC[0,1] 0≤s≤t

0≤s≤t

(14) for t ≥ 0.

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Proof. Since Wi (t), i = 1, 2, are strong solutions of (2), the function U (t) satisfies the equation 2

Utt + δUt + Uxxxx = −βUxx + κ (u1x ) Uxx + κ (u1x + u2x ) Ux u2xx −2γ (u1xx U + u2 Uxx ) − 2γ (u1x + u2x ) Ux .

(15)

Taking the inner product in H of (15) with 2Ut + θU and using the imbedding j H j+1 (Ω) ⊂ CB (Ω) for j = 0, 1, we can conduct an a priori estimate to reach the inequality 1 d kUt k2 + kUxx k2 − β kUx k2 + θ hUt , U i + θδkU k2 dt 2 D E 2 2 2 2 2 (16) +κ ku1x Ux k − 2γ u2 , (Ux ) + (2δ − θ) kUt k + θ kUxx k − θβ kUx k ≤ L1 kU (t)k + L2 kUx (t)kC[0,1] ,

where 0 < θ ≤ 1 and L1 = L1 (M ), L2 = L2 (M ) are two positive constants only depending on M . Define another functional D E 2 2 2 Θ(u, v) = Π(u, v) − β kux k + κ ku1x ux k − 2γ u2 , (ux ) , where u = u1 − u2 . Then we have d Θ (U (t), Ut (t)) + θΘ (U (t), Ut (t)) dt 1 2 2 2 2 ≤ θ kUt k + θ kUxx k + θ2 hU, Ut i + θ2 δkU k2 + (θ − 2δ) kUt k − θ kUxx k 2 D E +θκ ku1x Ux k2 − 2θγ u2 , (Ux )2 + L1 kU k + L2 kUx kC[0,1] .

Choose 0 < θ ≤ min{1, δ} and note that M is bounded in E. We get d Θ (U (t), Ut (t)) + θΘ (U (t), Ut (t)) ≤ L3 kU (t)k + L4 kUx (t)kC[0,1] , dt

(17)

where L3 and L4 are uniform constants. Integrating (17) and back to Π (U (t), Ut (t)), finally we can reach the inequality (14), with 0 < θ ≤ min{1, δ/2}. Theorem 2.1. There exists a global attractor A for the solution semigroup S(t) generated by the abstract Boussinesq equation (2) on E. Proof. By Lemma 2.1, with θ = δ/2 in (14), we see that Π(u, v) is equivalent to

2 the norm square (u, v)T E in the following sense,

1

(u, v)T 2 ≤ Π(u, v) ≤ max 1 + 3 δ 2 λ−1 , 3 (u, v)T 2 , (18) 1 E E 2 8 2

in which λ1 is the smallest eigenvalue of the operator A. Then (14) and (18) imply that there exist two constants C > 0 and L > 0 only depending on the original

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parameters in (1), such that kS(t)w10 − S(t)w20 k ≤ Ce−δt/4 kw10 − w20 kE 1/2 1/2 +L sup kU (s)k + sup kUx (s)kC[0,1] . 0≤s≤t

(19)

0≤s≤t

For each given t > 0, define ρt (w10 , w20 ) = ρ1t (w10 , w20 ) + ρ2t (w10 , w20 ) where ρ1t (w10 , w20 ) = L sup kU (s)k1/2 = L sup ku1 (s) − u2 (s)k 0≤s≤t

1/2

0≤s≤t

and 1/2

1/2

ρ2t (w10 , w20 ) = L sup kUx (s)kC[0,1] = L sup k∂x u1 (s) − ∂x u2 (s)kC[0,1] . 0≤s≤t

0≤s≤t

Now we show that ρt is a precompact pseudometric on E = V × H. First we show that ρ1t is a precompact pseudometric on E for each given t > 0. T Suppose that {w0n } is a bounded sequence in E and let wn (t) = (un (t), ∂t un (t)) be n the solution of (2) with w0 . From the boundedness of solutions shown in Theorem 1.1, there is a constant K > 0 such that sup kun (t)kH 2 (0,1) + sup k∂t un (t)kL2 (0,1) ≤ K. t≥0

0

(20)

t≥0

Hence, {un (s)} for each s ∈ [0, t] is in a compact set of H = L2 (0, 1). On the other hand, since {∂t un (t)} is uniformly bounded, by the imbedding V = H02 (0, 1) ⊂ C 1 [0, 1], and by the Leibniz formula, for a given t > 0, there is a constant K1 > 0 such that kun (t1 ) − un (t2 )k ≤ K1 |t1 − t2 | , for any t1 , t2 ∈ [0, t]. Therefore by the Arzel´ a-Ascoli theorem in Banach space, we can assert that {un (·)} is a precompact set in C [0, t]; L2 (0, 1) ,

so that there exists a convergent (also Cauchy) subsequence with respect to the pseudometric ρ1t . By definition, cf. Sell and You, 10 ρ1t is precompact on E. Next we prove that ρ2t is also a precompact pseudometric. Let {wn } be as above. We want to show that {∂x un (·)} is a precompact set in C([0, t]; C[0, 1]).

(21)

Note that C([0, t]; C[0, 1]) = C([0, t] × [0, 1]) = C([0, 1]; [0, t]) and their norms are equivalent. It reduces to proving {∂x un (·)} is a precompact sequence in C([0, 1]; [0, t]).

(22)

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By (20) and the Leibniz formula, there is a constant K2 > 0 such that k∂x un (σ1 ) − ∂x un (σ2 )kC[0,t] ≤ K2 |σ1 − σ2 | , for any σ1 , σ2 (spatial variable) in [0, 1]. Moreover, for each given 0 ≤ σ ≤ 1, we claim that {∂x un (·)} is precompact in C[0, t].

(23)

In order to verify (23), it suffices to check the following two properties. (i) For each given 0 ≤ σ ≤ 1, there exists a constant b1 (σ, t) such that |∂x un (σ, s)| ≤ b1 (σ, t), for all s ∈ [0, t] and for n ≥ 1. This property is satisfied because |∂x un (σ, s)| ≤ k∂x un (σ, ·)kC[0,t] ≤ const sup kun (·, s)kV ≤ C(t), 0≤s≤t

for all s ∈ [0, t], n ≥ 1, based on the boundedness of {w0n } and the absorbing property. Then we can take b1 (σ, t) = C(t), which is even independent of σ ∈ [0, 1]. (ii) For each given 0 ≤ σ ≤ 1, there exists a constant b2 (σ, t) such that |∂x un (σ, s1 ) − ∂x un (σ, s2 )| ≤ b2 (σ, t) |s1 − s2 | , for all s1 , s2 ∈ [0, 1] and for n ≥ 1. Below we prove this inequality. Indeed, Z t wn (t) = e−Gt w0n + e−G(t−s) F (wn (s)) ds,

t ≥ 0,

(24)

0

in which e−Gt is a differential semigroup as said earlier. It follows that e−Gt w0n is a classical solution of the homogeneous equation and Z t −Gt e w0 − w 0 = −Ge−Gs w0 ds, t ≥ 0, w0 ∈ E. (25) 0

{w0n } is a bounded sequence in E, from (25) it of e−Gt w0n is uniformly Lipschitz continuous

Since yn (t) Thus there exists a constant d1 (t) such that

is seen that the first component on any compact time interval.

kyn (s1 ) − yn (s2 )kV ≤ d1 (t) |s1 − s2 | , for any s1 , s2 ∈ [0, t],

n ≥ 1.

(26)

Substituting (26) into (24) and noting that F ∈ CLip (E), we can deduce that there exists a constant d2 (t) > 0 such that kun (·, s1 ) − un (·, s2 )kV ≤ d2 (t) |s1 − s2 | , for any s1 , s2 ∈ [0, t],

n ≥ 1.

In this step, Gronwall inequality is used. Then it follows that the inequality claimed in this item (ii) is valid with b2 (σ, t) = d2 (t), again independent of σ ∈ [0, 1]. Finally we can synthesize what has been proved: The items (i) and (ii) ensure that (23) holds, which together with the equicontinuity implies that (22) and (21) are valid. Hence, ρ2t is a precompact pseudometric on E. It follows that ρt is a precompact pseudometric on E. By Lemma 22.5 in Sell and You10 , (19) then

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implies that the solution semiflow {S(t)}t≥0 of the abstract Boussinesq equation (2) is κ-contracting. According to Theorem 23.12 in Sell and You 10 on the existence of a global attractor, we can conclude that a global attractor A exists in E for this solution semiflow. 3. Hausdorff Dimension of the Global Attractor Since the global attractor A ⊂ E is a nonempty, compact, invariant set, one has S(t) A = A for any t ≥ 0. A straightforward but tedious processing can verify that the nonlinear mapping S(t) of the solution semigroup of the equation (2) is continuously Fr´echet differentiable for each t ≥ 0. Here we use the technique of isomorphic re-installation of solutions. Let ϕ = (u, v)T , v = ut + u, where is an arbitrary positive constant. Then (2) can be rewritten as dϕ + Λϕ = F (ϕ), t > 0, dt (27) T ϕ(0) = ϕ0 = (u0 , u1 + u0 ) , where Λ=

I A + ( − δ) I

−I (δ − ) I

: D(A) × V → E

and F : E → E remains the same as in (2). For constants σ1 , σ2 > 0 and ϕ ∈ D(Λ) = D(A) × V , we can calculate hΛϕ, ϕiE − σ1 kϕk2E − σ2 kvk2 2

−1/2

≥ ( − σ1 ) kuxx k − λ1

|( − δ)| kuxx k kvk + (δ − − σ1 − σ2 ) kvk2 .

If we choose , σ1 , σ2 such that δ ≥ 2,

and σ1 = σ2 =

1 , 4

then for sufficiently small > 0, one can get hΛϕ, ϕiE ≥ σ1 kϕk2E + σ2 kvk2 .

(28)

Now the solution mapping SE (t) : E → E defined by the equation (27) is relative to the solution mapping S(t) defined by (2) by the following reversible transformation S (t) = R S(t) R− ,

(29)

where R is the isomorphism of E given by R : (u, v)T → (u, v+u)T . Consequently R A is the global attractor of the semiflow S (T ), t ≥ 0. Moreover, we can check that A and R A have the same Hausdorff dimension. The linear variational equation of (27) is dΨ = (−Λ + DF (ϕ)) Ψ, dt Ψ(0) = (ξ, η)T ∈ E,

(30)

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where Ψ = (U, V )T . In (30), ϕ = (u, v)T is a particular solution of (27), and 0 DF (ϕ) Ψ = . −βUxx + 2κuxxux Ux + κ (ux )2 Uxx − 2γ(uU )xx From (29) it is clear that for t > 0, S (t) is Fr´echet differentiable on E. Its T Fr´echet derivative at ϕ0 = (u0 , u1 + u0 ) is the linear solution operator: (ξ, η)T → (U (t), V (t))T of the variation equation (30). Lemma 3.1. Let Φ = {Φ1 , Φ2 , . . . , Φ` } be a set of ` orthonormal vectors in E. If the following condition is satisfied, sup sup

` X

Φ⊂E ϕ∈R A j=1

h(−Λ + DF (ϕ)) Φj ; Φj iE ≤ 0,

(31)

then the Hausdorff dimension of the global attractor R A for the semiflow S (t), t ≥ 0, is less than or equal to `. Proof. This is a direct consequence of Theorem V.3.3 in Temam

12

.

Theorem 3.1. The Hausdorff dimension dH (R A) of the global attractor R A for the semiflow S (t) generated by (27) on E satisfies   `  1X M02  dH (R A) ≤ min ` ∈ N : , (32) λj ≥  ` j=1 4σ1 σ2  for some constant M0 , where λj , j = 1, . . . , `, are the first ` eigenvalues of the operator A, and constants σ1 , σ2 satisfy (28).

Proof. Let ` ∈ N be fixed. Consider ` solutions Ψ1 , Ψ2 , . . . , Ψ` of the IVP (30). At a given time τ , let Q` (τ ) be the orthogonal projection of E onto the subspace T spanned by Ψ1 (τ ), Ψ2 (τ ), . . . , Ψ` (τ ). Let Φj (τ ) = (ξj , ηj ) ∈ E, j = 1, . . . , `, be an orthonormal basis of Q` (τ ) E. For ϕ ∈ R A, study ` X

h(−Λ + DF (ϕ)) Φj , Φj iE = −

j=1

` X

hΛΦj , Φj iE +

j=1

` X

h(DF (ϕ)) Φj , Φj iE .

j=1

From (28), we have 2

2

− hΛΦj , Φj iE ≤ −σ1 kξjxx k − σ2 kηj k . Besides we have

(33)

2 hDF (ϕ) Φj , Φj iE ≤ −βξjxx + 2κuxuxx ξjx + κ (ux ) ξjxx − 2γ (uξj )xx kηj k = kDB(u) ξj k kηj k ,

where 2

B(u) = −βuxx + κ (ux ) uxx − γ u2

xx

+f

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and DB(u) is the Fr´echet derivative of B at u. Since F ∈ CLip (E) ∩ CF1 (E), where the notation is explained in Sell and You

10

, so that

B ∈ CLip (V, H) ∩ CF1 (V, H), and R A is a bounded set, there exist a constant M0 > 0 such that for ϕ ∈ R A, kDB(u) ξj k ≤ M0 kξjxx k . Hence we get hDF (ϕ) Φj , Φj iE ≤

M02 2 2 kξjxx k + σ2 kηj k . 4σ2

(34)

Using (33) and (34), we obtain h(−Λ + DF (ϕ)) Φj , Φj iE ≤ 2

M02 2 − σ1 kξjxx k 4σ2

2

where kξjxx k ≤ kΦj kE = 1, and on the other hand ` X

2

kξjxx k =

j=1

` X

hAξj , ξj i ≥

j=1

` X

λj .

j=1

Therefore, ` X

h(−Λ + DF (ϕ)) Φj , Φj iE ≤ −σ1

j=1

` X

λj +

j=1

M02 `. 4σ2

Now if we choose ` ∈ N such that `

1X M02 λj ≥ , ` j=1 4σ1 σ2 we then get ` X

h(−Λ + DF (ϕ)) Φj , Φj iE ≤ 0.

(35)

j=1

By Lemma 3.1, (35) implies that (32) holds. Theorem 3.2. The Hausdorff dimension dH (A) of the global attractor A for the abstract Boussinesq equation (2) in E admits the same finite upper bound as given by (32). This conclusion follows from Theorem 3.1 and the fact dH (A) = dH (R A).

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4. The Relative Lattice Dynamics In the recent decade, the dynamics of lattice systems have been studied by many authors, cf. Chow et al. 4 and references therein. Lattice differential equations arise in a variety of applications such as in image processing, pattern recognition, material science, and molecular and cell biology. As another important aspect, lattice dynamical systems also emerge from spatial discretization of evolutionary PDEs. Here we shall study a lattice dynamical system relative to the Boussinesq equations. The objective remains to explore the existence of a global attractor for this lattice system and to address the upper semicontinuity with respect to finitedimensional approximate systems. Consider the following second-order nonlinear lattice differential equation, dui d 2 ui +δ + (ui+2 − 4ui+1 + 6ui − 4ui−1 + ui−2 ) dt dt i 1 h 3 3 +β (ui+1 − 2ui + ui−1 ) + λui − κ (ui+1 − ui ) − (ui − ui−1 ) = fi , 3

i ∈ Z, (36)

with the initial conditions ui (0) = ui0 ,

u˙ i (0) = ui1 ,

i ∈ Z,

(37)

where δ, λ, and κ are positive constants, β is a real constant, f = (fi ) ∈ `2 . The equation (36) can be regarded as a spatial discretization of the damped Boussinesq equation with γ = 0 and on the unbounded domain Ω = (−∞, ∞). We can allow γ 6= 0 but the treatment will be more complicated. Here we assume that λ > 4 |β|.

(38)

To formulate the IVP of the lattice differential equation (36)–(37) into an abstract evolutionary equation and IVP in the Hilbert space `2 , we define the following linear operators from `2 into itself: for u = (ui ) i ∈ Z ∈ `2 , (DU )i = ui+1 − ui ,

(D∗ U ) = ui − ui−1 ,

(BU )i = ui+1 − 2ui + ui−1 , (AU )i = ui+2 − 4ui+1 + 6ui − 4ui−1 + ui−2 ,

i ∈ Z.

We can check that all these are bounded linear operators on `2 , and we have A = B2,

B = DD∗ = D∗ D.

For u = (ui )i∈Z , v = (vi )i∈Z in `2 , both bilinear forms h·, ·i and h·, ·iλ defined by X hu, vi = ui vi , hu, viλ = hBu, Bvi + λhu, vi i∈Z

are inner-products of the space `2 . Moreover, their induced norms are equivalent λkuk2 ≤ kuk2λ ≤ (16 + λ) kuk2 .

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We shall denote the Hilbert space `2 with the inner-product h·, ·iλ and the norm k · kλ specifically by `2λ . Define E = `2λ × `2 , which is endowed with the inner-product and the norm as follows: for ϕj = u(j) , v (j) ∈ E, j = 1, 2, E E D D hϕ1 , ϕ2 iE = u(1) , u(2) + v (1) , v (2) λ X (1) (2) (1) (2) (2) (1) + λui ui + vi vi , Bu = Bu i

i

i∈Z

and kϕk2E = hϕ, ϕiE , for ϕ ∈ E. With the above defined, (36)–(37) can be written as the following equation in E, d2 u du 1 +δ + Au + βBu + λu − κD (D∗ u)3 = f, t > 0, 2 dt dt 3 u(0) = (ui0 )i∈Z = u0 , u(0) ˙ = (ui1 )i∈Z = u1 ,

(39)

where u = (ui )i∈Z and f = (fi )i∈Z . Let v = u+u, ˙ where > 0 is a small parameter. We can choose > 0 sufficiently small such that 2 3 δ + − δ ≤ 0, (1 + δ) + 4|β| − λ ≤ 0, and − 2 ≥ 0, 2 2 2

(40)

due to (38). The IVP (39) can be further formulated into the following IVP in the space E, ϕ˙ + Γϕ = R(ϕ),

ϕ(0) = ϕ0 = (u0 , v0 )T = (u0 , u1 + u0 )T ∈ E,

(41)

where ϕ = (u, v)T , v = u˙ + u, u − v Γ(ϕ) = , Au + λu + (δ − )(v − u) and R(ϕ) =

0 , g(u)

with 1 g(u) = −βBu + κD (D∗ u)3 + f. 3 Lemma 4.1. For any given initial data ϕ0 = (u0 , v0 )T ∈ E, there exists a unique local solution ϕ(t) = (u(t), v(t))T of the IVP (41), such that ϕ ∈ C([0, T ), E) for some T > 0.

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Proof. By using the inequality (a + b)2 ≤ 2 a2 + b2 , one can verify that kBuk ≤ 4kuk,

and

kAuk ≤ 16kuk,

kDuk ≤ 2kuk,

2 X

3 2 (ui+1 − 2ui + ui−1 )

D (D∗ u) = i∈Z

h

2

× (ui+1 − ui ) + (ui+1 − ui ) (ui − ui−1 ) + (ui − ui−1 )

2

i2

(42)

.

Since ui → 0 as i → ∞, there exists an integer N = N (u) such that 2

2

(ui+1 − ui ) + (ui+1 − ui ) (ui − ui−1 ) + (ui − ui−1 ) ≤ 1, for |i| ≥ N + 1. Then we have

2 2 X

3 3 D (D∗ u) + 16kuk2.

D (D∗ u) ≤ i

(43)

|i|≤N

Moreover, we can check that there exists a uniform constant L1 = L1 () > 0 such that kΓϕ1 − Γϕ2 kE ≤ L1 kϕ1 − ϕ2 kE

for any ϕ1 , ϕ2 ∈ E.

(44)

For the nonlinear mapping R, we have

2

kR (ϕ1 ) − R (ϕ2 )k2E = g u(1) − g u(2)

2 2 3 3

2

∗ (1) ∗ (2)

≤ 2β 2 B u(1) − u(2) + κ2 D D u − D u

9

3 3 2 2 8 X (1) (1) (2) (2) 2 2 (1) (2) ≤ 2β B u − u ui − ui−1 − ui − ui−1 ,

+ κ 9 i∈Z

where

3 3 2 X (1) (1) (2) (2) ui − ui−1 − ui − ui−1 i∈Z

2 2 2 2 2 X (1) (2) (1) (2) (1) (1) (2) (2) ≤6 ui − u i + ui−1 − ui−1 ui − ui−1 + ui − ui−1 . i∈Z

For any given bounded set S ⊂ E, there is a constant L2 = L2 (S) such that for any ϕ = (u, v)T ∈ S, one has kϕkE ≤ L2 . From the above two inequalities it follows that

2 2

kR (ϕ1 ) − R (ϕ2 )k2E ≤ L3 B u(1) − u(2) + λ u(1) − u(2) (45) 2 ≤ L3 kϕ1 − ϕ2 kE , where L3 = max 2β 2 , 342 κ2 λ−1 L42 .

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Thus both Γ and R are locally Lipschitz continuous. Then by the existence theory of local solutions of abstract differential equations in Banach spaces, the conclusion holds. Lemma 4.2. Assuming (38) and (40) are satisfied, then for any f ∈ `2 and any initial data ϕ0 ∈ E, the solution ϕ(t) of the IVP (41) exists globally for t ≥ 0. Furthermore, there exists an absorbing ball for this solution semiflow S (t), t ≥ 0. Proof. Note that D, B, and A satisfy the following properties: for u, v ∈ `2 , hDu, vi = − hu, D∗ vi , hBu, vi = −hDu, Dvi, hAu, vi = hBu, Bvi. Let ϕ(t) = (u(t), v(t))T be a solution of the equation (41), where v(t) = u(t)+u(t), ˙ and > 0 satisfies (40). Taking the inner-product of (39) with v(t) in `2 , we obtain d M (t) + Q(t) = 0, dt

t > 0,

(46)

where M (t) =

2 1 1 β λ 1

kuk ˙ 2 + kBuk2 − kDuk2 + kuk2 + κ (D∗ u)2 2 2 2 2 12 δ −hf, ui + hu, ˙ ui + kuk2 2

and Q(t) = (δ − ) kuk ˙ 2 + kBuk2 − β kDuk2 + λ kuk2

2

+ κ (D∗ u)2 − hf, ui. 3

With the choice of which satisfies (40), we have 3 β 2 M (t) − Q(t) = − δ kuk ˙ − kBuk2 + kDuk2 2 2 2

2 + (δ − λ) kuk2 − κ (D∗ u) + 2 hu, ˙ ui 4 2 2 3 2 + − δ kuk ˙ + ((1 + δ) + 4 |β| − λ) kuk2 ≤ 0. ≤ 2 2 2

(47)

From (46) and (47), it follows that M (t) ≤ M (0) e−t ,

t ≥ 0.

For any given α > 0, we have 1 1 2 M (t) ≥ kuk ˙ + kBuk2 + 4 2

λ δ α 2 2 + − 2 − − 2 |β| kuk2 − kf k2. 2 2 8 α

(48)

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Since (38) implies that there is a constant α > 0 such that α λ 1− − 2α − 4 |β| ≥ 0, 4λ one can deduce that 1 1 2 2 M (t) ≥ kuk ˙ + kBuk2 + λα kuk2 − kf k2 , 4 2 α where the last inequality of (40) is used. Hence, 2 kBuk2 + λ kuk2 + kuk ˙ ≤ max 4, α−1 M (t) + 2α−1 kf k2 .

(49)

(50)

Combining (48) and (50), we end up with

2 kϕ(t)k2E = kBuk2 + λ kuk2 + kuk ˙ ≤ max 4, α−1 M (0) e−t + 2α−1 kf k2 ,

and finally

lim sup kϕ(t)k2E ≤ (const)kf k2 .

(51)

t→∞

Therefore, for any given ϕ0 ∈ E, the solution ϕ(t) of the IVP (41) exists globally for t ≥ 0. The ball Or0 centered at the origin and of radius r0 > (const)kf k2 is an absorbing ball in E for this solution semiflow S (t), t ≥ 0. Next we adopt the approach of “tail ends” estimation to show the asymptotic compactness of this solution semigroup. This approach was introduced in Bates et al.1 and used in Zhou, 15 which seems effective in dealing with the semiflows on unbounded domains. Lemma 4.3. Under the same assumptions as in Lemma 4.2, let Or0 be the bounded absorbing ball given in the proof of Lemma 4.2. Then for any η > 0, there exist constants K(η) > 0 and T (η) > 0 such that for any initial data ϕ0 =(u0 , v0 )T = T

(u0 , u1 + u0 ) ∈ Or0 , the solution ϕ(t) = (ϕi (t))i∈Z = (ui (t), vi (t))

T

i∈Z

satisfies the following estimate, X X 2 2 2 2 |ϕi (t)|E , |(Bu(t))i | + λ |ui (t)| + |vi (t)| ≤ η, i≥K(η)

of (41) (52)

i≥K(η)

for all t ≥ T (η). Proof. Take a truncating function θ ∈ C 1 (R+ ) which satisfies θ(t) = 0,

for 0 ≤ t ≤ 1,

0 ≤ θ(t) ≤ 1,

for 1 ≤ t ≤ 2,

θ(t) = 1, for t ≥ 2. T For a solution ϕ(t) = (ϕi (t))i∈Z = (ui (t), vi (t)) u˙ i (t) + ui (t). Let m be a fixed integer. Set

w = (wi )i∈Z , where wi (t) = θ

|i| m

i∈Z

of (41), one has vi (t) =

ui (t),

i ∈ Z.

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By taking the inner product of the equation (41) with w˙ + w, we can get d Mθ (t) + Qθ (t) = 0, dt in which

t > 0,

(53)

X |i| 1 1 β λ 2 2 2 2 Mθ (t) = θ |u˙ i | + |(Bu)i | − |(Du)i | + |ui | m 2 2 2 2 i∈Z δ κ 4 2 ∗ |ui | , + |(D u)i | − fi ui + u˙ i ui + 12 2

and

X |i| h 2 2 2 Qθ (t) = θ (δ − ) |u˙ i | + |(Bu)i | − β |(Du)i | m i∈Z i κ +λ |ui |2 + |(D∗ u)i |4 − fi ui . 3

Similarly, as we did in the proof of Lemma 4.2, we obtain that for t ≥ 0, X |i| θ |(Bu(t))i |2 + λ |ui (t)|2 + |u˙ i (t)|2 m i∈Z ≤ max 4, α−1 Mθ (0)e−t + 2α−1 kf k2 ,

(54)

where α > 0 is the same constant chosen in (49). On the other hand, we have

1 |β| λ κ 1

∗ 2 2 2 2 2 2 kDu0 k + ku0 k + Mθ (0) ≤ |u1 k + kBu0 k +

(D u0 ) 2 2 2 2 12 (55) X |i| 2 1 δ 2 2 2 2 + θ |fi | + |u0i | + ku1 k + ku0 k + ku0 k . m 4 2 2 2 i∈Z

T

Since ϕ0 = (u0 , u1 + u0 ) ∈ Or0 , there is a constant K1 > 0 depending on the absorbing ball Or0 , such that max{1, λ + } ku0 k + ku1 k + kBu0 k ≤ K1 . Consequently, kDu0 k ≤ 2 K1 ,

∗ 2

(D u) ≤ 4 K12 .

It follows that there is a uniform constant K2 > 0 depending on K1 , such that X |i| 2 Mθ (0) ≤ K2 + θ |fi | . (56) m i∈Z

θ

Let ϕ (t) =

ϕθi (t) i∈Z

where ϕθi (t)

s |i| = θ ϕi (t). m

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From (54) and (56) we can get ! X |i| 2 θ |fi | , m

θ 2

ϕ (t) ≤ K3 e−t + K4 E

t ≥ 0,

(57)

i∈Z

where K3 and K4 are positive, uniform constants. For any given η > 0, there exists a T0 (η) > 0 such that K3 e−t ≤

η , 2

for all t ≥ T0 (η).

There exists a constant m > 0 such that X η K4 |fi |2 ≤ . 2

(58)

(59)

|i|≥m

As an absorbing set, there is a constant T1 = T1 (Or0 ) > 0 such that S (t) Or0 ⊂ Or0 ,

for all t ≥ T1 .

(60)

Let K(η) = 2m and T (η) = T0 (η) + T1 . Then by (57)–(60), it follows that (52) holds. Theorem 4.1. Under the condition (38) for any given f ∈ `2 , there exists a global attractor in E for the solution semiflow {S(t)}t≥0 generated by the lattice differential equation (36). Proof. We now prove that with (38) and (40) satisfied, the solution semiflow {S (t)}t≥0 generated by (41) is asymptotically compact in E. Given a bounded sequence {ϕn } ⊂ E and tn → ∞, such that kϕn kE ≤ r, n ≥ 1. Let O = Or0 be the absorbing ball shown in Lemma 4.2. Then there exists a constant Tr > 0 such that S (t) {ϕn } ⊂ O,

for all t ≥ Tr .

Hence, there is N1 (r) > 0 such that whenever n ≥ N1 (r) one has tn ≥ Tr , so that S (tn ) {ϕn } ⊂ O,

for n ≥ N1 (r).

(61)

Since a bounded set in a Hilbert space E is weakly precompact, there exists a subsequence of {S (tn ) ϕn }, which is relabeled as the same, such that S (tn ) ϕn → ϕˆ ∈ E weakly, as n → ∞.

(62)

Here we show that S (tn ) ϕn also converges to ϕˆ strongly in E. For any η > 0, by Lemma 4.3 and (61), there exist constants K(η), T (η) > 0 such that X η2 2 |(S (t + Tr ) ϕn )i |E ≤ , for t ≥ T (η), 4 |i|≥K(η)

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where | · |E is defined by (52). Since tn → ∞, there exists N2 (η) such that tn ≥ Tr + T (η) for n ≥ N2 (η). Thus, X X η2 2 2 |(S (tn ) ϕn )i |E = |(S (tn − Tr ) S (Tr ) ϕn )i |E ≤ . (63) 4 |i|≥K(η)

|i|≥K(η)

ˆ On the other hand, since ϕˆ ∈ E, there is a constant K(η) > 0 such that X η2 2 |ϕˆi |E ≤ . 4

(64)

ˆ |i|≥K(η)

n o ˆ Let K0 (η) = max K(η), K(η) . The weak convergence (62) implies that as n → ∞, (S (tn ) ϕn )i → ϕˆi ,

for each i with |i| ≤ K0 (η).

Thus there exists N3 (η) > 0 such that X η2 2 |(S (tn ) ϕn )i − ϕˆi |E ≤ , 2

(65)

for all n ≥ N3 (η).

(66)

|i|≤K0 (η)

Finally, for n ≥ N (η) = max {N1 (r), N2 (η), N3 (η)}, we have X |(S (tn ) ϕn )i − ϕˆi |2E kS (tn ) ϕn − ϕk ˆ 2E ≤ |i|≤K0 (η)

X

+

2

|(S (tn ) ϕn )i − ϕˆi |E

|i|>K0 (η)



2

≤

η +2 2

X

2

|(S (tn ) ϕn )i kE +

X

|i|>K0 (η)

|i|>K0 (η)

2

≤η ,

2



(67)

|ϕˆi |E 

because of (63), (64) and (66). This shows that the solution semiflow {S (t)}t≥0 is asymptotically compact in E. According to Theorem 23.12 in Sell and You, 10 based upon the fact that S (t) has an absorbing set in E and is asymptotically compact in E, there exists a global attractor A in E for the solution semiflow S (t) generated by (41). Note that the original solution semiflow S(t) generated by the lattice differential equation (36) is relative to the semiflow S (t) by S(t) = J− S (t) J , where J =

I

0 I

t ≥ 0,

on E0 = `2 × `2 .

Hence it follows that A = J− A

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is a global attractor in E0 for the semiflow S(t). However, E0 and E are the same Hilbert space with the equivalent norms. So, we have proved that there exists a global attractor A of the semiflow S(t) in E. Furthermore, one can prove that the global attractor A of the solution semiflow S(t) is upper semicontinuous in the sense that lim dE (An , A) = 0

n→∞

with respect to the Hausdorff distance dE in E, where An is the global attractor for the solution semiflow Sn (t) generated by the approximate (2n + 1)-dimensional ODE system which consists of the ith lattice equations, |i| ≤ n, with the cyclic cut-off terms. The details are omitted. References 1. P. W. Bates, K. Lu, and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11(1), 143–153 (2001). 2. J. Boussinesq, Theorie des ondes et des remous qui se propagent le long dun canal, J. Math. Pures Appl., Ser. 2, 17, 55–108 (1872). 3. F. Chen, B. Guo, and P. Wang, Long time behavior of strongly damped nonlinear wave equations, J. Differential Equations, 147(2), 231–241 (1998). 4. S.-N. Chow, R. Conti, R. Johnson, J. Mallet-Paret, and R. Nussbaum, Dynamical systems, Lecture Notes in Mathematics, Vol. 1822, Springer-Verlag, Berlin, 2003. Lectures from the C.I.M.E. Summer School held in Cetraro, June 19–26, 2000. 5. P. A. Clarkson, New exact solutions of the Boussinesq equation, European J. Appl. Math., 1(3), 279–300 (1990). 6. R. Hirota, Solutions of the classical Boussinesq equation and the spherical Boussinesq equation: the Wronskian technique, J. Phys. Soc. Japan, 55(7), 2137–2150 (1986). 7. Y. Li and Q. Chen, Finite-dimensional global attractor for dissipative SchrodingerBoussinesq equations, J. Math. Anal. Appl., 205(1), 107–132 (1997). 8. J. W. Miles, Solitary waves, in: Annual review of fluid mechanics, Vol. 12, pages 11–43. Annual Reviews, Palo Alto, Calif., 1980. 9. A. Nakamura, Exact solitary wave solutions of the spherical Boussinesq equation, J. Phys. Soc. Japan, 54(11), 4111–4114 (1985). 10. G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, Vol. 143, Springer-Verlag, New York, 2002. 11. B. Straughan, Global nonexistence of solutions to some Boussinesq type equations, J. Math. Phys. Sci., 26(2), 155–164 (1992). 12. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, second edition, 1997. 13. V. Varlamov, Asymptotic behavior of solutions of the damped Boussinesq equation in two space dimensions, Int. J. Math. Math. Sci., 22(1), 131–145 (1999). 14. Y. You, Nonlinear exponential stabilization of Boussinesq equations, in: Analysis and Optimization of Systems (Antibes, 1990), Lecture Notes in Control and Inform. Sci., Vol. 144, pages 642–651. Springer, Berlin, 1990. 15. S. Zhou, Attractors for second-order lattice dynamical systems, J. Differential Equations, 179(2), 605–624 (2002).

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RECENT PROGRESS ON NONLINEAR WAVE EQUATIONS VIA KAM THEORY∗

XIAOPING YUAN School of Mathematical Sciences, Fudan University, Shanghai 200433, CHINA Email: [email protected]

In this paper the present author gives out his some works on nonlinear wave equation using KAM theory. In Section 1, some backgrounds are given out for solving the nonlinear wave (NLW) equation via KAM theory. In Section 2, Theorem 2.1 shows that there are many invariant tori for the NLW equation with any non-vanishing prescribed potential. In Section 3, Theorems 3.1 and 3.2 show that there are many invariant tori of any dimension for the completely resonant NLW equation. This answers an open problem proposed by several authors. Keywords: KAM theory, invariant tori, nonlinear wave equation, quasi-periodic solution, completely resonant. 2000 Mathematics Subject Classification: Primary 37K55; Secondary 35L70.

1. Introduction The existence of solutions, periodic in time, for non-linear wave (NLW) equations has been studied by many authors. A wide variety of methods such as bifurcation theory and variational techniques have been brought on this problem. See Ref. 11 and the references therein, for example. There are, however, relatively less methods to find the quasi-periodic solutions of NLW or other PDE’s. The KAM theory is a very powerful tool in order to construct families of quasi-periodic solutions, which are on an invariant manifold, for some nearly integrable Hamiltonian systems of finite many degrees of freedom. In the 1980’s,the celebrated KAM theory has been successfully extended to infinitely dimensional Hamiltonian systems of short range so as to deal with certain class of Hamiltonian networks of weakly coupled oscillators. Vittot & Bellissard, 27 Frohlich, Spencer & Wayne 15 showed that there are plenty of almost periodic solutions for some weakly coupled oscillators of short range. In Ref. 30, it was also shown that there are plenty of quasi-periodic solutions for some weakly coupled oscillators of short range. ∗ Supported

by National Natural Science Foundation of China (10231020). 387

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Because of the restriction of short range, those results obtained in Refs. 27, 15 do not apply to PDE’s. In the 1980-90’s, the KAM theory has been significantly generalized, by Kuksin, 17,18,19 to infinitely dimensional Hamiltonian systems without being of short range so as to show that there is quasi-periodic solution for some class of partial differential equations. Also see P¨ oschel. 24 Let us focus our attention to the following nonlinear wave equation utt − uxx + V (x)u + u3 + h.o.t. = 0,

(1)

subject to Dirichlet and periodic boundary conditions on the space variable x. 1. Dirichlet boundary condition. In 1990, Wayne 28 obtained the time-quasiperiodic solutions of (1), when the potential V is lying on the outside of the set of some “bad” potentials. In Ref. 28, the set of all potentials is given some Gaussian measure and then the set of “bad” potentials is of small measure. Kuksin 17 assumed the potential V depends on n-parameters, namely, V = V (x; a1 , ..., an ), and showed that there are many quasi-periodic solutions of (1) for “most” (in the sense of Lebesgue measure) parameters a’s. However, their results exclude the constant-value potential V (x) ≡ m ∈ R+ , in particular, V (x) ≡ 0. When the potential V is constant, the parameters required can be extracted from the nonlinear term u3 . In order to use the KAM theorem, it is necessary to assume that there are some parameters in the Hamiltonian corresponding to (1). When V (x) ≡ m > 0, these parameters can be extracted from the nonlinear term u3 by Birkhoff normal form 25 , or by regarding (1.1) as a perturbation of sine-Gordon/sinh-Gordon equation 4 . And it was then shown that, for a prescribed potential V (x) ≡ m > 0, there are many elliptic invariant tori which are the closure of some quasi-periodic solutions of (1). By Remark 7 in Ref. 25, the same result holds also true for the parameter values −1 < m < 0. When m ∈ (−∞, −1) \ Z, it is shown in Ref. 29 that there are many invariant tori for (1). In this case, the tori are partially hyperbolic and partially elliptic. 2. periodic boundary condition. In this case, the eigenvalues of the linear d2 operator − dx 2 + V (x) are double (at least, asymptotically double). This results in some additional difficulties in applying KAM technique since the normal frequencies are double. According to our knowledge, the difficulty arising from the multiple normal frequencies (including double ones) was overcome in Ref. 9 in the year of 1969 when the multiplicity is bounded, although Hamiltonian systems are not considered. A key point is to bound the inverse of some matrix by requiring the determinant of the matrix is nonzero. Using Lyapunov-Schmidt decomposition and Newton’s iteration, Craig and Wayne 12 showed that for an open dense set of V (x) there exist time periodic solutions of (1) subject to periodic (also Dirichlet) boundary condition. (The equation considered by them contains more general form than (1). By developing Craig-Wayne’s method, in 1994, Bourgain 7 showed there are many quasi-periodic solutions of (1) for “most” parameters σ ∈ Rn where V = V (x; σ). In 2000, a similar result was obtained by KAM technique in Ref. 13. When the potential V ≡ m 6= 0, the existence of the quasi-periodic solutions was also obtained

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in Ref. 5 via the renormalization group method. 2. Nonlinear wave equation with a prescribed potential However, from the works mentioned above one does not know whether there is any invariant tori for prescribed (not random) non-constant-value potential V (x). Recently, the present author has shown that there are many invariant tori for any prescribed non-zero potential V (x) such as sin x and cos x. To give the statement of our results, we need to introduce some notations. We study equation (1) as an infinitely dimensional Hamiltonian system. Following P¨ oschel, 25 the phase space one may take, for example, the product of the usual Sobolev spaces W = H01 ([0, π])× L2 ([0, π]) with coordinates u and v = ut . The Hamiltonian is then H=

1 1 1 hv, vi + hAu, ui + u4 2 2 4

where A = d2 /dx2 − V (x) and h·, ·i denotes the usual scalar product in L2 . The Hamiltonian equation of motions are ∂H ∂H = v, −vt = = Au + u3 . ∂v ∂u Our aim is to construct time-quasi-periodic solutions of small amplitude. Such quasi-periodic solutions can be written in the form ut =

u(t, x) = U (ω1 t, · · · , ωn t, x), where ω1 , · · · , ωn are rationally independent real numbers which are called the basic frequency of u, and U is an analytic function of period 2π in the first n arguments. Thus, u admits a Fourier series expansion X √ u(t, x) = e −1hk,ωit Uk (x), k∈Zn

P

where hk, ωi = j kj ωj and Uk ∈ L2 [0, π] with Uk (0) = Uk (π). Since the quasi-periodic solutions to be constructed are of small amplitude, Equation (1) may be considered as the linear equation utt = uxx − V (x)u with a small nonlinear perturbation u3 . Let φj (x) and λj (j = 1, 2, ...) be the eigenfunctions and eigenvalues of the Sturm-Liouville problem −Ay = λy subject to Dirichlet boundary conditions y(0) = y(π) = 0, respectively. Then every solution of the linear system is the superposition of their harmonic oscillations and of the form X p u(t, x) = qj (t)φj (x), qj (t) = yj cos( λj t + φ0j ) j≥1

with amplitude yj ≥ 0 and initial phase φ0j . The solution u(t, x) is periodic, quasiperiodic or almost periodic depending on whether one, finitely many or infinitely many modes are excited, respectively. In particular, for the choice Nd = {j1 , j2 , · · · , jd } ⊂ N,

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of finitely many modes there is an invariant 2d-dimensional linear subspace ENd that is completely foliated into rational tori with frequencies λj1 , · · · , λjd : ENd = {(u, v) = (qj1 φj1 + · · · + qjd φjd , q˙j1 φj1 + · · · + q˙jd φjd )} [ = Tj (y), y∈P¯d

where Pd = {y ∈ Rd : yj > 0

for 1 ≤ j ≤ d} is the positive quadrant in Rd and

2 TNd (y) = {(u, v) : qj2k + λ−2 jk q˙jk = yk ,

for

1 ≤ k ≤ d}.

Upon restoring the nonlinearity u3 the invariant manifold ENd with their quasiperiodic solutions will not persist in their entirety due to resonance among the modes and the strong perturbing effect of u3 for large amplitudes. In a sufficiently small neighborhood of the origin, however, there does persist a large Cantor subfamily of rotational d-tori which are only slightly deformed. More exactly, we have the following theorem: Theorem R2.1. (Ref. 30) Assume that V (x) is sufficiently smooth in the interval π [0, π], and 0 V (x) dx 6= 0. Let K and N be positive constants large enough. Let Nd = {ip ∈ N : p = 1, 2, · · · , d} with min Nd > N K, max Nd ≤ C0 dN K, and K1 ≤ |ip − iq | ≤ K2 ,

for

p 6= q,

where C0 > 1 is an absolute constant and K1 , K2 , positive constants large enough, depending on K instead of N . Then, for given compact set C ∗ in Pd with positive Lebesgue measure, there is a set C ⊂ C ∗ with meas C > 0, a family of d-tori [ TNd (C) = TNd (y) ⊂ ENd y∈C

over C, and a Lipschitz continuous embedding Φ : TNd [C] ,→ H01 ([0, π]) × L2 ([0, π]) = W, which is a higher order perturbation of the inclusion map Φ0 : ENd ,→ W restricted to TNd [C], such that the restriction of Φ to each TNd (y) in the family is an embedding of a rotational invariant d-torus for the nonlinear equation (1.1). Rπ Basic idea of the proof. It is observed that when 0 V (x) dx 6= 0 the eigenvalues λi ’s satisfy p p p p | λi ± λj ± λk ± λl | ≥ Cm min(i, j, k, l)−1 , if min(i, j, k, l) 1,

unless trivial relations like

p p p p λi − λi + λk − λk ,

where Cm is a constant depending on m. This estimate implies that in the neighborhood of the origin the equation (1) can be put to the Borkhoff normal form 6

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up to terms of the fourth order. Then the application of KAM theorem for PDEs implies that there are many invariant tori for (1) in the neighborhood of the origin. Rπ Remark 2.1. The assumption 0 V (x) dx 6= 0 is not essential. One has p c1 c2 cn 1 λj = j + + 2 +···+ n +O , j j j j n+1 Rπ 1 where cj ’s are some constants depending on V , in particular, c1 = − 2π V (x) dx. 0 Rπ Then the assumption 0 V (x) dx 6= 0 is equivalent to c1 6= 0. The assumption c1 6= 0 is used just only in making Birkhoff normal form. By overcoming more technical trouble one can still get the normal form true under conditions R π c1 = 0, · · · , ck−1 = 0 and ck 6= 0 for some 1 ≤ k ≤ n. Therefore the assumption 0 V (x) dx 6= 0 can be nearly replaced by V (x) 6= 0 in Theorem 2.1. Remark 2.2. Theorem 2.1 still holds true for the following equation X utt = uxx − V (x)u ± u3 + ak u2k+1 m≥k≥2

where m is a positive integer and ak ’s are some real numbers. Remark 2.3. The method in proving Theorem 2.1 can be applied to NLS equation: √ −1ut − uxx + V (x)u ± u3 = 0 subject to Dirichlet boundary conditions. Remark 2.4. If λ1 > 0, then the obtained invariant tori are elliptic. If λ1 < 0, then the tori are hyperbolic-elliptic. Remark 2.5. We can give the measure estimate of the set C: meas C ≥ meas C ∗ · (1 − O(1/13 )). 3. Completely resonant nonlinear wave equation Naturally one can ask: Is there any invariant torus for (1) when V (x) ≡ 0? In this case, the equation is called completely resonant by P o ¨schel. 25 This problem remains open for a relatively long time, which was proposed by many authors, such as P¨ oschel, 25 Craig and Wayne, 12 Kuksin, 18 and Marmi and Yoccoz. 23 The present author answers this question: Theorem 3.1. (Ref. 31) Assume v(x) ≡ 0. For any d ∈ N, the equation (1) subject to the periodic boundary condition possesses many d + 1-dimensional invariant tori in the neighborhood of the equilibrium u ≡ 0. The motions on the tori are quasiperiodic.

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Basic idea of the proof. Assume V (x) ≡ 0. Let u0 (t) be a non-zero solution of the equation u ¨0 + u30 = 0. We will construct the invariant tori or quasi-periodic solutions in the neighborhood of the solution u0 (t) which is uniform in space and periodic in time. To that end, inserting u = u0 + u into (1) we get utt − uxx + 3u20 (t)u + · (h.o.t.) = 0,

x ∈ S 1.

(2)

In considerably rough speaking, by the averaging method we reduce this equation to c2 (0)u + · (h.o.t.) = 0, x ∈ S 1 , utt − uxx + 3u (3) 0 R c2 (0) = 1 2π u2 (t) dt 6= 0. Then we construct the invariant tori or quasiwhere u 0 0 2π 0 periodic solutions of (3) by advantage of c u2 (0) 6= 0. At this time, we should deal 0

with (3) by the same way as in Ref. 25. Unfortunately, one of the frequencies of the Hamiltonian corresponding to (3) is zero (see (6) below). This causes the “integrable” part of the Hamiltonian serious degenerate, incurring great expanse in using KAM technique. Firstly one can easily find the periodic solution u0 (t) of u ¨0 + u30 = 0 with its c2 (0) 6= 0. Then consider a family of Hamiltonian frequency ω and show that u 0 functions p 3u2 (t) Hn = λn zn z¯n + √0 (zn + z¯n )2 , λn 6= 0, n ∈ N = {1, 2, ...}. (4) 4 λn Notice that their equation of motion is linear. By the reducing theory from KAM theory 9,14 we reduce (4) to p 3 2 Hn = µn zn z¯n , µn = λn + ω + O(ω 23/9 /n), n ∈ N, (5) nπ

where λn is the eigenvalues of the Sturm-Liouville problem −y 00 = λy, x ∈ S 1 . At the same time by the Floquet theory 22 one can reduce the Hamiltonian H0 =

1 2 λ0 2 3u20 (t) 2 y + x0 + x0 , 2 0 2 2

λ0 = 0,

(6)

to 1 c0 ω 2 p 2 , (7) 2 where c0 is a constant. In order to exclude the multiplicity of the eigenvalues λn , one can find a solution which is even in the space variable x. That is, one can write P u(t, x) = n≥0 xn (t) cos nx. From this we get a Hamiltonian corresponding to (2) which reads as 1X 2 3 H= yn + λn x2n + u20 (t)x2n + G3 + 2 G4 , (8) 2 2 H0 (q, p) =

n≥0

where G3 (and G4 , resp. ) is a polynomial of order 3 (and 4, resp. ) in variables x0 , x1 , .... Introducing the complex variables we re-write (8):

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H = 12 y02 + 32 u20 (ϑ)x20 + +G3 + 2 G4 .

P

n≥1

√

λn zn z¯n +

2 (t) 3u √0 4 λn (zn

+ z¯n )2

Applying (4)-(7) we get a symplectic transformation Ψ such that X 1 ˜4. ˜ 3 + 2 G H := H ◦ Ψ = (c0 ω 2 )p2 + µn zn z¯n + G 2 n>0

(9)

(10)

˜ 3 and G ˜ 4 involve the time t. Let ϑ = ωt be an angle-variable and Notice that G J =Const. be an action-variable. Then (10) can read as X 1 ˜4, ˜ 3 + 2 G (11) H := H ◦ Ψ = (c0 ω 2 )p2 + Jω + µn zn z¯n + G 2 n>0 ˜ 3 and the non-resonant which is autonomous. One can now kill the perturbations G ˜ 4 by Birkhoff normal form. The one gets part of the perturbation G P H = 21 (c0 ω 2 )p2 + 2 c1 q 4 + 3 O(q 5 ) + n6=0 µn zn z¯n + Jω P + j c2 2 q 2 + 3 O(q 3 ) zj z¯j P + i∈Nd ,j∈N 2 c3 + 3 O(q) zi z¯i zj z¯j + small perturbation ,

(12)

After introducing action-angle variables (I0 , φ0 ) corresponding to (q, p), then (12) reads as P 4/3 H = 2/3 c4 I0 + j>0 µj zj z¯j + Jω P P (13) +Γ(I0 , φ0 ) + j>0 Γj (I0 , φ0 )zj z¯j + j∈N,i∈Nd Γij zi z¯i zj z¯j P + j∈N,i∈Nd 2 c5 zi z¯i zj z¯j + small perturbation .

The ci ’s are constants. Using the averaging method we remove the dependence of Γ, Γj and Γij on the angle variable φ0 . After this, we get a Hamiltonian H = H0 +small perturbation, where H0 is integrable and “twist”. The “twist” property can provide the parameters which we need in using KAM technique. Finally, one gets the invariant tori for (1) with V (x) ≡ 0, by making use of KAM theorem. Remark 3.1. Bourgain, 8 Bambusi-Paleari, 10 Berti-Bolle 2,3 and GentileMastropietro-Procesi 16 construct countably many families of periodic solutions for the nonlinear wave equation utt − uxx ± u3 + h.o.t. = 0. See also Refs. 3, 11, 21 and the references therein for the related problems. More recently, Procesi 26 and Baldi 1 constructed quasi-periodic solutions of 2-dimensional frequency and of Lebesgue measure 0 for the completely resonant nonlinear wave equations. Their construction of quasi-periodic solutions is concise and elegant. Theorem 3.2. (Ref. 31) For any d ∈ N, the equation utt − uxx − u3 = 0

(14)

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subject to the Dirichlet boundary condition u(t, 0) = u(t, π) = 0

(15)

possesses many d-dimensional hyperbolic-elliptic invariant tori in the neighborhood of the equilibrium u ≡ 0. The motions on the tori are quasi-periodic. Basic idea of the proof. Let u0 (t, x) ≡ u0 (x) solve ODE uxx + u3 = 0 with b.c. (15). let u = u0 + ˜ u.

(16)

Inserting (16) into (14) we get u ˜ obeys the following equation and b.c. u ˜tt − u ˜xx − 3u20 u ˜ − 3u0u ˜2 − 2 u ˜3 = 0, (17) u ˜(t, 0) = u ˜(t, π) = 0. Rπ Let V (x) = −3u20 . It is easy to verify 0 V (x) dx 6= 0. By the method similar to that of Theorem 2.1 we can show that the existence of invariant tori. References 1. P. Baldi, Quasi-periodic solutions of the equation utt − uxx + u3 = f (u), preprint. 2. M. Berti and P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearity, Preprint. 3. M. Berti and P. Bolle, Cantor families of periodic solutions for completely resonant nonlinear wave equations, Preprint. 4. A. I. Bobenko and S. Kuksin, The nonlinear Klein-Gordon equation on an interval as perturbed sine-Gordon equation, Comm. Math. Helv. 70, 63–112 (1995). 5. J. Bricmont, A. Kupiainen, and A. Schenkel, Renormalization group and the Melnikov Problems for PDE’s, Commun. Math. Phys. 221, 101–140 (2001). 6. G. D. Birkhoff, Dynamical systems, Am. Math. Soc. Colloq. Publ. IX. New York: American Mathematical Society. VIII, 1927. 7. J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and application to nonlinear pde, Int. Math. Research Notices 11, 475–497 (1994). 8. J. Bourgain, Periodic solutions of nonlinear wave equations, Harmonic analysis and partial equations, Chicago Univ. Press, 69–97 (1999). 9. N. N. Bogoljubov, Yu. A. Mitropoliskii, and A. M. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer-Verlag, New York (1976) [Russian Original: Naukova Dumka, Kiev, 1969]. 10. D. Bambusi and S. Paleari, Families of periodic orbits for resonant PDE’s, J. Nonlinear Science, 11, 69–87 (2001). 11. H. Br´ezis, Periodic solutions of nonlinear vibrating strings and duality principles, Bull. AMS 8, 409–426 (1983). 12. W. Craig and C. Wayne, Newton’s method and periodic solutions of nonlinear wave equation, Commun. Pure. Appl. Math., 46, 1409–1501 (1993). 13. L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys., 211, 497–525 (2000). 14. L. H. Eliasson, Reducibility and point spectrum for linear quasi-periodic skewproducts, Proceedings of ICM 1998, Vol. II, Doc. Math. J. DMV, 779–787.

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15. J. Frohlich, T. Spencer, and C. E. Wayne, Localization in Disordered, Nonlinear Dynamical Systems, J. Statistical Physics, 42, 247–274 (1986). 16. G. Gentile, V. Mastropietro, and M. Procesi, Periodic solutions for completely resonant nonlinear wave equations, Comm. Math. Phys., 256, 437–490 (2005). 17. S. B. Kuksin, Nearly integrable infinite-dimensional Hamiltonian systems, Lecture Notes in Math. 1556, Springer-Verlag, New York (1993). 18. S. B. Kuksin, Elements of a qualitative theory of Hamiltonian PDEs, Proceedings of ICM 1998, Vol. II, Doc. Math. J. DMV, 819–829 (1998). 19. S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funct. Anal. Appl., 21, 192–205 (1987). 20. S. B. Kuksin and J. P¨ oschel, Invariant Cantor manifolds of quasiperiodic oscillations for a nonlinear Schr¨ odinger equation, Ann. Math., 143, 149–179 (1996). 21. B. V. Lidskij and E. Shulman, Periodic solutions of the equation utt − uxx + u3 = 0, Funct. Anal. Appl., 22, 332–333 (1988). 22. K. R. Meyer and G. R. Hall, Introduction to Hamiltonian dynamical systems and the N -body problem, Applied Math. Sciences, 90, Springer-Verlag, New York (1992). 23. S. Marmi and J.-C. Yoccoz, Some open problems related to small divisors, Lecture Notes in Math. 1784, Springer-Verlag, New York (2002). 24. J. P¨ oschel, A KAM-theorem for some nonlinear PDEs, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., IV Ser. 15, 23, 119–148 (1996). 25. J. P¨ oschel, Quasi-periodic solutions for nonlinear wave equation, Commun. Math. Helvetici, 71, 269–296 (1996). 26. M. Procesi, Quasi-periodic solutions for completely resonant nonlinear wave equations in 1D and 2D, Discr. and Cont. Dyn. Syst., 13, 541–552 (2005). 27. M. Vittot and J. Bellissard, Invariant tori for an infinite lattice of coupled classical rotators. Preprint, CPT-Marseille (1985). 28. C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Commun. Math. Phys., 127, 479–528 (1990). 29. X. Yuan, Invariant manifold of hyperbolic-elliptic type for nonlinear wave equation, Int. J. Math. Math. Science, 18, 1111–1136 (2003). 30. X. Yuan, Quasi-periodic solutions of nonlinear wave equations with a prescribed potential, Discr. and Cont. Dyn. Syst., 3, 615–634 (2006). 31. X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Diff. Eqns., 230, 213–274 (2006).