Applied Analysis and Differential Equations

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aPPLIED ANALYSIS AND DIFFERENTIAL

EQUATIONS

This page intentionally left blank

EDITORS

Ovidiu Carja Loan l Vrabie AlCuza University, O Mayer Msthematics Institute of the Romanian Academy, Romania

aPPLIED ANALYSIS

AND DIFFERENTIAL

EQUATIONS lasi, Romania 4 – 9 september 2006

World Scientiﬁc NEW JERSEY LONDON * SINGAPORE * BElJlNG - 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.

APPLIED ANALYSIS AND DIFFERENTIAL EQUATIONS 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-13 978-981-270-594-5 ISBN-10 981-270-594-5

Printed in Singapore.

EH - Appl Analysis & Diff Eqns.pmd

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PREFACE The International Conference of Applied Analysis and Differential Equations was held September 4-9, 2006 at the “Al. I. Cuza” University of Ia¸si and the “Octav Mayer” Mathematical Institute of the Romanian Academy of Ia¸si, Romania. It was supported by the Project CERES 4-194/2004. The conference was devoted to recent advances in Nonsmooth Analysis and Optimizations, ODE, PDE, Control Theory, Stochastic Analysis and was well attended by mathematicians all over the world. There were 40 minutes plenary invited lectures, 20 minutes talks and a poster session. This volume includes 29 selected articles, many of them written by leading specialists in their fields and covering the main topics of the conference. We take this opportunity to express our gratitude to all who have chosen to publish their contributions in this volume as well as to the referees of the submitted papers for their promptitude and high exigence. Furthermore, we thank Ms. Carmen Savin for her excellent secretarial work, as well as Ms. Elena Mocanu for her extremely efficient and skilful assistance in preparation of the camera-ready copy. Finally, we express our warmest thanks to World Scientific for the very pleasant and fruitful cooperation during the publication of this volume.

Ia¸si, December 19, 2006

Ovidiu Cˆarj˘a and Ioan I. Vrabie

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ORGANIZING COMMITTEE SERIES EDITORIAL BOARD

ORGANIZING COMMITTEE of the International Conference on Applied Analysis and Differential Equations

O. Cˆarj˘a (Chairman)

—

University ”Al.I. Cuza”, Ia¸si and ”O. Mayer” Mathematics Institute of the Romanian Academy, Ia¸si, Romania

V. Barbu

—

University ”Al.I.Cuza”, Ia¸si and ”O. Mayer” Mathematics Institute of the Romanian Academy, Ia¸si, Romania

M. Durea (Secretary)

—

University ”Al.I. Cuza”, Ia¸si, Romania

M. Necula

—

University ”Al.I. Cuza”, Ia¸si, Romania

A. R˘a¸scanu

—

University ”Al.I. Cuza”, Ia¸si and ”O. Mayer” Mathematics Institute of the Romanian Academy, Ia¸si, Romania

I.I. Vrabie

—

University ”Al.I. Cuza”, Ia¸si and ”O. Mayer” Mathematics Institute of the Romanian Academy, Ia¸si, Romania

C. Z˘alinescu

—

University ”Al.I. Cuza”, Ia¸si and ”O. Mayer” Mathematics Institute of the Romanian Academy, Ia¸si, Romania

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CONTENTS

Preface Organizing Committees A Stability Criterion for Delay Differential Equations with Impulse Effects

v vii

1

J.O. Alzabut On the Use of Korn’s Type Inequalities in the Existence Theory for Cosserat Elastic Surfaces with Voids

11

M. Bˆırsan Time Periodic Viscosity Solutions of Hamilton–Jacobi Equations

21

M. Bostan and G. Namah A Viability Result for Semilinear Reaction-diffusion Systems

31

M. Burlic˘ a and D. Ro¸su Necessary Optimality Conditions for Hyperbolic Discrete Inclusions

45

A. Cernea Controllability of Damped Second-Order Initial Value Problem for a Class of Differential Inclusions with Nonlocal Conditions on Noncompact Intervals

55

D.N. Chalishajar Averaging of Evolution Inclusions in Banach Spaces

69

T. Donchev Influence of Variable Permeability on Vortex Instability of a Horizontal Combined Free and Mixed Convection Flow in a Saturated Porous Medium A.M. Elaiw, F.S. Ibrahim and A.A. Bakr

79

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Analyticity of Stable Invariant Manifolds for Ginzburg–Landau Equation

93

A.V. Fursikov Approximate Solutions for Non-linear Autonomous ODEs on the Basis of PWL Approximation Theory

113

A.G. Garcia, S.I. Biagiola, J.L. Figueroa, O.E. Agamennoni and L.S. Castro On the Influence of a Subquadratic Convection Term in Singular Elliptic Problems

127

M. Ghergu and V. R˘ adulescu Existence and Uniqueness Results in the Micropolar Mixture Theory of Porous Media

139

I.-D. Ghiba Approximate Controllability for Linear Stochastic Differential Equations with Control Acting on the Noise

153

D. Goreac Numerical Solutions of Two-point Boundary Value Problems for Ordinary Differential Equations Using Particular Newton Interpolating Series

165

G. Groza and N. Pop On the Integral and Asymptotic Representation of Singular Solutions of Elliptic Boundary Value Problems Near Components of Small Dimensions of Boundary

177

N. Jitara¸su An Existence Result for a Class of Nonlinear Differential Systems

185

R. Luca-Tudorache Well-posedness for a Non-autonomous Model of Fast Diffusion G. Marinoschi

199

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Backward Stochastic Generalized Variational Inequality

xi

217

L. Maticiuc and A. R˘ a¸scanu Trichotomy for Linear Skew-product Semiflows

227

M. Megan, C. Stoica and L. Buliga Feedback Control of Constrained Parabolic Systems in Uncertainty Conditions via Asymmetric Games

237

B.S. Mordukhovich and T.I. Seidman Implicit Parallel Solvers in Computational Electrocardiology

255

M. Munteanu and L.F. Pavarino Fan’s Inequality in the Context of MP -Convexity

267

C.P. Niculescu and I. Rovent¸a Range Condition in Optimization and Optimal Control

275

N.H. Pavel Limits of Solutions to the Initial Boundary Dirichlet Problem for Semilinear Hyperbolic Equation with Small Parameter

281

A. Perjan Well-posedness of the Fixed Point Problem for Multivalued Operators

295

A. Petru¸sel and I.A. Rus Some Remarks on a Nonlinear Semigroup Acting on Positive Measures

307

E. Popa Integral Inclusions in Banach Spaces Using Henstock-Type Integrals

319

B. Satco Good Intermediate-rank Lattice Rules Based on the Weighted Star Discrepancy V. Sinescu and S. Joe

329

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On the Minimum Energy Problem for Linear Systems in Hilbert Spaces

343

A.I. Vieru Author Index

351

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A STABILITY CRITERION FOR DELAY DIFFERENTIAL EQUATIONS WITH IMPULSE EFFECTS J.O. ALZABUT Department of Mathematics and Computer Science C ¸ ankaya University, 06530 Ankara, Turkey E-mail: [email protected] In this paper, we prove that if a delay differential equation with impulse effects of the form x0 (t) = A(t)x(t) + B(t)x(t − τ ) , t 6= θi , ∆x(θi ) = Ci x(θi ) + Di x(θi−j ), i ∈ N, verifies a Perron condition then its trivial solution is uniformly asymptotically stable. Keywords: Impulse; Delay; Adjoint; Perron; Uniform asymptotic stability.

1. Introduction and Preliminaries Delay differential equations with impulse effects can suitably model various evolutionary processes that exhibit both delay and impulse characteristics. In particular, they provide a natural description of the motion of several real world processes which, on one hand, depends on the processes history that often turns out to be the cause of phenomena substantially affecting the motion and, on other hand, is subject to short time perturbations whose duration is almost negligible. Such processes are often investigated in various fields of science and technology, such as physics, population dynamics, ecology, biological systems, optimal control, etc., see Refs. 1–11 and reference quoted therein. It is well known in the theory of ordinary differential equations (see e.g. Ref. 12 [p. 120]) that if for every continuous function f (t) bounded on [0, ∞), the solution of the equation x0 (t) = A(t)x(t) + f (t), satisfying x(0) = 0 is bounded on [0, ∞), then the trivial solution of the corresponding homogeneous equation

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x0 (t) = A(t)x(t) is uniformly asymptotically stable. This result is referred as Perron theorem Ref. 13. Later, Perron theorem has been extended to delay differential equations Ref. 12 [p. 371]. Indeed, it was shown that if for every continuous function f (t) bounded on [0, ∞), the solution of the equation x0 (t) = A(t)x(t) + B(t)x(t − τ ) + f (t), t > 0 satisfying x(t) = 0 for t ∈ [−τ, 0] is bounded on [0, ∞), then the trivial solution of the equation x0 (t) = A(t)x(t) + B(t)x(t − τ ), is uniformly asymptotically stable. For more related materials, see the papers Refs. 14,15. In this paper, we carry out the above result to a type of linear delay differential equations with impulse effects. Indeed, we consider equation of the form x0 (t) = A(t)x(t) + B(t)x(t − τ ),

t 6= θi ,

∆x(θi ) := x(θi+ ) − x(θi ) = Ci x(θi ) + Di x(θi−j ), i ∈ N,

(1)

and show that its trivial solution is uniformly asymptotically stable under a Perron condition. Our equation differs from the previous ones, see also Refs. 16–19, not only it is more general but also it allows delay terms in the impulse conditions. Such impulse conditions are more natural for delay differential equations. With regard to equation (4) it is assumed that (i) A and B are n×n continuous bounded matrices, τ > 0 is a positive real number; (ii) Ci and Di are n × n bounded matrices, j ∈ N is fixed; (iii) {θi } is an increasing sequence of real numbers with lim θi = ∞. i→∞

We also assume that det(I + Ci ) 6= 0 and that there exist a positive real numbers ρ and ν such that kDi k ≤ ρ and k(I +Ci )−1 k ≤ ρ and θi −θi−j ≤ ν for all i ∈ N. k · k denotes any matrix norm. Definition 1.1. Equation (4) is said to verify Perron condition if for every continuous bounded on [0, ∞) function f (t) and every bounded sequence

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βi the solution of x0 (t) = A(t)x(t) + B(t)x(t − τ ) + f (t), ∆x(θi ) = Ci x(θi ) + Di x(θi−j ) + βi ,

t 6= θi ,

i ∈ N,

(2)

satisfying x(t) = 0 for t ∈ [−τ, 0] is bounded on t ∈ [0, ∞). By a solution of (6) on an interval J, we mean a function x defined on J such that x is continuous on J except possibly at θi ∈ J for i ∈ N, where x(θi+ ) : lim+ x(t) and x(θi− ) := lim− x(t) exist, x(θi− ) := x(θi ), and that x t→θi

t→θi

satisfies (6) on J. Clearly, if f ≡ 0 and βi = 0 for all i ∈ N then (6) reduces to (4). Let P LC([−τ, 0], Rn ) denote the set of piecewise left continuous functions φ : [−τ, 0] → Rn having a finite number of discontinuity points of the first kind. Under the above conditions, one can easily show that for given σ ≥ 0 and φ ∈ P LC([−τ, 0], Rn ) there is a unique solution x(t) of (6) such that x(t + σ) = φ(t),

t ∈ [−τ, 0].

(3)

2. Preparatory Lemmas The following lemmas, see Ref. 12 for delay differential equations without impulse effects, are essential in proving the main result of this paper. Lemma 4.1 is needed to define an adjoint equation of (4), Lemma 2.2 provides representation of solutions, and Lemma 2.3 is concerned with the boundedness of fundamental matrices of (4). Consider the equation y 0 (t) = −AT (t)y(t) − B T (t + τ )y(t + τ ),

t 6= θi ,

+ T ∆y(θi ) − (I + CiT )−1 CiT y(θi ) − (I + CiT )−1 Di+j y(θi+j ), i ∈ N.

(4)

We claim that equation (9) is an adjoint of (4) with respect to a function resembles the one used by Halanay in Ref. 12 [p. 371]. It turns out that this function has the form Z t+τ y T (s)B(s)x(s − τ )ds < y(t), x(t) >= y T (t)x(t) + t

+

X

y T (θk+ )Dk x(θk−j ),

n(t)≤k
where n(t) = min{i ∈ N : θi ≥ t}.

(5)

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Lemma 2.1. If x(t) is a solution of (4) and y(t) is a solution of (9) then hy(t), x(t)i = c = constant, where h , i is defined by (10). Proof. Let t ∈ (θi , θi+1 ). Then d hy(t), x(t)i = −y T (t)A(t)x(t) − y T (t + τ )B(t + τ )x(t) dt + y T (t + τ )B(t + τ )x(t) − y T (t)B(t)x(t − τ ) + y T (t)A(t)x(t) + y T (t)B(t)x(t − τ ) = 0, and hence hy(t), x(t)i = ci = constant for t ∈ (θi , θi+1 ). We may claim that ci = c for i ∈ N. Indeed, since ci+1 − ci = ∆ hy(t), x(t)i |t=θi , by (10) we have ci+1 − ci = y T (θi+ )x(θi+ ) − y T (θi )x(θi ) X + y T (θk+ )Dk x(θk−j ) n(θi+ )≤k
−

X

y T (θk+ )Dk x(θk−j ).

n(θi )≤k
Since n(θi+ ) = i + 1 and n(θi ) = i, we have ci+1 − ci = y T (θi+ )x(θi+ ) − y T (θi )x(θi ) − y T (θi+ )Di x(θi−j ) + + y T (θi+j )Di+j x(θi ).

Using the impulse conditions in (4) x(θi+ ) = (I + Ci )x(θi ) + Di x(θi−j ) and the impulse conditions in (9) + y T (θi+j )Di+j = y T (θi ) − y T (θi+ ) − y T (θi+ )Ci

we deduce that ci+1 − ci = 0 for all i ∈ N and thus hy(t), x(t)i = c. Remark 2.1. It is easy to verify also that the adjoint of (9) is (4), i.e they are mutually adjoint of each other.

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Definition 2.1. A matrix solution X(t, α) of (4) satisfying X(α, α) = I and X(t, α) = 0 for t < α is called a fundamental matrix of (4). Definition 2.2. A matrix solution Y (t, α) of (9) satisfying Y (α, α) = I and Y (t, α) = 0 for t > α is said to be a fundamental matrix of (9). Lemma 2.2. Let X(t, α) be a fundamental matrix of (4) and σ ≥ 0 a real number. If x(t) is a solution of (6), then x(t) = X(t, σ)x(σ) Z σ Z t + X(t, α + τ )B(α + τ )x(α)dα + X(t, α)f (α)dα σ−τ σ X X + + X(t, θk+j )Dk+j x(θk ) + X(t, θk+ )βk . (6) n(σ)−j≤k
n(σ)≤k
Proof. Multiplying the differential equation in (6) by the matrix Y T (α, t) and integrating with respect to α from σ to t, we obtain Z t x(t) = Y T (σ, t)x(σ) − Y T (α + τ, t)B(α + τ )x(α)dα σ

Z

Z

t

+

T

t

Y (α, t)B(α)x(α − τ )dα + σ

Y T (α, t)f (α)dα

σ

X

+

h i Y T (θk+ , t)x(θk+ ) − Y T (θk , t)x(θk ) .

n(σ)≤k
Replacing α by α+τ in the second integral and using the impulse conditions in (6) and (9), we have Z σ x(t) = Y T (σ, t)x(σ) + Y T (α + τ, t)B(α + τ )x(α)dα σ−τ

X

+

+ Y T (θk+j , t)Dk+j x(θk )

n(σ)−j≤k
Z

t

+ σ T

Y T (α, t)f (α)dα +

X

Y T (θk+ , t)βk .

(7)

n(σ)≤k
Since X(t, σ) = Y (σ, t), which can be seen by replacing x(t) by the fundamental matrix X(t, σ) in (7) with f ≡ 0 and βi = 0 for all i ∈ N, (7) is the same as (6).

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Corollary 2.1. Let X(t, α) be a fundamental matrix of (4) and Y (t, α) be a fundamental matrix of (9). Then X(t, α) = Y T (α, t). Lemma 2.3. If (4) verifies Perron condition, then kX(t, α)k < M for t ≥ α ≥ 0. Proof. We first claim that there exists a constant d such that Z t X + kX(t, α)kdα + kX(t, θm )k < d for t ≥ 0. 0

(8)

0≤m
Define the space Π = CB × S, where CB is the set of bounded functions f ∈ C([0, ∞), Rn ) and S is the set of bounded sequences β = {βm }, βm ∈ Rn , m ∈ N. The elements are represented by the pair (f, β) supplied by the norm k(f, β)k = sup kf (t)k + sup kβm k. Consider the operator U t∈[0,∞)

m∈N

defined on the Banach space Π by Z t U (f, βm ) = X(t, α)f (α)dα + 0

X

+ X(t, θm )βm .

0≤m
We may use the Banach Steinhaus theorem Ref. 20 by employing similar arguments developed in Ref. 12 to arrive at (8). Now let us consider (9) satisfied by Y (α, t). Integrating both sides from σ to t leads to Z Y T (σ, t) = I +

t

Z Y T (α, t)A(α)dα +

σ

t

Y T (α + τ, t)B(α + τ )dα

σ

−

X

∆Y T (θi , t).

n(σ)≤i
+ Observing that Y T (θi , t) = Y T (θi+ , t)(I + Ci ) + Y T (θi+j , t)Di , we obtain

X n(σ)≤i
It follows that

k∆Y T (θi , t)k ≤ 2ρ

X 0≤i
kY T (θi+ , t)k.

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Z

t

kY T (σ, t)k ≤ 1 + 2γ

X

kY T (α, t)kdα + 2ρ

0

7

kY T (θi+ , t)k,

0≤i
where ½ ¾ γ = max sup kA(t)k, sup kB(t)k . t≥0

t≥0

Replacing Y T (σ, t) by X(t, σ) and using inequality (8) result in the desired conclusion. 3. The Main Result Theorem 3.1. If equation (4) verifies Perron condition then its trivial solution is uniformly asymptotically stable. Proof. Let x(t; σ, φ) denote the solution of (4) satisfying (3). From Lemma 2.2, Z

0

x(t; σ, φ) = X(t, σ)φ(0) +

X(t, α + σ + τ )B(α + σ + τ )φ(α)dα −τ

+

X

+ X(t, θk+n(σ)+j )Dk+n(σ)+j φ(θk ).

−j≤k<0

By Lemma 2.3, there exists M > 0 such that kX(t, r)k < M . Hence kx(t; σ, φ)k ≤ M (1 + τ γ + jρ)kφk0 = M1 kφk0 , where M1 = M (1 + τ γ + jρ) and kφk0 =

sup kφ(r)k. r∈[−τ,0]

Thus, the zero solution of (4) is uniformly stable. To complete the proof we need to show that lim x(t; σ, φ) = 0 uniformly with respect to σ and φ.

t→∞

(9)

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For our purpose, let µ ≥ σ. It is clear that x(t) = x(t; σ, φ) satisfies Z µ x(t; σ, φ) = X(t, µ)x(µ; σ, φ) + X(t, α + τ )B(α + τ )x(α; σ, φ)dα µ−τ

+

X

+ X(t, θk+j )Dk+j x(θk ; σ, φ).

n(µ)−j≤k
Integrating both sides from σ to t and then changing the order of integrations and the order of summation and the integral, we have Z

Z

σ

α+τ

(t − σ)x(t; σ, φ) =

X(t, α + τ )B(α + τ )x(α; σ, φ)dµdα σ−τ

Z

σ

t−τ

Z

α+τ

+

X(t, α + τ )B(α + τ )x(α; σ, φ)dµdα σ

Z

α

t

+

X(t, µ)x(µ; σ, φ)dµ σ

+

Z

X

σ

n(σ)−j≤k
+

X

Z

n(σ)≤k
θk+j

θk+j

θk

+ X(t, θk+j )Dk+j x(θk ; σ, φ)dµ

+ X(t, θk+j )Dk+j x(θk ; σ, φ)dµ.

We easily see from above that (t − σ)kx(t; σ, φ)k ≤ γτ 2 M M1 kφk0 + jρνM M1 kφk0 hZ t + M1 max{τ γ, νρ, 1}kφk0 kX(t, s)kds 0

+

X

i kX(t, θr )k .

(10)

0≤r
In view of (8), the right side of (10) is bounded. Hence kx(t; σ, φ)k ≤

M2 kφk0 , t−σ

where M2 is chosen so that M2 < M M1 (γτ 2 + jρν) + M1 max{τ γ2 , νρ, 1}d. Obviously, (9) follows from (11).

(11)

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References 1. S.H. Saker and J.O. Alzabut, Existence of periodic solutions, global attractivity and oscillation of impulsive delay population model, Nonlinear Anal. Real World Appl., to appear. 2. S.H. Saker and J.O. Alzabut, Periodic solutions, global attractivity and oscillation of impulsive delay host macroparasite model, Math. Comput. Modelling, to appear. 3. J. Yan, A. Zhao, L. Peng, Oscillation of impulsive delay differential equations and applications to population dynamics, ANZIAM J., (46) 4 (2005), 545– 554. 4. W. Zhang and M. Fan, Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays, Math. Comput. Modelling, 39 (2004), 479–493. 5. S. Tang and L. Chen, Global attractivity in a ”Food-Limited” Population Model with Impulsive Effect, J. Math. Anal. Appl., 292 (2004), 211–221. 6. X. Liu and G. Ballinger, Boundedness for impulsive delay differential equations and applications to population growth models, Nonlinear Anal., (53) 7-8 (2003), 1041–1062. 7. G. Ballinger and X. Liu, Practical Stability of Impulsive Delay Differential Equations and Applications to Control Problems, Optimization Methods and Applications, 3–21, Appl. Optim., 52, (Kluwer Acad. Publ., Dordrecht, 2001). 8. J.J. Nieto, Impulsive resonance periodic problems of first order, Appl. Math. Lett., 15 (2002), 489–493. 9. M. Akhmetov and R. Sejilova, The control of the boundary value problem for linear impulsive integro-differential systems, J. Math. Anal. Appl., (236) 2 (1999), 312–326. 10. J. Yan, A. Zhao and W. Yan, Existence and global attractivity of periodic solution for an impulsive delay differential equation with allee effect, J. Math. Anal. Appl., (309) 2 (2005), 489–504. 11. J.R. Yan, Existence and Global Attractivity of positive periodic solution for an impulsive Lasota-Wazewska Model, J. Math. Anal. Appl., 279 (2003), 111–120. 12. A. Halanay, Differential Equations: Stability, Oscillation, Time Lags, (Academic Press Inc., 1966). 13. O. Perron, Die Stabilitatsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703–728. 14. V.A. Tyshkevich, A perturbations-accumulation problem for linear differential equation with time lag, Differential Equations, 14 (1978), 177–186. 15. N.V. Azbelev, L.M. Berezanskii, P.M. Simonov and A.V. Chistyakov, Stability of linear systems with time lag, Differential Equations, 23 (1987), 493–500. 16. G. Ballinger and X. Liu, Existence and Uniqueness Results for Impulsive Delay Differential Equations, Dyn. Contin. Discrete Impuls. Syst., 5 (1999), 579–591.

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17. W. Li and H. Huo, Existence and global attractivity of positive periodic solutions of functional differential equations with impulses, Nonlinear Anal., 59 (2004), 857–877. 18. X. Li, X. Lin, D. Jiang and X. Zhang, Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effect, Nonlinear Anal., 62 (2004), 683–701. 19. K. Gopalsamy and B.G. Zhang, On the delay differential equations with impulses, J. Math. Anal. Appl., 139 (1989), 110–122. 20. E. Kreyszig, Introductory Functional Analysis with Applications, (Wiley, 1989).

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ON THE USE OF KORN’S TYPE INEQUALITIES IN THE EXISTENCE THEORY FOR COSSERAT ELASTIC SURFACES WITH VOIDS M. BˆIRSAN Faculty of Mathematics, “A.I. Cuza” University of Ia¸si, Romania E-mail: [email protected] We study the differential equations which govern the linear deformation of thin elastic shells made from a porous material. In our approach, we employ the theory of Cosserat elastic surfaces with voids. On the basis of inequalities of Korn’s type for Cosserat surfaces, we prove existence and uniqueness results for the solution of the boundary-value problems, in the equilibrium theory. For the dynamic equations, we investigate the boundary-initial-value problems for porous Cosserat shells using the semigroup of linear operators theory and we establish the existence, uniqueness and continuous dependence of solution. Keywords: Elastic shell; Cosserat surface; Material with voids; Existence; Uniqueness.

1. Introduction This paper is concerned with the linear theory of Cosserat elastic shells with voids. In this context, we employ the inequalities of Korn’s type to study the existence and uniqueness of solution to the equations which govern the deformation of shells. The inequalities of Korn are well-known in the three-dimensional theory of elasticity and their usefulness for the study of the field equations is well-established (see e.g. Refs. 1,2). In the classical linear shell theory, the inequalities of Korn’s type have been presented by Ciarlet in his treatise Ref. 3 (see also Ref. 4). In our approach of the problem, we consider thin elastic shells modeled as Cosserat surfaces. The foundations of the theory of Cosserat surfaces have been discussed in the monograph of Naghdi5 . The basic idea of the Cosserat theory for shells is to model thin bodies by means of a twodimensional continuum (i.e., a surface) endowed with a deformable vector (called director) assigned to each point of the surface. A modern approach

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of this theory, together with some applications, are presented in the book of Rubin Ref. 6. The inequalities of Korn’s type within the context of Cosserat theory of shells have been established in Ref. 7. In the present work, we deal with shells made from a porous elastic material. To describe the porosity of the material, we employ the NunziatoCowin theory for media with voids Refs. 8,9. This theory is based on the assumption that the bulk mass density ρ∗ of the porous elastic solid can be decomposed as the product of two fields: the matrix material mass density γ ∗ and the volume fraction field ν ∗ , such that ρ∗ = γ ∗ ν ∗ (0 < ν ∗ ≤ 1). In this way, an additional degree of kinematical freedom (namely, the volume fraction field) is introduced. The theory of porous elastic shells modeled as Cosserat surfaces with voids has been introduced in Ref. 10. In what follows, we investigate the equations for porous Cosserat shells established in Ref. 10. In Section 2, we recall the geometrical relations, constitutive assumptions and balance equations which govern the linear deformation of anisotropic and inhomogeneous shells. In Section 3, we restrict our attention to the equilibrium theory and show that the variational problems associated to our boundary-value problems can be treated using the inequalities of Korn’s type for Cosserat surfaces. In Section 4, we consider the dynamic theory and prove existence, uniqueness and continuous dependence results for the boundary-initial-value problems associated to the deformation of porous Cosserat shells. To this aim, we write the boundaryinitial-value problem in the form of an abstract Cauchy problem in a Hilbert space and we employ the theory of semigroup of linear operators. Once again, the Korn’s type inequality for Cosserat surfaces proves useful to establish the existence of solution. 2. Equations of the Linear Theory for Porous Cosserat Shells The theory of Cosserat elastic shells with voids has been established in Ref. 10. Following the approach of the Nunziato-Cowin theory for elastic media with voids, the porosity of the material is modeled using two volume fraction fields ν and χ, where ν is the average volume fraction through the thickness of the shell and χ is the average volume fraction gradient through the shell’s thickness (see Ref. 10). Thus, the deformation of a porous Cosserat shell is described by the functions ¡ ¢ ¡ ¢ r = r θ1 , θ2 , t , d = d θ1 , θ2 , t , (1) ¯ ν = ν(θ1 , θ2 , t), χ = χ(θ1 , θ2 , t), (θ1 , θ2 ) ∈ Σ, t∈T.

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Here, (θ1 , θ2 ) is a curvilinear material coordinate system on the reference surface, r and d are the position vector and the deformable director assigned to every point of the Cosserat surface, T = [0, ∞) is the time interval and Σ is a domain in R2 (i.e. an open, bounded and connected subset of R2 ) with a Lipschitz-continuous boundary Γ = ∂Σ. We employ the usual notation convention pertaining to summation over repeated indices. Also, the Greek indices and superscripts take the values {1, 2}, while the Latin indices and superscripts take the values {1, 2, 3}. Boldface letters denote tensor-valued or matrix-valued functions and their associated function spaces. A superposed dot stands for the material time derivative (holding θα fixed) and a comma preceding a subscript designates partial differentiation with respect to the corresponding coordinate. Let us denote by R, D, ν0 and χ0 the values of the functions (1) written for the reference configuration of the shell (ν0 and θ0 are assumed to be ¯ be the reference surface, which coincides with the constant). Let S = R(Σ) initial configuration. Thus, we have R = R(θα ) = r (θα , 0) ,

D = D(θα ) = d (θα , 0) ,

¯ (2) (θ1 , θ2 ) ∈ Σ.

¯ R is injective and Assume that the mappings R and D are of class C 2 (Σ), D is nowhere tangent to S. We introduce the notations Aα =

∂R , ∂θα

Aαβ = Aα · Aβ , Ai · Aj = δji ,

A3 =

A1 × A2 , |A1 × A2 |

Bαβ = A3 · Aβ,α , Aαβ = Aα · Aβ ,

A = det(Aαβ ),

(3)

Bαβ = Aβγ Bαγ ,

where Aα are the covariant base vectors (assumed to be linearly independent), A3 is the unit normal to the surface S, Aαβ and Bαβ are the first and second fundamental forms of S and δij is the Kronecker symbol. In the linear theory of porous Cosserat shells, we consider the infinitesimal displacement vectors u, δ and the infinitesimal changes in volume fraction fields ϕ, ψ given by u = r − R = ui Ai = ui Ai , δ = d − D = δi Ai = δ i Ai , ϕ = ν − ν0 , ψ = χ − χ0 .

(4)

For the case of shells with uniform thickness in the reference configuration (characterized by D = A3 ), the linear strain measures are ¢ 1¡ uα|β + uβ|α − Bαβ u3 , γα = δα + u3,α + Bαβ uβ , eαβ = 2 (5) γ3 = δ3 , ρβα = δβ|α − Bαγ uγ|β + Bαγ Bβγ u3 , ρ3α = δ3,α ,

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where a subscript vertical bar stands for covariant differentiation with respect to the metric tensor Aαβ . The contact force N , the contact director force M and the equilibrated stresses h, H (acting per unit length of curves c in S) satisfy the relations ¡ ¢ ¡ ¢ N = N αβ Aβ + V α A3 να , M = M αi Ai να , (6) h = hα να , H = H α να . Here, ν = να Aα is the unit (outward) normal to c tangent to the surface S. In this work, we consider anisotropic and inhomogeneous Cosserat elastic shells with voids. The constitutive equations are

N 0αβ g=

W = W (eαβ , γi , ρiα , ϕ, ψ, ϕ,γ , ψ,γ ) , µ ¶ 1 ∂W ∂W ∂W ∂W = + , Vi = , M αi = , 2 ∂eαβ ∂eβα ∂γi ∂ρiα ∂W , ∂ϕ

hα =

∂W , ∂ϕ,α

G=

∂W , ∂ψ

Hα =

∂W , ∂ψ,α

(7)

(8)

where the tensor N 0αβ is given by the expression N 0αβ = N αβ + Bγβ M γα ,

(9)

while g and G denote the internal equilibrated body forces. The function W represents the strain energy density per unit area of S and it is a quadratic form of its arguments. The constitutive coefficients are bounded measurable functions of (θµ ) which belong to H 1 (Σ). We assume that the quadratic form W is positive definite, i.e. there ¯ we have exists a constant c1 > 0 such that, for all (θµ ) ∈ Σ, W (eαβ , γi , ρiα , ϕ, ψ, ϕ,γ , ψ,γ ) ≥ ¡ ¢ (10) ≥ c1 eαβ eαβ + γi γi + ρiα ρiα + ϕ2 + ψ 2 + ϕ,γ ϕ,γ + ψ,γ ψ,γ . The equations of motion for Cosserat shells are N αβ |α − Bαβ V α + ρ0 f β = ρ0 u ¨β ,

V α |α + Bαβ N αβ + ρ0 f 3 = ρ0 u ¨3 ,

M αi |α − V i + ρ0 li = ρ0 α ¯ δ¨i ,

(11)

and the equations of equilibrated force can be written as hα |α − g + ρ0 p = ρ0 κ1 ϕ¨ ,

H α |α − G + ρ0 P = ρ0 κ2 ψ¨ .

(12)

Here, ρ0 is the mass density in the reference configuration, while α ¯ , κ1 and κ2 represent inertia coefficients, which are positive prescribed functions ¯ Also, f i and li designate the components of the assigned of class C 1 (Σ). force and assigned director force vectors, and p, P stand for the assigned equilibrated body forces.

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3. Boundary-value Problems of the Equilibrium Theory For the static theory, the functions defined previously do not depend on the time t. Thus, the equations of equilibrium are expressed by N αβ |α − Bαβ V α + ρ0 f β = 0,

V α |α + Bαβ N αβ + ρ0 f 3 = 0,

M αi |α − V i + ρ0 li = 0, h

α

|α

− g + ρ0 p = 0,

H

α

|α

(13)

− G + ρ0 P = 0.

Let Γ0 be a measurable subset of Γ = ∂Σ such that length(Γ0 )> 0, and let Γ1 = Γ − Γ0 . For the sake of simplicity, we consider the following homogeneous boundary conditions u = δ = 0, ϕ = ψ = 0 on

Γ0 ,

N = M = 0, h = H = 0 on

Γ1 .

(14)

Thus, the boundary-value problem consists in the equilibrium equations (13), the geometrical relations (5), the constitutive equations (8) and the boundary conditions (14). We denote by k · km the usual norm in the Sobolev space H m (Σ) and by | · |0 the usual norm in L2 (Σ). Let V (Σ) be the subspace of H 1 (Σ) given by © ª V (Σ) = y = (ui , δi , ϕ, ψ) ∈ H 1 (Σ); ui = δi = ϕ = ψ = 0 on Γ0 . Assume that f i , li , p, P ∈ L2 (Σ). We define the (symmetric) bilinear form B on V (Σ) × V (Σ) and the linear functional F on V (Σ) by Z £ 0αβ B(y, z) = N (z)eαβ (y) + V i (z)γi (y) + M αi (z)ρiα (y) Σ i√ + g(y)ϕˆ + hα (y)ϕˆ,α + G(y)ψˆ + H α (y)ψˆ,α A dθ1 dθ2 , (15) Z ¡ ¢√ F (y) = ρ0 f i ui + li δi + pϕ + P ψ A dθ1 dθ2 , Σ

ˆ ∈ V (Σ). for every y = (ui , δi , ϕ, ψ), z = (ˆ ui , δˆi , ϕ, ˆ ψ) Using the classical procedure, we call weak solution of the boundaryˆ ∈ V (Σ) such that value problem a function z = (ˆ ui , δˆi , ϕ, ˆ ψ) B(y, z) = F (y),

∀ y = (ui , δi , ϕ, ψ) ∈ V (Σ).

(16)

We observe that all the hypotheses of the Lax-Milgram lemma are satisfied for the variational problem (16), provided we prove that B is V (Σ)elliptic, i.e. there exists a constant c0 > 0 such that B(y, y) ≥ c0 kyk21 ,

∀ y ∈ V (Σ).

(17)

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By the definition (15), we deduce that Z √ B(y, y) = 2 W (eαβ (y), γi (y), ρiα (y), ϕ, ψ, ϕ,γ , ψ,γ ) A dθ1 dθ2 , (18) Σ

for any y = (ui , δi , ϕ, ψ) ∈ V (Σ). Using (18) and the inequality (10), we derive that there exists a constant c2 > 0 such that Z B(y, y) ≥ c2 [eαβ (y)eαβ (y) + γi (y)γi (y) + ρiα (y)ρiα (y) (19) Σ ¤ 1 2 2 2 + ϕ + ψ + ϕ,γ ϕ,γ + ψ,γ ψ,γ dθ dθ , ∀ y ∈ V (Σ). In order to show the validity of the inequality (17), we need to prove the following result, which establishes an inequality of Korn’s type “without boundary conditions”. ¯ Theorem 3.1. Assume that the injective mapping R is of class C 3 (Σ). 1 For any v = (ui , δi ) ∈ H (Σ), let eαβ (v), γi (v) and ρiα (v) be the linear strain measures. Then, there exists a constant c3 = c3 (Σ, R) > 0 such that Z [ui ui + δi δi + eαβ (v)eαβ (v)+γi (v)γi (v)+ρiα (v)ρiα (v)] dθ1 dθ2 ≥ (20) Σ ∀ v = (ui , δi ) ∈ H 1 (Σ). ≥ c3 kvk21 , The proof of Theorem 3.1, which is presented in Ref. 7, follows the same lines as the proof of Korn’s inequality in three-dimensional elasticity (see e.g., Ref. 2). By virtue of Theorem 3.1 and the infinitesimal rigid displacement lemma for Cosserat surfaces (see Ref. 7), we obtain the inequality of Korn’s type “with boundary conditions”, in the next theorem. Theorem 3.2. Assume that the hypotheses of Theorem 3.1 are satisfied. Then, there exists a constant c4 = c4 (Σ, Γ0 , R) > 0 such that Z [eαβ (v)eαβ (v) + γi (v)γi (y) + ρiα (v)ρiα (v)] dθ1 dθ2 ≥ c4 kvk21 , (21) Σ

for all v = (ui , δi ) ∈ H 1 (Σ) with ui = δi = 0 on Γ0 . Theorem 1.2 is proved in Ref. 7, using the same arguments as in the case of the Korn’s type inequality on a general surface (see Refs. 3,4). We are now in a position to establish the inequality (17). Indeed, if we substitute the relation (21) into the inequality (19), then we obtain that (17) holds. Hence, the bilinear form B is V (Σ)-elliptic and we can apply the Lax-Milgram lemma to deduce the following result.

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Theorem 3.3. Assume that the hypotheses of Theorem 3.1 are satisfied. Then, there exists a unique solution z ∈ V (Σ) to the variational problem (16). The solution z can be characterized as the minimizer on the space V (Σ) of the following functional Z √ J(y) = W(eαβ (y), γi (y), ρiα (y), ϕ, ψ, ϕ,γ , ψ,γ ) A dθ1 dθ2 ZΣ ¡ ¢√ − ρ0 f i ui + li δi + pϕ + P ψ A dθ1 dθ2 , y = (ui , δi , ϕ, ψ), Σ

where W is the strain energy density of the Cosserat shell and f i , li , p and P are the assigned body forces. This theorem states the existence and uniqueness of weak solution to the boundary-value problem for the static deformation of porous shells. 4. Boundary-initial-value Problems in the Dynamic Theory For the dynamic problems, we shall employ the theory of semigroup of linear operators to establish existence, uniqueness and continuous dependence results for Cosserat shells with voids. Although more general types of boundary conditions could be considered, we restrict our attention to the following boundary conditions u = δ = 0,

ϕ=ψ=0

on

Γ×T.

(22)

The initial conditions are given by u(θα , 0) = u0 (θα ), ˙ α , 0) = w0 (θα ), δ(θ ψ(θα , 0) = ψ 0 (θα ),

˙ α , 0) = v 0 (θα ), u(θ ϕ(θα , 0) = ϕ0 (θα ), ˙ α , 0) = µ0 (θα ), ψ(θ

δ(θα , 0) = δ 0 (θα ), ϕ(θ ˙ α , 0) = λ0 (θα ), on Σ.

(23)

where u0 , v 0 , δ 0 , w0 , ϕ0 , λ0 , ψ 0 and µ0 are prescribed functions. We consider the boundary-initial-value problem consisting in the equations of motion (11), the equations of equilibrated force (12), the geometrical relations (5), the constitutive equations (8), the boundary conditions © (22) and the initial conditions (23). Let Z be the Banach space ª Z = U = (ui , vi , δi , wi , ϕ, λ, ψ, µ); ui , δi , ϕ, ψ ∈ H01 (Σ), vi , wi , λ, µ ∈ L2 (Σ) endowed with the usual (product) norm k · kH given by kU k2H =

3 X ¡

¢ kui k21 + |vi |20 + kδi k21 + |wi |20 +kϕk21 +|λ|20 +kψk21 +|µ|20 . (24)

i=1

Let us define on Z the scalar product

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hU , V iZ =

1 2

Z

£ ¤√ 1 ˆ + κ2 µˆ ρ0 v i vˆi + α ¯ wi w ˆi + κ1 λλ µ A dθ1 dθ2 + B(x, z), 2 Σ

ˆ ψ, ˆ µ where U = (ui , vi , δi , wi , ϕ, λ, ψ, µ), V = (ˆ ui , vˆi , δˆi , w ˆi , ϕ, ˆ λ, ˆ) and x = ˆ ˆ (ui , δi , ϕ, ψ), z = (ˆ ui , δi , ϕ, ˆ ψ). By virtue of the inequalities of Korn’s type for Cosserat surfaces (see Section 3), the norm k · kZ induced by the scalar product is equivalent to the norm k · kH on Z, and hence (Z, h· , ·iZ ) is a Hilbert space. We define the linear operator A : D(A) ⊂ Z → Z by AU = (vi , Bi U , wi , Ci U , λ, DU , µ, EU ) ,

(25)

for every U = (ui , vi , δi , wi , ϕ, λ, ψ, µ) ∈ D(A), where ³ ´ ¢ 1 1 ¡ α Bα U = Aαγ N βγ |β −Bβγ V β (x), B3 U = V |α +Bαβ N αβ (x), ρ0 ρ0 ¡ βγ ¢ ¢ 1 ¡ α3 1 Aαγ M |β − V γ (x), C3 U = M |α − V 3 (x), (26) Cα U = ρ0 α ¯ ρ0 α ¯ ¢ ¢ 1 ¡ α 1 ¡ α DU = h |α − g (x), EU = H |α − G (x) ρ0 κ1 ρ0 κ2 and x = (ui , δi , ϕ, ψ). The domain D(A) of the operator A is defined by D(A) = {U ∈ Z;

AU ∈ Z} .

(27)

Then, the boundary-initial-value problem for Cosserat shells can be written in the form of the following abstract Cauchy problem in the Hilbert space Z: dU = AU (t) + F (t), dt where

U (0) = U 0 ,

³ fi li p P ´ F = 0, 0, 0, , 0, 0, 0, , 0, , 0, , ρ0 ρ0 α ¯ ρ0 κ1 ρ0 κ2 ¡ ¢ U 0 = u0i , vi0 , δi0 , wi0 , ϕ0 , λ0 , ψ 0 , µ0 .

(28)

(29)

Some properties of the operator A are stated by the following theorem. Theorem 4.1. Let A : D(A) ⊂ Z → Z be the operator defined by the relations (25)–(27). Then: (i) D(A) is dense in Z; (ii) The operator A satisfies the equality hAU , U iZ = 0,

∀ U ∈ D(A);

(30)

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(iii) A verifies the condition Range(I − A) = Z ,

(31)

where I is the identity operator; (iv) The operator A is the infinitesimal generator of a semigroup of contractions in the Hilbert space Z. Proof. (i) We notice that © ª U ∈ Z; ui , δi , ϕ, ψ ∈ H01 (Σ) ∩ H 2 (Σ), vi , wi , λ, µ ∈ H01 (Σ) ⊂ D(A) and hence, D(A) is dense in Z. (ii) In view of the geometrical relations (5) and the equation (9), we obtain by a straightforward calculation that Z h ¡ ¢ 2hAU , U iZ = vα N βα |β +v3 V α |α +wi M αi |α +λhα |α +µH α |α Σ (32) ¡ ¢ i√ βα + vα|β N + v3|α V α + wi|α M αi + λ|α hα + µ|α H α A dθ1 dθ2 , for any U = (ui , vi , δi , wi , ϕ, λ, ψ, µ) ∈ D(A). Taking into account the boundary conditions (22), from (32) we deduce the relation (30). (iii) In order to prove that the range condition (31) holds true, we employ the same arguments as in the proof of Lemma 3 from Ref. 7. (iv) By virtue of (i)–(iii), we can apply the Lumer-Phillips theorem (see e.g. Ref. 11, page 14) to obtain the statement of the theorem. ¤ The main result asserts the existence and uniqueness of solution. Theorem 4.2. Let t0 ∈ T and assume that F (t) ∈ C 1 ([0, t0 ], L2 (Σ)) and U 0 ∈ D(A). Then, there exists a unique solution U (t) ∈ C 1 ([0, t0 ], Z) ∩ C 0 ([0, t0 ], D(A)) to the problem (28). The solution U (t) satisfies Z t kU (t)kZ ≤ kU 0 kZ + kF (s)kZ ds, t ∈ [0, t0 ]. (33) 0

Proof. Let {S(t); t ≥ 0} be the semigroup of contractions generated by the operator A. In view of Theorem 2.2, we can apply the general results of the semigroup of linear operators theory (see e.g. Ref. 12, Sect. 8.1) to deduce the existence of a unique solution U (t) to the problem (28), given by Z t U (t) = S(t)U 0 + S(t − s)F (s) ds, t ∈ [0, t0 ]. 0

Hence, we derive the inequality (33).

¤

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We mention that the estimate (33) expresses the continuous dependence of the solution upon the initial data {u0 , v 0 , δ 0 , w0 , ϕ0 , λ0 , ψ 0 , µ0 } and the external body loads {f i , li , p, P }.

References 1. G. Duvaut & J.L. Lions, Inequalities in Mechanics and Physics, (SpringerVerlag, Berlin, 1976). 2. P.G. Ciarlet, Mathematical Elasticity. Vol. I: Three-dimensional Elasticity, (North-Holland, Amsterdam, 1988). 3. P.G. Ciarlet, Mathematical Elasticity. Vol. III: Theory of Shells, (NorthHolland, Amsterdam, 2000). 4. P.G. Ciarlet, Introduction to Linear Shell Theory, Ser. Applied Mathematics no. 1, (Gauthier-Villars, 1998). 5. P.M. Naghdi, The Theory of Shells and Plates, Handbuch der Physik, Vol. VI a/2, (Springer-Verlag, Berlin Heidelberg New York, 1972), 425–640. 6. M.B. Rubin, Cosserat Theories: Shells, Rods, and Points, (Kluwer Academic Publishers, Dordrecht, 2000). 7. M. Bˆırsan, Inequalities of Korn’s type and existence results in the theory of Cosserat elastic shells, submitted. 8. J.W. Nunziato & S.C. Cowin, A nonlinear theory of elastic materials with voids, Arch. Rational Mech. Anal. 72 (1979), 175–201. 9. S.C. Cowin & J.W. Nunziato, Linear elastic materials with voids, J. Elasticity 13 (1983), 125–147. 10. M. Bˆırsan, On the theory of elastic shells made from a material with voids, Int. J. Solids Struct. 43 (2006), 3106–3123. 11. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Ser. Applied Mathematical Sciences no. 44, (SpringerVerlag, New York, 1983). 12. I.I. Vrabie, C0 -Semigroups and Applications, Ser. Mathematics Studies no. 191, (North-Holland, Elsevier, Amsterdam, 2003).

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TIME PERIODIC VISCOSITY SOLUTIONS OF HAMILTON-JACOBI EQUATIONS MIHAI BOSTAN and GAWTUM NAMAH Laboratoire de Math´ ematiques de Besancon, UMR CNRS 6623, Universit´ e de Franche-Comt´ e, 16 route de Gray, 25030 Besancon Cedex, France, Email: [email protected], Email: [email protected] We analyze the existence and uniqueness of time periodic viscosity solutions. We prove that the existence for time periodic problems can be reduced to the existence for stationary problems, obtained by averaging the source term. We investigate also the asymptotic behaviour of time periodic solutions for large frequencies. Keywords: Hamilton-Jacobi equations; Time periodic viscosity solutions.

1. Introduction We are mainly interested in time periodic solutions of first order HamiltonJacobi equations of the form ∂t u + H(x, u, Du) = f (t), (x, t) ∈ IRN × IR,

(1)

where the hamiltonian H and f are continuous functions, f being T periodic in t. We perform our study in the framework of viscosity solutions introduced by Crandall and Lions, cf. Refs. 1–4. The usual hypotheses on the hamiltonian H are ∀ R > 0, ∃γR > 0 : H(x, u, p) − H(x, v, p) ≥ γR (u − v),

(2)

for all x ∈ IRN , −R ≤ v ≤ u ≤ R, p ∈ IRN ; ∀R > 0, ∃mR : |H(x, u, p) − H(y, u, p)| ≤ mR ( |x − y| · (1 + |p|) ), N

(3)

N

for all x, y ∈ IR , −R ≤ u ≤ R, p ∈ IR , where lim mR (z) = 0 ; z→0

∀R > 0, lim H(x, u, p) = ∞, uniformly for (x, u) ∈ IRN × [−R, R] ; (4) |p|→∞

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∀ R > 0, H is uniformly continuous on IRN × [−R, R] × B R ;

(5)

∃M > 0 : H(x, −M, 0) ≤ f (t) ≤ H(x, M, 0), ∀ x ∈ IRN , t ∈ IR.

(6)

The hypotheses (2) (with γR ∈ IR, ∀R > 0), (3), (5), (6) ensure existence and uniqueness results for the Cauchy problem ( ∂t u + H(x, u, Du) = f (t), (x, t) ∈ IRN ×]0, +∞[, (7) u(x, 0) = u0 (x), x ∈ IRN . The coercivity condition (4) guarantees the Lipschitz regularity of the solution. The condition (2) is crucial for the uniqueness result. A much more difficult situation is those of hamiltonians which are just nondecreasing with respect to u H(x, u, p) − H(x, v, p) ≥ 0, ∀x ∈ IRN , v ≤ u, p ∈ IRN .

(8)

The key point is to consider also the stationary averaged problem Z 1 T H(x, U, DU ) = hf i := f (t) dt, x ∈ IRN , T 0

(9)

and to observe that (1) has time periodic viscosity solutions iff (9) has viscosity solutions. Indeed, let us consider the ode x0 (t) + g(x(t)) = f (t), t ∈ IR,

(10)

where f, g : IR → IR are continuous functions. We have the following result (see Ref. 5) Proposition 1.1. Assume that f : IR → IR is continuous, T periodic and g : IR → IR is continuous, nondecreasing. Then (10) admits a T periodic (classical) solution iff there exists x0 which solves g(x) = hf i. A similar result holds for evolution equations. We can prove, cf. Ref. 5 Proposition 1.2. Assume that A : D(A) ⊂ H → H is a linear, maximal monotone and symmetric operator on a Hilbert space H and f : IR → H is a T periodic function. Then there is a periodic solution of x0 (t) + Ax(t) = f (t), t ∈ IR, iff hf i ∈ Range (A).

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Motivated by the above propositions, we expect that a similar result holds for Hamilton-Jacobi equations Theorem 1.1. Let H = H(x, z, p) be a hamiltonian verifying (8), (4), (5) and sup{|H(x, 0, 0)| : x ∈ IRN } < +∞, f ∈ C(IR) be a continuous time periodic function. Then there is a bounded Lipschitz time periodic viscosity solution of (1) iff there is a bounded continuous viscosity solution of (9). Another interesting problem is the asymptotic behaviour of time periodic viscosity solutions for high frequencies. For example, in the case of (10), let us introduce fn (t) = f (nt), ∀t ∈ IR, n ≥ 1, which is Tn periodic and has the same average as f . Suppose that hf i ∈ g(IR) and let xn be a Tn periodic solution of x0n (t) + g(xn (t)) = fn (t), t ∈ IR, such that supn kxn k∞ < +∞. We inquire about the convergence of (xn )n . Consider the change of variable yn (t) = xn ( nt ). The functions yn are T periodic and solves n · yn0 (t) + g(yn (t)) = f (t), t ∈ IR, n ≥ 1. can guess that RWe R T (yn )n converges uniformly to a constant y0 and since T g(y (t)) dt = f (t) dt, ∀ n ≥ 1 we deduce that g(y0 ) = hf i. We obtain n 0 0 that (xn )n converges towards a solution of g(x) = hf i. The same result holds in the setting of (1), (9) and we show that the rate of convergence is in O(1/n). 2. Preliminaries Let us recall the standard comparison result for semi continuous viscosity sub/supersolutions (see Ref. 6). Proposition 2.1. Let u and v be bounded u.s.c. subsolution of ∂t u+H(x, u, Du) = f (x, t) in IRN ×]0, T [, resp. l.s.c. supersolution of ∂t v + H(x, v, Dv) = g(x, t) in IRN ×]0, T [ where f, g ∈ BU C(IRN × [0, T ]). We assume that (2), (3), (5) hold (with γR ∈ IR not necessarily positive), lim (u(x, t)−u(x, 0))+ = lim (v(x, t)−v(x, 0))− = 0,

t&0

t&0

uniformly for x∈IRN ,

(12)

(here (·)± denotes the positive/negative part a± = max(±a, 0), ∀ a ∈ IR) and u(·, 0) ∈ BU C(IRN ) or v(·, 0) ∈ BU C(IRN ).

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Then we have for all t ∈ [0, T ] eγt k(u(·, t) − v(·, t))+ kL∞ (IRN ) ≤ k(u(·, 0) − v(·, 0))+ kL∞ (IRN ) (14) Z t + eγs k(f (·, s) − g(·, s))+ kL∞ (IRN ) ds, 0

³ where γ = γR0 , R0 = max

sup

u, −

´ v . Moreover, the hypothe-

inf

IRN ×[0,T ] N

IRN ×[0,T ] 1,∞

(IR ×]0, T [) or v ∈ W 1,∞ (IRN ×]0, T [).

sis (3) can be replaced by u ∈ W

Actually the above version is not enough for our purposes, but we can establish the following slightly improved comparison result Proposition 2.2. Under the hypotheses of Proposition 2.1 (with γR ≥ 0, ∀ R > 0) we have for all t ∈ [0, T ] sup (u(x, t) − v(x, t)) ≤ e−γt sup (u(x, 0) − v(x, 0))+

x∈IRN

x∈IRN Z t

+ sup 0≤s≤t

sup (f (x, σ) − g(x, σ)) dσ,

³ where γ = γR0 , R0 = max

sup IRN ×[0,T ]

(15)

s x∈IRN

u, −

inf

IRN ×[0,T ]

´ v . If the hypotheses (12),

(13) are not verified, we have Z sup (u(x, t) − v(x, t)) ≤ 2R0 · e−γt + sup

sup (f (x, σ) − g(x, σ)) dσ.

0≤s≤t

x∈IRN

t s x∈IRN

Moreover, the hypothesis (3) can be replaced by u ∈ W 1,∞ (IRN ×]0, T [) or v ∈ W 1,∞ (IRN ×]0, T [). Remark 2.1. The main difference between (14) and (15) is that in the right hand side of (15) we have now sup(f (·, σ) − g(·, σ)) and not sup(f (·, σ) − g(·, σ))+ .

IRN

IRN

Proof. We give here a formal proof. We assume that for all t ∈ [0, T ] there is z(t) ∈ IRN such that M (t) := sup (u(x, t) − v(x, t)) = u(z(t), t) − x∈IRN

v(z(t), t). Moreover we suppose that the functions u, v, z are smooth. Since Du(z(t), t) = Dv(z(t), t) =: p(t), ∀t ∈ [0, T ] we have d (u − v)(z(t), t) = ∂t u(z(t), t) − ∂t v(z(t), t). dt

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By using the viscosity equations we have for all t ∈ [0, T ] d {(u−v)(z(t), t)}+H(z(t), u(z(t), t), p(t))−H(z(t), v(z(t), t), p(t)) ≤ h(t), dt where h(t) = (f − g)(z(t), t), ∀t ∈ [0, T ]. Since H(x, z, p) − γ · z is nondecreasing with respect to z ∈ [−R0 , R0 ] we have for some r(t) ≥ 0 H(z(t), u(z(t), t), p(t)) − H(z(t), v(z(t), t), p(t)) = (γ + r(t))(u − v)(z(t), t). Finally one gets dM + (γ + r(t)) · M (t) ≤ h(t), t ∈ [0, T ]. dt Assume that r(·) is integrable and denote by A(t) the function A(t) = Rt {γ + r(σ)} dσ, t ∈ [0, T ]. After integration one gets 0 µ ¶ Z t −A(t) A(s) M (t) ≤ e M (0) + e · h(s) ds , t ∈ [0, T ]. 0

Rt We introduce also the primitive F (s) = − s h(σ) dσ, s, t ∈ [0, T ]. By integration by parts we can write Z t Z t eA(s) h(s) ds = eA(s) F 0 (s) ds 0 0 Z t Z t Z t = h(σ) dσ + eA(s) A0 (s) h(σ) dσ ds 0 0 s Z t Z t ³ ´ ≤ h(σ) dσ + eA(t) − 1 sup h(σ)dσ. 0≤s≤t

0

s

Finally we deduce that for all t ∈ [0, T ] we have sup (u(x, t) − v(x, t)) = M (t) ≤ e−γt sup (u(x, 0) − v(x, 0))+

x∈IRN

x∈IRN Z t

+ sup 0≤s≤t

sup (f (x, σ) − g(x, σ)) dσ. s x∈IRN

Corollary 2.1. Let u be a bounded time periodic viscosity u.s.c subsolution of ∂t u + H(x, u, Du) = f (x, t) in IRN × IR and v a bounded time periodic viscosity l.s.c. supersolution of ∂t v + H(x, v, Dv) = g(x, t) in IRN × IR, where f, g ∈ BU C(IRN × IR) and H are T periodic such that (2), (3), (5) hold (with γR > 0, ∀ R > 0). Then we have Z t sup (u(x, t) − v(x, t)) ≤ sup sup (f (x, σ) − g(x, σ)) dσ. x∈IRN

s≤t

s x∈IRN

Moreover, the hypothesis (3) can be replaced by u ∈ W 1,∞ (IRN × IR) or v ∈ W 1,∞ (IRN × IR).

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Proof. By using the time periodicity and Proposition 2.2 we have sup (u(x, t) − v(x, t)) = sup (u(x, t + nT ) − v(x, t + nT )) x∈IRN

x∈IRN

≤ e−γ(t+nT ) (kukL∞ + kvkL∞ ) (16) Z t+nT + sup sup (f (x, σ) − g(x, σ)) dσ. 0≤s≤t+nT

Observe that Z Z t+nT sup (f (x, σ) − g(x, σ)) dσ = s

x∈IRN

x∈IRN

s

t+nT

sup (f (x, σ − nT ) − g(x, σ − nT )) dσ

s x∈IRN Z t

=

sup (f (x, σ) − g(x, σ)) dσ s−nT x∈IRN Z t

≤ sup r≤t

sup (f (x, σ) − g(x, σ)) dσ. r x∈IRN

The conclusion follows by letting n → +∞ in (16). 3. Time Periodic Viscosity Solution when γR > 0 In this section we assume that the hypothesis (2) holds. In this case we obtain immediately the following comparison result for time periodic viscosity sub/supersolutions. Proposition 3.1. Let u, v ∈ BU C(IRN × IR) be bounded time periodic viscosity subsolution, resp. supersolution of (1) where H ∈ C(IRN ×IR×IRN ) is time periodic. We assume that (2), (3), (5) hold (with γR > 0, ∀ R > 0). Then we have u(x, t) ≤ v(x, t), ∀(x, t) ∈ IRN × IR. Moreover, the hypothesis (3) can be replaced by u ∈ W 1,∞ (IRN × IR) or v ∈ W 1,∞ (IRN × IR). Proof. By Proposition 2.1 we have eγT k(u(·, T ) − v(·, T ))+ kL∞ (IRN ) ≤ k(u(·, 0) − v(·, 0))+ kL∞ (IRN ) , where γ = γR0 > 0, R0 = max(kukL∞ (IRN ×IR) , kvkL∞ (IRN ×IR) ). By periodicity we have u(·, T ) − v(·, T ) = u(·, 0) − v(·, 0) and since eγT > 1 we deduce that (u(x, 0)−v(x, 0))+ ≤0, ∀x∈IRN. The conclusion follows by Proposition 2.1.

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Corollary 3.1. Under the assumptions of Proposition 3.1 there is at most one time periodic viscosity solution of (1). We investigate now the existence of time periodic viscosity solution. Proposition 3.2. Let u0 ∈ BU C(IRN ) and assume that (2), (3), (5), (6) hold (with γR > 0, ∀ R > 0). Denote by u ∈ C(IRN × [0, +∞[) the unique viscosity solution of (7) and let un (x, t) = u(x, t + nT ), (x, t) ∈ IRN × [0, T ] for n ≥ 0. Then (un )n converges uniformly on IRN × [0, T ] towards a time periodic viscosity solution of (1). Proof. By hypothesis (6) we deduce that −M (resp. M ) is subsolution (resp. supersolution ) of (7) and by Proposition 2.1 we have −M −k(−u0 −M )+ k∞ ≤u(x, t)≤k(u0 −M )+ k∞ +M, (x, t) ∈ IRN ×[0, +∞[. Let γ=γR0 > 0 where R0 =kukL∞ (IRN ×]0,+∞[) . Consider v(x, t)=u(x, t+T ), ∀(x, t) ∈ IRN ×[0, +∞[. By the periodicity of H we deduce that v is viscosity solution of ∂t v + H(x, v, Dv) = f (t), (x, t) ∈ IRN × [0, +∞[. By using Proposition 2.1 we have ku(·, t + T ) − u(·, t)kL∞ (IRN ) = kv(·, t) − u(·, t)kL∞ (IRN ) ≤ e−γt kv(·, 0) − u(·, 0)kL∞ (IRN ) ≤ 2e−γt kukL∞ (IRN ×]0,+∞[) . In particular, by taking t = s + nT , s ∈ [0, T ] we deduce that kun+1 (·, s) − un (·, s)kL∞ (IRN ) ≤ 2e−nT γ kukL∞ (IRN ×]0,+∞[) , and therefore there is w ∈ C(IRN × IR), T periodic such that un → w|IRN ×[0,T ] uniformly on IRN × [0, T ]. By using the stability result (see Ref. 2,6) we deduce that w is viscosity solution of (1). 4. Time Periodic Viscosity Solution when γR = 0 In this section we investigate the more general case of hamiltonians H satisfying (8). In the case of coercif hamiltonians i.e., hamiltonians verifying (4), Theorem 1.1 gives a necessary and sufficient condition for the existence of time periodic viscosity solution. Proof. (of Theorem 1.1) Assume that there is a bounded viscosity solution V of (9). Since the hamiltonian satisfies (4) we can prove as usual that V is a Lipschitz function. For any α > 0 take 1 Mα = kV kL∞ (IRN ) + (C + kf kL∞ (IR) ) α

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and observe that for any (x, t) ∈ IRN × IR we have α(−Mα − V (x)) + H(x, −Mα , 0) ≤ f (t) ≤ α(Mα − V (x)) + H(x, Mα , 0). Therefore we can construct the stationary viscosity solution Vα of α(Vα − V (x)) + H(x, Vα , DVα ) = hf i, x ∈ IRN , and the time periodic viscosity solution vα of α(vα − V (x)) + ∂t vα + H(x, vα , Dvα ) = f (t), (x, t) ∈ IRN × IR.

(17)

In fact we have Vα = V , ∀α > 0 and by using Corollary 2.1 we obtain |vα (x, t)−V (x)| = |vα (x, t)−Vα (x)|≤kf −hf ikL1 (0,T ) , ∀(x, t) ∈ IRN ×IR, (18) which implies that (vα )α is uniformly bounded kvα kL∞ (IRN ×IR) ≤ kV kL∞ (IRN ) + kf − hf ikL1 (0,T ) , ∀α > 0. In order to extract a subsequence which converges uniformly on compact sets we prove that (vα )α are uniformly Lipschitz on IRN × IR. For this note that wα (x, t) = vα (x, t + h), (x, t) ∈ IRN × IR is time periodic viscosity solution of α·(wα −V (x))+∂t wα +H(x, wα , Dwα ) = f (t+h), (x, t) ∈ IRN ×IR. (19) By using Corollary 2.1 we have Z t vα (x, t + h) − vα (x, t) ≤ sup (f (σ + h) − f (σ)) dσ s≤t

s

(Z

= sup s≤t

Z

t+h

s+h

f (σ) dσ − t

(20) ) f (σ) dσ

s

≤ 2|h| · kf kL∞ (IR) , ∀ (x, t) ∈ IRN × IR, h ∈ IR. Since the hamiltonian satisfies (4) we can prove as usual that vα are uniformly Lipschitz with respect to x. Therefore the functions (vα )α>0 are uniformly Lipschitz and we can extract a subsequence which converges uniformly on compact sets of IRN × IR to a bounded Lipschitz function v. By using the stability result for continuous viscosity solutions we deduce that v is a viscosity solution of (1). The converse implication follows in a similar way. Corollary 4.1. Let H = H(x, z, p) be a hamiltonian verifying (8), (4), (5) and f be a continuous time periodic function. Assume that there is M > 0 such that H(x, −M, 0) ≤ hf i ≤ H(x, M, 0), ∀x ∈ IRN .

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Then there is a bounded Lipschitz time periodic viscosity solution v of (1) satisfying −M −kf −hf ikL1 (0,T ) ≤ v(x, t) ≤ M +kf −hf ikL1 (0,T ) , ∀(x, t) ∈ IRN ×IR. 5. Asymptotic Behaviour for Large Frequencies Consider f : IR → IR a T periodic continuous function and denote by fn the Tn periodic functions given by fn (t) = f (nt), ∀t ∈ IR. Assume that the hamiltonian H satisfies (2), (3), (5), (6). Under the above hypotheses, for any n ≥ 1, there is a unique Tn periodic viscosity solution un of ∂t un + H(x, un , Dun ) = fn (t), (x, t) ∈ IRN × IR.

(22)

A natural question arising in this context concerns the convergence of the sequence (un )n . This convergence can be justified at least formally by introducing the fast variable s = nt and by using the asymptotic expansion 1 un (x, t) = u0 (x) + u1 (x, nt) + ... (23) n Plugging the ansatz (23) into (22) we obtain 1 1 ∂s u1 (x, s)+...+H(x, u0 (x)+ u1 (x, s)+..., Du0 (x)+ Du1 (x, s)+...) = f (s). n n Since un is Tn periodic with respect to t, we are looking for a T periodic function u1 (x, s) with respect to s. By integration over [0, T ] one gets Z 1 1 1 T H(x, u0 (x)+ u1 (x, s)+..., Du0 (x)+ Du1 (x, s)+...)ds=hf i, x ∈ IRN. T 0 n n After passing to the limit for n → +∞ we deduce formally that limn→+∞ un = u0 , where u0 solves the cell problem (see Ref. 7) H(x, u0 , Du0 ) = hf i, x ∈ IRN . Theorem 5.1. Let H = H(x, z, p) be a hamiltonian satisfying (2), (3), (5), (6), where f is a T periodic continuous function. Denote by U , un the stationary, resp. time periodic viscosity solution of (9), resp. (22). Then the sequence (un )n converges uniformly on IRN × IR towards U and we have kun − U kL∞ (IRN ×IR) ≤

1 kf − hf ikL1 (0,T ) , ∀n ≥ 1. n

Proof. We introduce also wn (x, t) = un (x, nt ), (x, t) ∈ IRN × IR, which is T periodic. We deduce that wn satisfies in the viscosity sense n ∂t wn + H(x, wn , Dwn ) = f (t), (x, t) ∈ IRN × IR,

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which we can rewrite as 1 1 ∂t wn + H(x, wn , Dwn ) = f (t), (x, t) ∈ IRN × IR. n n Recall also that we have in the viscosity sense

(24)

1 1 H(x, U, DU ) = hf i, x ∈ IRN . (25) n n By using Corollary 2.1 we deduce that Z 1 t 1 wn (x, t) − U (x) ≤ sup (f (σ) − hf i) dσ ≤ kf − hf ikL1 (0,T ) , n s≤t n s and similarly U (x) − wn (x, t) ≤ n1 kf − hf ikL1 (0,T ) , ∀n ≥ 1. We obtain for all n ≥ 1 ¯ µ ¯ ¶ ¯ ¯ ¯un x, t − U (x)¯ ≤ 1 kf − hf ikL1 (0,T ) . ¯ ¯ n n Finally we deduce that kun − U kL∞ (IRN ×IR) ≤ n ≥ 1.

1 n kf

− hf ikL1 (0,T ) for all

References 1. M.G. Crandall and P.-L. Lions, Condition d’unicit´e pour les solutions g´en´eralis´ees des ´equations de Hamilton-Jacobi du premier ordre, C.R. Acad. Sci. Paris, S´er. I Math. 292 (1981), 183–186. 2. M.G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277(1983), 1–42. 3. M.G. Crandall, L.C. Evans and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487–502. 4. P.-L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, (Research Notes in Mathematics, Pitman, 1982). 5. M. Bostan, Periodic solutions for evolution equations, (Electronic J. Differential Equations, Monograph 3 2002), 41 pp. 6. G. Barles, Solutions de Viscosit´e des Equations de Hamilton-Jacobi, (SpringerVerlag, 1994). 7. P.-L. Lions, G. Papanicolaou and S.R.S. Varadhan, Homogeneization of Hamilton-Jacobi equations, preprint.

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A VIABILITY RESULT FOR SEMILINEAR REACTION-DIFFUSION SYSTEMS ˘ and DANIELA ROS MONICA BURLICA ¸ U∗ Department of Mathematics, ”Gh.Asachi” University Ia¸si, Romania E-mail:[email protected] [email protected] We consider a reaction-diffusion system of the form  0  u (t) = Au(t) + F (u(t), v(t)), t ≥ 0 v 0 (t) = Bv(t) + G(u(t), v(t)), t ≥ 0  u(0) = ξ, v(0) = η, where X and Y are real Banach spaces, K is a nonempty and locally closed subset in X × Y, A : D(A) ⊆ X → X, B : D(B) ⊆ Y → Y are the generators of two C0 -semigroups, {SA (t) : X → X; t ≥ 0} and {SB (t) : Y → Y ; t ≥ 0} respectively, F : K → X, G : K → Y, are continuous such that A + F and B + G are of compact type. We prove a necessary and sufficient condition in order that for each (ξ, η) ∈ K, problem above has at least one mild solution (u, v) : [ 0, T ] → K. Keywords: C0 -semigroup; Compact semigroup; Reaction-diffusion system; Viable set; Tangency condition; β-compact function.

1. Introduction The goal of this paper is to present a viability result referring to a class of semilinear reaction-diffusion system of the form:  0  u (t) = Au(t) + F (u(t), v(t)), t ≥ 0 (1) v 0 (t) = Bv(t) + G(u(t), v(t)), t ≥ 0  u(0) = ξ, v(0) = η where (X, k · kX ) and (Y, k · kY ) are real Banach spaces, K ⊆ X × Y, F : K → X, G : K → Y, and ξ ∈ X, η ∈ Y. ∗ This

work was supported by CNCSIS Grant A 1159/2006

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We suppose that A : D(A) ⊆ X → X and B : D(B) ⊆ Y → Y generate two C0 -semigroups, {SA (t) : X → X; t ≥ 0} and {SB (t) : Y → Y ; t ≥ 0} respectively, and K is a nonempty and locally closed subset in X × Y. We recall that the subset K ⊆ X × Y is locally closed if for each ζ = (ξ, η) ∈ K there exists ρ > 0 such that DX×Y (ζ, ρ) ∩ K is closed, where, as usual, DX×Y (ζ, ρ) denotes the closed ball with center ζ and radius ρ in X × Y. Each subset in X × Y which is either open or closed is locally closed. Moreover, each subset K in X × Y which is closed relative to some open subset D, i.e. for which there exists a closed subset C ⊂ X × Y such that K = C ∩ D, is locally closed in X × Y. We are interested in finding sufficient conditions in order that for each (ξ, η) ∈ K there exists T > 0 such that (1) has at least one mild solution (u, v) : [ 0, T ] → K. As an application, we include a comparison result for a reaction-diffusion system. For a systematic study of problems of this kind see Ref. 1. 2. Preliminaries and the Main Results We assume familiarity with the basic concepts and results concerning infinitesimal generators of C0 -semigroups and mild solutions for Cauchy problems in Banach spaces and we refer to Refs. 2 and 3 for details. In order to prove our theorems we need the following technical results, concerning the Hausdorff measure of necompactness and uniqueness functions . Let (X, k · k) be a Banach space and B(X) the family of all bounded subsets of X. Definition 2.1. The function β : B(X) → R+ , defined by   n(ε)   [ D(xi , ε) β(B) = inf ε > 0; ∃ x1 , x2 , ...xn(ε) ∈ X, B ⊆   i=1

is called the Hausdorff measure of necompactness on X. Remark 2.1. We have β(B) = 0 if and only if B is relatively compact set. So, if X is finite dimensional β ≡ 0, because in this case the class of relatively compact subsets of X coincides with B(X). Lemma 2.1. Let X be a separable Banach space and {Fm ; m ∈ N } a subset in L1 (τ, T ; X ) for which there exists l ∈ L1 (τ, T ; R+ ) such that kFm (s)k ≤ l(s)

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for each m ∈ N and a.e. for s ∈ [τ, T ]. Then the mapping s 7→ β({Fm (s); m ∈ N} ) is integrable on [τ, T ], and, for each t ∈ [τ, T ], we have µ½Z t ¾¶ Z t β Fm (s) ds; m ∈ N ≤ β( {Fm (s); m ∈ N } ) ds. τ

τ

See Ref. 4. Lemma 2.2. Let (un )n be a bounded sequence in X such that lim β({un ; n ≥ k } ) = 0. k

Then {un ; n ∈ N } is relatively compact. For details, see Ref. 5. We recall now: Definition 2.2. A function ω : R+ → R+ which is continuous, nondecreasing and the only solution of the Cauchy problem ½ 0 x (t) = ω(x(t)) x(0) = 0. is x ≡ 0 is called a uniqueness function. Remark 2.2. If ω : R+ → R+ is a uniqueness function, then, for each k > 0, kω is a uniqueness function too. Lemma 2.3. Let ω : R+ → R+ be a uniqueness function and let (γk )k be strictly decreasing to 0. Let (xk )k be a bounded sequence of measurable functions, from [ 0, T1 ] to R+ , such that Z t xk (t) ≤ γk + ω(xk (s)) ds, 0

for k = 1, 2, ... and for each t ∈ [ 0, T1 ]. Then there exists T ∈ ( 0, T1 ] such that limk xk (t) = 0 uniformly for t ∈ [ 0, T ]. For details see Ref. 5. First, let us consider the Cauchy problem ½ 0 u (t) = Au(t) + f (u(t)) u(0) = ξ,

(2)

where A : D(A) ⊆ X → X is the infinitesimal generator of C0 -semigroup {S(t) : X → X; t ≥ 0}, K is a nonempty subset in X and f : K → X is a given function.

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Definition 2.3. The set K ⊆ X is viable with respect to A + f if for each ξ ∈ K, there exists T > 0 such that the Cauchy problem (2) has at least one mild solution u : [ 0, T ] → K. Definition 2.4. We say that η ∈ X is a A-tangent to K at ξ ∈ K if lim inf h↓0

1 dist (S(h)ξ + hη; K) = 0. h

(3)

In other words, η ∈ X is A-tangent to K at ξ ∈ K if for each δ > 0 and each neighborhood V of 0 there exist h ∈ (0, δ) and p ∈ V such that S(h)ξ + h(η + p) ∈ K.

(4)

The set of all A-tangent elements to K at ξ ∈ K is denoted by TKA (ξ). We notice that if A ≡ 0, then TKA (ξ) is the contingent cone at ξ ∈ K in the sense of Bouligand6 . Proposition 2.1. A necessary and sufficient condition in order that η ∈ TKA (ξ) is 1 lim inf dist h↓0 h

Ã

Z

!

h

S(h)ξ +

S(h − s)η ds; K

= 0.

0

For the proof see Ref. 5. Definition 2.5. Let A : D(A) ⊂ X → X be the infinitesimal generator of a C0 -semigroup {S(t) : X → X ; t ≥ 0} and f : K → X be a given function. We say that A + f is locally of β-compact type if, for each ξ ∈ K, there exist ρ > 0, a continuous function ` : R+ → R+ and a uniqueness function ω : R+ → R+ , such that β(S(t)f (C)) ≤ `(t)ω(β(C)) for each t > 0 and C ⊆ D(ξ, ρ) ∩ K. We present now some necessary and sufficient conditions in order that the set K be viable with respect to A + f. These conditions improve the main results in Refs. 7 and 8. Theorem 2.1. Let X be a Banach space, A : D(A) ⊂ X → X the infinitesimal generator of a C0 -semigroup {S(t) : X → X; t ≥ 0}, K a nonempty and locally closed subset in X and f : K → X such that A + f is locally of β-compact type. Then K is viable with respect to A + f if and only if for each ξ ∈ K, the next tangency condition is satisfied:

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lim inf h↓0

1 dist (S(h)ξ + hf (ξ); K) = 0. h

35

(5)

Remark 2.3. Under the general hypotheses of Theorem 2.1, K is viable with respect to A + f if and only if for each ξ ∈ K, we have f (ξ) ∈ TKA (ξ). Theorem 2.2. Let A : D(A) ⊂ X → X be the infinitesimal generator of a compact C0 -semigroup, {S(t) : X → X; t ≥ 0}, K a nonempty and locally closed subset in X and f : K → X a continuous function. Then K is viable with respect to A + f if and only if, for each ξ ∈ K, the tangency condition (5) is satisfied. Since Theorem 2.2 is a consequence of Theorem 2.1 and the necessity of the latter is obvious, we will show next how to prove the sufficiency. Lemma 2.4. Let X be a Banach space, A : D(A) ⊂ X → X the infinitesimal generator of a C0 -semigroup, {S(t) : X → X; t ≥ 0 }, K a nonempty and locally closed subset in X and f : K → X a continuous function satisfying the condition (5). Let ξ ∈ K be arbitrary and let r > 0 be such that D(ξ, r) ∩ K is closed. Then, there exist ρ ∈ (0, r ] and T > 0 such that, for each ε ∈ (0, 1), there exist σ : [ 0, T ] → [ 0, T ] nondecreasing, θ : { (t, s); 0 ≤ s < t ≤ T } → [ 0, T ] measurable, g : [ 0, T ] → X Riemann integrable and u : [ 0, T ] → X continuous such that: (i) (ii) (iii) (iv)

s − ε ≤ σ(s) ≤ s for each s ∈ [ 0, T ]; u(σ(s)) ∈ D(ξ, ρ) ∩ K for each s ∈ [ 0, T ] and u(T ) ∈ D(ξ, ρ) ∩ K; kg(s)k ≤ ε for each s ∈ [ 0, T ]; θ(t, s) ≤ t for each 0 ≤ s < t ≤ T and t 7→ θ(t, s) is nonexpansive on ( s, T ]; Z t Z t (v) u(t) = S(t)ξ + S(t − s)f (u(σ(s))) ds + S(θ(t, s))g(s) ds 0

for each t ∈ [ 0, T ]; (vi) ku(t) − u(σ(t)) k ≤ ε for each t ∈ [ 0, T ].

0

Definition 2.6. A 4-uple (σ, θ, g, u ) which satisfies (i)-(vi) in Lemma 2.4 is called an ε-approximate solution of the Cauchy problem (2) on [ 0, T ]. Proof. Here we include only a sketch of proof. The details will appear in a forthcoming paper. Let ξ ∈ K be arbitrary and let r > 0 be such that D(ξ, r)∩K is closed. Let us choose ρ ∈ (0, r ], N > 0, M ≥ 1 and a > 0 such that kf (x)k ≤ N for every x ∈ D(ξ, ρ)∩K and kS(t)kL(X) ≤ M eat for every t ≥ 0. Since t 7→ S(t)ξ is continuous at t = 0 and S(0)ξ = ξ, we may find a

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sufficiently small T > 0 such that sup kS(t)ξ − ξk + T M eaT (N + 1) ≤ ρ. t∈[ 0,T ]

First, we show the existence of an ε-approximate solution on an interval [0, δ] with δ∈(0, T ]. As, for every ξ∈K, f satisfies the tangency condition (5), from Proposition 2.1, it follows that for each ε ∈ (0, 1) there exist δ∈(0, T ] Z δ and p ∈ X with kpk ≤ ε, such that S(δ)ξ + S(δ − s)f (ξ)ds + δp ∈ K. 0

Let σ : [0, δ] → [0, δ], g : [0, δ] → X and θ : {(t, s); 0 ≤ s < t ≤ δ} → [0, δ] be given by σ(s) = 0, g(s) = p, θ(t, s) = 0 and let u : [0, δ] → X be Z t defined by u(t) = S(t)ξ + S(t − s)f (ξ) ds + tp, for each t ∈ [0, δ]. Simple 0

computational arguments show that (σ, θ, g, u) is an ε-approximate solution to the Cauchy problem (2) on the interval [ 0, δ ]. In the second step we will prove the existence of an ε-approximate solution for (2) defined on the whole interval [ 0, T ]. Let us denote by D(c) the set D(c) = [ 0, c ] × {(t, s); 0 ≤ s < t ≤ c} × [ 0, c ] × [ 0, c ], with c ≥ 0 and by S the set of all ε-approximate solutions to the problem (2), defined on D(c), with c ≤ T . On S we introduce a preorder “¹”as follows: we say that (σ1 , θ1 , g1 , u1 ), defined on D(c1 ), and (σ2 , θ2 , g2 , u2 ), defined on D(c2 ), satisfy (σ1 , θ1 , g1 , u1 ) ¹ (σ2 , θ2 , g2 , u2 ) if c1 ≤ c2 , σ1 (t) = σ2 (t), g1 (t) = g2 (t) for t ∈ [ 0, c1 ] and θ1 (t, s) = θ2 (t, s) for each 0 ≤ s < t ≤ c1 . Let us define the function N : S → R ∪ {+∞} by N ((σ, θ, g, u)) = c, where D(c) is the domain of definition of (σ, θ, g, u). Then N satisfies the hypotheses of Brezis–Browder Theorem 2.1.1, in Ref. 9, so S contains at least one N maximal element (σ, θ, g, u) whose domain is D(c). Finally, one may prove by contradiction that c = T. Proof. Sufficiency of Theorem 2.1. We will prove that there exists at least one sequence (εn )n ↓ 0 for which the sequence ((σn , θn , gn , un ))n , has the property that (un )n is uniformly convergent on [ 0, T ] to a function u : [ 0, T ] → K which is a mild solution of (2). Let ξ ∈ K be arbitrary and let r > 0 be such that D(ξ, r) ∩ K be closed. Let us choose ρ ∈ ( 0, r ], N > 0, M ≥ 1 a > 0 and T > 0 as in Lemma 2.4. Since A + f is locally of β-compact type, diminishing ρ ∈ (0, r ] and T > 0 if necessary, there exist ` : R+ → R+ continuous and a uniqueness function ω : R+ → R+ such that β(S(t)f (C) ) ≤ `(t)ω(β(C) ), for each C ⊆ D(ξ, ρ ) ∩ K, t ≥ 0 and all conclusions of Lemma 2.4 be satisfied. Let (εn )n ↓ 0 be a sequence in (0, 1) and let ((σn , θn , gn , un ))n be a sequence of εn -approximate solutions defined on [ 0, T ] whose existence is

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ensured by Lemma 2.4. From (v), we have Z t Z t S(t − s)f (un (σn (s) ) ) ds + S(θn (t, s) )gn (s) ds (6) un (t) = S(t)ξ + 0

0

for each n ∈ N and t ∈ [ 0, T ]. Let Y ⊆ X be the closed subspace spanned by Y1 ∪ Y2 ∪ Y3 , where   Y1 = {S(r)ξ; r ∈ [ 0, T ] ∩ Q}, Y = {S(r)f (un (σn (s))); n = 1, 2, . . . , r, s ∈ [ 0, T ] ∩ Q, s ≤ r},  2 Y3 = {S(θn (r, s))gn (s); n = 1, 2, . . . , r, s ∈ [ 0, T ] ∩ Q, s ≤ r}. Clearly Y is separable. Since (t, x) 7→ S(t)x, f and un are continuous, and gn Z t are integrable (n = 1, 2, ..., ) it follows that S(t)ξ, S(t−s)f (un (σn (s)))ds, 0 Z t S(θn (t, s))gn (s)ds and thus un (t) belong to Y for n=1, 2, ... and 0

t ∈ [0, T ]. From Lemma 2.1, (iii), (vi), and (6) we get β({un (t); n ≥ k}) ≤ βY ({un (t); n ≥ k}) µ½Z t ¾¶ ≤ βY S(t − s)f (un (σn (s))) ds; n ≥ k 0 µ½Z t ¾¶ +βY S(θn (t, s))gn (s) ds ; n ≥ k Z

0 t

≤2 β ({S(t − s)f (un (σn (s))); n ≥ k}) ds 0 Z t +2 β ({S(θn (t, s))gn (s) ; n ≥ k}) ds 0 Z t ≤2 `(t − s)ω(β ({un (σn (s)); n ≥ k})) ds + 2T M eaT εk 0 Z t ≤2 sup `(θ)ω(β({un (s); n≥k}+{un (σn (s))−un (s); n≥k}))ds+2T M eaT εk 0 θ∈[0,T ]

Z ≤2

t

sup `(θ)ω(β ({un (s); n ≥ k}) + εk ) ds + 2T M eaT εk .

0 θ∈[ 0,T ]

Set xk (t) = β({un (t); n ≥ k}) + εk , for k = 1, 2, . . . and t ∈ [ 0, T ], ω0 (x) = 2 supθ∈[ 0,T ] `(θ)ω(x), for x ∈ R+ , and γk = (2T M eaT + 1)εk . We deduce that Z t xk (t) ≤ γk + ω0 (xk (s)) ds, 0

for k = 1, 2, . . . and t ∈ [ 0, T ].

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From Remark 2.2, ω0 is a uniqueness function, so, by Lemma 2.3, diminishing T, if necessary, we have limk xk (t) = 0, uniformly for t ∈ [ 0, T ], which means that limk β( {un (t); n ≥ k} ) = 0, uniformly for t ∈ [ 0, T ]. Now, we can apply Lemma 2.2 to obtain that for each t ∈ [ 0, T ], {un (t); n ∈ N } is relatively compact in X. But, Theorem 2.3.1, p.45 in Ref. 10 proves that there exists u ∈ C( [ 0, T ]; X ) such that, on a sequence at least, we have limn un (t) = u(t), uniformly for t ∈ [ 0, T ]. Taking into account of (ii) and (vi) we get limn un (σn (t) ) = u(t), uniformly for t ∈ [ 0, T ] and u(t) ∈ D(ξ, ρ) ∩ K for each t ∈ [ 0, T ], because D(ξ, ρ) ∩ K is closed. Passing to the limit in (6) and using (iii), we obtain Z t u(t) = S(t − s) + S(t − s)f (u(s) ) ds, 0

for each t ∈ [ 0, T ]. We may now pass to a viability result referring to a reaction-diffusion system of the form (1). Let (X, k·kX ) and (Y, k·kY ) two real Banach spaces, K ⊆ X × Y, F : K → X, G : K → Y, and ξ ∈ X, η ∈ Y. We assume that the operators A : D(A) ⊆ X → X and B : D(B) ⊆ Y → Y are the generators of two C0 -semigroups. Definition 2.7. The set K is viable with respect to (A + F, B + G) if for each (ξ, η) ∈ K there exists T > 0 such that (1) has at least one mild solution (u, v) : [ 0, T ] → K. Remark 2.4. The system (1) can be written as a semilinear equation in a product space. Let X = X × Y endowed with the norm k · k, defined by k(x, y)k = kxkX + kykY , for each (x, y) ∈ X . Let A = (A, B) : D(A) ⊆ X → X be defined by A(x, y) = ( Ax, By ) for each (x, y) ∈ D(A) and let F : X → X , F(z) = ( F (z), G(z) ) for each z = (x, y) ∈ K. Hence, the system (1) can be written as ½ 0 z (t) = Az(t) + F(z(t)) (7) z(0) = ζ, where ζ = (ξ, η). In the hypotheses above, A is the infinitesimal generator of a C0 -semigroup { S(t) : X → X ; t ≥ 0}, given by S(t)(ξ, η) = (SA (t)ξ, SB (t)η) for each t ≥ 0 and (ξ, η) ∈ X . It is obvious that K is viable with respect to (A + F, B + G) in sense of Definition 2.7 if and only if K is viable with respect to A + F in sense of Definition 2.3, which means that for each ζ ∈ K, there exists T > 0 such that the system (7) has at least one mild solution z : [0, T ] → K.

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Definition 2.8. We say that (A + F, B + G) is locally of β-compact type if F, G are continuous functions and there exist ` : R+ → R+ continuous and a uniqueness function ω : R+ → R+ such that for each ζ = (ξ, η) ∈ K, there exists ρ > 0 such that for each set C ⊆ DX×Y (ζ, ρ) ∩ K and for each t ≥ 0, we have βX×Y ( (SA (t)F (C), SB (t)G(C) ) ) ≤ `(t)ω(βX×Y (C) ). Remark 2.5. One may easily see that (A + F, B + G) is locally of βcompact type if and only if A + F is locally of β-compact type in sense of Definition 2.5. Let us introduce the following hypotheses which we will use in the sequel: (H1 ) A : D(A) ⊆ X → X, B : D(B) ⊆ Y → Y are the generators of two C0 -semigroups, {SA (t) : X→X; t ≥ 0} and {SB (t) : Y →Y ; t ≥ 0} respectively; (H2 ) K ⊆ X × Y is nonempty and locally closed; (H3 ) (A + F, B + G) is locally of β-compact type. The main viability result is: Theorem 2.3. Assume that (H1 ), (H2 ), (H3 ) are satisfied. The necessary and sufficient condition in order that K be viable with respect to (A + F, B + G) is that, for every (ξ, η) ∈ K lim inf h↓0

1 dist((SA (h)ξ + hF (ξ, η), SB (h)η + hG(ξ, η)); K) = 0. h

(8)

In view of Remark 2.4, the Theorem 2.3 can be reformulated as: Theorem 2.4. Assume that (H1 ), (H2 ), (H3 ) are satisfied. The necessary and sufficient condition in order that K be viable with respect to A + F is that, for every ζ ∈ K lim inf h↓0

1 dist(S(h)ζ + hF(ζ); K ) = 0. h

(9)

Hence the proof of this theorem follows from the Theorem 2.1. We conclude this section with a nonautonomous variant of Theorem 2.3. Definition 2.9. Let A : D(A) ⊆ X → X and B : D(B) ⊆ Y → Y be the infinitesimal generators of two C0 -semigroups {SA (t) : X → X ; t ≥ 0} and {SB (t) : Y → Y ; t ≥ 0} respectively, K a nonempty subset in R × X × Y, F : K → X and G : K → Y two given functions. We say that (A+F, B +G) is locally of β-compact type with respect to the second argument, i.e. with

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respect to (u, v), if, for each (τ, ξ, η) ∈ K, there exist ρ > 0, a continuous function l : R+ → R+ and a uniqueness function ω : R+ → R+ , such that βX×Y (SA (t)F (C), SB (t)G(C)) ≤ l(t)ω(βX×Y (ΠX×Y C)) for each t > 0 and each C ⊆ DR×X×Y ((τ, ξ, η), ρ) ∩ K. Theorem 2.5. Assume that X and Y are Banach spaces and (H1 ) is satisfied. Let K ⊆ R×X ×Y be a locally closed set and let (F, G) : K → X ×Y be continuous. Let us assume that (A + F, B + G) is locally of β-compact type with respect the second argument. Then a necessary and sufficient condition in order that K be viable with respect to (A + F, B + G) is that, for each (τ, ξ, η) ∈ K lim inf h↓0

1 dist ((τ + h, SA (h)ξ + hF (τ, ξ, η), SB (h)η + hG(τ, ξ, η)); K) = 0. h

3. An Example for a Predator-pray System Let Ω ⊆ Rn , n = 1, 2, . . . , be a bounded domain with C 2 boundary Γ, let δi ≥ 0, i = 1, 2, p > 0, q > 0, let f : R × R → R+ and g : R × R → R− be two continuous functions and let us consider the following general predatorpray system  ut (t, x) = δ1 ∆u − pu(t, x) + f (u(t, x), v(t, x)) (t, x) ∈ Qτ,∞    vt (t, x) = δ2 ∆v + qv(t, x) + g(u(t, x), v(t, x)) (t, x) ∈ Qτ,∞ (10)  u(t, x) = v(t, x) = 0 (t, x) ∈ Στ,∞ ,   u(τ, x) = ξ(x), v(τ, x) = η(x) x ∈ Ω, where 0 ≤ τ < T ≤ ∞, Qτ,T = (τ, T ) × Ω, Στ,T = (τ, T ) × Γ and ξ, η ∈ L2 (Ω). Let fe : R × R → R+ and ge : R × R → R− be two continuous functions such that f (u, v) ≤ fe(u, v) and g(u, v) ≥ ge(u, v)

(11)

for each (u, v) ∈ R × R. Let us consider also the comparison predator-pray system  ut (t, x) = δ1 ∆u − pu(t, x) + fe(u(t, x), v(t, x)) (t, x) ∈ Q0,∞    vt (t, x) = δ2 ∆v + qv(t, x) + ge(u(t, x), v(t, x)) (t, x) ∈ Q0,∞ (12)  u(t, x) = v(t, x) = 0 (t, x) ∈ Σ0,∞ ,   u(0, x) = u0 (x), v(0, x) = v0 (x) x ∈ Ω, where u0 , v0 ∈ L2 (Ω), u0 (x) ≥ 0, v0 (x) ≥ 0 a.e. for x ∈ Ω. Let (e u, ve) : R+ × Ω → R+ × R+ be a mild solution of (12).

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Using the viability result of Theorem 2.5, we will prove a sufficient conditions in order that, for each (ξ, η) ∈ L2 (Ω) × L2 (Ω), with 0 ≤ ξ(x) ≤ u e(τ, x) and η(x) ≥ ve(τ, x)

(13)

a.e. for x ∈ Ω, the predator-pray system (2.1) has at least one solution (u, v) : R+ × Ω → R+ × R+ , such that, for each t ∈ [ τ, ∞), we have 0 ≤ u(t, x) ≤ u e(t, x) and v(t, x) ≥ ve(t, x) a.e. for x ∈ Ω. Let K ⊆ R × L2 (Ω) × L2 (Ω) be defined by © ª K = (t, u, v) ∈ R+ × L2 (Ω) × L2 (Ω); (u, v) satisfy (16) below . 0 ≤ u(x) ≤ u e(t, x) and v(x) ≥ ve(t, x)

(14)

(15) (16)

a.e. for x ∈ Ω. Theorem 3.1. Let Ω ⊆ Rn , n = 1, 2, ..., be a bounded domain with C 2 boundary Γ, let f : R × R → R+ , g : R × R → R− , fe : R × R → R+ and ge : R × R → R− be continuous such that (11) are satisfied. Assume that for each (u0 , v0 ) ∈ R × R, u 7→ fe(u, v0 ) and v 7→ ge(u0 , v) are nondecreasing, u 7→ ge(u, v0 ) and v 7→ fe(u0 , v) are nonincreasing and there exist the constants ci ≥ 0, i = 1, ..., 5 such that |fe(u, v)| ≤ c1 |u| + c2 , and |e g (u, v)| ≤ c3 |u| + c4 |v| + c5 for each (u, v) ∈ R × R. Let (u0 , v0 ) ∈ L2 (Ω) × L2 (Ω) with u0 (x) ≥ 0 and v0 (x) ≥ 0 a.e. for x ∈ Ω and let (e u, ve) : R+ → L2 (Ω)×L2 (Ω) be a global mild solution of (12) with u e(t, x) ≥ 0 for each t ≥ 0 and a.e. for x ∈ Ω. Let K be defined by (15). Then, for each (τ, ξ, η) ∈ K, the problem (2.1) has at least one global mild solution (u, v) : [ τ, ∞) → L2 (Ω) × L2 (Ω) satisfying for each τ < δ < T : (i) u, v ∈ C([ τ, T ]; L2 (Ω)) ∩ L2 (δ, T ; H 2 (Ω)) ∩ W 1,1 (δ, T ; H01 (Ω)) ; (ii) for each t ∈ [ τ, ∞), (t, u(t), v(t)) ∈ K. Proof. Since (i) is well known – see Theorem 11.6.1 in Ref. 3, p. 265, it remains to prove (ii). So, we have to show first that K is viable, in R+ × L2 (Ω) × L2 (Ω), with respect to (δ1 ∆ − pI + f, δ2 ∆ + qI + g) and second that every mild solution (u, v) : [ τ, T ) → L2 (Ω) × L2 (Ω), satisfying (t, u(t), v(t)) ∈ K for each t ∈ [ τ, T ), can be extended to a global one obeying the very same constraints. Let us denote by X = L2 (Ω). We rewrite (2.1) and (12) as an evolution systems in X. Let us define A : D(A) ⊆ X → X and B : D(B) ⊆ X → X by D(A) = H01 (Ω) ∩ H 2 (Ω) and Au = δ1 ∆u − pu for u ∈ D(A)

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and respectively by D(B) = H01 (Ω) ∩ H 2 (Ω) and Bv = δ2 ∆v + qv for v ∈ D(B). Let us define F : X × X → X and G : X × X → X by F (u, v)(x) = f (u(x), v(x)) and G(u, v)(x) = g(u(x), v(x)) respectively, for each (u, v) ∈ e by the same relations with F and X × X and a.e. for x ∈ Ω, and Fe and G e G but with f instead of f and ge instead of g. With the notations above, the problem (2.1) can be rewritten as the abstract evolution system ½ 0 u (t) = Au(t) + F (u(t), v(t)) (17) v 0 (t) = Bv(t) + G(u(t), v(t)), while (12) takes the abstract form ( u0 (t) = Au(t) + Fe(u(t), v(t)) e v 0 (t) = Bv(t) + G(u(t), v(t)).

(18)

e are wellSince fe and ge are continuous and have sublinear growth, Fe and G defined, continuous and have sublinear growth. From the very same reason F and G are well-defined, continuous and have sublinear growth too. Let e ⊆ R+ × L2 (Ω) × L2 (Ω) be defined by K © ª e = (t, u, v) ∈ R+ × L2 (Ω) × L2 (Ω); (u, v) satisfy (20) below K (19) u(x) ≤ u e(t, x), and v(x) ≥ ve(t, x)

(20)

a.e. for x ∈ Ω. To prove that K is viable with respect to (A + F, B + G) e is viable with respect to (A + F, B + G). This is it suffices to show that K a direct consequence of the maximum principle for parabolic equations — see Theorem 1.7.5. in Ref. 5 — combined with the fact that F and u e are nonnegative. Since both A and B generate compact semigroups {SA (t) : X → X, t ≥ 0} and {SB (t) : X → X, t ≥ 0} respectively, in view of e is viable with respect to (A + F, B + G), we Theorem 2.5, to show that K have merely to check the tangency condition lim inf h↓0

1 e = 0, (21) dist ((τ + h, SA (h)ξ + hF (ξ, η), SB (h)η + hG(ξ, η)); K) h

e To do this, it suffices to prove that for each (τ, ξ, η)∈K e for each (τ, ξ, η)∈K. e and and each h > 0 there exists (uh , vh )∈X × X with (τ + h, uh , vh ) ∈ K  1   inf kSA (h)ξ + hF (ξ, η) − uh k = 0  lim h↓0 h (22) 1    lim inf kSB (h)η + hG(ξ, η) − vh k = 0. h↓0 h

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e and let us define uh and vh by So, let (τ, ξ, η) ∈ K, Z τ +h uh = SA (h)ξ + SA (τ + h − s)F (ξ, η) ds τ Z τ +h + SA (τ + h − s)[ Fe(e u(s), ve(s)) − Fe(e u(τ ), ve(τ )) ] ds τ

and respectively by Z τ +h vh = SB (h)η + SB (τ + h − s)G(ξ, η) ds τ Z τ +h e u(s), ve(s)) − G(e e u(τ ), ve(τ )) ] ds. + SB (τ + h − s)[ G(e τ

Now let us observe that, inasmuch as ξ ≤ u e(τ ) and η ≥ ve(τ ) a.e. on Ω, we have both SA (h)ξ ≤ SA (h)e u(τ ) and SB (h)η ≥ SB (h)e v (τ ). See Theorem 1.7.5 in Ref. 5. Recalling that f ≤ fe and taking into account of the monotonicity properties of fe, we get F (ξ, η) ≤ Fe(ξ, η) ≤ Fe(e u(τ ), ve(τ )). Similarly, using the fact that g ≥ ge and the monotonicity properties of ge, e η) ≥ G(e e u(τ ), ve(τ )). We get both uh ≤ u we deduce G(ξ, η) ≥ G(ξ, e(τ + h) e On the other hand and vh ≥ ve(τ + h) and thus (τ + h, uh , vh ) ∈ K. Z

τ +h

kSA (h)ξ + hF (ξ, η) − uh k ≤ kSA (τ + h − s)F (ξ, η) − F (ξ, η)k ds τ Z τ +h +M eah kFe(e u(s), ve(s)) − Fe(e u(τ ), ve(τ ))k ds, τ

where M ≥ 1 and a > 0 are the growth constants of the C0 -semigroup {SA (t) : X → X, t ≥ 0}. Since Fe, u e and ve are continuous we conclude that the first equality in (22) holds. Similarly, we get the second equality, and e and consequently the viability this completes the proof of the viability of K of K. Let us remark that f and g have sublinear growth. In addition K satisfies: for each sequence ((tn , ξn , ηn )n ) in K with limn (tn , ξn ηn ) = (t, ξ, η) and t < TK , where TK = sup{t ∈ R; there exists (ξ, η)∈X×X, with (t, ξ, η)∈K}, it follows that (t, ξ, η) ∈ K. Then it follows that each mild solution (u, v) : [ τ, T ] → X × X of (2.1) satisfying (t, u(t), v(t)) ∈ K for each t ∈ [ τ, T ] can be continued up to a global one (u∗ , v ∗ ) : [ τ, TK ) → X × X satisfying the very same condition on [ τ, TK ). Since (e u, ve) is defined on R+ and (t, u e(t), ve(t)) ∈ K for each t ∈ [ 0, ∞) we conclude that TK = ∞ and this completes the proof.

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References 1. S. Ladde, V. Lakshmikantham, and A. S. Vatsalo, Monotone, Iterative Techniques for Nonlinear Differential Equations, Monographs, Advanced Texts and Surveys in Pure and Applied Mathematics, 27 (Pitman, 1985). 2. V. Barbu, Nonlinear Semigroups and Differential Equation in Banach Spaces, (Editura Academiei, Bucure¸sti, Noordhoff, 1976). 3. I. I. Vrabie, C0 -Semigroups and Applications, (North-Holland Mathematics Studies, 191 (2003). 4. H. M¨ onch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal., 4 (1980), 985–999. 5. O. Cˆ arj˘ a, M. Necula and I. I. Vrabie, Viability, Invariance and Applications, (North-Holland Mathematics Studies, 207), in print. 6. H. Bouligand, Sur les surfaces d´epourvues de points hyperlimit´es, Ann. Soc. Polon. Math., 9, (1930). 7. N. H. Pavel, Invariant sets for a class of semi-linear equations of evolution, Nonlinear Anal., 1 (1977), 187–196. 8. O. Cˆ arj˘ a and I. I. Vrabie, Some new viability results for semilinear differential inclusions, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 401– 424. 9. O. Cˆ arj˘ a and I. I. Vrabie, Differential Equations on Closed Sets, in Handbook of Differential Equations, Ordinary Differential Equations, 2, Edited by A. Ca˜ nada, P. Dr´ abek and A. Fonda, (Elsevier B.V., 2005), 147–238. 10. I. I. Vrabie, Compactness methods and flow-invariance for perturbed nonlinear semigroups, An. S ¸ tiint¸. Univ. Al. I. Cuza Ia¸si Sect¸. I a Mat., 27 (1981), 117–125.

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NECESSARY OPTIMALITY CONDITIONS FOR HYPERBOLIC DISCRETE INCLUSIONS AURELIAN CERNEA Faculty of Mathematics and Informatics, University of Bucharest, Academiei 14, 010014 Bucharest, Romania E-mail: [email protected] The aim of this paper is to present a short survey of several new results concerning optimization of hyperbolic discrete inclusions. We study an optimization problem given by a hyperbolic discrete inclusion with end point constraints and we present several approaches concerning first and second-order necessary optimality conditions for this problem. Keywords: Derived cone; Discrete inclusion; Necessary optimality conditions, Minimum Principle.

1. Introduction Consider the problem minimize g(xN N )

(1)

over the solutions of the multiparameter discrete inclusion xij ∈ Fij (xij−1 , xi−1j , xi−1j−1 ),

i = 0, 1, ..., N,

j = 0, 1, ..., N,

(2)

with end point constraints of the form xN N ∈ XN ,

(3)

where Fij (.) : R3n → P(Rn ), i, j = 0, ..., N , are given set-valued maps, XN ⊂ Rn and g(.) : Rn → R is also a given function. The aim of this paper is to announce several new results concerning first and second-order necessary optimality conditions for problem (1)–(3). At the beginning we obtain necessary optimality conditions for a solution x = (x0 , x1 , ..., xN ), xi = (xi0 , xi1 , ..., xiN ), i = 0, 1, ..., N to the problem (1)–(3) in terms of a variational inclusion associated to the problem

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(2) and in terms of the cone of interior directions (Dubovitskij-Miljutin tangent cone) to the set XN at xN N . Afterwards this result is improved by replacing the cone of interior directions with the concept of derived cone introduced by Hestenes1 and using a remarkable ”intersection property” of derived cones obtained by Mirica2 . Finally, we present an approach concerning second-order necessary optimality conditions for the problem (1)–(3). Optimal control problems for systems described by discrete inclusions have been studied by many authors (Refs. 3–8 etc.). In the framework of multivalued problems, necessary optimality conditions for problem (1)–(2) (i.e. without end point constraints) are obtained in Ref. 9. The idea in Ref. 9 is to use a special (Warga’s) open mapping theorem to obtain a sufficient condition for (2) to be locally controllable around a given trajectory and as a consequence, via a separation result, to obtain the maximum principle. In contrast with the approach in Ref. 9, even if the problem studied in the present paper is more difficult, due to end point constraints, the method in our approach seems to be conceptually very simple, relying only 2–3 clear-cut steps and using a minimum of auxiliary results. 2. Preliminaries Denote by SF the solution set of inclusion (2), i.e. SF := {x = (x0 , x1 , ..., xN ); i, j = 0, 1, ..., N,

xi = (xi0 , xi1 , ..., xiN ),

x is a solution of (2),

xij = 0

xij ∈ Rn ,

if i < 0 or j < 0}.

and by RFN := {xN N ; x ∈ SF } the reachable set of inclusion (2). We consider x = (x0 , x1 , ..., xN ) ∈ SF a solution of (2). Since the reachable set RFN is, generally, neither a differentiable manifold, nor a convex set, its infinitesimal properties may be characterized only by tangent cones in a generalized sense, extending the classical concepts of tangent cones in Differential Geometry and Convex Analysis, respectively. From the multitude of the intrinsic tangent cones in the literature, the contingent, the quasitangent and Clarke’s tangent cones, defined, respectively, by ½ ¾ xm − x n Kx X = v ∈ R ; ∃ sm → 0+, xm ∈ X : →v sm Qx X = {v ∈ Rn ; ∃ c(.) : [0, s0 ) → X, c(0) = x, c0 (0) = v} ½ ¾ ym − xm Cx X = v ∈ Rn ; ∀(xm , sm )→(x, 0+), xm ∈X, ∃ ym ∈X : →v sm

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seem to be among the most oftenly used in the study of different problems involving nonsmooth sets and mappings. The second-order quasitangent set to X at x relative to v ∈ Qx X is defined by Q2(x,v) X = {w ∈ Rn ; ∀sm → 0+, ∃wm → w : x + sm v + s2m wm ∈ X}. We recall that, in contrast with Kx X, Qx X, the cone Cx X is convex and one has Cx X ⊂ Qx X ⊂ Kx X. Another important tangent cone is the cone of interior directions (Dubovitskij-Miljutin tangent cone) defined by Ix X

:= {v ∈ Rn ; ∃ s0 , r > 0 : x + sB(v, r) ⊂ X ∀ s ∈ [0, s0 )},

B(v, r) := {w ∈ Rn ; kw − vk < r},

B(v, r) := cl B(v, r).

From the properties of the cone of interior directions we recall only the following: Qx X1 ∩ Ix X2 ⊂ Qx (X1 ∩ X2 ).

(4)

The concept of derived set was introduced for the first time by Hestenes1 . Definition 2.1. A subset M ⊂ Rn is said to be a derived set to X ⊂ Rn at x ∈ X if for any finite subset {v1 , ..., vk } ⊂ M , there exist s0 > 0 and a continuous mapping a(.) : [0, s0 ]k → X such that a(0) = x and a(.) is (conically) differentiable at s = 0 with the derivative col[v1 , ..., vk ] in the sense that Pk ka(θ) − a(0) − i=1 θi vi k lim = 0. kθk Rk + 3θ→0 A subset C ⊂ Rn is said to be a derived cone of X at x if it is a derived set and also a convex cone. For the basic properties of derived sets and cones we refer to Hestenes1 ; we recall that if M is a derived set then M ∪ {0} as well as the convex cone generated by M , defined by

cco(M ) =

( k X

) λj v j ;

λj ≥ 0, k ∈ N, vj ∈ M, j = 1, ..., k

i=1

is also a derived set, hence a derived cone. The fact that the derived cone is a proper generalization of the classical concepts in Differential Geometry and Convex Analysis is illustrated by the

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following results (Ref. 1): if X ⊂ Rn is a differentiable manifold and Tx X is the tangent space in the sense of Differential Geometry to X at x Tx X = {v ∈ Rn ;

∃ c : (−s, s) → X, of class C 1 , c(0) = x, c0 (0) = v}

then Tx X is a derived cone; also, if X ⊂ Rn is a convex subset then the tangent cone in the sense of Convex Analysis defined by T Cx X = cl{t(y − x);

t ≥ 0, y ∈ X}

is also a derived cone. By cl A we denote the closure of the set A ⊂ Rn . Since any convex subcone of a derived cone is also a derived cone, such an object may not be uniquely associated to a point x ∈ X; moreover, simple examples show that even a maximal with respect to set-inclusion derived cone may not be uniquely defined: if the set X ⊂ R2 is defined by [ X = C1 C2 , C1 = {(x, x); x ≥ 0}, C2 = {(x, −x), x ≤ 0} then C1 and C2 are both maximal derived cones of X at the point (0, 0) ∈ X. We recall that two cones C1 , C2 ⊂ Rn are said to be separable if there exists q ∈ Rn \{0} such that: hq, vi ≤ 0 ≤ hq, wi

∀v ∈ C1 , w ∈ C2 .

We denote by C + the positive dual cone of C ⊂ Rn C + = {q ∈ Rn ;

hq, vi ≥ 0,

∀ v ∈ C}

The negative dual cone of C ⊂ Rn is C − = −C + . The following ”intersection property” of derived cones, obtained by Miric˘a2 , is a key tool in the proof of necessary optimality conditions. Lemma 2.1. Let X1 , X2 ⊂ Rn be given sets, x ∈ X1 ∩ X2 , and let C1 , C2 be derived cones to X1 , resp. to X2 at x. If C1 and C2 are not separable, then Cl(C1 ∩ C2 ) = (Cl(C1 )) ∩ (Cl(C2 )) ⊂ Qx (X1 ∩ X2 ). For a mapping g(.) : X ⊂ Rn → R which is not differentiable, the classical (Fr´echet) derivative is replaced by some generalized directional derivatives. We recall only, in the case when g(.) is locally-Lipschitz at x ∈ int(X), Clarke’s generalized directional derivative, defined by: DC g(x; v) =

g(y + θv) − g(y) , θ (y,θ)→(x,0+) lim sup

v ∈ Rn .

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The first and second order uniform lower Dini derivative are defined as follows g(x + θv 0 ) − g(x) D↑ g(x; v) = 0 lim inf , θ (v ,θ)→(v,0+) D↑2 g(x, v; w) =

g(x + θv + θ2 w0 ) − g(x) − θD↑ g(x; v) . θ2 (w ,θ)→(w,0+) lim inf 0

When g(.) is of class C 2 one has 1 D↑2 g(x, v; w) = g 0 (x)T z + v T g 00 (x)v. 2 The results in the next section will be expressed, in the case where g(.) is locally-Lipschitz at x, in terms of the Clarke generalized gradient, defined by: D↑ g(x, v) = g 0 (x)T v,

∂C g(x) = {q ∈ Rn ;

< q, v >≤ DC g(x; v)

∀ v ∈ Rn }.

By P(Rn ) we denote the family of all subsets of Rn . Corresponding to each type of tangent cone, say τx X, one may introduce a set-valued directional derivative of a multifunction G(.) : X ⊂ Rn → P(Rn ) (in particular of a single-valued mapping) at a point (x, y) ∈ Graph(G) as follows τy G(x; v) = {w ∈ Rn ; (v, w) ∈ τ(x,y) Graph(G)},

v ∈ τx X.

Similarly one may define second-order directional derivatives of the setvalued map G(.). For example the second-order quasitangent derivative of G at (x, u) relative to (y, v) ∈ Q(x,u) (graph(G(.)) is the set-valued map Q2(u,v) G(x, y, .) defined by graphQ2(u,v) G(x, y; .) = Q2((x,u),(y,v)) (graphG(.)). We recall that a set-valued map, A(.) : Rn → P(Rn ) is said to be a convex (respectively, closed convex) process if Graph(A(.)) ⊂ Rn × Rn is a convex (respectively, closed convex) cone. In what follows, we shall assume the following hypothesis. Hypothesis 1.1. i) The set-valued maps Fij (.) have nonempty compact convex values ∀i, j ∈ {0, 1, ..., N } and there exists L > 0 such that Fij (.) is Lipschitz with the Lipschitz constant L, ∀i, j ∈ {0, 1, ..., N }.

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ii) There exists Aij (.) : R3n → P(Rn ), i, j = 0, 1, ..., N a family of closed convex processes such that Aij (u, v, w) ⊂ Qxij Fij ((xij−1 , xi−1j , xi−1j−1 ); (u, v, w)) ∀u, v, w ∈ Rn , ∀i, j ∈ {0, 1, ..., N }. Let A0 ⊂ Qx00 F00 (0, 0, 0) be a closed convex cone. To the problem (2) we associate the linearized problem wij ∈Aij (wij−1 , wi−1j , wi−1j−1 ), w00 ∈A0 , i, j=0, 1, ..., N, i+j > 0 with the boundary conditions wij = 0 for i < 0 or j < 0

(5)

N Denote by SA the solution set of inclusion (5) and by RA the reachable set of inclusion (5). We recall that if A : Rn → P(Rn ) is a set-valued map then the adjoint of A is the multifunction A∗ : Rn → P(Rn ) defined by

A∗ (p) = {q ∈ Rn ; hq, vi ≤ hp, v 0 i

∀(v, v 0 ) ∈ graphA(.)}.

The next result, due to Tuan and Ishizuka8 characterizes the positive N dual of the reachable set RA of problem (5). Lemma 2.2. Assume that Hypothesis 1.1 is satisfied and let r(.) : Rn → P(Rn ) be the set-valued map defined by r(α) := {wN N ; w = (w0 , ..., wN ) is a solution of (5), w00 = α}. Then, for all b ∈ Rn r∗ (b) = {u101 + u210 + u311 ; (u1ij , u2ij , u3ij ) ∈ A∗ij (u1ij+1 + u2i+1j + u3i+1j+1 ), i, j=0, 1, ..., N, 0 < i + j < 2N, (u1N N , u2N N , u3N N ) ∈ A∗N N (b)}. 3. Results N Using the property in (4), the fact that RA ⊂ QxN N RFN (Ref. 10) and Lemma 2.2 we obtain a Minimum Principle for problem (1)–(3).

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Theorem 3.1. Let XN ⊂ Rn be a closed set, let x ∈ SF be an optimal solution for problem (1)–(3) such that Hypothesis 1.1 is satisfied and let g(.) : Rn → R be a locally Lipschitz function. Then for any closed convex cone A0 ⊂ Qx00 F00 (0, 0, 0) and any convex cone C1 ⊂ IxN N XN there exist λ ∈ {0, 1}, q ∈ Rn and u1ij , u2ij , u3ij ∈ Rn such that (u1ij , u2ij , u3ij ) ∈ A∗ij (u1ij+1 + u2i+1j + u3i+1j+1 ), 0 < i + j < 2N, (u1N N , u2N N , u3N N ) ∈ A∗N N (q), q ∈ λ∂C g(xN N ) − C1+ , u101 + u210 + u311 ∈ A+ 0,

(6)

(7)

® ® 1 uij+1 + u2i+1j + u3i+1j+1 , xij = min{ u1ij+1 + u2i+1j + u3i+1j+1 , v ; v ∈ Fij (xij−1 , xi−1j , xi−1j−1 )}, 0 < i + j < 2N, hq, xN N i = min{hq, vi ; v ∈ FN N (xN N −1 , xN −1N , xN −1N −1 )}, 1 ® u01 + u210 + u311 , x00 = min{hv, x00 i ; v ∈ F00 (0, 0, 0)} λ + kqk > 0.

(8)

(9)

For the details of the proof see Ref. 10. In Theorem 3.1 an important hypothesis is that the terminal set XN is assumed to have a nonempty cone of interior directions. Such type of assumption may be overcome by using the concept of derived cone. Using the fact that if A0 is a derived cone to F00 (0, 0, 0) at x00 then N the reachable set RA is a derived cone to RFN at xN N (Ref. 11), Lemma 2.1 and Lemma 2.2 we have the next version of the Minimum Principle for problem (1)–(3). Theorem 3.2. Let XN ⊂ Rn be a closed set, let x ∈ SF be an optimal solution for problem (1)–(3) such that Hypothesis 1.1 is satisfied and let g(.) : Rn → R be a locally Lipschitz function. Then for any derived cones A0 of F00 (0, 0, 0) at x00 and C1 of XN at xN N there exist λ ∈ {0, 1}, q ∈ Rn and u1ij , u2ij , u3ij ∈ Rn such that (6)-(9) hold true.

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The proof can be found in Ref. 11. In particular, when Fij (., ., .) are expressed in the parametrized form Fij (xij−1 , xi−1j , xi−1j−1 ) = fij (xij−1 , xi−1j , xi−1j−1 , Uij ),

(10)

where Uij ⊂ Rmij are compact sets for 0 < i + j < 2N and F00 (0, 0, 0) = {x00 }, for a solution x of inclusion (2) and its corresponding control u = (u0 , u1 , ..., uN ), ui = (ui0 , ui1 , ..., uiN ), uij ∈ Uij , i = 0, 1, ..., N , j = 0, 1, ..., N we define ∂fij ∂fij A1ij = (xij−1 , xi−1j , xi−1j−1 , uij ), A2ij = (xij−1 , xi−1j , ∂xij−1 ∂xi−1j ∂fij xi−1j−1 , uij ), A3ij = (xij−1 , xi−1j , xi−1j−1 , uij ). ∂xi−1j−1 Corollary 3.1. Assume that the function fij (., ., ., uij ) is Lipschitz for every fixed uij ∈ Uij , the function fij (xij−1 , xi−1j , xi−1j−1 , .) is continuous for every fixed xij−1 , xi−1j , xi−1j−1 and the function fij (., ., ., uij ) is differentiable at (xij−1 , xi−1j , xi−1j−1 ). Let x = (x0 , x1 , ..., xN ) ∈ SF be an optimal solution for problem (1)–(3), with Fij defined by (10), such that Hypothesis 1.1 is satisfied. Consider XN ⊂ Rn a closed set and g(.) : Rn → R a locally Lipschitz function. Then for any derived cone C1 of XN at xN N there exist q ∈ Rn and a solution pij ∈ Rn such that pij = (A1ij+1 )∗ pij+1 + (A2i+1j )∗ pi+1j + (A3i+1j+1 )∗ pi+1j+1 , 0 < i + j < 2n, pij = 0

for

i>N

or

j > N,

pN N ∈ λ∂C g(xN N ) − C1+ , hpij , xij i = min{hpij , vi ; v ∈ Fij (xij−1 , xi−1j , xi−1j−1 )}, 0 < i + j. N Denote by RQ the reachable set of the discrete inclusion

wij ∈Qxij Fij ((xij−1 , xi−1j , xi−1j−1 ); (wij−1 , wi−1j , wi−1j−1 )), i+j>0 w00 ∈ Qx00 F00 (0, 0, 0),

wij = 0

for i < 0 or j < 0.

(11)

2 Let y = (y 0 , y 1 , ..., y N ) satisfy (10) and let RQ denote the reachable set of the discrete inclusion vij ∈ Q2(xij ,y ) Fij ((xij−1 , xi−1j , xi−1j−1 ), (y ij−1 , y i−1j , ij

y i−1j−1 ); (vij−1 , vi−1j , vi−1j−1 )), v00 ∈ Q2(x00 ,y

00 )

F00 (0, 0, 0),

i + j > 0,

vij = 0

for i < 0 or j < 0.

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In the next result we obtain second-order necessary optimality conditions for problem (1)–(3). Theorem 3.3. Assume that Hypothesis 1.1 is satisfied, let g(.) : Rn → R be a locally Lipschitz function, let A0 ⊂ Qx00 F00 (0, 0, 0) be a closed convex cone, let x = (x0 , x1 , ..., xN ) ∈ SF be an optimal solution for problem (1)–(3) and assume that the following constraint qualification is satisfied {−q ∈ Rn ;

∃ u1ij , u2ij , u3ij ∈ Rn

such that

(u1ij , u2ij , u3ij ) ∈

(Cxij Fij ((xij−1 , xi−1j , xi−1j−1 ); (., ., .)))∗ (u1ij+1 + u2i+1j + u3i+1j+1 ), i, j = 0, 1, ..., N,

0 < i + j < 2N,

(u1N N , u2N N , u3N N )∈(CxN N FN N ((xN N −1 , xN −1N , xN −1N −1 ); (., ., .)))∗ (q), + u101 + u210 + u311 ∈ A+ 0 } ∩ (CxN N XN ) = {0}. Then we have the first-order necessary condition D↑ g(xN N ; yN N ) ≥ 0

N ∀ yN N ∈ RQ ∩ QxN N XN .

Furthermore, if equality holds for some y N N , then we have the second-order necessary condition D↑2 g(xN N , y N N ; wN N ) ≥ 0

2 ∀ wN N ∈ RQ ∩ Q2(xN N ,yN N ) XN .

The proof, that can be found in Ref. 12, is based on a general (abstract) optimality condition formulated by Zheng13 and use also several first and second-order approximations of the reachable set RFN at xN N (Ref. 12). References 1. M.R. Hestenes, Calculus of Variations and Optimal Control Theory, (Wiley, New York, 1966). 2. S ¸ . Miric˘ a, New proof and some generalizations of the Minimum Principle in Optimal Control, J. Optim. Theory Appl., 74 (1992), 487–508. 3. V.G. Boltjanskii, Optimal Control for Discrete Systems, (Nauka, Moskow, 1973). 4. A.G. Kusraev, Discrete maximum principle, Math. Notes, 34 (1984), 617– 619. 5. V.N. Phat, Controllability of nonlinear discrete systems without differentiability assumption, Optimization, 19 (1988), 133–142. 6. A.I. Propoi, The maximum principle for discrete control systems, Automat. Remote Control, 26 (1965), 451–461. 7. R. Pytlak, A variational approach to discrete maximum principle, IMA J. Math. Control and Information, 9 (1992), 197–220.

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8. N.D. Yen and T. C. Dien, On local controllability of nondifferentiable discrete time systems with nonconvex constraints on control, Optimization, 20 (1989), 889–899. 9. H.D. Tuan and Y. Ishizuka, On controllability and maximum principle for discrete inclusions, Optimization, 34 (1995), 293–316. 10. A. Cernea, Minimum Principle for a class of discrete inclusions, Math. Reports, 8(58) (2006), to appear. 11. A. Cernea, Minimum principle and controllability for multiparameter discrete inclusions via derived cones, Discrete Dynamics in Nature and Society, 2006 (2006), ID 96505, 1–12. 12. A. Cernea, An approach to second-order necessary conditions for multiparameter discrete inclusions, Revue Roum. Math. Pures Appl., to appear. 13. H. Zheng, Second-order necessary conditions for differential inclusion problems, Appl. Math. Opt., 30 (1994), 1–14.

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CONTROLLABILITY OF DAMPED SECOND-ORDER INITIAL VALUE PROBLEM FOR A CLASS OF DIFFERENTIAL INCLUSIONS WITH NONLOCAL CONDITIONS ON NONCOMPACT INTERVALS D.N. CHALISHAJAR Department of Applied Mathematics, Sardar Vallabhbhai Patel Institute of Tech., Gujarat University, Vasad-388306. Gujaratr State, India E-mail: [email protected] In this short article, we investigate the controllability of damped second-order initial value problems for a class of differential inclusions with nonlocal conditions on unbounded real interval. We shall employ a theorem of Ma1 , which is an extension to multivalued maps on locally convex topological spaces, of Schaefer’s theorem. Example is provided to illustrate the theory. This work is motivated by the papers of Benchohra and Ntouyas2 and Benchohra, Gatsori and Ntouyas3 . Keywords: Controllability; Initial value problems; Convex multivalued map; Evolution Inclusion; Fixed point; Nonlocal conditions.

1. Introduction The IVP with nonlocal conditions is of significance since they have applications in many physical problems. For the importance of nonlocal conditions in different fields we refer to Ref. 4 and the references cited there in. Also, controllability of the second-order systems with local and nonlocal conditions has received much attention in the recent years. It is advantageous to treat the second-order abstract differential equations directly rather than to convert them to first-order system, (Fitzgibbon5 and Ball6 ). Fitzgibbon5 used the second-order abstract system for establishing the boundedness of solutions of the equation governing the transverse motion of an extensible beam. A useful tool for the study of abstract second-order equations is the theory of strongly continuous cosine families of operators (Refs. 7,8). Our aim in this paper is to obtain the sufficient conditions for the controllability of second-order initial value problems (IVP) for a class of damped

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differential inclusions with nonlocal conditions on noncompact intervals, whose existence was proved by Benchohra and Ntouyas9 . Consider the inclusion for the damped second-order system of the form y 00 (t) − Ay(t) ∈ Gy 0 (t) + Bu(t) + F (t, y(t), y 0 (t)), y(0) + g(y) = φ, y 0 (0) = y0 ,

(1)

where the state y(t) takes values in a real Banach space X with the norm k.k and the control u(.) is given in L2 (J, U ), a Banach space of admissible control function with U as a Banach space. B is a bounded linear operator from U to X, g : C(J, X) → X, φ : J → X, y0 ∈ X, J is an unbounded real interval. © A is a linear ªinfinitesimal generator of a strongly continuous cosine family C(t) : t ∈ R in a Banach space X. For the sack of simplicity we choose J = [0, +∞). Here G is bounded linear operator on X and F : J × X × X → 2X is a bounded, closed, convex multivalued map. The method we are going to use is to reduce the controllability problem of (1) to the search for fixed points of a suitable multivalued map on the Frechet space C(J, X). In order to prove the existence of fixed points, we shall rely on a theorem due to Ma1 , which is an extension to multivalued maps between locally convex topological spaces, of Schaefer’s theorem (Ref. 10). 2. Motivations The study of the dynamical buckling of the hinged extensible beam which is either stretched or compressed by axial force in a Hilbert space, can be modeled by the hyperbolic equation Z L ³ ´ ¯ ´ 2 ¯ ∂u ∂2u ∂4u ³ ¯ (ξ, t)¯2 dξ ∂ u + g ∂u , + − α + β (2) ∂t2 ∂t4 ∂t ∂x2 ∂t 0 where α, β, L > 0u(t, x) is the deflection of the point x of the beam at the time t, g is a nondecreasing numerical function, and L is the length of the beam. Equation (2) has its analogue in Rn and can be included in a general mathematical model ¡ 1 ¢ u00 + A2 u + M kA 2 uk2H Au + g(u0 ) = 0, (3) where A is a linear operator in a Hilbert space H, and M and g are real functions. Equation (2) was studied by Patcheu11 and (3) was studied by Matos and Pereira12 . These equations are the special cases of the following second order damped nonlinear differential equation in an abstract space u00 + Au + Gu0 = f (t, u, u0 ); u(0) = u0 , u0 (0) = u1 , where A, B are linear operators.

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3. Preliminaries In this section, we introduce notations, definitions, and preliminary facts from multivalued analysis which will be used throughout this paper. Let Jb be the compact real interval [0, b] (b ∈ N ). Let C(J, X) be the linear metric Frechet space of continuous functions from J into X with the metric (see Ref. 13) d(y, z) =

∞ X 2−b ky − zkb , 1 + ky − zkb

for each y, z ∈ C(J, X),

b=0

where

½

¾

kykb := sup ky(t)k : t ∈ Jb . Let B(X) denote the Banach space of bounded linear operators X into X with standard norm. A measurable function y : J → X is Bochner integrable if and only if kyk is Lebesgue integrable. Let L1 (J, X) denotes the Banach space of continuous functions y : J → X which are Bochner integrable normed by Z ∞ kykL1 = ky(t)kdt < ∞ for all y ∈ L1 (J, X). 0

U p denotes the neighbourhood of 0 in C(J, X) defined by n o U p := y ∈ C(J, X) : kykb < p . The convergence in C(J, X) is the uniform convergence in the compact intervals, i.e.yj → y in C(J, X) if and only if each b ∈ N, kyj − ykb → 0 in C(Jb , X) as j → ∞.M ⊆ C(J, X) is a bounded set if and only if there exists a positive function ξ ∈ C(J, R+ ) such that ky(t)k ≤ ξ(t)

for all t ∈ J

and all y ∈ M.

The Arzela-Ascoli theorem says that a set M ⊆ C(J, X) is compact if and only if for each b ∈ N , M is a compact set in the Banach space (C(Jb , X), k.kb ). ½ ¾ We say that one-parameter family

C(t) : t ∈ R

of bounded linear

operators in B(X) is a strongly continuous cosine family if and only if (1) C(0) = I, I is the identify operator on X; (2) C(t + s) + C(t − s) = 2C(t)C(s) for all s, t ∈ R; (3) The map t 7→ C(t)y is strongly continuous in t on R for each fixed y ∈ X.

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The strongly continuous sine family {S(t) : t ∈ R}, associated to the strongly continuous cosine family {C(t) : t ∈ R} is defined by Z t S(t)y = C(s)y ds, y ∈ X, t ∈ R. 0

Assume the following condition on A. (H1) A is the infinitesimal generator of a strongly continuous cosine family C(t), t ∈ R of bounded linear operators X into itself and the adjoint operator A∗ is densely defined i.e. D(A∗ ) = X ∗ (Ref. 14). The infinitesimal generator of a strongly continuous cosine family C(t), t ∈ R is the operator A : X → X defined by d2 C(t)y|t=0 , y ∈ D(A), dt2 where D(A) = {y ∈ X : C(t)y is twice continuously differentiable in t} i.e. Ay =

D(A) = {y ∈ X : C(.)y ∈ C 2 (R, X)}. Define X1 = {y ∈ X : C(t)y} is once continuously differentiable in t = {y ∈ X : C(.)y ∈ C 1 (R, X)}. For more details on strongly continuous cosine and sine family, we refer to the book of Goldstein15 and papers of Travis and Webb7,8 . We know that that if the multivalued map G1 : X → 2X is completely continuous with nonempty compact values, then G1 is u.s.c. if and only if G1 has a closed graph (i.e. xn → x0 , yn → y0 , yn ∈ G1 (xn ) imply y0 ∈ G1 (x0 )). We assume the following hypotheses: (H2) C(t), t > 0 is compact. (H3) t → B(u(t)) is continuous in t. (H4) The linear operator W : L2 (J, U ) → X defined by Z b Wu = S(t − s)Bu(s)ds 0

f −1 which takes the values in induces a bounded invertible operator W 2 f L (Jb , U )\ ker W, (for construction of W −1 , see Ref. 4), and there exist f −1 k ≤ M3 . positive constants M2 and M3 such that kBk ≤ M2 and kW 0 0 (H5) F : J × X × X → BCC(X); (t, y, y ) 7→ F (t, y, y ) is measurable with respect to t for each y ∈ X, u.s.c. with respect to y, for each t ∈ J and for each fixed y ∈ C(J, X) and z ∈ C(J1 , X) the set SF,y,z = {v ∈ L1 (J, X) : v(t) ∈ F (t, y(t), y 0 (t)) for a.e. t ∈ J} is nonempty.

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(H6) There exists a constant L such that kg(y)k ≤ L,

for y ∈ X.

(H7) kF (t, y, z)k = sup{kvk : v ∈ F (t, y, z)} ≤ p(t)ψ(kyk + ky 0 k) for almost all t ∈ J and all y ∈ X, where p ∈ L1 (J, R+ ) and ψ : R+ → (0, ∞) is continuous and increasing with Z b Z ∞ ds p(s)ds < = ∞, for each b > 0, s + ψ(s) 0 c where c = (M1 + M1∗ )[kφ(0)k + L] + kGkky(t)k + h i + (1 + b) M1 ky(0)k + M1 N b + M1 kGkkφ(0)k n o M1 = sup kC(t)k : t ∈ J , M1∗ = sup{kAS(t)k : t ∈ J} h N = M2 M3 ky1 k + (M1 + M1 b + kGk)kφ(0)k + M1 L + M1 bky0 k Z t Z b i + M1 kGk ky(s)kds + M1 b p(s)ψ(ky(s)k + ky 0 (s)k)ds . 0

0

2

(H8) For fixed u ∈ L (J, X), each neighbourhood U p of 0, y ∈ U p and t ∈ J, the set Z t {C(t) − S(t)G}φ(0) − C(t)g(y) + S(t)y0 + C(t − s)Gy(s)ds 0 Z t Z t + S(t − s)Bu(s)ds + S(t − s)v(s)ds; v ∈ SF,y,y0 } 0

0

is relatively compact. A mild solution of the system (1) is given by Ref. 14, Z t y(t) = {C(t) − S(t)G}φ(0) − C(t)g(y) + S(t)y0 + C(t − s)Gy(s)ds 0 Z t Z t + S(t − s)Bu(s)ds + S(t − s)v(s)ds, (4) 0

0 1

where v ∈ SF,y,y0 = {v ∈ L (J, X) : v(t) ∈ (F (t, y(t), y 0 (t)) for a.e. t ∈ J}. Definition 3.1. System (1) is said to be infinite controllable on J=[0, ∞) if for every φ(0)∈D(A), y0 ∈X1 and y1 ∈ X, there exists a control u∈L2 (Jb , U ) such that the mild solution y(.) of (1) satisfies y(b) + g(y) = y1 . The following lemmas will be used in the proof of our main theorem.

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Lemma 3.1. (Ref. 17) Let I = Jb be a compact real interval and X be a Banach space. Let F be a multivalued map satisfying (H5) and let Γ be a linear continuous mapping from L1 (I, X) to C(I, X), then the operator ΓoSF : C(I, X) → BCC(C(I, X))

y → (ΓoSF )(y) := Γ(SF,y )

is a closed graph operator in C(I, X) × C(I, X). Lemma 3.2. (Ref. 1) Let X be a locally convex space and N1 : X → 2X be a compact convex valued, u.s.c. multivalued map such that there exists a closed neighbourhood U p of 0 for which N1 (U p)is a relatively compact set for each p ∈ N . If the set ½ Ω :=

¾ y ∈ X : λy ∈ N1 (y)

for some λ > 1

is bounded, then N1 has a fixed point. 4. Main Result We now state and prove our main controllability result. Theorem 4.1. Assume that the hypotheses (H1 )–(H8 ) are satisfied. Let g : C(J, X) → X be a continuous function. Then the system (1) is controllable on J. Proof. For fixed b ∈ N consider the space Z = C 1 (J, X) with norm kykz = max{kyk, ky 0 k, t ∈ J}. Using the hypothesis (H5) for an arbitrary function y(·), we define the control " f −1 y1 − {C(b) − S(b)G}φ(0) + C(b)g(y) − S(b)y0 ub (t) = W y

Z

Z

b

−

C(b − s)Gy(s)ds − 0

b

# S(b − s)v(s)ds (t).

0

Using this control we shall show that the operator N1 : Z → 2Z defined by

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N1 y := h ∈ C(J, X) : h(t)  Z t    {C(t) − S(t)G}φ(0) − C(t)g(y) + S(t)y0 + C(t − s)Gy(s)ds    0  Z t h    f −1 y1 − {C(b) − S(b)G}φ(0) + C(b)g(y) − S(b)y0  S(t − η)B W  + =

0

Z b Z b i    − C(b − s)Gy(s)ds − S(b − s)v(s)ds (η)dη    0 0   Z t     S(t − s)v(s)ds : v ∈ SF,y,y0 ; t ∈ J, + 0

where

n o v ∈ SF,y,y0 = v ∈ L1 (J, X) : v(t) ∈ F (t, y(t), y 0 (t))for a.e. t ∈ J ,

has a fixed point. This fixed point is then a solution of equation (4). Clearly y1 − g(y) ∈ (N1 y)(b), which means that the control u steers the system from initial state φ(0) to y1 in time b, provided we obtain a fixed point of the nonlinear operator N1 . In order to study the controllability problem for system (1), we apply fixed point theorem due to Ma1 to the following system: y 00 (t)∈λ−1 Ay(t)+λ−1 Bu(t)+λ−1 Gy 0 (t)+λ−1 F (t, y(t), y 0 (t)), y(0) + g(y) = φ, y 0 (0) = y0 , t ∈ J.

(5)

Let y be a mild solution of system (5). Then for some λ ∈ (0, 1), y(t) = λ−1 {C(t) − S(t)G}φ(0) − λ−1 C(t)g(y) Z t +λ−1 S(t)y0 + λ−1 C(t − s)Gy(s)ds Z +λ−1

0

t

h f −1 y1 − {C(b) − S(b)G}φ(0) + C(b)g(y) S(t − η)B W

0

Z

−S(b)y0 − Z +λ−1

Z

b

C(b − s)Gy(s)ds − 0

b

(6)

i S(b − s)v(s)ds (η)dη

0

t

S(t − s)v(s)ds, v ∈ SF,y,y0 ; t ∈ J. 0

We shall show that N1 (Uq ) is relatively compact for each neighbourhood Uq of 0 ∈ C(J, X) with q ∈ N and the multivalued map N1 has bounded, closed and convex values and it is u.s.c. The proof will be given in several steps.

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Step 1. The set Ω := {y ∈ C(J, X) : λy ∈ N1 (y), λ > 1} is bounded. For that we obtain a priori bounds for the equation (6). We have ky(t)kb ≤ (M1 + M1 bkGk)kφ(0)k + M1 L + M1 bky0 k Z t Z t 2 +M1 kGk ky(s)kds+M1 N b +M1 b p(s)ψ(ky(s)k + ky 0 (s)k)ds. 0

0

Denote by r1 (t) the R.H.S. of the above inequality, we have r1 (0) = (M1 + M1 bkGk)kφ(0)k + M1 L + M1 bky0 k + M1 N b2 andky(t)k ≤ r1 (t), t ∈ Jb . Using the increasing character of ψ, we get r10 (t) ≤ M1 kGkky(t)k + M1 bp(t)ψ(ky(t)k + ky 0 (t)k). But

h i y 0 (t) = λ (AS(t) − C(t)G)φ(0) − AS(t)g(y) + C(t)y0 + Gy(t) Z t + AS(t − s)Gy(s)ds Z t h 0 f −1 y1 − {C(b) − S(b)G}φ(0) + C(b)g(y) − S(b)y0 +λ C(t − η)B W 0 Z b Z b Z t i − C(b − s)Gy(s)ds − S(b − s)v(s)ds (η)dη + λ C(t − s)v(s)ds. 0

0

0

Thus we have ky 0 (t)kb ≤ (M1∗ + M1 bkGk)kφ(0)k + M1∗ L + M1 ky0 k + kGkky(t)k Z t Z t +M1∗ kGk ky(s)kds + M1 N b + M1 b p(s)ψ(ky(s)k + ky 0 (s)k)ds. 0

0

Denote by r2 (t) the R.H.S. of the above inequality, we have r20 (t) ≤ kGkky 0 (t)k + M1∗ kGkky(t)k + M1 p(t)ψ(ky(t)k + ky 0 (t)k) and ky 0 (t)k ≤ r2 (t); t ∈ J. Let w(t) = r1 (t) + r2 (t), then c = w(0) = r1 (0) + r2 (0) = (M1 + M1∗ )[kφ(0)k + L] + kGkky(t)k h i +(1 + b) M1 ky(0)k + M1 N b + M1 kGkkφ(0)k .

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Also, 0 0 w0 (t) h = r1 (t) + r2 (t)

i ≤ (M1 + M1∗ )ky(t)k + ky 0 (t)k kGk + (1 + b)[M1 p(t)ψ(ky(t)k + ky 0 (t)k) ≤ (M1 + M1∗ )kGkr1 (t) + kGkr2 (t) + (1 + b)[M1 p(t)ψ(r1 (t) + r2 (t)) ³ ´ = pb(t) w(t) + ψ(w(t)) , t ∈ J, where pb(t) = max{(1 + b)M1 p(t), (M1 + M1∗ )kGk, kGk}. This implies that for each t ∈ Jb Z b Z ∞ Z w(t) ds ds pb(s)ds < ≤ = ∞. s + ψs s + ψs c 0 c This inequality implies that there exists a constant K such that r1 (t) + r2 (t) = w(t) ≤ K, t ∈ Jb . Then kykz = max{ky(t)k, ky 0 (t)k)} ≤ K, where K depends only on b and on the functions p and ψ. This shows that Ω is bounded. Step 2. N1 y is convex for each y ∈ C(J, X). Indeed if h1 , h2 ∈ N1 y then there exist v1 , v2 ∈ SF,y,y0 such that for each t ∈ J, let 0 ≤ α ≤ 1 we have (αh1 + (1 − α)h2 )(t)

Z

t

= {C(t) − S(t)G}φ(0) − C(t)g(y) + C(t − s)Gy(s)ds + S(t)y0 0 Z t + S(t − s)[αv1 (s) + (1 − α)v2 (s)]ds Z0 t h f −1 y1 − {C(b) − S(b)G}φ(0) − C(b)g(y) + S(t − η)B W 0 Z t Z b i +S(b)y0 + C(t−s)Gy(s)ds − S(b − s)αv1 (s) + (1−α)v2 (s)ds (η)dη. 0

0

Since SF,y,y0 is convex as F is convex, then v = αh1 + (1 − α)h2 ∈ SF,y,y0 and hence αh1 + (1 − α)h2 ∈ N1 y. Step 3. N1 (Uq ) is bounded in C(J, X) for each q ∈ N . Indeed, it is enough to show that there l1 ½ exists a positive constant ¾ such that for each h ∈ N1 y, y ∈ Uq = y ∈ Z : kyk∞ ≤ q , one has

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khk∞ ≤ l1 . In other words, we have to bound the sup-norm of both h and h0 . If h ∈ N1 y, then there exists v ∈ SF,y,y0 such that for each t ∈ Jb we have Z t C(t − s)Gy(s)ds h(t) = {C(t) − S(t)G}φ(0) − C(t)g(y) + S(t)y0 + 0 Z t Z t h f −1 y1 − {C(b) − S(b)G}φ(0) S(t − s)v(s)ds + S(t − η)B W + 0 0 Z b Z b i + C(b)g(y) − S(b)y0 − C(b − s)Gy(s)ds − S(b − s)v(s)ds (η)dη. 0

0

By (H4), (H6) and (H7), we have for each t ∈ Jb that Z

t

kh(t)kb ≤ (M1 + M1 bL)kφ(0)k + M1 L + M1 bky0 k + M1 kGk ky(s)kds 0 µZ ¶ Z t t ³ ´ h +M1 b sup ψ ky(t)k + ky 0 (t)k p(s)ds + kS(t − η)kM2 M3 ky1 k t∈[0,q] 0 0 Z b +(M1 + M1 bL)kφ(0)k + M1 L + M1 bky0 k + M1 kGk ky(s)kds 0 ÃZ ! b ³ ´ i +M1 b sup ψ ky(t)k + ky 0 (t)k p(s)ds dη. t∈[0,q]

0

Then for each h ∈ N1 (Uq ) we have khk∞ ≤ l1 . Also, kh0 (t)kb ≤ (M1∗ + M1 L)kφ(0)k + M1∗ kg(y)k Z t +M1 ky0 k + kGkky(t)k + M1∗ bkGk ky(s)kds 0

h +M1 M2 M3 b ky1 k + (M1 + M1 bL)kφ(0)k + M1 kg(y)k + M1 bky0 k Z +M1 kGk 0

+M1

b

³ ´³ Z ky(s)kds + M1 b sup ψ ky(t)k + ky 0 (t)k t∈[0,q]

b

´i p(s)ds

0

¶ ³ ´ µZ t 0 sup ψ ky(t)k + ky (t)k p(s)ds . t∈[0,q]

0

Then for each h ∈ N1 (Uq ) we have kh0 k∞ ≤ l2 . Step 4. N1 (Uq ) is equicontinuous sets of Uq ∈ Z for each q ∈ N . That is the family h ∈ N1 y : y ∈ Uq is equicontinuous. Let t1 , t2 ∈ Jb , 0 < t1 < t2 < b and Uq be a neighbourhood of 0 in Z for q ∈ N . For each y ∈ Uq and h ∈ N1 y, we have

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kh(t1 ) − h(t2 )k ≤ kC(t1 )φ(0) − C(t2 )φ(0)k + kC(t1 ) − C(t2 )kkGk + kS(t1 )y0 − S(t2 )y0 k Z b Z b Z t2 +M1 (t1 − t2 ) kGkky(s)k + M1 (t1 − t2 ) kv(s)kds + M1 b kv(s)kds 0 t1 Z0 b h £ ¤ M2 M3 ky1 k + M1 kφ(0)k + kGk + bky0 k + bkGkkφ(0)k +M1 (t1 − t2 ) 0 Z b Z t2 Z b i +M1 kGkky(s)kds + M1 b kv(s)kds dη + M1 b M2 M3 [ky1 k 0

0

t1

+M1 [kφ(0)k + kGk + bky0 k + bkGkkφ(0)k] Z b Z b +M1 kGkky(s)kds + M1 b kv(s)kds]dη. 0

0

By using (H6), (H7) and continuity of C(t) and S(t), we see that the right-hand side of the above inequality tends to zero as (t2 − t1 ) −→ 0. The compactness of C(t), S(t) for t > 0 implies the continuity in the uniform operator topology. The compactness of S(t) follows from that of C(t). In an analogous way one can obtain a similar estimate for kh0 (t1 ) − h0 (t2 )k. Thus N1 (U q) maps U q into an equicontinuous family of functions. It is easy to see that the family N1 (U q) is uniformly bounded. The above estimate implies the required equicontinuity. This also proves the relative compactness of N1 (Uq ). Now it remains to prove the uppersemicontinuity (u.s.c.) of N1 . Equivalently, it is enough to prove that N1 has a closed graph. We do this in the next step using Lemma 3.1. Step 5. N1 has a closed graph. Let yn −→ y ∗ , hn −→ h∗ and hn ∈ N1 (yn ). We shall prove that h∗ ∈ N1 y ∗ . Since hn ∈ N1 (yn ) means that there exists vn ∈ SF,yn ,yn0 such that Z t Z t f −1 hn (t)=C(t)[φ(0)−g(yn )]+S(t)y0 + S(t−s)vn (s)ds + S(t−η)B W 0 0 Z b h £ ¤ ¤ y1 − g(y) − C(b) φ(0) − g(yn ) − S(b)y0 − S(b − s)vn (s)ds (η)dη. 0

We must prove that there exists v ∗ ∈ SF,y∗ such that

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Z t £ ¤ S(t − s)v ∗ (s)ds h∗ (t) = C(t) φ(0) − g(y ∗ ) + S(t)y0 + 0 Z t h f −1 y1 − g(y) − C(b)[φ(0) − g(y ∗ )] − S(b)y0 + S(t − η)B W Z0 b i − S(b − s)v ∗ (s)ds (η)dη.

(7)

0

Then the idea is to use the facts (1) hn −→ h∗£; ¤ (2) hn − C(t) φ(0) − g(yn ) − S(t)y0 ∈ Γ(SF,yn ), where Γ : L1 (J, X) −→ C(J, X), defined by Z t Z t f −1 (Γv)(t) = S(t − s)v(s)ds + S(t − η)B W 0 0 Z b h i £ ¤ y1 − g(y) − C(b) φ(0) − g(y) − S(b)y0 − S(b − s)v(s)ds (η)dη 0

If Γ ◦ SF is a closed graph operator, we would be done. But we do not know whether Γ ◦ SF is a closed graph operator. So we cut the functions yn , hn − C(t)φ(0) + C(t)g(yn ) − S(t)y0 , gn and we consider them defined on the interval [k, k + 1] for any k ∈ N ∪ {0}. Then, using Lemma 3.1, in this case we are able to affirm that (7) is true on the compact interval [k, k + 1], i.e. Z t Z t ∗ ∗k ∗ h (t)|[k,k+1] =C(t)[φ(0)−g(y )]+S(t)y0 + S(t−s)v (s)ds + S(t−η) 0 0 Z b h i −1 ∗ ∗k f BW y1 −g(y)−C(b)[φ(0)−g(y )]−S(b)y0 − S(b−s)v (s)ds (η)dη. 0 ∗k

1

∗

for a suitable L − selection v of F (t, y (t)) on the interval [k, k + 1]. k At this point we can paste the functions v ∗ obtaining the selection v ∗ defined by k

v ∗ (t) = v ∗ (t),

for t ∈ [k, k + 1).

We obtain then that v ∗ is an L1 − selection and (7) will be satisfied. We give now the details. Clearly we have that °¡ ¢ ¡ ¢° ° ° ° hn −C(t)φ(0)+C(t)g(yn )−S(t)y0 − h∗ −C(t)Φ(0)+C(t)g(y ∗ )−S(t)y0 °

∞

−→ 0 as n −→ ∞. Now, we consider for all k ∈ N ∪ {0}, the mapping SFk : C([k, k + 1]; X) −→ L1 ([k, k + 1]; X)

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n o ¡ ¢ k u 7→ SF,u := f ∈ L1 [k, k+1]; X : f (t) ∈ F (t, u(t)) for a.e. t ∈ [k, k+1] . Also, we consider the linear continuous operators Γk : L1 ([k, k + 1], X) −→ C([k, k + 1]; X) Z v 7→ Γk (v)(t) =

Z

t

S(t − s)v(s)ds + 0

t

f −1 S(t − η)B W

0

Z h y1 − g(y) − C(b)[φ(0) − g(y)] − S(b)y0 −

b

i S(b − s)v(s)ds (η)dη.

0

Γk oSFk

is a closed graph operator for all From Lemma 3.1, it follows that k ∈ N. Moreover, we have that ³ ´ k hn (t) − C(t)φ(0) + C(t)g(yn ) − S(t)y0 |[k,k+1] ∈ Γk (SF,y ). n Since yn −→ y ∗ , it follows from Lemma 3.1 that ³ ´ h∗ (t) − C(t)φ(0) + C(t)g(y ∗ ) − S(t)y0 |[k,k+1] Z t Z t h k f −1 y1 −g(y)−C(b)[φ(0)−g(y ∗ )] = S(t−s)v ∗ (s)ds+ S(t−η)B W 0 0 Z b i ∗k −S(b)y0 − S(b−s)v (s)ds (η)dη 0 k

k ∗ for some v ∗ ∈ SF,y defined on J by ∗ . So the function v k

v ∗ (t) = v ∗ (t),

for t ∈ [k, k + 1)

is in SF,y∗ , since v ∗ (t) ∈ F (t, y ∗ (t)) for a.e. t ∈ J. Set X := C(J, X). As a consequence of Lemma 3.2, we deduce that N1 has a fixed point (in Z). This means that any fixed point of N1 is a mild solution of (1) on J satisfying (N1 y)(t) = y(t). Thus, system (1) is controllable on J. ¤ References 1. T.W. Ma, Topological degrees for set-valued compact vector fields in locally convex spaces, Dissertationess Math., 92 (1972), 1–43. 2. M. Benchohra and S.K. Ntouyas, Controllability for an infinite time horizone of second order differential inclusions in Banach spaces with nonlocal condition, J. Opt. Theory and Appl., 109, (2001), 85–98. 3. M. Benchohra, E.P. Gatsori and S.K. Ntouyas, Nonlocal quasilinear Damped Differential Inclusions, Electronic J. of Diff. Equations., 2002 (2002), 1–14.

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4. Byszewski, L., Theorems about the existence and uniqueness of solution of semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162 (1991), 494–505. 5. W.E. Fitzgibbon, Global existence and boundedness of solutions to the extensible beam equation, SIAM J. Math. Anal., 13 (1982), 739–745. 6. J. Ball, Initial boundary value problems for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61–90. 7. C.C. Travis and G.F. Webb, Cosine families and abstract nonlinear secondorder differential equations, Acta Math. Acad. Sci. Hung., 32 (1978), 75–96. 8. C.C. Travis and G.F. Webb, Second-order differential equations in Banach spaces, (Proc. Int. Symp. on Nonlinear Equations in Abstract spaces, Academic Press New-York, 1978), 331-361. 9. M. Benchohra and S.K. Ntouyas, Existence of mild solutions on noncompact intervals to second-order initial value problems for a class of differential inclusions with nonlocal conditions, Computers and Mathematics with Applications, 39 (2000), 11–18. 10. H. Schaefer, Uber die method der a priori Schranken, Math. Anal., 129 (1955), 415–416. 11. S.K. Patcheu, On the global solution and asymptotic behaviour for the generalized damped extensible beam equation, J. Differential Equations, 135 (1997), 299–314. 12. M. Matos and D. Periera, On a hyperbolic equation with strong damping, Funkcial. Ekvac., 34 (1991), 303–311. 13. J. Dugunji and A. Granas, Fixed Point Theory, (Monografic. Math. PWN. Warsaw, 1982). 14. J. Bochenek, An abstract nonlinear second-order differential equation, Annales Polonici Mathematici, 54 (1991), 155–166. 15. J.A. Goldstein, Semigroups of Linear Operators and Applications, (Oxford Univ. Press, New York, 1985). 16. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, (Springer-Verlag, New-York, 1983). 17. A. Lasota and Z. Opial, An application of the Kakutani-Ky-Fan theorem in the theory of ordinary differential equaitons, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 13 (1965), 781–786.

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AVERAGING OF EVOLUTION INCLUSIONS IN BANACH SPACES TZANKO DONCHEV Department of Mathematics University of Architecture and Civil Engineering 1 ”Hr. Smirnenski” str., 1046 Sofia, Bulgaria E-mail: [email protected] We study semi-linear evolution inclusion in a Banach space E with uniformly convex dual E ∗ . Averaging and partial averaging results on a finite interval is proved, when A is linear m-dissipative and F , G are one sided Lipschitz. When F (t, ·) is one sided Lipschitz with negative constant we prove such results on an infinite interval. Keywords: Averaging; One sided Lipschitz; Evolution inclusions.

1. Preliminaries We study the following evolution inclusion: x(t) ˙ ∈ Ax + F (t, x) + εG(t, x), x(0) = x0 , t ∈ [0, ∞),

(1)

where A is densely defined linear operator with closed graph, generating a continuous semigroup. The multi-functions F (·, ·) and G(·, ·) are almost upper hemicontinuous with convex weakly compact values. The state space E is Banach with uniformly convex dual E ∗ . Through the paper we suppose F and G are bounded on the bounded sets. We denote by Sol(1) the solution set of (1). We refer to Refs. 1,2 for the theory of semilinear differential equations and inclusions in Banach spaces (see also Refs. 3,4). Further we assume that the following limit: Z 1 t+T G0 (x) = lim G(τ, x) dτ (A) T →∞ T t exists uniformly on the bounded subsets of E. Consider the averaged evolution inclusion: x(t) ˙ ∈ Ax + F (t, x) + εG0 (x), x(0) = x0 , t ∈ [0, ∞).

(2)

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We want to show that the Hausdorff distance between the solution set Sol(ε) of (1) and the solution set Sol0 (ε) of (2) converges to 0. This averaging principle is the main result of the paper. The problem (2) is often much simpler that the problem (1). This method is very comprehensively studied (in case A ≡ 0 and F ≡ 0) in the works of Plotnikov at al. We note Refs. 5–8 and the references therein. In Refs. 1,9 the averaging method is studied in case F ≡ 0. Our goal here is first to extend the averaging principle to the more general case (1)–(2) and also to replace the commonly used Lipschitz assumption on F and G with the weaker one sided Lipschitz. Let A : D(A) ⊂ E → E be closed densely defined linear operator, where E is a Banach space with uniformly convex dual E ∗ . We assume that A is m-dissipative, i.e. hJ(x), Axi ≤ 0 for every x ∈ D(A) and the range R(λI − A) = E for some (equivalently for any) λ > 0. With J(x) = {l ∈ E ∗ hl, xi = |x|2 = |l|2 } we denote the duality map. It is well known that J(·) is single valued and uniformly continuous on the bounded sets when E ∗ is uniformly convex (see e.g. Ref. 10). Denote by B the open unit ball in E. Let A, B ⊂ E be nonempty closed bounded. We let Ex(A, B) = sup inf |a − b| and DH (A, B) = max{Ex(A, B), Ex(B, A)}. a∈A b∈B

The family T (t) : E → E of bounded linear operators satisfying: 1) T (0) = I (the identity operator) 2) T (t + s) = T (t)T (s) = T (s)T (t), ∀t, s ≥ 0 3) t → T (t)x is continuous ∀x ∈ E is called C0 semigroup. The semigroup {T (t)}t≥0 is called compact when for every t > 0 and every bounded B the set T (t)B is compact. The densely defined linear operator A : D(A) → E is said to be an infinitesimal generator of a C0 (compact) semigroup {T (t)}t≥0 if lim +

∆t→0

T (∆t) − I x = Ax, ∀x ∈ D(A). ∆t

Let B ⊂ E be bounded for l ∈ E ∗ denote by σ(l, B) = sup hl, ai the a∈B

support function. Let M be a Banach space, the multifunction F :M⇒E is said to be upper hemicontinuous (UHC) when the support function σ(l, F (·)) is upper semicontinuous as a real valued function. It is easy to see that if F is bounded on the bounded sets with nonempty convex weakly compact values then F (·) is UHC if and only if its graph Graph(F ) is (M–strong) × (E–weak) closed. F : I × E ⇒ E is said to be almost UHC

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if for every ε > 0 there exists Iε ⊂ I with meas(I \ Iε ) < ε such that F (·, ·) is UHC on Iε × E. Further we assume that F and G are bounded on [0, ∞) × B for every bounded B ⊂ E. Definition 1.1. The multifunction F : [0, T ] × E ⇒ E is said to be one sided Lipschitz (OSL) when there exists a constant L such that for every x, y ∈ E σ(J(x − y), F (t, x)) − σ(J(x − y), F (t, y)) ≤ L|x − y|2 . Definition 1.2. The continuous function y(·) is said to be mild solution of (1) when there exists a (strongly) measurable selection h(t) ∈ F (t, y(t)) + εG(t, y(t)) such that Z t y(t) = T (t)x0 + T (t − s)h(s) ds. 0

We will skip the word mild, i.e. the mild solutions will be called solutions. It is appropriate to check that the mild solutions (since A is linear) coincide with the integral solutions, i.e. x(0) = x0 and there exists a (strongly) measurable selection h of F + εG such that for every u ∈ D(A) and 0 ≤ s ≤ t: Z t |x(t) − u|2 ≤ |x(s) − u|2 + 2 hJ(x(τ ) − u), h(τ ) − Aui dτ. s

It is easy to see that h(t) = f (t) + εg(t), where f (t) ∈ F (t, x(t)) and g(t) ∈ G(t, x(t)) are (strongly) measurable. In the paper we study under some natural conditions the proximity between the solution sets of (1) and (2). This problem seems to be a particular case of the singularly perturbed problems (compare Ref. 11). However in our case the assumptions on F (·, ·) are weaker. 2. The Results In this section we prove the main results of the paper. First we prove the existence result when G(t, x) ≡ 0. Standing Hypotheses (SH): A1. A is densely defined closed m-dissipative linear operator, which is an infinitesimal generator of C0 semigroup. A2. A generates a compact semigroup.

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F1. F (·, ·) and G(·, ·) are almost UHC with nonempty convex weakly compact values, which satisfy the OSL condition with constants LF and LG respectively. Theorem 2.1. Under A1, F1 the differential inclusion: x(t) ˙ ∈ Ax + F (t, x(t)), x(0) = x0

(EI)

admits nonempty C([0, S], E) bounded closed set of (mild) solutions on every interval [0, S], where S > 0. Proof. The proof is the same as in the proof of Theorem 2.4 of Ref. 12 with only small obvious modifications. We have only to see that the solution set is uniformly bounded. So let x(·) be a solution. Due to A1, F1 we have that there exists a positive constant M such that |F (s, 0)| ≤ M ∀ s ∈ [0, S]. Further: Z t |x(t)|2 ≤ 2 (σ(J(x(s)), F (s, x(s))) − hJ(x(s)), A(0))ids 0 Z t ≤2 hσ(J(x(s) − 0), F (s, x(s))) − σ(J(x(s) − 0), F (s, 0)) 0

Z i.e. |x(t)|2 ≤ 2

t

¡

+σ(J(x(s)), F (s, 0)))i ds, ¢ L|x(s)|2 + M |x(s)| ds.

0

d |x(t)|2 ≤ 2L|x(t)|2 + 2M |x(t)| ≤ (2L + 1)|x(t)|2 + M 2 . dt The proof is complete thanks to Gronwall’s inequality. Thus

Denote by Sol(EI) the solution set of (EI). Corollary 2.1. Suppose A1, F1 are valid. If moreover, LF < 0 then lim+ DH (Sol(ε), Sol(EI)) = 0 with respect to the uniform metric in [0, ∞).

ε→0

Proof. For ε > 0 sufficiently small one has that the multifunction F (t, x) + εG(t, x) is OSL with negative OSL constant LF + εLG . It is standard to prove that since F and G are bounded on the bounded sets there exist constants K and M such that |x(t)| ≤ K and |F (t, x(t) + B)| + |G(t, x(t + B))| ≤ M . Let x(·) be a solution of (1). Then x(·) is a solution of x(t) ˙ = Ax + f (t) + εg(t), where f (t) ∈ F (t, x(t)) and g(t) ∈ G(t, x(t)). We define the following multifunction: H(t, y) = {u ∈ F (t, x) : hJ(x(t) − y), f (t) − ui ≤ LF |x(t) − y|2 .

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It is easy to see that H(·, ·) is almost UHC. Let y(·) be a solution of y(t) ˙ ∈ Ay + H(t, y(t)), y(0) = x0 . From Theorem 1.2 in Ref. 12 (taken there from Ref. 13) we know that: Z t |x(t) − y(t)|2 ≤ 2 hJ(x(s) − y(s)), x(s) ˙ − y(s)i ˙ ds. 0

Z

t

2

Z

t

2

LF |x(s) − y(s)| ds + 2ε

Consequently |x(t) − y(t)| ≤ 2 0

σ(J(x(s) − 0

y(s)), G(s, x(s)))ds. However, there exists a constant M such that Z t |G(t, x(t))| ≤ M . Hence |x(t) − y(t)|2 ≤ 2 LF |x(t) − y(t)|2 + 0 Z t d 2ε M |(x(s) − y(s)|ds. Therefore |x(t) − y(t)|2 ≤ (2LF + M ε)|x(t) − dt 0 y(t)|2 + M ε. Since 2LF + M ε < 0 for sufficiently small ε, one has that |x(t) − y(t)|2 ≤ M ε. S1. There exists δ > 0 and a bounded set D ⊂ E such that the solution sets of (1) and (2) are contained in D for every ε < δ. Let K = |D|. For every x, y ∈ D one has |x − y| ≤ 2K. As it was pointed out the normalized duality map J(·) is uniformly continuous on 2KB. Denote by ΩJ (δ) := sup {|J(x) − J(y)| : |x| ≤ K, |y| ≤ K} its |x−u|≤δ

module of continuity. Further we need the following variant of Filippov–Pliss lemma. Lemma 2.1. Assume A1, A2, F1, S1 hold. Let η > 0 be given. Then there exists a constant C(ε) such that if x(·) is a solution of x(t) ˙ ∈ Ax(t) + F (t, x(t) + ηB) + εG(t, x(t) + ηB), x(0) = x0 , then there exists a solution y(·) of y(t) ˙ ∈ Ay(t) + F (t, y(t)) + εG(t, y(t)), y(0) = x0 , p such that |x(t) − y(t)| ≤ C(ε) η + ΩJ (η) on [0, ε−1 ]. If LF ≤ 0, then C does not depend on ε. Proof. It is easy to see that there exist strongly measurable functions |f˜(t)| < η and |˜ g (t)| < η such that x(t) ˙ ∈ Ax(t) + F (t, x(t) + f˜(t)) + εG(t, x(t) + g˜(t)), x(0) = x0 . Moreover there exist strongly measurable selections f (t) ∈ F (t, x(t) + f˜(t)) and g(t) ∈ G(t, x(t) + g˜(t)). Define the multi-maps F1 (t, z) := {v ∈ F (t, z) : hJ(x(t) − z), f (t) − vi ≤ LF |x(t) − x|2 }

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and G1 (t, z) := {v ∈ G(t, z) : hJ(x(t) − z), g(t) − vi ≤ LG |x(t) − x|2 }. It is easy to see that F1 (·, ·) and G1 (·, ·) are almost UHC with nonempty convex weakly compact values. Therefore there exist a solution y(·) of y(t) ˙ ∈ Ay(t) + F1 (t, y(t)) + εG1 (t, y(t)), y(0) = x0 . It it easy to see that there exist strongly measurable selections f1 (t) ∈ F1 (t, y(t)) and g1 (t) ∈ G1 (t, y(t)) such that y(t) ˙ = Ay(t) + f1 (t) + εg1 (t), y(0) = x0 . Consequently hJ(x(t) − y(t) + f˜(t)), f (t) − f1 (t)i ≤ LF |x(t) + f˜(t) − y(t)|2 and hJ(x(t) − y(t) + g˜(t)), g(t) − g1 (t))i ≤ LG |x(t) + g˜(t) − y(t)|2 . Therefore hJ(x(t) − y(t)), f (t) − f1 (t)i ≤ LF |x(t) − y(t)|2 +|J(x(t) − y(t)) − J(x(t) − y(t) + f˜(t))| (|f (t)| + |fi (t)|) ¯ ¯ ¯ ¯ + ¯LF |x(t) − y(t)|2 − LF |x(t) − y(t) + f˜(t)|2 ¯ ≤ LF |x(t) − y(t)|2 + 2M ΩJ (η) + |LF |(4K + η)η. Analogously, hJ(x(t) − y(t) + g˜(t)), g(t)−g1 (t))i ≤ LG |x(t)−y(t)|2 + d |x(t) − y(t)|2 ≤ 2M ΩJ (η) + |LG |(4K + η)η. Consequently, dt 2 2(LF + εLG )|x(t) − y(t)| + 4M (1 + ε)ΩJ (η) + 2(|LF | + ε|LG |)(4K + η)η. Therefore there exists a constant C(ε) such that |x(t) − y(t)| ≤ p d C(ε) η + ΩJ (η) on [0, ε−1 ]. If LF ≤ 0, then |x(t) − y(t)|2 ≤ dt 2εLG |x(t) − y(t)|2 + 4M (1 + ε)ΩJ (η) + 2(|LF | + ε|LG |)(4K + η)η. Due to Gronwall’s inequality in this case C does not depend on ε (on [0, ε−1 ]). Now we are ready to prove our main result: Theorem 2.2. Let LF ≤ 0. Under SH and S1 the solution sets R1 of (1) and R2 of (2) are nonempty and compact. Furthermore for every δ > 0 there exists ε(δ) > 0 such that for every 0 < ε ≤ ε(δ) one has DH (R1 , R2 ) ≤ δ on [0, ε−1 ]. Remark 2.1. It is assumed in Ref. 11 (for more general singularly perturbed problem) that LF < 0 (this assumption is crucial) which does not cover the classical case F (t, x(t)) ≡ 0. Here we permit LF = 0. Proof. Notice that the proof must be given only in case LF = 0. If LF < 0 Theorem 1.2 trivially follows from Corollary 2.1 (in this case S1 is redundant). Since A generates a compact semigroup, one has that R1 and R2 are

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compact sets on [0, ε−1 ]. Furthermore the multifunction G0 (·) is also OSL with a constant LG . One can choose mε → ∞ with εmε → 0 as ε → 0 such 1 that, denoting one has: εmε ¶ µ Z 1 t+q G(s, x)ds = 0, [0, ε−1 ] × B. lim DH G0 (x), ε→0 q t Furthermore it is easy to see that for every δ > 0 there exists mε such that DH (Soli (ε, G), R1 ) < δ, as well as DH (Soli (ε, G0 ), R2 ) < δ. Here Soli (ε, G) is the solution set of x(t) ˙ ∈ Ax + F (t, x) + εG(t, xi ), xi = x(ti ), ti = iq, i = 1, . . . , [mε ]. Soli (ε, G0 ) is defined analogously, replacing G by G0 . Due to S1 every solution of (1) or (2) is uniformly continuous on [0, ∞). STEP 1. Let x(·) be a solution of (2) and let δ > 0 be given. There exists a strongly measurable selection g(t) ∈ G(x(t)) such that x(·) is a solution of x(t) ˙ = Ax(t) + F (t, x) + εg(t). i . εmε Let z(·) be a solution of z(t) ˙ ∈ Az+F (t, z)+εgi (t) such that z(ti ) = x(ti ) and gi (t) ∈ G0 (xi ) for i = 0, 1, · · · , mε . Due to S1 there exists M > 0 such that |F (t, x)| ≤ M and |G(x)| ≤ M on D for t ∈ [0, T ]. That is x(·) is uniformly continuous and hence for every δ > 0 one can choose ε and m that |x(t) − z(t)| ¯ εZsuch ¯ ≤ δ. Moreover one can choose gi (·) such that ¯ 1 ti+1 ¯ ¯ (gi (t) − g(t)) dt¯¯ < 2δ. Define the multifunction: ¯q Let mε be the numbers from the beginning of the proof. Denote ti =

ti

F˜ (t, u) := {v ∈ F (t, u) : σ(J(z(t) − u), F (t, z(t))) − hJ(z(t) − u), vi ≤ LF |z(t) − u|2 } Let fi (t) ∈ F˜ (t, z(t)) For i = 0, 2, · · · , mε define successively y(0) = x0 and y(t) ˙ = Ay + fi (t) + εgi (t)) for t ∈ [ti , ti+1 ]. Therefore |y(t) − z(t)| < 2δ. Hence |y(t) − x(t)| < 4δ. Consequently we have y(t) ˙ ∈ Ay(t) + F (t, y(t) + 4δB) + εG(t, y(t) + 4δB). Lemma 2.1 applies. Thus there exists a solution y˜(·) of (1) which is sufficiently closed to x(·) on [0, ε−1 ].

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STEP 2. Let y(·) be a solution of (1) and let δ > 0 be given. Let i mε be the numbers from the beginning of the proof. Denote ti = . εmε There exists a strongly measurable selection g˜(t) ∈ G(t, y(t)) such that y(t) ˙ ∈ Ay(t) + F (t, y(t)) + g˜(t), y(0) = x0 . Now one can continue as in the proof of previous step. Notice that Theorem 1.2 is not true for LF > 0: Example 2.1. Consider the following differential equation: ε x˙ = x + , x(0) = 0, t ≥ 0 t+1 ³ ´ with solution xε (t) ≥ ε2 et/ε−1 → ∞ as ε → 0 for every t > 0. The averaged system is: x˙ = x, x(0) = 0, t ≥ 0 with the only solution x(t) ≡ 0. When F ≡ 0 one can study the (asymptotic) behavior of x(t) ˙ ∈ Ax + εG(t, x), x(0) = x0

(DI)

x(t) ˙ ∈ Ax + εG0 (x), x(0) = x0 .

(ADI)

and respectively

In this case Theorem 1.2 has the form. Corollary 2.2. Under the conditions of Theorem 1.2 for every δ > 0 there exists ε(δ) > 0 such that for every 0 < ε ≤ ε(δ) one has DH (R(DI) , R(ADI) ) ≤ δ on [0, ε−1 ]. Theorem 2.3. Let all the assumptions of Theorem 1.2 hold. If LF < 0 then for any δ > 0 there exists ε(δ) > 0 such that for every 0 < ε ≤ ε(δ) one has DH (R1 , R2 ) ≤ δ on [0, ∞). Proof. The statements follows from Corollary 2.1. Remark 2.2. Theorems 1.2 and 2.1 hold true when (A) is replaced by lim DH (Graph(G0 ), Graph(GT )) = 0,

T →∞

where GT (x) = function G(·).

1 T

Z

t+T

G(τ, x) dτ and Graph(G) is the graph of the multit

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Using the same fashion as in the proof of Theorem 1.2 one can prove the following Filatov–Hapaev type result (cf. ch. II of Ref. 14). Theorem 2.4. Assume A1, A2, F1, S1 hold with LF ≤ 0. Ã Z ! 1 t+T a) If lim Ex G(τ, x) dτ, G0 (x) 7→ 0 uniformly on x ∈ D and T →∞ T t if G0 (·) is OSL (G(·, ·) – not necessarily), then for every δ > 0 there exists ε(δ) > 0 such that for every 0 < ε ≤ ε(δ) one has Ex(R1 , R2 ) ≤ δ on [0, ε−1 ]. Ã ! Z 1 t+T b) If lim Ex G0 (x), G(τ, x) dτ 7→ 0 uniformly on x ∈ D and T →∞ T t if G(·, ·) is OSL (G0 (·) – not necessarily), then for every δ > 0 there exists ε(δ) > 0 such that for every 0 < ε ≤ ε(δ) one has Ex(R2 , R1 ) ≤ δ on [0, ε−1 ]. Proof. The proof of a) is the same as the proof of Step 1 of Theorem 1.2 and the proof of b) is the proof of Step 2. References 1. M. Kamenski, V. Obukhovskii, P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, (De Gruyter, Berlin, 2001). 2. R. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, (Willey, New York, 1976). 3. S. Hu, N. Papageorgiou, Handbook of Multivalued Analysis, vol. I Theory, Kluwer, 1997, vol. II Applications, (Kluwer, Dordrecht, 2000). 4. I. Vrabie, Compactness Methods for Nonlinear Evolutions, Second Edition, Pitman Monographs and Surveys in Pure and Applied Mathematics 75, (Longman, 1995). 5. V. Plotnikov, A. Plotnikov, A. Vityuk, Differential Equations with Multivalued Right–Hand Side. Asymptotic Methods (in Russian), (Astro Print, Odessa, 1999). 6. V. Plotnikov, V. Savchenko, About averaging of differential inclusions (in Russian), Ukr. Math. Journal, 48 (1996), 1572–1575. 7. T. Janiak, E. Luczak-Kumorec, Method of Averaging for IntegralDifferential Inclusions, Studia Univ. Babes Bolyai, Math., 39 (1994). 8. T. Zverkova, Ground of the averaging method when the Lipschitz condition is violated (in Russian), Ann. Higher Schools Sofia, 23 (1987), 41–48. 9. J. Couchouron, M. Kamenski, An abstract topological point of view and a general averaging principle in the theory of differential inclusions, Nonlinear Analysis, 42 (2000), 625–651.

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10. K. Deimling, Multivalued Differential Equations, (De Gruyter, Berlin, 1992). 11. T. Donchev, A. Dontchev, Singular perturbations in infinite-dimensional control systems, SIAM J. Control Optim., 42 (2003), 1795–1812. 12. T. Donchev, Multivalued perturbations of m-dissipative differential inclusions in uniformly convex spaces, New Zeland J. Math., 31 (2002), 19–32. 13. V. Lakshmikantham, S. Leela, Nonlinear Differential Equations in Abstract Spaces, (Pergamon, Oxford, 1981). 14. O. Filatov, M. Hapaev, Averaging of Systems of Differential Inclusions (in Russian), (Moskow University Press, Moskow, 1998). 15. N. Papageorgiou, Mild solutions of semilinear evolution inclusions, Indian J. Pure Appl. Math., 26 (1995), 189–216.

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INFLUENCE OF VARIABLE PERMEABILITY ON VORTEX INSTABILITY OF A HORIZONTAL COMBINED FREE AND MIXED CONVECTION FLOW IN A SATURATED POROUS MEDIUM A.M. ELAIWa , F.S. IBRAHIMb and A.A. BAKRa a

Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt. E-mail: a m [email protected]

b

Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt.

A linear stability theory is used to analyze the vortex instability of combined free and mixed convection boundary layer flow in a saturated porous medium incorporating the variation of permeability and thermal conductivity due to paking particles with non-uniform temperature. Local Nusselt number for the base flow are presented for both uniform permeability (UP) and variable permeability (VP) cases. An implicit finite difference method is used to solve the base flow and the resulting eigenvalue problems are solved numerically. The critical Peclet and Rayleigh numbers and the associated wave number are obtained for both UP and VP cases. The results indicate that the inertia coefficients reduces the heat transfer rate and destabilizes the flow to the vortex mode of disturbance. The effect of variable permeability tends to increase the heat transfer rate and destabilize the flow to the vortex mode of disturbance. Keywords: Non Darcian; Vortex instability; Porous media; Variable permeability; Mixed convection; Free convection; Finite difference method.

Nomenclature A, m, n d, d

∗

constants in Eq. (7) and (9) constants defined in Eqs. (10) and (11)

f

dimensionless base state stream function

F

dimensionless disturbance stream function

g

gravitational acceleration

i

complex number

k

dimensionless wave number

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a

spanwise wave number

P

pressure

K

permeability of porous medium

N ux

local Nusselt number

Rax

local Rayliegh number

P ex

local Peclet number

F r, Re

Forchheimer coefficient numbers

T

fluid temperature

x, y, z

axial, normal and spanwise coordinates

u, v, w

volume averaged velocity in the x−, y−, z−direction

Greek symbols α

thermal diffusivity

β

volumetric coefficient of thermal expansion

ε

porosity of the saturated porous medium

η

similarity variable

θ

dimensionless base state temperature

Θ

dimensionless disturbance temperature

λf

thermal conductivity of the fluid

λs

thermal conductivity of the solid

λm

effective thermal conductivity of the saturated porous medium

µ

dynamic viscosity of the fluid

ρ

fluid density

σ

ratio of thermal conductivity of the solid to the fluid

ψ

stream function

Subscripts w

conditions at the wall

∞

conditions at the free stream

0

basic undisturbed quantities

1

disturbed quantities

Superscripts *

critical value

0

differentiation w.r.t η

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1. Introduction The problem of the vortex mode of instability in mixed and free convection flow over a heated plate in a saturated porous medium have recently received considerable attention. This is primarily due to a large number of technical applications, such as fluid flow in geothermal reservoirs, separation processes in chemical industries, storage of radioactive nuclear waste materials, transpiration cooling, transport processes in aquifers, etc. The instability mechanism is due to the presence of a buoyancy force component in the direction normal to the plate surface. Hsu and Cheng1 analyzed the vortex mode of instability for a horizontal mixed convection boundary layer flow in a saturated porous media. Hsu et al.2 and Hsu and Cheng3 analyzed the vortex mode of instability of horizontal and inclined natural convection flows in a uniform porosity medium. Jang and Lie4 studied the vortex instability of mixed convection flow over horizontal and inclined surfaces in a porous medium, where both the streamwise and normal components of the buoyancy force are retained in the momentum equations. All of these investigations are based on the Darcy flow. However, at high flow rates or in porous media of high permeability, there is a departure from Darcy’s law and the inertia effects become significant. The non-Darcian effect on vortex instability of free or mixed convection flow was investigated by Refs. 5–9 and 10. All of the works mentioned above are based on the uniform permeability formulation with uniform porosity. In some applications, such as fixed-bed catalytic reactors, packed bed heat exchangers and drying, the porosity is not uniform but has a maximum value at the wall and a minimum value away from the wall. This wall-channeling phenomenon has been reported by a number of investigators such as Refs. 11–13 and 14. It is shown that the variable porosity effect increases the temperature gradient adjacent to the wall resulting in the enhancement of the surface heat flux. The effect of variable permeability on vortex instability of a horizontal free or mixed convection boundary layer flow in a saturated porous medium was studied by Refs. 15,16 and 17. Jang and Chen18 examined the effect of variable porosity and thermal dispersion on vortex instability of natural convection boundary layer flow adjacent to an isothermal horizontal surface in a porous medium. The aim of this paper is to study the effect of variable permeability on vortex instability combined free and mixed convection boundary layer flow over a horizontal heated plate in a saturated porous medium. This is accomplished by considering the Forchheimer extended Darcy equation of

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motion. The wall temperature is a power function of the distance from the origin. The permeability of the medium is assumed to vary exponentially with distance from the wall. The analysis of the disturbance flow is based on linear stability theory. The disturbance quantities are assumed to be in the form of a stationary vortex roll that is periodic in the spanwise direction, with its amplitude function depending primarily on the normal coordinate and weakly on the streamwise coordinate. The resulting eigenvalue problems are solved by using a finite difference scheme. 2. Analysis 2.1. The main flow Consider the problem of free or mixed convection flow along a horizontal surface embedded in a fluid saturated porous medium. Here, x represents the distance along the plate from its leading edge, and y the distance normal to the surface. The horizontal plate assumed to be heated in such a way that its surface temperature varies in the power-law form Tw = T∞ + Axm , where A is constant. For the mathematical analysis of the problem we assume that: 1) the fluid and the porous solid matrix are in local thermodynamic equilibrium; 2) the porous medium is everywhere isotropic and homogeneous; 3) the fluid properties are constant accept for the density in the buoyancy force term; 4) the Boussinesq approximation is employed; 5) Forchheimer’s non-Darcy model is used for the momentum equation. With these assumptions, the governing equations are given by ∂u ∂v + = 0, ∂x ∂y

(1)

√ c K 2 K ∂P u+ u =− , ν µ ∂x

(2)

√ µ ¶ c K 2 K ∂P v+ v =− + ρg , ν µ ∂y

(3)

u

∂T ∂T ∂ +v = ∂x ∂y ∂y

µ α

∂T ∂y

¶ ,

(4)

ρ = ρ∞ [1 − β(T − T∞ )]. Here u and v are velocities in the x and y directions, respectively; P is the pressure; T is the temperature; ρ is the fluid density; µ is the dynamic

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viscosity; K is the permeability of the porous medium; g is the gravitational acceleration; β is the thermal expansion coefficient of the fluid and α represents the equivalent thermal diffusivity. Note that the second term on the left-hand side of equations (2) and (3) is the inertia force in Forchheimer’s model, c is the transport property related to the inertia effect. As c = 0, equations (2) and (3) reduce to Darcy model. The other symbols are defined in the nomenclature. The pressure terms appearing in equations (2) and (3) can be eliminated through cross-differentiation. The boundary layer assumption yields ∂/∂x ¿ ∂/∂y and v ¿ u. With ψ being a stream function such that u = ∂ψ/∂y and v = −∂ψ/∂x, the equations (1)–(4) become Ã ! " √ √ µ ¶2# 2c K ∂ψ ∂ 2 ψ 1 ∂ψ c K ∂ψ dK Kρ∞ gβ ∂T 1+ − + =− , (5) 2 ν ∂y ∂y K ∂y 2ν ∂y dy µ ∂x ∂ψ ∂T ∂ψ ∂T ∂2T dα ∂T − =α 2 + . ∂y ∂x ∂x ∂y ∂y dy ∂y

(6)

The boundary conditions are defined as follows ∂ψ = 0, T = Tw = T∞ + Axm , (7) ∂x ∂ψ = 0, T →T∞ (for free convection flow), (8) y→∞ u = ∂y ∂ψ y→∞ u = →U∞ = Bxn , T →T∞ (for mixed convection flow). (9) ∂y y=0 v=−

Here we consider that the porosity ε and permeability K vary exponentially from the wall (Ref. 12) ε = ε∞ (1 + de−y/γ ),

(10)

K = K∞ (1 + d∗ e−y/γ ),

(11)

where ε∞ and K∞ are the porosity and permeability at the edge of the boundary layer; d and d∗ are constants whose values are taken as 1.5 and 3 respectively, (see Chandrasekhara12 ). Further, α = λm /(ρ∞ cp )f also varies since it is related to the effective thermal conductivity of the saturated porous medium λm , where λm can be computed according to the following semi-analytical expression given by Nayagam et al19 : λm = ελf + (1 − ε)λs ,

(12)

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where λf and λs are the thermal conductivities of the fluid and solid respectively. Hence the expression for the thermal diffusivity has the form ´ i h ³ α = α∞ ε∞ 1 + de−y/γ + σ{1 − ε∞ (1 + de−y/γ )} , (13) where σ = λs /λf and α∞ = λf /(ρ∞ cp )f . 2.1.1. Free convection flow On applying the following transformations η(x, y) =

y 1/3 Ra , x x

ψ(x, y)

f (η) =

gβK∞ (T −T∞ )x is α∞ ν 1/3 x/Rax such that ε

where, Rax =

1/3 α∞ Rax

, θ(η) =

T − T∞ , Tw − T∞

(14)

the modified local Rayleigh number. We

and K are purely functions of η only. choose γ = Equations (5)–(9) can be nondimensionalized as follows ¸ h i 00 G0 · 0 1 2/3 0 1/2 2m−1 2/3 0 2 1/2 2m−1 3 3 1 + 2F r f G x f − f + Fr f G x = G 2 µ ¶ m−2 0 − G mθ + ηθ , (15) 3 µ

α α∞

¶

d θ + dη

µ

00

α α∞

η=0

¶ 0

0

θ = mf θ −

f = 0,

m+1 0 fθ , 3

θ = 1,

0

η → ∞ f = 0, θ = 0. 3/4

(16)

(17)

1/2

c3/2 gβK∞ Aα∞ is the Forchheimer number ν 5/2 expressing the relative importance of the inertia effects. The follow is governed by Darcy’s law when F r = 0. Obviously, equations (15) and (16) will be independent of x, if m = 0.5 that is, the surface temperature varies with x1/2 thus, equations (15)–(16) can be written in the form ¸ i 00 G0 · 0 1 h 0 0 0 1 1 + 2Ref G1/2 f − f + Ref 2 G1/2 = − M G(θ − ηθ ), (18) G 2 2

where, G = (1+d∗ e−η ), F r =

µ

α α∞

¶ 00

θ +

d dη

µ

α α∞

¶ 0

θ =

´ 0 1³ 0 f θ − fθ , 2

(19)

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with the boundary conditions η=0

f = 0, 0

η→∞

f = 0,

θ = 1, θ = 0.

(20)

In the above equations, the primes denote the derivatives with respect to η. 2.1.2. Mixed convection flow In this case we shall use the following transformations yp ψ(x, y) T − T∞ √ η(x, y) = P ex , f (η) = , θ(η) = . x Tw − T∞ α∞ P ex U∞ x x is the local peclet number. Here, we choose γ = √ . α P ex Equations (5)–(6) reduce to √ √ · ¸ ¸ 0 · 0 00 0 0 c K∞ G 1 c K∞ 1+2 Bxn f G1/2 f − f + Bxn f 2 G1/2 = ν G 2 ν ¶ µ n−1 0 ηθ , (21) − M G mθ + 2

where, P ex =

µ

µ ¶ 0 0 d α n+1 0 θ + θ = mf θ − fθ , (22) dη α∞ 2 √ ¶ µ c K∞ Rax is the mixed convection parameter, Re = U∞ where, M = 3/2 ν P ex is the Forchheimer coefficient dependent Reynolds number. It can be shown 3n + 1 that the similarity solutions to equations (21)–(22) exist if m = . 2 Obviously, equations (21) and (22) will be independent of x, if (n = 0, m = 0.5) that is, the surface temperature varies with x1/2 thus, equations (21)–(22) can be written in the form ¸ h i 00 G0 · 0 1 0 0 0 1 1 + 2Ref G1/2 f − f + Ref 2 G1/2 = − M G(θ − ηθ ), (23) G 2 2 α α∞

µ

¶

α α∞

00

¶ 00

θ +

d dη

µ

α α∞

¶ 0

θ =

´ 0 1³ 0 f θ − fθ , 2

(24)

with the boundary conditions η=0 η→∞

f = 0, 0

f = 1,

θ = 1, θ = 0.

(25)

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In terms of the new variables, it can be shown that the velocity components and the local Nusselt number are given by α∞ B 2 0 f x 0 α∞ B u(x, y) = − (f − ηf ) 2x 0 N ux = −Bθ (0) u(x, y) =

where, B =

√

1/3

P ex for mixed convection and B = Rax

for free convection.

2.2. The disturbance flow The standard linear stability method yields the following ∂v1 ∂w1 ∂u1 + + = 0, ∂x ∂y ∂z

(26)

√ 2c K K ∂P1 u1 + u0 u1 = − , ν µ ∂x

(27)

√ µ ¶ 2c K K ∂P1 v1 + v0 v1 = − − ρ∞ gβT1 , ν µ ∂y

(28)

w1 = −

K ∂P1 , µ ∂z

(29)

∂T1 ∂T1 ∂T0 ∂T0 ∂2T ∂ u0 + v0 + u1 + v1 =α 2 + ∂x ∂y ∂x ∂y ∂x ∂y

µ ¶ ∂T ∂2T α + α 2 , (30) ∂y ∂z

where the subscripts 0 and 1 signify the main flow and disturbance components, respectively. Following the method of order of magnitude analysis described in detail by Hsu and Cheng1 and Hsu et al.2 . The terms ∂u1 /∂x and ∂ 2 T1 /∂x2 in equations (26)–(30) can be neglected. The omission of ∂u1 /∂x in equation (26) implies the existence of a disturbance stream function Ψ1 , such as w1 =

∂Ψ1 , ∂y

v1 = −

∂Ψ1 . ∂z

(31)

Eliminating P1 from equations (27)–(29) with the aid of (31) leads to ! Ã √ ∂u1 ∂ 2 Ψ1 2c K u0 = , (32) 1+ ν ∂z ∂x∂y

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Ã

! √ 2c K ∂ 2 Ψ1 ∂ 2 Ψ1 1 dK ∂Ψ1 Kρ∞ gβ ∂T1 − 1+ v0 = − + , ν ∂z 2 ∂y 2 K dy ∂y µ ∂z ∂Ψ1 ∂T0 ∂T1 ∂T1 ∂T0 u0 + v0 + u1 − =α ∂x ∂y ∂x ∂z ∂y

µ

∂2T ∂2T + ∂y 2 ∂z 2

¶ +

(33)

dα ∂T . (34) ∂y ∂y

As in Hsu et al.1 , we assume that the three-dimensionless disturbances for neutral stability are of the form (Ψ1 , u1 , T1 ) = [Ψ(x, y), u(x, y), T (x, y)]eiaz ,

(35)

where a is the spanwise periodic wave number. Substituting equation (35) into equations (32)–(34) yields Ã ! √ ∂2Ψ 2c K ia 1 + u0 u = , (36) ν ∂x∂y Ã a2

! √ ∂2Ψ 2c K 1 dK ∂Ψ iaKρ∞ gβ T, 1+ v0 Ψ = − + ν ∂y 2 K dy ∂y µ

∂T ∂T ∂T0 ∂T0 u0 + v0 +u − iaΨ =α ∂x ∂y ∂x ∂y

µ

∂2T − a2 T ∂y 2

¶ +

dα ∂T . dy ∂y

(37)

(38)

Equations (36)–(38) are solved based on the local similarity approximation (see Ref. 20), wherein the disturbances are assumed to have weak dependence in the streamwise direction (i.e. ∂/∂x ¿ ∂/∂y), k=

ax ψ T , F (η) = , Θ(η) = , B iα∞ B Ax1/2

(39)

we obtain the following system of equations for the local similarity approximation: · 0 ³ ´¸ p 00 0 G 0 2 1/2 Re F −k 1−G f − ηf F− F = −kSG P ex Θ, (40) B G µ

α α∞

¶

·

µ

1 d Θ + f+ 2 dη 00

α α∞

µ

¶¸ 0

Θ −

¶ 1 0 α 2 f + k Θ+ 2 α∞

00

(41)

F (0) = F (∞) = Θ(0) = Θ(∞) = 0.

(42)

0

0 η(ηθ − θ)F £ ¤ = kBθ F, + 0 1/2 4kB 1 + 2Ref G

subject the boundary conditions

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In the above equations S = M and S = 1for mixed and free convection, respectively. Equations (40) and (41) along with its the boundary condition (42) constitutes a second-order system of linear ordinary differential equations for the disturbance amplitude distributions F (η) and Θ(η). For fixed values of d, d∗ , ε∞ , σ, k and M , the solution F and Θ is an eigenfunction for the eigenvalue P ex or Rax .

3. Numerical Scheme In this section, we compute the approximate value of Rax and P ex for equations (40) and (41) with the boundary conditions (42). An implicit finite difference method is used to solve first the base flow and the results are stored for a fixed step size h, which is small enough to predict accurate linear interpolation between mesh point. The domain is 0 ≤ η ≤ η∞ , where η∞ is the edge of the boundary layer of the basic flow. For a positive integer N , let h = η∞ /N and ηi = ih, i = 0, 1, ..., N . The problem is discretized with standard centered finite differences of order two, following Usmani21 . Solving eigenvalue problem is achieved by using the subroutine GVLRG of the IMSL library, see Ref. 22.

4. Results and Discussion Numerical results for the local Nusselt number, neutral stability curves, the critical Rayleigh and Peclet and numbers and the associated wave numbers at the onset of vortex instability are presented for both uniform permeability (UP), i.e. d = d∗ = 0 and variable permeability (VP), i.e. d, d∗ 6= 0 cases. For the purpose of numerical integration we have assumed d = 1.5, d∗ = 3, σ = 2, ε∞ = 0.4 (see Ref. 12). Figures 1 and 2 show the effect of Forchheimer coefficient numbers F r and Re, the mixed convection parameter M and the varaible permeability on the local Nusselt number. As expected, the local Nusselt number decrease as F r or Re (inertia coefficient) increases. Further, at a given value of Re as M increases the local Nusselt number increases. From this figures, it can be seen that a large discrepancy in the results exists between the variable permeability and uniform permeability. The local Nusselt number, predicated by the variable permeability, are much higher than those by the uniform permeability.

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Figures 3 and 4 show the neutral stability curves, in terms of Rax and P ex and the dimensionless wave number k, for selected values of F r and Re, respectively, for both UP and VP cases. It is observed that, as F r and Re increases, the neutral stability curves shift to lower Rax and P ex , respectively. When the variable permeability effect is considered, the neutral

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stability curves shift to lower Rayleigh and Peclet numbers and higher wave number k, indicating a destabilization of the flow. The critical Rayleigh and Peclet numbers Ra∗x , P e∗x and critical wave number k ∗ , which mark the onset of longitudinal vortices, can be found from the minima of the neutral stability curves. The critical wave number k ∗ and Rayleigh number Ra∗x are plotted as a function of F r in figures 5 and 6, respectively, for both UP and VP cases. We observe that, the critical Rayleigh number and critical wave number decrease as F r increase. The critical Peclet number and wave number are plotted as function of M for various values of Re in figures 7 and 8, respectively, for both UP and VP cases. We observe that, the critical Peclet number decreases and critical wave number increases as M increase. Also, we can see that, the critical Peclet number and critical wave number decrease as Re increase, indicating a destabilization of the flow to the vortex instability. Figures 5-8 show that the variable permeability effect tends to destabilize the flow to the vortex mode of disturbance. To assess the accuracy of our results, we have shown a comparison of our results with those Ref. 2 in the case of uniform permeability UP and darcy flow (i.e. F r = 0). A comparison of our results with those from the literature indicates that the agreement between the two calculations is good.

References 1. C.T. Hsu and P. Cheng, Vortex instability of mixed convection flow in a semiinfinite porous medium bounded by horizontal surface, Int. J. Heat Mass Transf., 23 (1980), 789–798. 2. C.T. Hsu, P. Cheng, P. and G.M. Homsy, Instability of free convection flow over a horizontal impermeable surface in a porous medium. Int. J. Heat Mass Transf., 21 (1978), 1221–1228. 3. C.T. Hsu and P. Cheng, The onset of longitudinal vortices in mixed convection flow over an inclined surface in a porous medium, J. Heat Transf., 102 (1980), 544-549. 4. J.Y. Jang and K.N. Lie, Vortex instability of mixed convection flow over horizontal and inclined surfaces in a porous medium,Int. J. Heat Mass Transf., 35 (1992), 2077–2085. 5. W.J. Chang and J.Y. Jang, Inertia effects on vortex instability of a horizontal natural convection flow in a saturated porous medium, Int. J. Heat Mass Transf., 32 (1989), 541–550. 6. K.N. Lie and J.Y. Jang, Boundary and inertia effects on vortex instability of a horizontal mixed convection flow in a porous medium, Numer. Heat Mass, A 23 (1993), 361–378. 7. J.Z. Zhao, T.S. Chen, Inertia effects on non-parallel thermal instability of

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11. 12.

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18.

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20. 21.

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natural convection flow over horizontal and inclined plates in porous media, Int. J. Heat Mass Transf., 45 (2002), 2265–2276. J.Y. Jang and K.N. Lie, The influence of surface mass flux on vortex instability of a horizontal mixed convection flow in a saturated porous medium, Int. J. Heat Mass Transf., 38 (1995), 3305–3311. I.A. Hassanien, A.A. Salama and N.M. Moursy, Non-Darcian effects on vortex instability of mixed convection over horizontal plates in porous medium with surface mass flux, Int. Comm. Heat Mass Transf., 31 (2004), 231–240. I.A. Hassanien, A.A. Salama and N.M. Moursy, Inertia effect on vortex instability of horizontal natural convection flow in a saturated porous medium with surface mass flux, Int. Comm. Heat Mass Transf., 31 (2004), 741–750. K. Vafai, Convection flow and heat transfer in variable porosity media, J. Fluid Mech., 147 (1984), 233–259. B.C. Chandrasekhara, Mixed convection in the presence of horizontal impermeable surfaces in saturated porous media with variable permeability, Waerme - Stoffuebertrag, 19 (1985), 195–201. F.S. Ibrahim and I.A. Hassanien, Influence of variable permeability on combined convection along a non-isothermal wedge in saturated porous medium, Transport in Porous Media, 39 (2000), 57–71. I.A. Hassanien and A.M. Elaiw, Variable permeability effect on buoyancyinduced boundary layer flow in a saturated porous medium with variable wall temperature, Heat Mass Transf., (2006) in press. J. Jang and J. Chen, Variable porosity effect on vortex instability of a horizontal mixed convection flow in a saturated porous medium, Int. J. Heat Mass Transf., 32(1993), 1573–1582. I.A. Hassanien, A.A. Salama and A.M. Elaiw, Variable permeability effect on vortex instability of mixed convection flow in a semi-infinite porous medium bounded by a horizontal surface, Appl. Math. and Comp., 146 (2003), 829– 847. I.A. Hassanien, A.A. Salama and A.M. Elaiw, Variable permeability effect on vortex instability of a horizontal natural convection flow in a saturated porous medium with variable wall temperature, ZAMM. Z. Angew. Math. Mech., 84 (2004), 39–47. J.Y. Jang and J.L. Chen, Variable porosity and thermal dispertion effects on vortex instability of a horizontal natural convection flow in a saturated porous medium, Warme und Stoffubertrangung, 29 (1994), 153–160. M. Nayagam, P. Jain, and G. Fairweather, The effect of surface mass transfer on buoyancy-induced flow in a variable porosity medium adjacent a horizontal heated plate, Int. Commun. Heat Mass Transf., 14 (1987), 495–506. E.M. Sparrow, H. Quack, and C.T. Boerner, Local nonsimilarity boundary layer solutions. AIAA J., 8 (1970) 1936-1942. R.A. Usmani, Some new finite difference methods for computing eigenvalues of two-point boundary value problems, Comput. Math. Appl., 9 (1985), 903– 909. IMSL, References Manual, IMSL, Inc., Houston, TX, (1990).

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ANALYTICITY OF STABLE INVARIANT MANIFOLDS FOR GINZBURG-LANDAU EQUATION∗ A.V. FURSIKOV Department of Mechanics & Mathematics Moscow State University, Moscow, Russia 119992 E-mail: [email protected] This paper is devoted to prove analyticity of stable invariant manifold in a neighbourhood of an unstable steady-state solution for Ginzburg-Landau equation defined in a bounded domain of dimension not more than three. This investigation is made for possible applications in stabilization theory for semilinear parabolic equation. Keywords: Ginzburg-Landau equation; Stable invariant manifold; Analyticity; Stabilization.

1. Introduction In this paper we prove analyticity of stable invariant manifold M− near unstable steady-state solution of Ginzburg-Landau equation. This result can be used in stabilization theory for semilinear parabolic PDE defined in a bounded domain Ω with feedback Dirichlet control given on the boundary ∂Ω or on its open part. This theory for general quasilinear parabolic equation and for NavierStokes system was built in Refs. 1–3. We have to emphasize that the main reason to develop stabilization theory is to provide reliable stable algorithms for numerical stabilization. To construct such algorithms it is very desirable to have a simple description for infinite-dimensional invariant manifold M− allowing to calculate it easily in arbitrary point. Just such description gives functional-analytic decomposition of M− . Using classical description of M− by means of a map F (y− ) (see Refs. 4,5), one can look for this map as a series ∗ The

work has been fulfilled by RAS programm ”Theoretical problems of modern mathematics‘”’, project ”‘Optimization of numerical algorithms of Mathematical Physics problems”’. Author was supported in part by RFBI Grant #04-01-00066.

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F (y− ) =

∞ X

Fk (y− )

k=2

where Fk (y− ) are power operators in y− of order k. Using special differential equation in variational derivatives for map F it is possible to obtain recurrent formulae for Fk . These recurrent formulae allow us to prove convergence of series for F (y− ). First step in realization of this plan has been made in Ref. 6 where analyticity of stable invariant manifold in a neghborhood of zero steady state solution was proven in the case of one-dimensional semilinear parabolic equation. Moreover, obtained recourrence relations were succesfully used in Ref. 7 for numerical calculations. Note that under assumptions of Ref. 6 the linearization near steady state solution of the space part of semilinear equation is ordinary differential equation with constant coefficients. Therefore its eigenfunctions are sin kx. This circumstance were used essentially in Ref. 6. The methods of present paper do not use explicit form of eigenfunctions and therefore can be applied to situation when aforementioned linearization is an elliptic operator with variable coefficients defined in arbitrary bounded domain. 2. Stable Invariant Manifold In this section we recall certain notions connected with stable invariant manifolds for Ginzburg-Landau equation. 2.1. Ginzburg-Landau equation Let G ⊂ IRn , n = 1, 2, 3 be a bounded domain with C ∞ -boundary ∂G. We consider Ginzburg-Landau equation ∂t v(t, x) − ν∆v(t, x) − v(t, x) + v 3 (t, x) = f (x),

x ∈ G, t > 0

(1)

with boundary and initial conditions v(t, ·)|x∈∂G = 0, v(t, x)|t=0 = v0 (x),

x ∈ G,

(2) (3)

where ∂t v=∂v/∂t, ν > 0 is a parameter, f (x)∈L2 (G), v0 (x)∈H 2 (G)∩H01 (G) are given functions. Recall that H k (G) is the Sobolev space of functions belonging to L2 (G) together with all their derivaties up to the order k, H01 (G) = {u(x) ∈ H 1 (G) : u|∂G = 0}.

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As a phase space of the dynamical system generated by (1), (2) we take the functional space V ≡ V (G) = H 2 (G) ∩ H01 (G).

(4)

Let vb(x) ∈ V be a steady-state solution of (1), (2), i.e. a solution of the problem −ν∆b v (x) − vb(x) + vb3 (x) = f (x),

x ∈ G,

vb|∂G = 0.

(5)

To study the structure of the dynamical system (1), (2) in a neighborhood of vb(x) we make the change of unknown functions in (1), (2): v(t, x) = vb(x) + y(t, x).

(6)

After substituting (6) into (1)–(3) and taking into account (5) we get: ∂t y(t, x)−ν∆y(t, x)+q(x)y(t, x)+B(x, y(t, x)) = 0, x ∈ G, t > 0,

(7)

y(t, ·)|x∈∂G = 0,

(8)

y(t, x)|t=0 = y0 (x) = v0 (x) − vb(x),

x ∈ G,

(9)

where q(x) = 3b v 2 (x) − 1,

B(x, y) = y 3 + 3b v (x)y 2

(10)

Let {ek (x), λk },

λ1 ≤ λ2 ≤ ... ≤ λk → ∞

as k → ∞

(11)

be the eigenfunctions and the eigenvalues of the spectral problem Ae ≡ −ν∆e(x) + q(x)e(x) = λe(x), x ∈ G

e|∂G = 0.

(12)

We assume that eigenvalues λk of the spectral problem (12) satisfy the condition: λ1 ≤ ... ≤ λN < 0 < λN +1 ≤ ... ≤ λk ≤ ...

(13)

Since operator A is symmetric in L2 (G), the set (11) of its eigenfunctions {ek } forms orthogonal basis in L2 (G). We can assume (have done normalization) that {ek } is orthonormal basis in L2 (G). It is well-known that usual Sobolev H 2 -norm in V = H 2 (G) ∩ H01 (G) is equivalent to the norm Z ∞ ∞ X X kvk2V = λ2j |vj |2 , where vj = v(x)ej (x)dx, and v(x) = vj ej (x). (14) j=1

G

j=1

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Evidently, {ej } forms orthogonal basis in V with respect to scalar product defined by norm (27). Below we suppose that the phase space V is supplied with the norm (27). In virtue of (13), the solutions e−λk t ek (x) of the linear equation ∂y + Ay = 0 (15) ∂t tend to infinity as t → ∞ for k = 1, ..., N , and tend to zero as t → ∞ for k > N. We introduce the subspaces V+ ≡ V+ (G) = [e1 , ..., eN ],

V− ≡ V− (G) = [eN +1 , eN +2 ...]

(16)

of unstable and stable modes for equation (15). Since eigenfunctions (11) form orthogonal basis in the phase space (4), the following relation is true: V+ (G) ⊕ V− (G) = V (G).

(17)

2.2. Stable invariant manifold It is well-known, that for each y0 ∈ V there exists a unique solution y(t, x) ∈ C(0, T ; V (G)) of problem (7)–(10), where T > 0 is arbitrary fixed number. We denote by S(t, y0 ) the solution operator of the boundary value problem (7)–(10): S(t, y0 ) = y(t, ·)

(18)

where y(t, x) is the solution of (7)–(10). Recall now some commonly used definitions of stable invariant manifold (see Chapter V in Ref. 4) adopted for our case. The origin of the phase space V , i.e. the function y(x) ≡ 0, is, evidently, a steady-state solution of problem (7)–(10). Definition 2.1. The set M− ⊂ V defined in a neighborhood of the origin is called the stable invariant manifold if for each y0 ∈ M− the solution S(t, y0 ) is well-defined and belongs to M− for each t > 0, and kS(t, y0 )kV ≤ ce−rt

as

t→∞

(19)

where 0 < r < λN +1 . The stable invariant manifold can be defined as a graph in the phase space V = V+ ⊕ V− by the formula M− = {y− + F (y− ), y− ∈ O(V− )}

(20)

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where O(V− ) is a neighborhood of the origin in the subspace V− , and F : O(V− ) → V+

(21)

is a certain map satisfying kF (y− )kV+ /ky− kV− → 0 as

ky− kV− → 0.

(22)

So, in order to construct the invariant manifold M− we have to calculate the map (21), (22). 3. Preliminaries To get functional-analytic decompozition of the map F that defines stable invariant manifold, we have to derive differential equation for F . First of all we recall derivation of well-known equation for map (21) that determines invariant manifold M− . After that we recall definitions of certain notions that we use later. 3.1. Equation for F Let us introduce several notations. We rewrite equations (7), (10) using definition (12) of operator A as follows: ∂t y(t) + Ay(t) + B(·, y(t)) = 0.

(23)

Define the orthoprojectors P+ : V → V+ ,

P − : V → V−

(24)

and introduce notations P+ y=y+ , P− y = y− , P+ S(t, y0 )=S+ (t, y0 ), P− S(t, y0 )=S− (t, y0 ). (25) Taking into account that the spaces V+ , V− are invariant with respect to acting of operator A and using notations (11) we can rewrite (9) as follows: ∂t y+ (t) + Ay+ (t) + P+ B(·, y+ (t) + y− (t)) = 0 ∂t y− (t) + Ay− (t) + P− B(·, y+ (t) + y− (t)) = 0.

(26)

Let y0 ∈ M− . Then by (20) it has the form y0 = y− +F (y− ). By definition of an invariant manifold for each t ∈ R+ S(t, y0 ) ∈ M− or, what is equivalent S+ (t, y− + F (y− )) = F (S− (t, y− + F (y− ))).

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We differentiate this equation with respect to t and express t-derivatives with help of equations (12). As a result we get: AS+ (t, y− + F (y− )) + P+ B(·, S(t, y− + F (y− ))) = hF 0 (S− (t, y− + F (y− ))), AS− (t, y− + F (y− ))

(27)

+ P− B(·, S+ (t, y− + F (y− )) + S− (t, y− + F (y− )))i where by hF 0 (z), hi we denote the value of derivative F 0 (z) on vector h. Passing to limit in (13) as t → 0 we get the desired equation for F : AF (y− ) + P+ B(·, y− + F (y− ))=hF 0 (y− ), Ay− + P− B(·, y− + F (y− ))i. (28) 3.2. Analytic maps Let Hi be Hilbert spaces with the scalar products (·, ·)i and the norms k · ki where i = 1, 2. We denote by (H1 )k = H1 × ... × H1 (k times) the direct product of k copies of H1 and define by Fk : (H1 )k → H2 a k-linear operator Fk (h1 , ..., hk ), i.e. the operator that is linear with respect to each variable hi , i = 1, ..., k. Then kFk k =

sup

kFk (h1 , ..., hk )k2 .

(29)

khi k1 =1,i=1,...,k

Restriction of k-linear operator Fk (h1 , ..., hk ) to diagonal h1 = ... = hk = h is called power operator of order k: Fk (h) = Fk (h, ..., h).

(30)

Using derivatives one can restore k-linear operator Fk (h1 , ..., hk ) by power operator Fk (h). Denote by O(H1 ) a neighbourhood of origin in the space H1 . The map F : O(H1 ) → H2

(31)

is called analytic if it can be decomposed in the series F (h) = F0 +

∞ X

Fk (h)

(32)

k=1

where F0 ∈ H2 and Fk (h) are power operators of order k. Series (18) ∞ X kFk (h)k2 converges. converges if the numerical series kF0 k2 + k=1

Proposition 3.1. Let norms (15) of power operator Fk (h) from (18) satisfy kFk k ≤ γρ−k

(33)

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where γ > 0, ρ > 0 do not depend on k. Then series (18) converges for each h belonging to the ball Bρ (H1 ) = {h ∈ H1 : khk1 < ρ}. Proof. There exists ε > 0 such that khk1 ≤ ρ − ε. Then using (15), (19) we get ¶k ∞ ∞ µ X X ρ−ε k kF (h)k2 ≤ kF0 k2 + < ∞. ¤ kFk kkhk1 ≤ γ ρ k=1

k=1

3.3. Operators from equation for F and their kernels We consider here operators from equation (14). 3.3.1. Subspapces V± and projectors P± Subspaces V+ , V− of V are defined in (16), and projectors P± are defined in (10). Orthogonality of decomposition (17) as well as orthogonality of projectors (10) take place with respect to the scalar product corresponding to norm (27). Therefore kP+ k ≤ 1,

kP− k ≤ 1.

(34)

Kernels Pb± of operators P± , i.e. distributions on G × G such that Z (P± v)(x) = Pb± (x, ξ)v(ξ)dξ ∀v(ξ) ∈ V

(35)

G

are defined as follows: Pb+ (x, ξ) =

N X

ek (x)ek (ξ),

Pb− (x, ξ) = δ(x − ξ) −

k=1

N X

ek (x)ek (ξ),

(36)

k=1

where δ(x − ξ) is Dirac δ-function. Note that integral in (29) in the case Pb− (x, ξ) is understood (at each fixed x) as value of distribution N X δ(x − ξ) − ek (x)ek (ξ) on the test function v(x). Such notation for values k=1

of distributions will be often used below without additional explanations. 3.3.2. Analyticity of the map B(·, y) We intend to decompose operator B(·, y) defined in (10) in series (18). For this we use that the phase space V is the algebra, i.e. in this space the operation of multiplication of functions is well-defined.

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Define the operator of multiplication Γk as follows: Γk : V k −→ V,

Γk (v1 , ..., vk )(x) = v1 (x) · ... · vk (x)

(37)

where V k = V × ... × V (k times). Lemma 3.1. Let V = H 2 (G)∩H01 (G), G ⊂ IRn , n = 1, 2, 3. Then operator Γk defined in (31) is k-linear bounded operator. Moreover, there exists a constant γ > 0 such that for each k kΓk (v1 , ..., vk )kV ≤ γ k−1 kv1 kV · ... · kvk kV .

(38)

Proof. Since norm (27) is equivalent to the norm of Sobolev space H 2 (G), we can use H 2 -norm. Taking into account that embeddings H 2 (G) ⊂ C(G) and H 2 (G) ⊂ W41 (G) are continuous we get:  1/2 X Z kv1 · v2 kH 2 (G) =  |Dα (v1 (x)v2 (x))|2 dx kα|≤2

≤ kv1 kH 2 kv2 kC + kv1 kC kv2 kH 2 + 2kv1 kW4! kv2 kW41 ≤ γkv1 kH 2 kv2 kH 2 . Using this inequality we obtain (31) by induction in k. ¤ It follows from Lemma 3.1 and (10) that for y ∈ V B(x, y(x)) = Γ3 (y, y, y)(x) + 3b v (x)Γ2 (y, y)(x).

(39)

Therefore operator B is analytic, and relation (33) is its decomposition in series (18). The kernels of operators from (33) are as follows: b 3 (x; ξ1 , ξ2 , ξ3 ) = δ(x − ξ1 )δ(x − ξ2 )δ(x − ξ3 ) Γ

(40)

b 2 (x; ξ1 , ξ2 ) = 3b 3b v (x)Γ v (x)δ(x − ξ1 )δ(x − ξ2 ).

(41)

3.4. Series for operator F Let us consider the special case when H1 = V− , H2 = V+ with Hilbert spaces V− , V+ defined in (16). In this case analytic map (17), (18) can be rewritten as follows: F : O(V− ) → V+ ,

F (y− ) =

∞ X

Fk (y− ).

(42)

k=2

We assume that F0 = 0, F1 = 0 because by (22) the map F defining stable invariant manifold M− has just this form.

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Since F (y− ) ∈ V+ , it is a function depending on argument x: F (y− ) ≡ F (x; y− ). kernels Fbk (x; ξ1 , ..., ξk )

Now we define Fk (·; y1 , ..., yk ), yj ∈ V− , j = 1, ..., k. Let

of

k-linear

operator

V ⊂ L2 (G) ⊂ V 0

(43)

where V 0 is the space dual to V with respect to duality generated by scalar product in L2 (G). Define Z V−0 = {u(x) ∈ V 0 : u(x)ϕ(x)dx = 0 ∀ϕ ∈ V+ } = V+⊥ . (44) Below we use the following notation: ξ k = (ξ1 , ..., ξk ),

dξ k = dξ1 ...dξk ,

where ξj ∈ G, j = 1, ..., k,

(45)

y− (ξ k ) = y− (ξ1 ) · ... · y− (ξk ), y(j k ; ξ k ) = yj1 (ξ1 ) · ... · yjk (ξk ). (46) The kernel Fbk (x; ξ k ), x ∈ G, ξ k ∈ Gk ≡ G × ... × G (k times) belongs to the k k space V+ ⊗ (⊗ V−0 ) where ⊗ V−0 = V 0 ⊗ ... ⊗ V 0 (k times), i.e. Fbk (x; ξ k ) is a distribution on Gk with values in V+ , such that for each yj ∈ V, j = 1, ..., k the value Z Fk (x; y1 , ..., yk ) = Fbk (x; ξ k )y(j k ; ξ k ) dξ k (47)

of distribution Fbk (x; ξ k ) on test function y(ξ1 ) · ... · y(ξk ) is well-defined. Moreover, if yj ∈ V+ at least for one j ∈ {1, ..., k} then right hand side of equality (40) equals zero. Moreover, since Fk (·, y1 , ..., yk ) is symmetric with respect to (y1 , ..., yk ), i.e. Fk (·, y1 , ..., yk ) = Fk (·, yj1 , ..., yjk ) for each permutation (j1 , ..., jk ) of (1, ..., k), we can assume that the distribution Fbk (x; ξ k ) is symmetric with respect to (ξ1 , ..., ξk ) Now using (40) and (41) we can rewrite the series from (36) in the form: ∞ Z X F (x, y− ) = Fbk (x; ξ k )y− (ξ k )dξ k . (48) k=2

In accordance with (15) we define the norm kFk k of Fbk (x; ξ k ) ∈ k

V+ ⊗ (⊗ V−0 ) by the following way: kFk k =

sup kyj kV− =1 j=1,...,k

=

sup

kFk (·, y1 , ..., yk )kV+

sup

ky+ kV+ =1 kyj kV− =1 j=1,...,k

Z y+ (x)Fbk (x; ξ k )y(j k ; ξ k ) dx dξ k .

(49)

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For each function or distribution K(η1 , ..., ηr ) defined on Gr we determine the function σηr K(η1 , ..., ηr ) which is simmetric with respect to arbitrary permutation (ηj1 , ..., ηjr ) of variables (η1 , ..., ηr ) by the formula: 1 X K(ηj1 , ..., ηjr ) (50) σηr K(η1 , ...ηr ) = r! (j1 ,...,jr )

where the sum in the r.h.s. of (50) performs over all permutations (j1 , ..., jr ) of the set (1, ..., r). Lemma 3.2. Let K(η1 , ..., ηr ) be defined on Gr . Then (a) The following equality is true: Z Z ¡ ¢ K(η r )h(j r ; η r ) dη r = σηr K(η r ) h(j r ; η r ) dη r

(51)

for any h(j r ; η r ) such that the series in the l.h.s. converges. (b) For any function G(η1 , ..., ηr ) simmetric in its arguments G(η r )σηr K(η r ) = σηr [G(η r )K(η r )].

(52)

k

(c) If all distributions Fk (x; η k ) ∈ V+ ⊗ (⊗ V−0 ) from (42) are symmetric in their arguments η k then these distributions are defined uniquely by values of analytic functions F (y− ), y ∈ V− from (42). The proof of this Lemma is evident. 4. Formal Construction of the Map F We look for the map defining stable invariant manifold in the form of a series (42). In this section we find recurrence relations for kernals Fbk (x; ξ k ). 4.1. Calculation of Fb2 (x; ξ 2 ) Below using k-linear and power operators Fk (x; y, ..., y) = Fk (x; y) we omit sometimes variable x writing Fk (y). After substituting (42) into (14) we get taking into account (10) that for each y = y− ∈ V−   ∞ 3 k X X X AFq (y) + P+  ak Ckj F j (y)y k−j  q=2

=

∞ X q=2



k=2

j=0



qFq y, ..., y, Ay + P− 

3 X

k=2

ak

k X j=0

 Ckj F j (y)y k−j 

(53)

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where Ckj = k!/(j!(k − j)!) and a3 ≡ a3 (x) ≡ 1,

a2 ≡ a2 (x) = 3b v (x).

(54)

Let us equate the terms from (20) of the second order with respect to y: AF2 (y, y) + 3P+ (b v y 2 ) = 2F2 (y, Ay). Using kernels of bilinear operator F2 (y, y) we can rewrite this relation as follows: Z [Fb2 (x; ξ 2 )(Aξ1 + Aξ2 )y(ξ 2 )) − Ax Fb2 (x; ξ 2 )y(ξ 2 )]dξ 2 = 3P+ (b v y 2 )(x) (55) where subscript of operator A indicates independent variable of a function to that this operator A is applied. We will use notation: Aξk =

k X

Aξj .

(56)

j=1

Carrying operator Aξ2 from y(ξ 2 ) to Fb2 (x; ξ 2 ) and using operator (31) in right side of (21) we get: Z Z 2 2 2 b b 2 (η; ξ 2 )y(ξ 2 )dηdξ 2 . (57) (Aξk − Ax )F2 (x; ξ )y(ξ )dξ = 3 Pb+ (x, η)b v (η)Γ Since y ∈ V− and subspaces V+ , V−0 are invariant with respect of operator A, we obtain from (23) the relation determining Fb2 : Z −1 2 b 2 (η; ζ 2 )Pb− (ζ 2 ; ξ 2 )dζ 2 , (58) b Pb+ (x, η)b v (η)Γ F2 (x; ξ ) = 3(Aξ2 − Ax ) where Pb− (ζ 2 , ξ 2 ) is defined below, in (19). Note that operator (Aξk −Ax )−1 is well-defined. Moreover, the following k

assertion hold (recall that we define the norm of the space V+ ⊗ (⊗ V−0 ) by (49)): Lemma 4.1. Operator k

k

(Aξk − Ax )−1 : V+ ⊗ (⊗ V−0 ) −→ V+ ⊗ (⊗ V−0 )

(59)

is well-defined and bounded, and for its norm the following estimate holds: k(Aξk − Ax )−1 k ≤ b(k + 1)−1 with certain constant b > 0.

(60)

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The proof of this Lemma will be presented in some other place. At last we write down the recurrence relation for the kernel Fb(x; ξ 3 ) that can be obtained similarly to the formula (24) ·Z −1 3 b b 3 (η; ζ 3 )Pb− (ζ 3 ; ξ 3 )dηdζ 3 F3 (x; ξ ) = (Aξk − Ax ) Sξ3 Pb+ (x, η)Γ Z + 6 Pb+ (x, η)b v (η)Fb2 (η; ζ 2 )δ(η − ζ3 )Pb− (ζ 3 ; ξ 3 )dηdζ 3 (61) ¸ Z b 2 (s; ζ 2 )Pb− (ζ 2 ; ξ 2 )dη 2 dsdζ 2 . − 2 Fb2 (x; η 2 )Pb− (η1 , ξ3 )Pb− (η2 , s)b v (s)Γ

4.2. Calculation of Fbq (x; ξ q )

Let us rewrite equality (20) as follows: ∞ X

" AFq (y) + P+

q=2

=

∞ X

# ak y

k

+ I1

k=2

Ã qFq

3 X

"

y, ..., y, Ay + P−

q=2

3 X

(62)

#! ak y

k

+ I2

k=2

where  I1 =P+

3 X

ak

k=2

I2 =

∞ X

k X

Ckj y k−j

∞ X

Fm1 (y)...

m1 =2

j=1





qFqy, ..., y, P−

q=2

3 X

ak

k=2

∞ X

 Fmj (y)

mj =2 k X j=1

Ckj y k−j

∞ X

∞ X

Fm1 (y)...

m1 =2

(63) Fmj (y).

mj =2

Writing operators with help of their kernels we get Ã P+

3 X k=2

! ak y

k

(x) =

3 Z X k=2

b k (s; ξ k )y(ξ k ) ds dξ k Pb+ (x, s)ak (s)Γ

(64)

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where, recall, we use notations (39), (41). Similarly we obtain: I1 ≡ I1 (x) =

∞ k X 3 X X

Z

X

Ckj

Pb+ (x, s)ak (s)(Fbm1 (s; ·)...

k=2 j=1 p=2j m1 +...+mj =p

b k−j (s; ·))(ξ p+k−j )y(ξ p+k−j ) ds dξ p+k−j ...Fbmj (s; ·)Γ (doing change of variables (k, j, p) −→ (q, j, p) : q = k − j + p) Z ∞ X X X j = Cq−p+j Pb+ (x, s)aq−p+j (s)(Fbm1 (s; ·)... q=3 (j,q)∈Qq m1 +...+mj =p

(65)

b q−p (s; ·))(ξ q )y(ξ q ) ds dξ q ...Fbmj (s; ·)Γ

where Qq = {(j, p) ∈ IN2 : 2 ≤ q − p + j ≤ 3, 1 ≤ j ≤ q − p + j, p ≥ 2j} = (if q ≥ 4) {(j, p) ∈ IN2 : (1, q − 2), (1, q − 1), (2, q − 1), (2, q), (3, q), (3, q + 1)}, Q3 = {(j, p) ∈ IN2 : (1, 2), (2, 2), (2, 3), (3, 3), (3, 4)}.

(66)

Besides, we get ∞ X

qFq (x; y, ..., y, Ay) =

q=2

∞ Z X

=

∞ Z X

Fbq (x; ξ q )Aξq y(ξ q ) dξ q

q=2

(67)

Aξq Fbq (x; ξ q )y(ξ q ) dξ q .

q=2

Using notation Z q−1 ) = \ F q P− (x; s, ζ

Fbq−1 (x; η, ζ q−2 )Pb− (η, s)dη

we can write Ã ! ∞ 3 X X k qFq x; y, ..., y, P− ak y q=2

=

∞ X 3 Z X

n−1 )a (s)Γ(s; b η m )y(ζ n−1 )y(η m ) ds dζ n−1 dη m \ nF m n P− (x; s, ζ

n=2 m=2

=

∞ X

(68)

k=2

X

q=3 n+m=q+1 n≥2,m=2,3

Z q q q b \ nam (s)(F n P− (x; s, ·)Γm (s; ·))(ξ )y(ξ ) ds dξ .

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At last I2 (x) =

∞ X 3 X k X r=2 k=2 j=1

Ckj

∞ X

X

Z r−1 )a (s) \ rF r P− (x; s, ζ k

p=2j m1 +...+mj =p

b k−j (s; ·))(ξ p+k−j )y(ζ r−1 )y(ξ p+k−j )dsdζ r−1 dξ p+k−j (Fbm1 (s; ·)...Fbmj (s; ·)Γ (changing variables (k, r, j, p)−→(q, r, j, p) : q = p + r + k − j − 1) Z ∞ X X X j = Cq−p−r+j+1 aq−p−r+j+1 (s)

(69)

q=4 (r,p,j)∈Rq m1 +...+mj =p q q q b b b \ × r(F r P− (x; s, ·)Fm1 (s; ·)...Fmj (s; ·)Γq−p−r+1 (s; ·))(ξ )y(ξ ) ds dξ

where Rq ={(r, p, j)∈IN3 : 1≤q−p−r+j≤2, 1≤j≤q−p−r+j+1, p≥2j, r≥2}. (70) After substituting (14), (17), (18) into (12) and doing some simple transformation we get ∞ Z X

(Aξq − Ax )Fbq (x; ξ q )y(ξ q ) dξ q

q=2

= −

3 Z X k=2 ∞ X

b k (s; ξ k )y(ξ k )dsdξ k Pb+ (x, s)ak (s)Γ X

(71)

Z q q q b \ nam (s)(F n P− (x; s, ·)Γm (s; ·))(ξ )y(ξ )dsdξ +I1 (x)−I2 (x)

q=3 n+m=q+1 n≥2,m=2,3

where I1 (x), I2 (x) are defined in (15),(69). In order to derive from (71) recurrence relation for Fbq (x; ξ q ) we i) make the change y(ξ q ) = Pb− (ξ q ; ζ q )z(ζ q ) with arbitrary z(ζ) ∈ V where we use the notation Pb− (ζ k ; ξ k ) = Pb− (ζ1 , ξ1 ) · ... · Pb− (ζk , ξk )

(72)

for kernel of tensor product P− ⊗ ... ⊗ P− (k times) for projection operator P− ; ii) apply symmetrization operator σζ q and avoid z(ζ q ); and iii) using Lemma 4.1 invert operator Aξq − Ax . As a result (renaming coordinates) we get the recurrence relation for Fbq (x; ξ q ) with q ≥ 4: Fbq (x; ξ q ) = (Aξq − Ax )−1 σξq (J1 (x; ξ q ) − J2 (x; ξ q ) − J3 (x; ξ q ))

(73)

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where X

J1 (x; ξ q ) =

X

j Cq−p+j

(74)

(j,q)∈Qq m1 +...+mj =p

Z

b q−p (s; ·))(ζ q )Pb− (ζ q ; ξ q ) ds dζ q × Pb+ (x, s)aq−p+j (s)(Fbm1 (s; ·)...Fbmj (s; ·)Γ J2 (x; ξ q ) X =

(75)

Z q b q q q b \ nam (s)(F n P− (x; s, ·)Γm (s; ·))(ζ )P− (ζ ; ξ ) ds dζ

n+m=q+1 n≥2,m=2,3

J3 (x; ξ q ) =

X

X

Z j Cq−p−r+j+1

aq−p−r+j+1 (s)

(76)

(r,p,j)∈Rq m1 +...+mj =p q b q q q b b b \ ×r(F r P− (x; s, ·)Fm1 (s; ·)...Fmj (s; ·)Γq−p−r+1 (s; ·))(ζ )P− (ζ ; ξ ) ds dζ .

Thus we have proven the following Theorem: Theorem 4.1. The kernels Fbq (x; ξ q ) from decopmposition (42) of the map F (x; y) defining stable invariant manifold are defined in (24) (for q = 2), in recurrence relation (27) (for q = 3) and in (20)–(23) (for q ≥ 4).

5. Analyticity of the Map F In this section we prove convergence of series (42) for map F (x; y) defining stable invariant manifold.

5.1. Estimate of norm for Fbq (x; ξ q ) k

First of all we have to recall that the norm of the space V+ ⊗ (⊗ V−0 ) is defined by relation (49). In virtue of (20), (50) and Lemma 4.1, kFq k ≡ kFbq k ≤ b(q + 1)−1 (kJ1 k + kJ2 k + kJ3 k) for

q≥4

(77)

where kernels J1 , J2 , J3 are defined in (21), (22), (23). Let us estimate now these kernels. Using that P+ y+ = y+ , for y+ ∈ V+ , P− y− = y− , for

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y− ∈ V− , and taking into account Lemmas 3.1, 3.2 we get from (21) X X j sup Cq−p+j × kJ1 k = sup ky+ kV+ =1 kyr kV− =1 (j,q)∈Qq m1 +...+mj =p r=1,...,q

Z

y+ (x)aq−p+j (x)Fm1 (x; y1 , ...)...Fmj (x; ..., yp )yp+1 (x)...yq (x) dx X X j ≤ Cq−p+j kaq−p+j kV kFm1 k...kFmj kγ q−p+j .

(78)

(j,q)∈Qq m1 +...+mj =p

Similarly, we get from (22) Z X kJ2 k = sup y+ (x)nFn (x; y1 , ..., yn−1 , P− (am yn ...yq )) dx ky+ kV+ =1 n+m=q+1 kyr kV− =1 n≥2,m=2,3 r=1,...,q

X

≤

(79) m

nkFn kkam kV γ .

n+m=q+1 n≥2,m=2,3

At last we obtain from (23) X kJ3 k = sup

X

j Cq−p−r+j+1

kykV+ =1 (r,p,j)∈Rq m1 +...+mj =p kyr kV− =1 r=1,...,q

Z

× rFr (x; P− (y1 ...yq−p−r+1 aq−p−r+j+1 Fm1 (·; yq−p−r+2 , ...)

(80)

...Fmj (·; ..., yq−r+1 )), yq−r+2 , ..., yq )y(x)dx ≤

X

j Cq−p−r+j+1 rkaq−p−r+j+1 kV kFr kkFm1 k...kFmj kγ q−p−r+j+1 .

(r,p,j)∈Rq m1 +...+mj =p

Let kak kV ≤ b0 /b,

k = 2, 3.

Then summarizing (32)–(35) we obtain the following estimate for kFq k when q ≥ 4: Ã X X b0 j Cq−p+j kFm1 k...kFmj kγ q−p+j kFq k ≤ q+1 m +...+m =p (j,q)∈Qq

+

X

nkFn kγ

n+m=q+1 n≥2,m=2,3

m

1

X

j

X

! (81)

Cκj rkFr kkFm1 k...kFmj kγ κ + (r,p,j)∈Rq m1 +...+mj =p

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where κ = q − p − r + j + 1. Similarly we get from (24), (27): kF2 k ≤ b0 γ 2 /3

kF3 k ≤ b0 (γ 3 + 4γ 2 kF2 k)/4.

(82)

Thus, we have proven the following lemma. Lemma 5.1. The norms kF2 k, kF3 k satisfy inequality (37), and the norms kFq k for q ≥ 4 satisfy estimate (36). 5.2. Convergence of series for F (x, y) Define coefficients ϕq of the series ϕ(λ) =

∞ X

ϕ q λq

(83)

q=2

by the relations ϕ2 = b0 γ 2 /3

ϕ3 = b0 (γ 3 + 4γ 2 ϕ2 )/4,

(84)

and for q ≥ 4 Ã ϕq = b0 (q + 1)

+

X n+m=q+1 n≥2,m=2,3

X

X

(j,q)∈Qq

m1 +...+mj =p

−1

nϕn γ

m

+

X

j Cq−p+j ϕm1 ...ϕmj γ q−p+j

X

!

(85)

Cκj rϕr ϕm1 ...ϕmj γ κ

(r,p,j)∈Rq m1 +...+mj =p

with κ = q − p − r + j + 1. Evidently, kFq k ≤ ϕq

for each q ≥ 2

(86)

and therefore to prove convergence of series (42) we have to prove convergence of series (38). Theorem 5.1. The series (38) with coefficients ϕq defined in (39), (85) converges for sufficiently small |λ|. Proof. Multiplying both parts of equality (85) on (q + 1)λq , and both parts of equalities (39) on the same multiplier with q = 2, 3, and summing obtained equalities over q from 2 to ∞ we get the equality: ∂ (λϕ(λ)) = b0 (S1 (λ) + S2 (λ) + S3 (λ)) ∂λ

(87)

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where 3 X S1 (λ) = (γλ)k +

(88)

k=2

+

∞ X

X

λq

j Cq−p+j ϕm1 ...ϕmj γ q−p+j ,

(j,q)∈Qq m1 +...+mj =p

q=3

∞ X S2 (λ) = λq q=3

X

nϕn γ m

(89)

n+m=q+1 n≥2,m=2,3

∞ X S3 (λ) = λq q=4

X

X

X

j Cq−p−r+j+1 rϕr ϕm1 ...ϕmj γ q−r+1 . (90)

(r,p,j)∈Rq m1 +...+mj =p

Taking into account definition (16) of the set Qq and doing change of variables (q, j, p) → (k, j, p) : k = q + j − p in (88) we get:

S1 (λ) =

3 X

(γλ)k +

k=2

=

3 X k=2

=

3 X

k

(γλ) +

3 X

(γλ)k

k

(γλ)

k X j=1

k=2

Ckj

µ Ckj

∞ X

X

ϕm1 ...ϕmj λp−j

p=2j m1 +...+mj =p

j=1

k=2 3 X

k X

ϕ(λ) λ

¶j =

3 X

µ ¶k ϕ(λ) (γλ) 1 + λ k

(91)

k=2

γ k (λ + ϕ(λ))k .

k=2

Changing order of summation in (89) we obtain:

S2 (λ) =

∞ X 3 X n=2 m=2

m n+m−1

nϕn γ λ

0

= ϕ (λ)

3 X

(γλ)m .

(92)

m=2

Changing variables (k, r, j, p) → (q, r, j, p) : k = q − p − r + j + 1 in (90) with help of definition (70) of the set Rq we get:

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S3 (λ) =

k ∞ X 3 X X

Ckj

=

rϕr λ

r−1

r=2

= ϕ0 (λ)

X

3 X

(γλ)

k

k X

(γλ)k

j=1

k=2

k X

Ã

Ckj

µ Ckj

∞ X

!j ϕm λ

m

/λj

(93)

m=1

j=1

k=2 3 X

λp+r+k−j−1 rϕr ϕm1 ...ϕmj γ k

p=2j m1 +...+mj =p

r=2 k=2 j=1 ∞ X

∞ X

111

¶j

ϕ(λ) λ

.

Relations (92), (93) imply: S2 (λ)+S3 (λ)=ϕ0 (λ)

3 X

µ ¶k 3 X ϕ(λ) k (γλ)k 1 + = ϕ0 (λ) γ k (λ+ϕ(λ)) . (94) λ

k=2

k=2

Therefore we get from (87), (91), (94): Ã ! 3 X ∂ k 0 k (λϕ(λ) = b0 (1 + ϕ (λ)) γ (λ + ϕ(λ)) . ∂λ

(95)

k=2

Doing in (95) the change of functions ψ(λ) = λ + ϕ(λ)

(96)

we obtain the equality ∂ ∂ (λψ(λ) − λ2 ) = b0 ∂λ ∂λ

Ã

3 X γk k+1 (ψ(λ)) k+1

! .

(97)

k=2

Since in (97) both expressions under sign of derivative equal zero at λ = 0, we derive that Ã 3 ! X γk k+1 2 λψ(λ) − λ = b0 (ψ(λ)) . (98) k+1 k=2

Doing in (98) the change ψ(λ) = λz(λ) we obtain the relation Φ(z(λ), λ) ≡ z(λ) − 1 − b0

Ã

3 X γ k λk−1

k=2

k+1

(99) ! k+1

z(λ)

=0

(100)

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where the first equality is definition of the function Φ(z, λ). Since (96), (99) imply that z(0) = 1, we get from definition (100) that function Φ(z, λ) satisfies: Φ(z, λ)| z=1 = 0, λ=0

Φ0z (z, λ)| z=1 = 1. λ=0

Therefore by Implicite Function Theorem there exists a function z(λ) analytic in a neighborhood of origin such that z(0) = 1 and Φ(z(λ), λ) ≡ 0. Hence, by (96), (99) the function ϕ(λ) = λ(z(λ) − 1) defined in (38) is also analytic. ¤ Theorem 5.1 and inequalities (86) imply Theorem 5.2. Let F (·, y− ) be the map (21), (22) that defines the stable invariant manifold (20). Then decomposition of this map in series (40) converges in a neighborhood of origin of the space V− . References 1. A.V. Fursikov, Stabilizability of quasi linear parabolic equation by feedback boundary control. Sbornik Mathematics, 192, 4 (2001), 593–639. 2. A.V. Fursikov, Stabilizability of Two-Dimensional Navier-Stokes equations with help of a boundary feedback control. J. of Math. Fluid Mech., 3 (2001), 259–301. 3. A.V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete and Cont. Dyn. Syst., 10, 1–2 (2004), 289–314. 4. A.V. Babin and M.I. Vishik, Attractors of Evolution equations, (NorthHolland, Amsterdam, London, New-York, Tokyo, 1992). 5. D. Henry, Geometric Theory of Semilinear Parabolic Equations, (Lecture Notes in Mathematics 840, Springer-Verlag, Berlin-Heidelberg-New York 1981). 6. A.V. Fursikov, Analyticity of Stable Invariant Manifolds of 1D-Semilinear Parabolic Equations. AMS Contemporary Mathematics Series. Proc. of Joint Summer Research Conference on Control Methods in PDE-Dynamical Systems. (AMS Providence, 2007) (to appear). 7. A.B. Kalinina, Numerical realization of the method of functional-analytic series for projection on a stable manifold. Numerical methods and programming, 7, 1 (2006), 65-72, http://num-meth.srcc.msu.su (in Russian).

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APPROXIMATE SOLUTIONS FOR NON-LINEAR AUTONOMOUS ODEs ON THE BASIS OF PWL APPROXIMATION THEORY A.G. GARCIAa , S.I. BIAGIOLA, J.L. FIGUEROA, O.E. AGAMENNONI∗ Departamento de Ingenier´ıa El´ ectrica y de Computadoras, Universidad Nacional del Sur, Bah´ıa Blanca, Buenos Aires B8000CPB, Argentina a E-mail: [email protected] L.R. CASTRO Departamento de Matem´ atica, Universidad Nacional del Sur, Bah´ıa Blanca, Buenos Aires B8000CPB, Argentina E-mail: [email protected] In this paper a general approach to approximate the solutions of Ordinary Differential Equations (ODEs) is presented. The methodology uses a Piecewise Linear (PWL) approximation of the ODEs vector field which describes the dynamics of a system. Once the number of Simplices to include in the PWL approximation of the nonlinear vector field has been decided, a measure of the dynamics approximation error is used to estimate upper and lower bounds for the error between the actual and the approximated trajectories. Several examples are analyzed to illustrate the method. A comparison with a classical numerical solution is provided to show some advantages of the proposed technique. Keywords: Piecewise Linear (PWL); ODEs.

1. Introduction Finding approximate solutions (ASs) for nonlinear dynamics modelled by Ordinary Differential Equations (ODEs) is a relevant problem in the field of systems. Applications related to dynamics and control, involve computation of ASs for these ODEs, which is usually time-consuming and demands a significant computational effort. ∗ S.

Biagiola and J. Figueroa are also with CONICET. O. Agamennoni is also with CIC.

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Many research works focus on discretization methods for processes which are originally continuous-time and mathematically described by a system of nonlinear ODEs. The discretization approach used to describe process dynamic behavior usually aims at digital process simulation (Ref. 1). Most of classical methods for solving ODEs (e.g. Runge-Kutta) find ASs on the basis of time domain discretization. An alternative approach is to compute ASs using a discretization of the state space. In this sense, Girard2 developed a method to solve ODEs by replacing the vector field by a PWL approximation. Although there has been much research work dedicated to error estimators for numerical integration methods (Ref. 3 and references therein), to the best of the authors knowledge, an approximate solution method for nonlinear ODEs with an a priori determined error-bound is a novel approach. The present work introduces a new approach to the problem of describing the dynamic behavior of nonlinear autonomous systems. A general method to approximate the solutions of a set of ODEs is presented. The technique makes use of a PWL approximation of the ODEs vector field which describes the dynamics of a system. A measure of the dynamics approximation error is used to estimate upper and lower bounds for the error between the real and the approximate trajectories (i.e. the real and approximate solutions to the ODE). The nonlinear ODEs solution approach herein introduced, can be straightforwardly implemented for real-time applications. Due to recent advances in microtechnology such as the integrated circuits, i.e. electronic device that can operate as a high-speed analogical computer of very small size due to the high compression density (see for instance Ref. 4), it is possible to easily implement continuous-time algorithms, which is especially appealing for simulation and control purposes. The inclusion of a piecewise linear (PWL) approximation of the nonlinear dynamics yields a substantial reduction regarding computational time and effort demands. The proposed solution approach can be used in many applications. For instance, in the field of physical and (bio)chemical processes that exhibit nonlinear behavior. These ones, are good examples since many of them are modelled by systems of ODEs in continuous-time domain. Moreover, it could be used in the solution of dynamic optimization problems with significant advantages as regards computational time consumption. For instance, the AS method can be implemented to efficiently solve back-off algorithms immerse in control applications, to determine the operating point which guarantees feasibility even under disturbances effects

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(Refs. 5,6). Besides, the method could be also applied in many control strategies of nonlinear processes, such as in model predictive control, where a long-term process behavior has to be determined in order to decide the next manipulated variable movement (Ref. 7). 2. An Error Bound for PWL Solutions: The Scalar Case Let us consider an ODE given by: x(t) ˙ = f (x),

(1)

where f (x) : < → < a continuous function, x(t) : D ⊂ <+ → < with D a compact set. We approximate the function f using a PWL approximation as defined in Ref. 8. If we note fPi W L the approximation to f on each simplex, we have: x˙ i (t) = fPi W L (xi ) = ai xi (t) + bi , i = 1, · · · , r, r being the number of simplices considered. Let us assume that the error bound on each simplex is given by: ¯ ¯ ¯f (x) − fPi W L (x)¯ ≤ εi , i = 1, .., r,

(2)

(3)

then: ¯ ¡ ¢¯ ¯x(t) ˙ − ai x(t) + bi ¯ ≤ εi , i = 1, · · · , r.

(4)

E i (t) = x(t) − xi (t), i = 1, · · · , r,

(5)

If we note:

we can write equation (4) in the following way: ¯ ¡ ¡ ¢ ¢¯ ¯x(t) ˙ − ai xi (t) + E i (t) + bi ¯ ≤ εi ⇒

(6)

¯ ¯ ¯x(t) ˙ − ai xi (t) − ai E i (t) − bi ¯ ≤ εi ⇔

(7)

¯ ¯   ¯ ¯ ¯ ¯ ¯  i i ¯ i i i ˙ − a x (t) + b  − a E (t)¯ ≤ εi ⇔ ¯x(t) ¯ ¯ | {z } ¯ ¯ x˙ i (t)

(8)

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ˙ − x˙ i (t) − ai E i (t)¯ ≤ εi . ¯x(t) ¯| ¯ {z } ¯ ¯ ˙i

(9)

E (t)

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The last step in all this development is to evaluate the differential inequality (9) and for that purpose, we can write it as follows: −εi ≤ E˙ i (t) − ai E i (t) ≤ εi , i = 1, .., r,

(10)

−ai ·t

and multiplying every side in (10) by exponential e , we get: ³ ´ i i i −e−a ·t εi ≤ e−a ·t E˙ i (t) − ai E i (t) ≤ e−a ·t εi , i = 1, .., r, whence: −e−a ·t εi ≤ i

i d ³ −ai ·t i ´ e E (t) ≤ e−a ·t εi , i = 1, .., r. dt

(11)

(12)

Integrating:

¡ i ¢−1 i i ai ·t E i (t) ≥ (a−1 a )ε + ea ·t E i (0), (13) i −e ³ ¡ ¢−1 ´ ¡ ¢ i i −1 E i (t) ≤ − ai + ea ·t ai · εi + ea ·t E i (0), i = 1, .., r. (14)

3. From Single to Multivariable Bounds As we were discussing in the last section, it is not so difficult to obtain upper and lower bounds when approximating the scalar functions with PWL. However, the multivariable case requires a more detailed analysis, and such a problem can be stated as follows: Let: £ ¤T ˙ X(t) = f (X), f (X) = f1 (X) f2 (X) · · · fn (X) , fi (.) ∈ U ⊂
(15) (16)

where T stands for transpose. Let us consider U divided in r simplices using a boundary configuration H as described in Ref. 8. Let £ i ¤T i i fPi W L (X) = f1PWL (X)f2PWL (X) · · · fnPWL (X) (17) be the PWL approximation of f on the ith -simplex. In what follows we use the idea presented in Ref. 8, p. 100, where the maximum norm was used in order to characterize the size of the error in the following way: ¯  ¯¯ i i ¯ f1 (X) − f1P  W L (X) ≤ ε1   ¯ ¯   i i ¯  ¯f2 (X) − f2P W L (X) ≤ ε2 , i = 1, .., r. (18)  ..   . ¯  ¯ ¯ fn (X) − f i (X)¯ ≤ εin nP W L

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We are looking for an error bound such that: ° ° °X − X i ° ≤ hi (εi ), i = 1, . . . , r, ∞ with

117

(19)

£ ¤T εi = εi1 εi2 · · · εin ,

where hi :
that is:

(22)

(23)

j

¯ i ¯ ¯e˙ j − Aij E i ¯ ≤ εij , £

(21)

(24)

¤T

where E i = ei1 ei2 · · · ein , i = 1, .., r. We can rewrite the expression (24) as follows: ¯  ¯¯ i i i¯ E ≤ εi1 e ˙ − A  1 1   ¯ ¯   ¯e˙ i − Ai E i ¯ ≤ εi 2 2 2 , i = 1, .., r, (25) ..   .   ¯  ¯¯ i e˙ n − Ain E i ¯ ≤ εin whence:

 −εi1 ≤ e˙ i1 − Ai1 · E i ≤ εi1      −εi ≤ e˙ i − Ai · E i ≤ εi 2 2 2 2 , ..   .    i −εn ≤ e˙ in − Ain · E i ≤ εin

i = 1, .., r.

(26)

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Using matrix notation, (26) can be written in the following way: −εi ≤ E˙ i − Ai · E i ≤ εi , £ ¤T £ ¤ i0 i0 T where εi = εi1 εi2 · · · εin , Ai = Ai0 , i = 1, .., r. 1 A2 · · · An i Pre-multiplying by the exponential matrix e−A ·t , we obtain: −e−A

i

·t

· εi ≤ e−A

i

·t

(27)

i i · E˙ i − e−A ·t · Ai · E i ≤ e−A ·t · εi , i = 1, .., r. (28)

Then, we have: i i d h −Ai ·t e · E i (t) ≤ e−A ·t · εi , i = 1, .., r. dt Integrating each member of (29), Z t i i − e−A ·σ · dσ · εi ≤ e−A ·t · E i (t) − E i (0) 0 Z t i ≤ e−A ·σ · dσ · εi , i = 1, .., r, −e−A

i

·t

· εi ≤

(29)

(30) (31)

0

whence, the following upper and lower bounds are obtained Z t i Ai ·t i Ai ·t e · E (0) − e · e−A ·σ · dσ · εi ≤ E i (t) 0Z t i Ai ·t i Ai ·t ≤e · E (0) + e · e−A ·σ · dσ · εi , i = 1, .., r.

(32)

0

Finally, we must note that, in order to compute (32) it is necessary to i i evaluate not only the matrix eA ·t but also the matrix e−A ·t , which requires an extra computational effort, then the idea is to transform (32) into an i equivalent equation which only contains eA ·t . This can be done using the following equality: Z t Z t eA·t · e−A·σ · dσ = eA·σ · dσ. (33) 0

0

Finally, using (33) we can write (32) as follows: Z t i i eA ·t · E i (0) − eA ·σ · dσ · εi ≤ E i (t) 0Z t i Ai ·t i eA ·σ · dσ · εi , i = 1, .., r. ≤e · E (0) +

(34)

0

It is simple to verify that (34) is valid for the scalar case presented in Section 2 when the Ai , i = 1, .., r are invertible, we can write: h ³ ¡ ¢−1 ´ ³¡ ¢−1 ´i i i i i ·ε , − −Ai E i (t) ≤ eA ·t ·E i (0) + eA ·t · e−A ·t · − Ai (35) i = 1, .., r.

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In this way: E i (t) ≤ eA

i

·t

h ¡ ¢−1 i i ¡ i ¢−1 i · E i (0) + Ai ·ε − A · ε , i = 1, .., r,

119

(36)

which is completely analogous to (14). The case which Ai , i = 1, .., r are not invertible will be developed in next section. 4. Further Analysis of the Bound Although the obtained bound (32) is valid in general, for practical purposes an expression that can be directly implemented is needed. To do this we can use any of the methodologies presented in Ref. 9 where a complete survey of the methodologies for computing the exponential matrix are presented and classified by categories: in particular we use here the eight methodology (see Ref. 9), since the optimal computational method is beyond the scope of this work. In this way we can write: eA·t = y0 (t) · I + y1 (t) · A + · · · + yn−1 (t) · An−1 ,

(37)

where yi (t), i = 1, .., n − 1 is the ith solution of the ODE given by: 0 = y (n) (t) + an−1 · y (n−1) (t) + · · · + a0 · y(t), n

n−1

(38)

(i)

with λ + an−1 · λ + · · · + a0 · λ = det(λ · I − A) and y (t), i = 0, .., n stands for the ith time derivative of y(t). Replacing (38) into (32) and considering only the superior bound (the inferior one is analogous), we get: £ ¤ E(t) ≤ y0 (t) · I + y1 (t) · A + · · · + yn−1 (t) · An−1 · E(0) Z t (39) £ ¤ + y0 (t) · I + y1 (t) · A + · · · + yn−1 (t) · An−1 · dσ · ε, 0

where E, ε and A stands for the variables on the ith simplex. This explicit form for the error bound allows to analyze how the structure of the matrix A impacts on each value of vector ε ∈
(40) −1

We first use a PWL approximation of this vector field (− tan 2 Simplices for x ∈ [0, 1] as shown in Figure 1-a.

(x)) with

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−0.4

−0.6

−0.8

0

0.1

0.2

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0.3 b)

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PWL approximation vs Nonlinear vector field. PWL trajectories vs real ones.

In this way a simple calculation shows that the vertex is allocated at x1 = 0.33333. Now we set the initial condition to x0 = 0.4. In order to establish the motion direction inside this simplex [0, 0.5] all we need to do is to analyze the closed-form expression for the PWL trajectory in this simplex: µ ¶ b b x(t) = − + x0 + ea·t , (41) a a (

Simplex 1: x ∈ [0, 0.5] ⇒ a1 = −0.9273, b1 = 0 Simplex 2: x ∈ [0.5, 1] ⇒ a1 = −0.6435, b1 = −0.1419

(42)

Then, for Simplex 1: x1 (t) = x0 e−0.9273·t ,

(43)

which is in fact a decreasing function of the time. This means that the solution remains in the present simplex since the evolution is going to an attractor allocated there (x = 0). In Figure 1 approximate and true trajectories are shown while the error and bounds on it are depicted in Figure 2. Notice that the prediction in Section 2 is verified in Figure 2 where the true error is between the upper and lower bounds.

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0.2 Error Upper Bound Lower Bound 0.15

0.1

0.05

0

−0.05

−0.1

−0.15

−0.2

0

1

2

3

4

5

6

7

8

9

10

t[secs]

Fig. 2.

Error and error bounds (2 simplices).

The last comment regarding this ODE using only 2 simplices is that the PWL approximation for each simplex was obtained using the Least Squared Error (LSE) criterion and gives ε1 = 0.14, ε2 = 0.12. It turns out that for each simplex (in this case only one) the trajectory was calculated using a time step of 0.01 seconds. For this case, the following comparative results were obtained:   Time Consumed by the PWL method=0.078 secs.  Time Consumed by ODE45 =0.1880 secs.

(44)

Clearly, the PWL methodology exhibits an outstanding advantage regarding computational effort. In order to improve the quality of the upper and lower error bounds we increase the number of simplices to 5, obtaining the error bounds shown in Figure 3. In this case the values for εi , i = 1, .., 5 are: ε1 = 0.015, ε2 = 0.023, 3 ε = 0.025, ε4 = 0.023, ε5 = 0.02.The execution times were:   Time Consumed by the PWL method=0.079 secs.  Time Consumed by ODE45 =0.1880 secs.

(45)

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0

−0.005

−0.01

−0.015

0

1

2

3

4

5

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7

8

9

10

t[secs]

Fig. 3.

Error and error bounds (5 simplices).

5.2. Example 2: A batch bioprocess This example shows how the bound actually works in a real case. We consider the following bioprocess:  µm · S(t)  ˙  = · X(t)  X(t) K + S(t) , X ∈ <+ , S ∈ <+ , (46)  µm · S(t)  ˙  S(t) = −α · · X(t) K + S(t) where µm , K and α are real and positive constants. This is a stable system and we expect that the bounds given in Section 3 are tight for simplices in a region shown in Figure 4 where only two simplices are considered. Figure 5 shows a plot for the real vector field we are going to work with. In Figure 4, the eigenvalues corresponding to the linear approximations on each simplex are given. The corresponding trajectories are drawn in Figure 6. If we increase the number of simplices to 4 we get the domain partition shown in Figure 7. With this new subdivision we improve the approximation and bounds as shown in Figure 8. As expected the new bounds are much better than the previous ones (compare Figures 6 and 8).

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0.3 y

0.25

0.2 λ1=0.43034 0.15 λ2=−2.7756e−017 0.1

0.05

0 0.5

0.55

0.6

Fig. 4.

0.65

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0.75 x

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Domain Division (2 simplices).

Surface for µm.S.X/(K+S) 1

0.5

0 1

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0.2

S

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X

Surface for −α.µm.S.X/(K+S) 0

−0.5 1 0.8 0.6 0.4 0.2 0 S

Fig. 5.

0

0.2

0.4

0.6

0.8

1

X

Both Components of the Vector Field of the second example.

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0.04

0.4 0.02

0.2 0 0.5

0.6

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0

1

0

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Fig. 6.

20 t[secs]

x

Trajectories, errors and bounds (2 simplices).

0.5 Phase−Portrait Simplex Division 0.45 λ1=0 0.4 λ2=−0.77114

λ1=2.2204e−016

0.35 λ2=−0.55679 0.3

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0.25 λ1=0.070832 0.2 λ2=5.5511e−017 0.15 λ1=0.46889 0.1

λ2=2.7756e−017

0.05

0 0.5

0.55

0.6

0.65

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x

Fig. 7.

Domain Division (4 simplices).

1

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α=1, K=1, µm=1 1

1 PWL. Real.

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Error for X1. Upper bound for X1. Lower bound for X1.

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t[secs] t[secs]

Trajectories, errors and bounds (4 simplices).

6. Conclusions A methodology for obtaining closed-form expressions for approximate solutions to a given set of ODEs has been presented. This methodology is endowed with an error bound which allows to evaluate how precise could be our approximate trajectories, which is a great advantage when compared with classical methodologies such as Runge-Kutta and Adam’s (see for instance Ref. 10 for details on how these methods obtain approximations for the errors). Using an example in <1 , the validity of the bound was analyzed and contrasted using two different partitions of the domain. A significant improveness of the error bounds is observed as the number of simplices increases. Finally, a real batch bioprocess example was presented in order to show the potential applications of our approach.

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Acknowledgments This work was partially supported by ANPCyT, CONICET, CIC, SECyTUNS. References 1. N. Kazantis, N. Huynh and T. Good, A model-based characterization of the long-term asymptotic behavior of nonlinear discrete-time processes using invariance functional equations, Computer and Chemical Engineering, 29 (2005), 2346–2354. 2. A. Girard, Approximate solutions of ODEs using piecewise linear vector fields, Computer Algebra in Scientific Computing, (2002) 107–120. 3. J. de Swart and G. S¨ oderling, On the construction of error estimators for implicit Runge-Kutta methods, Journal of Computational and Applied Mathematics, 2 (1997), 347–358. 4. T. Roska and L. O. Chua, The CNN universal machine: An analogic array computer, IEEE Transactions on Circuits and Systems-II, 40 (1993), 163– 173. 5. S. Biagiola, A. Bandoni and J. L. Figueroa, Back-off application for dynamic optimization and control of nonlinear processes, European Symposium on Computer Aided Process Engineering, (2005), 1267–1272. 6. J. L. Figueroa and A. Desages, Dynamic back-off analysis: use of piecewise linear approximations, Optimal Control Applications and Methods, 24 (2003), 103–120. 7. M.A. Henson, Nonlinear model predictive control: current status and future directions, Computers and Chemical Engineering, 23 (1998), 187. 8. P. Juli´ an, A High Level Canonical Piecewise Linear Representation: Theory and applications, PhD dissertation, (Universidad Nacional del Sur, 1999). 9. C. Moler and C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, SIAM Review, 20 (1978), 801–837. 10. D. Kincaid and W. Cheney, Numerical Analysis. (Brooks/Cole Publishing Company, 1991).

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ON THE INFLUENCE OF A SUBQUADRATIC CONVECTION TERM IN SINGULAR ELLIPTIC PROBLEMS b ˘ MARIUS GHERGUa and VICENT ¸ IU RADULESCU a

Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. BOX 1-764, RO-014700 Bucharest, Romania E-mail: [email protected] b Department of Mathematics, University of Craiova, RO-200585 Craiova, Romania E-mail: [email protected]

We establish some existence results for the singular elliptic equation −∆u = g(u) + λ|∇u|a + µf (x, u) either in a smooth bounded domain Ω ⊂ RN or in the whole space. We suppose that λ and µ are positive parameters, 0 < a ≤ 2, f is a nondecreasing function which is sublinear with respect to the second variable, and g ∈ C 1 (0, ∞) is a decreasing function such that lim g(s) = +∞. s&0

The analysis we develop in this paper emphasizes the central role played by the convection term |∇u|a . Keywords: Singular elliptic equation; Convection term; Maximum principle.

1. Introduction We are concerned in this paper with singular elliptic equations of the type −∆u = g(u) + λ|∇u|a + µf (x, u),

u>0

in Ω,

(1)

where Ω ⊂ RN (N ≥ 2) is either a smooth bounded domain or the whole space, 0 < a ≤ 2 and λ, µ ≥ 0. We suppose that g ∈ C 1 (0, ∞) is a positive nonincreasing function such that (g1) lim g(s) = +∞. s&0

We also assume that f : Ω × [0, ∞) → [0, ∞) is a H¨older continuous function such that f > 0 on Ω × (0, ∞) and is sub-linear with respect to the second variable, that is, f (x, s) is nonincreasing for all x ∈ Ω; (f 1) the mapping (0, +∞) 3 s 7−→ s

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(f 2) lim

s→∞

f (x, s) = 0, uniformly for x ∈ Ω. s

Problems of this type arise in the study of non-Newtonian fluids, boundary layer phenomena for viscous fluids, chemical heterogeneous catalysts, as well as in the theory of heat conduction in electrically conducting materials. Our general setting includes some simple prototype models from boundary-layer theory of viscous fluids (see Ref. 1). If λ = 0 and µ = 0, problem (1) is called the Lane-Emden-Fowler equation. Problems of this type, as well as the associated evolution equations, describe naturally certain physical phenomena. For example, super-diffusivity equations of this type have been proposed by de Gennes2 as a model for long range Van der Waals interactions in thin films spreading on solid surfaces. This equation also appears in the study of cellular automata and interacting particle systems with self-organized criticality (see Ref. 3), as well as to describe the flow over an impermeable plate (see Refs. 4,5). The main feature of this paper is the presence of the convection term |∇u|a . As remarked in Refs. 6,7, the requirement that the nonlinearity grows at most quadratically in |∇u| is natural in order to apply the maximum principle. In the case where λ = 0, the problem (1) subject to Dirichlet boundary condition has a unique solution for all µ ≥ 0 (see Refs. 8–11 and the references therein). If λ > 0, the following problem was considered in Zhang and Yu12  −α a in Ω,  −∆u = u + λ|∇u| + σ (2) u>0 in Ω,  u=0 on ∂Ω, where Ω is a smooth bounded domain, λ, σ ≥ 0, α > 0, and a ∈ (0, 2]. By using the change of variable v = eλu − 1 in the case a = 2, it is proved in Ref. 12 that problem (2) has classical solutions if λσ < λ1 , where λ1 is the first eigenvalue of −∆ in H01 (Ω). This will be used to deduce the existence and nonexistence in the case 0 < a < 2. If f (x, u) depends on u, the above change of variable does not preserve the sublinearity condition (f 1)–(f 2) and the monotony of the nonlinear term g in (1). In turn, if f (x, u) does not depend on u and a = 2, this method successfully applies to our study and we will be able to give a complete characterization of (1). Due to the singular term g(u) in (1), we cannot expect to have solutions

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in C 2 (Ω) for (1). As it was pointed out in Ref. 12, if α > 1 then the solution of (2) is not in C 1 (Ω). We are seeking in this paper classical solutions of (1), that is, solutions u ∈ C 2 (Ω) ∩ C(Ω) that verify (1).

2. Singular Elliptic Equations in Bounded Domains We present in this section some existence results for  a  −∆u = g(u) + λ|∇u| + µf (x, u) u>0  u=0

the problem in Ω, in Ω, on ∂Ω.

(3)

Theorem 2.1. Assume that conditions (f 1)–(f 2), (g1) are fulfilled and 0 < a ≤ 1. Then for all λ, µ ≥ 0 the problem (3) has at least one solution. Proof (Sketch). The proof relies on the sub and super-solution argument. Let us first notice that, by Ref. 13, there exists v ∈ C 2 (Ω) ∩ C(Ω) a solution of the problem  in Ω,  −∆v = g(v) (4) v>0 in Ω,  v=0 on ∂Ω. Then uλµ = v is a sub-solution of (3). The main point is to find a supersolution uλµ ∈ C 2 (Ω) ∩ C(Ω) of (3). This will be done separately for 0 < a < 1 and a = 1. Since g is decreasing, we can easily obtain that uλµ ≤ uλµ in Ω so (3) has at least one solution. Case 0 < a < 1. Let h ∈ C 2 (0, η] ∩ C[0, η] be such that  00 for all 0 < t < η,  h (t) = −g(h(t)), h(0) = 0,  h > 0 in (0, η].

(5)

The existence of h follows by classical arguments of ODE. Since h is concave, there exists h0 (0+) ∈ (0, +∞]. By taking η > 0 small enough, we can assume that h0 > 0 in (0, η], so h is increasing on [0, η]. We also have Lemma 2.1. R1 (i) h ∈ C 1 [0, η] if and only if 0 g(s)ds < +∞; (ii) If 0 < p ≤ 2, then there exist c1 , c2 > 0 such that (h0 )p (t) ≤ c1 g(h(t)) + c2 ,

for all 0 < t < η.

(6)

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Now, we construct a super-solution in the form uλµ = M h(cϕ1 ) for M > 1 large enough and c > 0 sufficiently small, where ϕ1 represents the first eigenfunction of −∆ in H01 (Ω). Case a = 1. This case was left as an open problem in Ref. 14. Note that the method used in Case 0 < a < 1 applies here only for small values of λ and µ. Let R > 0 be large enough such that Ω ⊂ BR (0), where BR (0) = {x ∈ RN ; |x| < R}. We consider the problem  |x| < R,  −∆u = g(u) + λ|∇u| + µf (x, u) (7) u>0 |x| < R,  u=0 |x| = R. In order to provide a super-solution for (7) let us first consider the problem  |x| < R,  −∆u = g(u) + λ|∇u| + 1 (8) u>0 |x| < R,  u=0 |x| = R. We need the following auxiliary result. Lemma 2.2. Problem (8) has at least one solution. Proof. We are looking for radially symmetric solution u of (8), that is, u = u(r), 0 ≤ r = |x| ≤ R and  N −1 0  u (r) = g(u(r)) + λ|u0 (r)| + 1 0 ≤ r < R,  −u00 − r (9) u>0 0 ≤ r < R,   u(R) = 0. This implies −(rN −1 u0 (r))0 ≥ 0 for all 0 ≤ r < R, which yields u0 (r) ≤ 0 for all 0 ≤ r < R. Then (9) gives ¶ µ N −1 0 0 00 u (r) + λu (r) = g(u(r)) + 1, 0 ≤ r < R. − u + r We obtain −(eλr rN −1 u0 (r))0 = eλr rN −1 (g(u(r)) + 1),

0 ≤ r < R.

(10)

From (10) we get Z t Z r u(r)=u(0) − e−λt t−N +1 eλs sN −1 (g(u(s)) + 1)dsdt, 0 ≤ r < R. (11) 0

0

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131

Let w ∈ C 2 (BR (0)) ∩ C(B R (0)) be the unique radial solution of the problem  |x| < R,  −∆w = g(w) + 1 (12) w>0 |x| < R,  w=0 |x| = R. Clearly, w is a sub-solution of (8). As above we get Z r Z t −N +1 w(r) = w(0) − t sN −1 (g(w(s)) + 1)dsdt, 0

0 ≤ r < R.

(13)

0

We claim that there exists a solution v ∈ C 2 [0, R) ∩ C[0, R] of (11) such that v > 0 in [0, R). Let A = w(0) and define the sequence (vk )k≥1 inductively by  Z r Z t  −λt −N +1  e t eλs sN −1 (g(vk−1 (s)) + 1)dsdt,  vk (r) = A − 0 0 (14) 0 ≤ r < R, k ≥ 1,    v0 = w. Note that vk is decreasing in [0, R) for all k ≥ 0. From (13) and (14) we easily check that v1 ≥ v0 and by induction we deduce vk ≥ vk−1 for all k ≥ 1. Hence w = v0 ≤ v1 ≤ ... ≤ vk ≤ ... ≤ A

in BR (0).

Thus, there exists v(r) := lim vk (r), for all 0 ≤ r < R and v > 0 in [0, R). k→∞

We can now pass to the limit in (14) in order to get that v is a solution of (11). By classical regularity arguments we also obtain v ∈ C 2 [0, R)∩C[0, R]. This proves the claim. We have obtained a super-solution v of (8) such that v ≥ w in BR (0). Hence, the problem (8) has at least one solution and the proof of our Lemma is now complete. ¤ Let u ∈ C 2 (Ω) ∩ C(Ω) be a solution of the problem (8). For M > 1 we have −∆(M u) ≥ g(M u) + λ|∇(M u)| + M in Ω. Since f is sublinear, we can choose M = M (µ) > 1 such that M ≥ µf (x, M |u|∞ ) in BR (0). Then uλµ := M u is a super-solution for (1). This finishes the proof of Theorem 2.1. ¤ In the case 1 < a ≤ 2 we prove the following result.

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Theorem 2.2. Assume µ = 1 and f , g satisfy (f 1)–(f 2) and (g1) respectively. Then there exists λ∗ > 0 such that (1) has at least one classical solution for 0 ≤ λ < λ∗ and no solutions exist if λ > λ∗ . Proof. For small values of λ > 0 we can construct a super-solution of (3) in the same manner as in the proof of Theorem 2.1. Set A = { λ ≥ 0 : problem (1) has at least one classical solution}. From the above arguments, A is nonempty. Let λ∗ = sup A. First we claim that if λ ∈ A, then [0, λ) ⊆ A. For this purpose, let λ1 ∈ A and 0 ≤ λ2 < λ1 . If uλ1 is a solution of (1) with λ = λ1 , then uλ1 is a super-solution for (1) with λ = λ2 while v defined in (4) is a sub-solution. Hence, the problem (1) with λ = λ2 has at least one classical solution. This proves the claim. Since λ ∈ A was arbitrary chosen, we conclude that [0, λ∗ ) ⊂ A. Let us prove that λ∗ < +∞. For this purpose we use the following result Lemma 2.3. (see Ref. 15). If a > 1, then there exists a real number σ ¯>0 such that the problem ½ −∆u ≥ |∇u|a + σ in Ω, (15) u=0 on ∂Ω, has no solutions for σ > σ ¯. Set

³ τ :=

inf

´ g(s) + f (x, s) .

(x,s)∈Ω×(0,+∞)

Since lim g(s) = +∞ and the mapping (0, +∞) 3 s 7−→ min f (x, s) is s&0

x∈Ω

positive and nondecreasing, we deduce that m is positive. Let λ > 0 be such that (3) has a solution uλ . If w = λ1/(p−1) uλ , then v verifies  p 1/(a−1) τ in Ω,  −∆w ≥ |∇w| + λ (16) w>0 in Ω,  w=0 on ∂Ω. By Lemma 2.3 it follows that λ1/(a−1) τ ≤ σ ¯ which gives λ ≤ (¯ σ /τ )a−1 . ∗ This means that λ is finite. This completes the proof. ¤ Theorems 2.1 and 2.2 show the importance of the convection term λ|∇u|a in (3). Indeed, according to Ref. 10, Theorem 1.3, for any µ > 0,

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the boundary value problem  −α a β  −∆u = u + λ|∇u| + µu u>0  u=0

in Ω, in Ω, on ∂Ω,

133

(17)

has a unique solution, provided λ = 0 and α, β ∈ (0, 1). The above theorems show that if λ is not necessarily 0, then the following situations may occur: (i) problem (17) has solutions if a ∈ (0, 1] and for all λ ≥ 0; (ii) if a ∈ (1, 2) then there exists λ∗ > 0 such that problem (17) has a solution for any λ < λ∗ and no solution exists if λ > λ∗ . To better understand the dependence between λ and µ in (3), let us consider the special case f ≡ 1 and let m := lim g(s) ∈ (0, +∞). s→∞

In this case the result concerning (3) is the following. Theorem 2.3. Assume that a = 2 and f ≡ 1. Then the following properties hold: (i) The problem (1) has a solution if and only if λ(m + µ) < λ1 ; λ1 (ii) Assume µ > 0 is fixed and let λ∗ = . Then (1) has a unique m+µ solution uλ for every 0 < λ < λ∗ and the sequence (uλ )0<λ<λ∗ is increasing with respect to λ. Moreover, if lim sup sα g(s) < +∞, for s&0

some α ∈ (0, 1), then the sequence of solutions (uλ )0<λ<λ∗ has the following properties: (ii1) uλ ∈ C 1,1−α (Ω) ∩ C 2 (Ω); (ii2) lim∗ uλ = +∞ uniformly on compact subsets of Ω. λ%λ

Remark. The assumption lim sup sα g(s) < +∞, for some α ∈ (0, 1), has s&0

been used in Ref. 8,10 and it implies the following Keller-Osserman-type growth condition around the origin ¶−1/2 Z 1 µZ t g(s)ds dt < +∞. (18) 0

0

As proved by B´enilan, Brezis and Crandall in Ref. 16, condition (18) is equivalent to the property of compact support, that is, for any h ∈ L1 (RN ) with compact support, there exists a unique u ∈ W 1,1 (RN ) with compact support such that ∆u ∈ L1 (RN ) and −∆u = g(u) + h a.e. in RN .

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Proof of Theorem 2.3. With the change of variable v = eλu − 1, the problem (1) takes the form   −∆v = Ψλµ (x, u) in Ω, (19) v>0 in Ω,  v=0 on ∂Ω, where Ψλµ (x, s) = λ(s + 1)g

µ

¶ µ ¶ 1 1 ln(s + 1) + λµ(s + 1)f x, ln(s + 1) , λ λ

for all (x, s) ∈ Ω × (0, ∞). The existence and nonexistence results follow now from Ref. 14, Theorem 2.4. In order to prove the asymptotic behavior of the solution near λ∗ we use the following alternative which is due to H¨ormander (see Ref. 17, Theorem 4.1.9). Proposition 2.1. Let (uλ )0<λ<λ∗ be a sequence of positive super-harmonic functions which are increasing with respect to λ. Then the following alternative holds: (i) either uλ converges in L1loc (Ω); (ii) or uλ → ∞ uniformly on compact subsets of Ω. 3. Ground State Solutions for Singular Elliptic Problems We consider in this section the following singular problem  a in RN , (N ≥ 3),  −∆u = p(x)(g(u) + f (u) + |∇u| ) u>0 in RN ,  u(x) → 0 as |x| → ∞,

(20)

where f and g satisfy (f 1)–(f 2) and (g1), 0 < a < 1, and p : RN → (0, ∞) is a H¨older continuous function of exponent γ ∈ (0, 1). We are concerned here with ground state solutions, that is, positive solutions defined in the whole space and decaying to zero at infinity. The case f ≡ 0 and a = 0 was considered in Lair and Shaker18 . More exactly, it was proved in Ref. 18 that a necessary condition in order to have solution for the problem  in RN ,  −∆u = p(x)g(u) (21) u>0 in RN ,  u(x) → 0 as |x| → ∞,

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is

Z

135

∞

tψ(t)dt < ∞,

(22)

1

where ψ(r) = min p(x), r ≥ 0. Note that condition (22) is also necessary |x|=r

for our problem (20), since any solution of (20) is a super-solution of (21). The sufficient condition for existence supplied in Ref. 18 is Z ∞ tϕ(t)dt < ∞, (23) 1

where ϕ(r) = max p(x), r ≥ 0. Hence, when p is radially symmetric, the |x|=r Z ∞ problem (21) has solutions if and only if tp(t)dt < ∞ (see Ref. 18). 1

Our result concerning the problem (20) is the following. Theorem 3.1. Assume that (f 1)–(f 2), (g1) and (23) are fulfilled. Then problem (20) has at least one solution. 2,γ Proof. The solution of problem (20) is obtained as a limit in Cloc (RN ) of a monotone sequence of solutions associated to (20) in smooth bounded domains. Let Bn := {x ∈ RN ; |x| < n}. According to Theorem 2.1, for all n ≥ 1 there exists un ∈ C 2,γ (Bn ) ∩ C(Bn ) such that  a in Bn ,  −∆un = p(x)(g(un ) + f (un ) + |∇un | ) (24) u >0 in Bn ,  n un = 0 on ∂Bn .

We extend un by zero outside of Bn . We claim that un ≤ un+1 in Bn . Assume by contradiction that the inequality un ≤ un+1 does not hold throughout Bn and let ζ(x) =

un (x) , un+1 (x)

x ∈ Bn .

Clearly ζ = 0 on ∂Bn , so that ζ achieves its maximum in a point x0 ∈ Bn . At this point we have ∇ζ(x0 ) = 0 and ∆ζ(x0 ) ≤ 0. This yields ³ ´ −div(u2n+1 ∇ζ)(x0 ) = − div(u2n+1 )∇ζ + u2n+1 ∆ζ (x0 ) ≥ 0. A straightforward computation shows that −div(u2n+1 ∇ζ) = −un+1 ∆un + un ∆un+1 . Hence

³

´ − un+1 ∆un + un ∆un+1 (x0 ) ≥ 0.

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The above relation produces µ ¶ g(un ) + f (un ) g(un+1 ) + f (un+1 ) − (x0 )+ un un+1 µ ¶ |∇un |a |∇un+1 |a + − (x0 ) ≥ 0. un un+1

(25)

g(t) + f (t) is decreasing on (0, ∞) and un (x0 ) > un+1 (x0 ), t from (25) we obtain µ ¶ |∇un |a |∇un+1 |a − (x0 ) > 0. (26) un un+1 Since t 7−→

On the other hand, ∇ζ(x0 ) = 0 implies un+1 (x0 )∇un (x0 ) = un (x0 )∇un+1 (x0 ). a−1 Furthermore, relation (26) leads us to ua−1 n (x0 ) − un+1 (x0 ) > 0, which is a contradiction since 0 < a < 1. Hence un ≤ un+1 in Bn which means that

0 ≤ u1 ≤ · · · ≤ un ≤ un+1 ≤ . . .

in RN .

The main point is to find an upper bound for the sequence (un )n≥1 . To this aim, set Z r Φ(r) = r1−N tN −1 ϕ(t)dt, for all r > 0. 0

Using the assumption (23) and L’Hˆopital’s rule, we get lim Φ(r) = r→∞

lim Φ(r) = 0 and

r&0

1 lim Φ(r) = r→∞ N −2

Z

∞

rϕ(r)dr < ∞. 0

Let k > 2 be such that k 1−a ≥ 2 max Φa (r) and define r≥0

Z

∞

Φ(t)dt,

ξ(x) = k

for all x ∈ RN .

|x|

Then ξ satisfies   −∆ξ = kϕ(|x|) ξ>0  ξ(x) → 0

in RN , in RN , as |x| → ∞.

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Z

137

t

1 ds ∈ [0, ∞) is bijective, we 0 g(s) + 1 → (0, ∞) by

Since the mapping [0, ∞) 3 t 7−→

can implicitly define w : RN Z w(x) 1 dt = ξ(x), g(t) +1 0

for all x ∈ RN .

It is easy to see that w ∈ C 2 (RN ) and w(x) → 0 we have  a  −∆w ≥ p(x)(g(w) + 1 + |∇w| ) w>0  w(x) → 0

as |x| → ∞. Furthermore, in RN , in RN , as |x| → ∞.

(27)

Using the assumption (f 1), we can find M > 1 large enough such that M > f (M w) in RN . Multiplying by M in (27) we deduce that v := M w satisfies  a in RN ,  −∆v ≥ p(x)(g(v) + f (v) + |∇v| ) v>0 in RN ,  v(x) → 0 as |x| → ∞. With the same proof as above we deduce that un ≤ v in Bn , for all n ≥ 1. This implies 0 ≤ u1 ≤ · · · ≤ un ≤ v in RN . Thus, there exists u(x) = lim un (x), for all x ∈ RN and un ≤ u ≤ v in RN . Since v(x) → 0 n→∞

as |x| → ∞, we deduce that u(x) → 0 as |x| → ∞. A standard bootstrap 2,γ argument (see Gilbarg and Trudinger19 ) implies that un → u in Cloc (RN ) and that u is a solution of problem (20). This completes the proof of Theorem 3.1. ¤ Acknowledgments. The authors have been supported by Grant 2-CEx0611-18/2006. References 1. J.S.W. Wong, On the generalized Emden-Fowler equation, SIAM Review, 17 (1975), 339–360. 2. P.G. de Gennes, Wetting: statics and dynamics, Review of Modern Physics, 57 (1985), 827–863. 3. J.T. Chayes, S.J. Osher and J.V. Ralston, On singular diffusion equations with applications to self-organized criticality, Comm. Pure Appl. Math., 46 (1993), 1363, 1377. 4. A. Callegari and A. Nashman, Some singular nonlinear equations arising in boundary layer theory, J. Math. Anal. Appl., 64 (1978), 96–105.

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5. A. Callegari and A. Nashman, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math., 38 (1980), 275– 281. 6. Y. Choquet-Bruhat and J. Leray, Sur le probl`eme de Dirichlet quasilin´eaire d’ordre 2, C. R. Acad. Sci. Paris, Ser. A, 274 (1972), 81–85. 7. J. Kazdan and F.W. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math., 28 (1975), 567–597. 8. F.-C. Cˆırstea, M. Ghergu, and V. R˘ adulescu, Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of LaneEmden-Fowler type, J. Math. Pures Appliqu´ees, 84 (2005), 493–508. 9. M.M. Coclite and G. Palmieri, On a singular nonlinear Dirichlet problem, Comm. Partial Differential Equations, 14 (1989), 1315–1327. 10. M. Ghergu and V. R˘ adulescu, Sublinear singular elliptic problems with two parameters, J. Differential Equations, 195 (2003), 520–536. 11. J. Shi and M. Yao, On a singular nonlinear semilinear elliptic problem, Proc. Royal Soc. Edinburgh Sect. A, 128 (1998), 1389–1401. 12. Z. Zhang and J. Yu, On a singular nonlinear Dirichlet problem with a convection term, SIAM J. Math. Anal., 4 (2000), 916–927. 13. M.G. Crandall, P. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2 (1977), 193–222. 14. M. Ghergu and V. R˘ adulescu, Multiparameter bifurcation and asymptotics for the singular Lane-Emden-Fowler equation with convection term, Proc. Royal Soc. Edinburgh Sect. A, 135 (2005), 61–84. 15. N.E. Alaa and M. Pierre, Weak solutions of some quasilinear elliptic equations with data measures, SIAM J. Math. Anal., 24 (1993), 23–35. 16. P. B´enilan, H. Brezis, and M. Crandall, A semilinear equation in L1 (RN ), Ann. Scuola Norm. Sup. Pisa, 4 (1975), 523–555. 17. L. H¨ ormander, The Analysis of Linear Partial Differential Operators I, (Springer-Verlag, Berlin Heidelberg New York, 1983). 18. A.V. Lair and A.W. Shaker, Classical and weak solutions of a singular semilinear elliptic problem, J. Math. Anal. Appl., 211 (1997), 371–385. 19. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., (Springer-Verlag, Berlin Heidelberg New York, 1983).

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EXISTENCE AND UNIQUENESS RESULTS IN THE MICROPOLAR MIXTURE THEORY OF POROUS MEDIA IONEL-DUMITREL GHIBA “Octav Mayer” Mathematics Institute, Romanian Academy of Science, Ia¸si Branch, Bd. Carol I, nr. 8, 700506-Ia¸si, Romania E-mail: ghiba [email protected] In this paper we study the existence and uniqueness of solutions for the initialboundary value problem of a micropolar binary mixture. We use some results of the semigroup theory of linear operators to obtain an existence and uniqueness theorem for the initial value problem with homogeneous Dirichlet boundary conditions. Continuous dependence of the solutions upon the initial data and supply terms is also established. Keywords: Micropolar mixture; Micropolar elastic solid; Incompressible micropolar viscous fluid; Existence; Uniqueness; Continuous dependence.

1. Introduction In Ref. 1 Eringen has developed a continuum theory for a mixture of a micropolar elastic solid and a micropolar viscous fluid. All materials, whether natural or synthetic, possess microstructures. The material points of porous solids and dirty fluids undergo translation and rotations. Classical elasticity and fluid dynamics ignore the rotational degrees of freedom. In the micropolar continuum theory, the rotational degrees of freedom play a central role. Thus, we have six degrees of freedom, instead of the three degrees of freedom considered in classical elasticity and fluid mechanics. A material point of micropolar continuum is endowed with three rigid directors whose rotations provide the intrinsic rotations of the material point (see Refs. 2,3). The continuum theory of mixtures has been a subject of intensive study in literature. A presentation of the work on the subject can be found in a review article by Bowen4 . In classical theories of mixtures, the rotational degrees of freedom are ignored.

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The microstructural motions are observed to produce new effects that cannot be accounted for the classical translatory degrees of freedom alone that are used to formulate classical mixture theories. A large class of engineering materials, as well as soils, rocks, granular materials, sand and underground water mixtures may be modeled more realistically by means the theory proposed by Eringen1 . Consolidation problems in the building industry, earthquake problems and oil exploration problems fall into the domain of this theory. For the classical isothermal theory of a mixture consisting of three component: an elastic solid, a viscous fluid and a gas some results concerning uniqueness and continuous dependence have been studied by Gale¸s5 . Some results have been established in Ref. 6 concerning the existence and uniqueness of solutions in the case when the thermal effects are present, but the time derivative of temperature is not present into the set of independent variables. We outline that the general theorems concerning the uniqueness and continuous data dependence of solutions in the thermal theory of a mixture consisting of three component have been established by Chirit¸˘a7 . We recall that the existence and uniqueness results in the classical theories of mixtures have been obtained using the semigroup theory by Mart´ınez and Quintanilla8 and Gale¸s6 . In the present paper we consider the linear theory of a binary homogeneous mixture of an isotropic micropolar elastic solid with an incompressible micropolar viscous fluid and we formulate the basic initial-boundary value problem. Within this theory some uniqueness and continuous dependence results have been established in Ref. 9. The main purpose of this paper is to establish some existence and uniqueness results for the solutions of the initial-boundary problems associated with the theory in question. The paper also investigates the continuous dependence of solution with respect to the initial data and supplies terms. We transform the initial boundary value problem in an abstract equation in an appropriate Hilbert space and then we use the results of the semigroups theory of linear operators (see Refs. 10,11) in order to obtain the existence, uniqueness and continuous dependence results. 2. Preliminaries We consider a binary homogeneous mixture of an isotropic micropolar elastic solid with an incompressible micropolar viscous fluid. Both continuous bodies are considered to occupy a bounded domain Ω of the threedimensional space, whose boundary is regular surface ∂Ω.

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We refer the motion of a continuum to a fixed system of rectangular Cartesian axes Oxk (k = 1, 2, 3). The Latin subscripts are understood to range over the integers 1, 2, 3 and superscripts s, f denote respectively, the micropolar elastic solid and the incompressible micropolar fluid. Summation over repeated subscripts and other typical conventions for differential operations are implied such as a superposed dot or comma followed by a subscript to denote the partial derivative with respect to the time or the corresponding cartesian coordinate. Let ρα 0 denote the density of the αth constituent and ρ0 the density of the mixture at time t = 0. We denote by α uα i the displacement vector fields and by ϕi the rotation vector fields of the αth constituent. The fundamental equations for mechanical behaviour of the mixture in the framework of the linearized theory developed by Eringen1 are — balance of momentum α α α ˆα ¨α ρα 0u i , i = tji,j + ρ0 fi + p X α pˆi = 0,

(α = f, s), (1)

α=s,f

tα ji ,

where fiα and pˆα i are, respectively, the stress tensor, the body force density and the force exerted on the αth constituent from the other constituent; — balance of moment of momentum α α α α α ˆα ¨i = mα ρα 0j ϕ i , ji,j + εijk tjk + ρ0 li + m X α m ˆ i = 0,

(α = f, s), (2)

α=s,f α α where j α , ϕα ˆα i , mji , li , m i are respectively, microinertia density, rotation rate (gyration), couple stress, body couple density, the couple exerted on the αth constituent from the other constituent and εijk denotes the permutation symbol;

— energy conservation K −C0 T˙ + (−β0 T0 + ζ)u˙ si,i + T,ii + ρ0 h = 0, T0

(3)

where h, T, T0 are respectively, the energy source, the temperature change and the initial temperature for the mixture and C0 , β0 , ζ, K are the micropolar thermoelastic coefficients; — incompressibility condition u˙ fi,i = 0.

(4)

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The constitutive equations for a homogeneous mixture of an isotropic micropolar elastic solid and an incompressible micropolar viscous fluid are tsji = −β0 T δji + λs usk,k δji + µs (usj,i + usi,j ) + k s (usi,j + εijk ϕsk ), msji = αs ϕsk,k δji + β s ϕsj,i + γ s ϕsi,j , tfji = −π f δji + µf (u˙ fj,i + u˙ fi,j ) + k f (u˙ fi,j + εijk ϕ˙ fk ),

(5)

mfji = αf ϕ˙ fk,k δji + β f ϕ˙ fj,i + γ f ϕ˙ fi,j , pˆsi = −ˆ pfi = −ξ(u˙ si − u˙ fi ) −

ζ T,i , T0

m ˆ si = −m ˆ fi = −$(ϕ˙ si − ϕ˙ fi ), where π f is the micropolar fluid pressure, β0 , λs , µs , k s , αs , β s , γ s are the micropolar thermoelastic coefficients for the isotropic micropolar elastic solid, µf , k f , αf , β f , γ f are the micropolar fluid viscosities and ζ, K, ξ, $ are the micropolar thermoelastic coefficients of the mixture. The local form of the Clausius-Duhem inequality (see Ref. 1) implies that the dissipation potential ˙ s , u˙ f , ϕ ˙ f , T ), (u˙ s , ϕ ˙ s , u˙ f , ϕ ˙ f , T )] = Φ[(u˙ s , ϕ = µf u˙ fj,i u˙ fi,j + (µf + k f )u˙ fi,j u˙ fi,j + 2k f εijk u˙ fi,j ϕ˙ fk + 2k f ϕ˙ fi ϕ˙ fi + + αf ϕ˙ fk,k ϕ˙ fi,i + β f ϕ˙ fj,i ϕ˙ fi,j + γ f ϕ˙ fi,j ϕ˙ fi,j + ξ(u˙ si − u˙ fi )(u˙ si − u˙ fi )+ T,i K + 2ζ(u˙ si − u˙ fi ) + $(ϕ˙ si − ϕ˙ fi )(ϕ˙ si − ϕ˙ fi ) + 2 T,i T,i , T0 T0

(6)

must be a positive semidefinite quadratic form in term of u˙ fi,j , ϕ˙ fi , ϕ˙ fi,j , T,i and ϕ˙ si − ϕ˙ fi . u˙ si − u˙ fi , T0 This is true if and only if ξK − ζ 2 ≥ 0,

K ≥ 0,

2µf + k f ≥ 0, f

f

f

k f ≥ 0,

3α + β + γ ≥ 0,

f

(7) f

γ + β ≥ 0,

f

f

γ − β ≥ 0,

$ ≥ 0.

It is convenient to introduce the following dimensionless quantities t uα x ¯α = , ϕ ¯α = ϕα , x ¯ = , t¯= , u L τ L (8) T T0 L3 α α ¯ ¯ T = , T0 = , ρ¯0 = ρ , (α = s, f ), m 0 Tˆ Tˆ where L, τ, Tˆ and m are constants with dimension of length, time, temperature and mass respectively. With these quantities we can write the system (1)–(4) in the form

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¯s + µ ¯u ¨¯si = (λ ρ¯s0 u ¯s )¯ usj,ij + (¯ µs + k¯s )¯ usi,jj + k¯s εijk ϕ¯sk,j − ξ( ¯˙ si − u ¯˙ fi )− µ ¯ ¶ ζ − ¯ + β¯0 T¯,i + ρ¯s0 f¯is , T0 ρ¯s ¯j s ϕ¨¯s = (¯ αs + β¯s )ϕ¯s + γ¯ s ϕ¯s + k¯s (εijk u ¯s − 2ϕ¯s )− 0

i

¨¯fi = ρ¯f0 u

j,ij i,jj f s s s ¯ − $( ¯ ϕ¯˙ i − ϕ¯˙ i ) + ρ¯0 li , −¯ π,if + (¯ µf + k¯f )u ¯˙ fi,jj +

k,j

i

¯u k¯f εijk ϕ¯˙ fk,j + ξ( ¯˙ si − u ¯˙ fi )+

ζ¯ + ¯ T¯,i + ρ¯f0 f¯if , T0 ρ¯f0 ¯j f ϕ¨¯fi = (¯ αf + β¯f )ϕ¯˙ fj,ij + γ¯ f ϕ¯˙ fi,jj + k¯f (εijk u ¯˙ fk,j − 2ϕ¯˙ fi )+ + $( ¯ ϕ¯˙ s − ϕ¯˙ f ) + ρ¯f ¯lf , i

i

(9)

0 i

¯ K ¯ ¯u 0 = −C¯0 T¯˙ + (−β¯0 T¯0 + ζ) ¯˙ si,i + ¯ T¯,ii + ρ¯0 h, T0 u ¯˙ fi,i = 0, where subscripts preceded by comma denote partial differentiation respect to x ¯, superposed dot denotes partial differentiation respect to t¯ and 2 Lτ 2 s ¯s Lτ 2 s ¯ s = Lτ λs , µ λ ¯s = µ , k = k , m m m Lτ f ¯f Lτ f µ ¯f = µ , k = k , m m 2 2 τ τ τ2 s α ¯s = αs , β¯s = β s , γ¯ s = γ , Lm Lm Lm τ f ¯f τ f τ f α ¯f = α , β = β , γ¯ f = γ Lm Lm Lm L3 τ Lτ 2 Lτ ξ¯ = ξ, ζ¯ = ζ, $ ¯ = $, m m m 3 Lτ 2 Tˆ Lτ 2 Tˆ ¯ = Kτ , β¯0 = β0 , C¯0 = C0 , K m m Lm 2 2 1 τ τ α α α α α ¯j = j , f¯i = f , ¯li = 2 li , (α = s, f ), L2 L i L 3 2 2 τ L τ ¯= h h, π ¯f = π f , ρ¯0 = ρ¯s0 + ρ¯f0 . L2 m

(10)

To these equations we have to adjoin the following boundary conditions ¯ u ¯α ¯α i = 0, ϕ i = 0, T = 0 on ∂Ω × [0, t1 ), (α = s, f )

(11)

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and the following initial conditions u ¯α x, 0) = 0 uα x), u ¯˙ α x, 0) = 0 u˙ α x), T¯(¯ x, 0) = 0 T (¯ x), i (¯ i (¯ i (¯ i (¯ α α α α ¯ (α = s, f ), ϕ¯ (¯ x, 0) = 0 ϕ (¯ x), ϕ¯˙ (¯ x, 0) = 0 ϕ˙ i (¯ x), x ¯ ∈ B, i

i

(12)

i

α α α where aα ˙α ˙α i , 0 ϕi , 0 ϕ i and 0 T are prescribed continuous funci , ϕi , Q, 0 ui , 0 u tions. From the relations (10) we can remark that if the constitutive constants of mixture satisfy the inequalities (7), then the corresponding dimensionless constitutive constants satisfy the same type of inequalities. We define the following dimensionless quantities corresponding to the stress tensor and the couple stress of the two constituents and to the force and the couple exerted on the αth constituent from the other constituent ¯su t¯s = −β¯0 T¯δji + λ ¯s δji + µ ¯s (¯ us + u ¯s ) + k¯s (¯ us + εijk ϕ¯s ), ji

j,i

k,k

i,j

=α ¯

= −¯ π f δji + µ ¯f (u ¯˙ fj,i + u ¯˙ fi,j ) + k f (u ¯˙ fi,j + εijk ϕ¯˙ fk ),

¯ˆ s m i

ϕ¯sk,k δji

i,j

t¯fji m ¯ fji p¯ˆsi

s

=α ¯ f ϕ¯˙ fk,k δji + β¯f ϕ¯˙ fj,i + γ¯ f ϕ¯˙ fi,j , ζ¯ = −p¯ˆfi = −ξ(u ¯˙ si − u ¯˙ fi ) − ¯ T¯,i , T0 f f s ¯ = −m ˆ = −$( ¯ ϕ¯˙ − ϕ¯˙ ), i

k

+ β¯s ϕ¯sj,i + γ¯ s ϕ¯si,j ,

m ¯ sji

i

(13)

i

With these quantities we can write the equations (8) in the same forms with the equations (1)–(4). In this paper we shall assume that the densities ρ¯s0 , ρ¯f0 are strictly positive and ¯ s + 2¯ 3λ µs + k¯s ≥ 0, 2¯ µs + k¯s ≥ 0, k¯s ≥ 0, (14) 3¯ αs + β¯s + γ¯ s ≥ 0, γ¯ s + β¯s ≥ 0, γ¯ s − β¯s ≥ 0, C¯0 > 0. In what follows we will study the existence and uniqueness of solution of the initial-boundary value problem defined by the equations (9), the boundary conditions (11) and the initial conditions (12). 3. Existence and Uniqueness Results In this section we use the results of the semigroup theory of linear operators to establish an existence theorem to the equations that characterize the linear theory of a binary homogeneous mixtures of an isotropic micropolar elastic solid with an incompressible micropolar viscous fluid. Firstly, we transform our initial boundary value problem defined by the relations (9), (11) and (12) to an abstract problem on an appropriate Hilbert space.

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In order to simplify the notation, we suppress the superposed bar throughout in what follows. Let us define X = {w=(us , vs , ϕs , ν s , vf , ν f , T ); us ∈H10 (Ω), vs ∈L2 (Ω), ϕs ∈H10 (Ω), ν s ∈L2 (Ω), vf ∈H(Ω), ν f ∈L2 (Ω), T ∈L2 (Ω)},

(15)

where L2 (Ω) = [L2 (Ω)]3 , H10 (Ω) = [H01 (Ω)]3 , H(Ω) is the closure of the 2 space Y = {u ∈ C∞ 0 (Ω); ui,i = 0 on Ω, ui ni = 0 on ∂Ω} in L (Ω) (see Ref. 12) and H01 (Ω) is the well known Sobolev space (see Ref. 13). Because Ω is open, bounded, connected of class C 2 , we have (see Ref. 12) L2 (Ω) = H ⊕ H1 ⊕ H2 ,

(16)

where H, H1 , H2 are mutually orthogonal spaces, H1 = {u ∈ L2 (Ω); u = grad p, p ∈ H1 (Ω), ∆p = 0}

(17)

and H2 = {u ∈ L2 (Ω); u = grad q, q ∈ H10 (Ω)}.

(18)

Further, we introduce the operators s As1 i w = vi , · 1 s2 Ai w = s (λs + µs )usj,ij + (µs + k s )usi,jj + k s εijk ϕsk,j − ρo µ ¶ ¸ ζ f s −ξ(vi − vi ) − + β0 T,i , T0 s As3 i w = νi , 1 s s s s s s s s As4 i w = f f [(α + β )ϕj,ij + γ ϕi,jj + k (εijk uk,j − 2ϕi )− ρ0 j

−$(νis − νif )], · ¸ 1 ζ f1 f f f f f f s Ai w = f P (µ + k )vi,jj + k εijk νk,j + ξ(vi − vi ) + T,i , T0 ρ0 1 f f f + γ f νi,jj + k f (εijk vk,j − νif )+ Afi 2 w = f [(αf + β f )νj,ij ρ0 j f +$(νis − νif )], ·µ ¶ ¸ ζ K T0 s T −β0 + vi,i + 2 T,ii , Ai w = C0 T0 T0

(19)

where P : L2 (Ω) → H(Ω) is the Leray Projector. Let A be the operator A = (As1 , As2 , As3 , As4 , Af 1 , Af 2 , AT )

(20)

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with the domain D(A) = {w = (us , vs , ϕs , ν s , vf , ν f , T ) ∈ X; Aw ∈ X, vf = 0, ν f = 0, T = 0 on ∂Ω}.

(21)

∞ ∞ ∞ ∞ ∞ We note that C∞ 0 (Ω)×C0 (Ω)×C0 (Ω)×C0 (Ω)×V(Ω)×C0 (Ω)×C0 (Ω), where V(Ω) = H10 (Ω) ∩ H2 (Ω) ∩ H(Ω), is a dense subset of X (see Ref. 12) which is contained in D(A). The boundary initial value problem (9), (11) and (12) can be transformed into the following abstract equation in the Hilbert space X

dw (t) = Aw(t) + F (t), w(0) = w0 , dt where

µ F (t) =

0, f s , 0,

1 s 1 ρ0 T0 l , P (f f ), f lf , h js j C0

(22) ¶

and ˙ s , 0 u˙ f , 0 ϕ ˙ f , 0 T ). w0 = (0 us , 0 u˙ s , 0 ϕs , 0 ϕ Because the problem is linear, if vf and ν f are known the displacement and the rotation of fluid can be obtained by a simple integration. We introduce the following inner product in X es , ϕ e s, ν es, v ef , ν e f , Te)iX = h(us , vs , ϕs , ν s , vf , ν f , T ), (e us , v ½ Z C0 e = ρs0 vis veis + ρf0 vif veif + ρs0 j s νis νeis + ρf0 j f νif νeif + TT+ T0 Ω ¾ s s s es +E[(u , ϕ ), (e u , ϕ )] dv,

(23)

where e s )] = λs usk,k u E[(us , ϕs ), (e us , ϕ esk,k + µs usj,i u esj,i + (µs + k s )usi,j u esi,j + + k s εijk (usi,j ϕ esk + u esi,j ϕsk ) + 2k s ϕsi ϕ esi +

(24)

+ αs ϕsk,k ϕ esi,i + β s ϕsj,i ϕ esi,j + γ s ϕsi,j ϕ esi,j . In view of the assumptions (14) we have E[(us , ϕs ), (us , ϕs )] ≥ σm (usi,j usi,j + ϕsi ϕsi + ϕsi,j ϕsi,j )

(25)

σM (usi,j usi,j + ϕsi ϕsi + ϕsi,j ϕsi,j ) ≥ E[(us , ϕs ), (us , ϕs )],

(26)

and

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where σm = min{2µs + k s , k s , 3λs + 2µs + k s , 3αs + β s + γ s , β s + γ s }, σM = max{2µs + k s , k s , 3λs + 2µs + k s , 3αs + β s + γ s , β s + γ s }. As a consequence, we have that the norm induced by E is equivalent to the usual norm in H10 (Ω) × H10 (Ω). Thus h , i defines a norm equivalent to the usual norm in X. Lemma 3.1. The operator A is dissipative, that is hAw, wiX ≤ 0, for all w ∈ D(A).

(27)

Proof. First of all we note that, in view of the relations (10)–(12), we can find p ∈ H1 (Ω) with ∆p = 0 and q ∈ H10 (Ω) so that · ¸ ζ f f P (µf + k f )vi,jj + k f εijk νk,j + ξ(vis − vif ) + T,i = T0 (28) ζ f f f f f f s = (µ + k )vi,jj + k εijk νk,j + ξ(vi − vi ) + T,i − p,i − q,i . T0 Using the divergence theorem and the boundary conditions we find that Z ½ hA w, wiX = vis (tsji,j + pbsi ) + νis (msji,j + εijk tsjk + m b si )+ Ω f f + vi (tji,j

+ pbfi − p,i − q,i ) + νif (mfji,j + εijk tfjk + m b fi )+ ·µ ¸ ¶ ζ K s +T −β0 + vi,i + 2 T,ii + T0 T ¾ 0 + E[(vs , ν s ), (us , ϕs )] dv = Z = − Φ[(vs , ν s , vf , ν f , T ), (vs , ν s , vf , ν f , T )]dv.

(29)

Ω

Thus, by using the assumption (7) we obtain the relation (27) and the proof is complete. ¤ Lemma 3.2. The operator A satisfies the range condition Range(I − A) = X.

(30)

Proof. Assume that w∗ = (u∗s , v∗s , ϕ∗s , ν ∗s , v∗f , ν ∗f , T ∗ ) ∈ X. We must show that the equation w − Aw = w∗ ,

(31)

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has a solution in D(A). From the definition of the operator A we obtain the system ∗s s s2 ∗s usi − As1 i w = ui , vi − Ai w = vi , ∗s s s4 ∗s ϕsi − As3 i w = ϕi , νi − Ai w = νi ,

vif − Afi 1 w = vi∗f , νif − Afi 2 w = νi∗f ,

(32)

T − ATi w = T ∗ . By eliminating the functions vis and ϕsi , we obtain for the determination of the functions usi , ϕsi , vif , νif and T the following system · 1 s1 s Li y = ui − s (λs + µs )usj,ij + (µs + k s )usi,jj + k s εijk ϕsk,j − ρ0 µ ¶ ¸ ζ f s − ξ(ui − vi ) − + β0 T,i = gi∗s1 , T0 1 s s s s s s s s s Ls2 i y = ϕi − s s [(α + β )ϕj,ij + γ ϕi,jj + k (εijk uk,j − 2ϕi )− ρ0 j −$(ϕsi − νif )] = gi∗s2 , · 1 f1 f s s + k f εijk νk,j + Li y = vi − f − p,i − q,i + (µf + k f )vi,jj ρ0 ¸ ζ + ξ(usi − vif ) + T,i = gi∗f 1 , T0 1 f f2 f f f s Li y = νi − f [(α + β f )νj,ij + γ f νi,jj + k f (εijk vk,j − 2νif )+ ρ0 j f

(33)

+ $(ϕsi − νif )] = gi∗f 2 , ¸ ·µ ¶ K ζ T02 s T T,ii = g ∗T , − β0 + u + L y =T− C0 T0 i,i T02 where y = (us , ϕs , vf , ν f , T ), p ∈ H 1 (Ω) with ∆p = 0, q ∈ H01 (Ω) and ξ ∗s u , ρs0 i $ = νi∗s + s s ϕ∗s , ρ0 j i

gi∗s1 = vi∗s + gi∗s2

ξ ∗s ui , ρf0 $ = νi∗f − f ϕ∗s i , ρ0 j f µ ¶ ζ T2 u∗s . = T ∗ − 0 − β0 + C0 T0 i

gi∗f 1 = vi∗s − gi∗f 2 g ∗T

(34)

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We study this system in the following Hilbert space Z = (H10 (Ω) ∩ H2 (Ω)) ×(H10 (Ω) ∩ H2 (Ω))×V(Ω)× ×(H10 (Ω) ∩ H2 (Ω))×(H01 (Ω) ∩ H 2 (Ω)). We introduce the bilinear form B : Z × Z → R ¿ e ) = ( Ls1 y, Ls2 y, Lf 1 y, Lf 2 y, LT y), B(y, y µ ¶À C0 e s , ρf0 v e f , ρf0 j f ν e f , 2 Te e s , ρs0 j s ϕ ρs0 u = T0 L2 ×L2 ×L2 ×L2 ×L2 Z ½ C0 = ρs0 usi u esi + ρs0 j s ϕsi ϕ esi + ρf0 vif veif + ρf0 j f νif νeif + 2 T Te+ T0

(35)

(36)

Ω

e s, u ef , ϕ e f , Te)]+ + Φ[(us , ϕs , uf , ϕf , T ), (e us , ϕ ¾ fs )] dv, +E[(us , ϕs ), (e us , ϕ e = (e e s, v ef , ν e f , Te) and the linear operator l : Z → R where y us , ϕ ¿ l(e y) = (g∗s1 , g∗s2 , g∗f 1 , g∗f 2 , g ∗T ), ¶À µ f f f C0 e s es s s e s f ef e , 2T = ρ 0 u , ρ 0 j ϕ , ρ0 v , ρ 0 j ν T0 2 2 2 2 2 ¶ µ L ×L ×L ×L¶ ×L Z ½ µ ξ $ = ρs0 vi∗s + s u∗s veis + ρs0 j s νi∗s + s s ϕ∗s ϕ esi + i ρ0 ρ0 j i Ω ! Ã µ ¶ $ ∗s ξ ∗s f f f ∗f f ∗f vei + ρ0 j νi − s s ϕi νeif + +ρ0 vi − f ui ρ0 j ρ0 · µ ¶ ¸ ¾ 2 C0 T ζ + 2 T ∗ − 0 −β0 + u∗i Te dv. T0 C0 T0

(37)

Using the relations (7), (14), (19) and the Schwarz inequality we see that B is bounded, that is e ) ≤ M1 kykZ ke B(y, y ykZ , M1 = positive constant

(38)

B(y, y) ≥ M2 kyk2Z , M2 = positive constant

(39)

and coercive

and the linear operator l is bounded l(e y) ≤ M3 ke ykZ , M3 = positive constant.

(40)

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The Lax-Milgram theorem proves the existence of solution of system (33) in Z and, as a consequence, we have the solution of system (32). The proof is complete. ¤ Theorem 3.1. The operator A defined by the relation (14) generates a C0 contractive semigroup in X. Proof. The proof follows using the previous lemmas and the LumerPhillips corollary to Hille-Yosida theorem (see Ref. 10). ¤ Theorem 3.2. Assume that fis , lis , fif , lif , h ∈ C 1 ([0, t1 ); L2 (Ω)) and w0 ∈ D(A). Then, there exists a unique solution w ∈ C1 ((0, t1 ); X) ∩ C0 ([0, t1 ); D(A)) of the boundary value problem (16). Proof. The proof result from the results concerning the abstract Cauchy problem (see Refs. 10,11). ¤ Corollary 3.1. If w ∈ C1 ((0, t1 ); X) ∩ C 0 ([0, t1 ); D(A)) is the solution of boundary value problem (16) then exist π f = p + q with p ∈ H01 (Ω), ∆p = 0, q ∈ H 1 (Ω) so that · ¸ dvif 1 ζ f f f f f f s = f (µ + k )vi,jj + k εijk νk,j + ξ(vi − vi ) + T,i − dt T0 ρ0 (41) 1 f − f π,i + fif . ρ0 Proof. We obtain the conclusion of this corollary using the relation (10) and the definition of Leray Projector. ¤ Corollary 3.2. In the hypothesis of the Theorem 3.2 we have the following estimate Zt ¡ s kw(t)kX ≤ kw0 (t)kX + M4 kf (τ )kL2 (Ω) + kls (τ )kL2 (Ω) + (42) 0 ¢ + kP (f f (τ ))kL2 (Ω) + klf (τ )kL2 (Ω) + kh(τ )kL2 (Ω) dτ, where M4 = positive constant. Proof. For the proof of this Corollary we use the fact that the semigroup generated by A is contractive. ¤ 4. Concluding Comments The main purpose of this paper was to study the existence, uniqueness and continuous dependence of solution for the approach developed by Erin-

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gen1 for describing the behaviour of a binary homogeneous mixture of an isotropic micropolar elastic solid with an incompressible micropolar viscous fluid. The mechanical interpretation of condition (7) and (14) is obvious. The assumptions (7) results by the entropy inequality and the assumption (14), usual in the study of well posed problems, are in agreement with the restrictions imposed in Ref. 9 for the study of the uniqueness and continuous dependence of solution. The present results show that the general approach of a binary homogeneous mixture of an isotropic micropolar elastic solid with an incompressible micropolar viscous fluid is well posed. The uniqueness result is in concordance with that obtained for a Newtonian fluid (see Ref. 14). Acknowledgments The author acknowledges support from the Romanian Ministry of Education and Research through CEEX program, contract CERES-2-Cex06-1112/25.07.2006. References 1. A.C. Eringen, Micropolar mixture theory of porous media, Journal of Applied Physics, 94 (2003), 4184–4190. 2. A.C. Eringen, Microcontinuum Field Theories. Foundations and Solids, (Springer, New York, 1999). 3. A.C. Eringen, Microcontinuum Field Theories, II, Fluent Media, (Springer, New York, 2001). 4. R.M. Bowen, Theory of Mixtures, Continuum Physics, III, edited by A.C. Eringen, (Academic Press, New York, 1976). 5. C. Gale¸s, Some uniqueness and continuous dependence results in the theory of swelling porous elastic soils, Int. J. Eng. Sci., 40 (2002), 1211–1231. 6. C. Gale¸s, Existence and uniqueness results in the theory of swelling porous elastic soils, An. St. Univ. Ia¸si, Sect. Matematica, 49 (2003), 161–174. 7. S. Chirit¸a ˘, On the uniqueness and continuous data dependence of solutions in the theory of swelling porous thermoelastic soils, Int. J. Eng. Sci., 41 (2003), 2363–2380. 8. F. Mart´ınes, R. Quintanilla, Some qualitative results for the linear theory of binary mixtures of thermoelastic solids, Collect. Math., 46 (1995), 263–277. 9. I.D. Ghiba, Some uniqueness and continuous dependence results in the micropolar mixture theory of porous media, Int. J. Eng. Sci., 44 (2006), 1269–1279. 10. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, (Springer Verlag, New York, Berlin, Heidelberg, Tokyo, 1983).

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11. I. Vrabie, Semigrupuri de operatori liniari ¸si aplicat¸ii, (Ed. Univ. ”Al. I. Cuza”, Ia¸si, 2001). 12. R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis. With an appendix by F. Thomasset. 3rd edition. Studies in Mathematics and its Aplications, 2 (North-Holland Publishing Co., Amsterdam, 1984). 13. R.A. Adams, Sobolev Space, (Academic Press, New York, 1975). 14. M.E. Gurtin, An Introduction to Continuum Mechanics, (Academic Press. Inc., San Diego-New York-Berkeley-Boston-London-Sydney-Tokyo-Toronto, 1981).

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APPROXIMATE CONTROLLABILITY FOR LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS WITH CONTROL ACTING ON THE NOISE DAN GOREAC Laboratoire de Math´ ematiques, Unit´ e CNRS UMR 6285, Universit´ e de Bretagne Occidentale, 6, av. Victor LeGorgeu, B.P. 809, 29200 Brest cedex, France E-mail: [email protected] In this paper we study approximate controllability for a linear stochastic differential equation dy(t) = (Ay(t) + Bu(t)) dt + (Cy(t) + Du(t)) dW (t), for the case when the control acts also on the noise. This may be considered as a generalization of the work of Buckdahn, Quincampoix and Tessitore where the problem is solved for D = 0 and of Peng for D of full rank. We prove, using the dual BSDE and Riccati methods that approximate controllability is equivalent to the local in time viability for a suitable set. Finally, an invariance criterion is given. Keywords: Stochastic control; Controllability; Controllability under constraints; Backward stochastic differential equation; Riccati equation.

1. Introduction Given a complete probability space (Ω, F, P ) and a standard Brownian motion (W (t), t ≥ 0) on this space, we consider the natural complete filtration (Ft )t≥0 generated by W. We let A = A(Ω, F, P ; W ) be the set of all (Ft )progressively measurable processes v(·) taking their values in Rd and such hR i T that E 0 |v(s)|2 ds < ∞ for all T > 0. A process v(·) ∈ A will be referred as an admissible control process. We consider the following linear stochastic differential equation dy(t) = (Ay(t) + Bu(t)) dt + (Cy(t) + Du(t)) dW (t), 0 ≤ t ≤ T, n

(1)

n

governed by the control process u(·) ∈ A, where A, C ∈ L(R , R ), and B, D ∈ L(Rd , Rn ). The initial condition is y(0) = x ∈ Rn .

(2)

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For all x ∈ Rn and all admissible control u(·), the equation (1) admits a unique predictable solution y(·, x, u) with continuous trajectories such that y(0, x, u) = x. Furthermore, this solution satisfies, for all T > 0, " # E

sup |y(s)|2 < ∞. s∈[0,T ]

In Refs. 1 and 2, Peng (respectively Liu and Peng) have studied the ”exact controllability” and ”exact terminal controllability” for (1). They show that exact terminal controllability is equivalent with the condition that D has full rank. Moreover, algebraic conditions of Kalman type give a characterization of exact controllability. The case where the control u(·) does not act on the noise (i.e. D = 0 ) has been studied by Buckdahn, Quincampoix and Tessitore in Ref. 3. Since D is not of full rank, exact terminal controllability must be weakened to ”approximate controllability”. Algebraic Riccati equation methods are used in Ref. 3 in order to obtain a characterization of approximately-controllable stochastic linear equations. This criterion says that, in order to have approximate controllability, the only locally in time viable subset of a certain linear space has to be the trivial set. The objective of our paper is to extend the results in Refs. 3 and 1 to the general case where the control is allowed to act on the noise (i.e. rank(D) ≥ 0) without necessarily having D of full rank. We give a characterization of approximate controllability using the notion of local in time viability of a linear space V ⊂ Rn conditioned to another linear space U ⊂ Rn (a generalization of the concept of local in time viability (l.i.t.v.) introduced in Ref. 3). We prove that the study of the linear stochastic differential equation (1) can be reduced to an equation of the form dy(t) = (Ay(t) + B1 u0 (t) + B2 u00 (t))dt + (Cy(t) + D1 u0 (t))dW (t), n

n

r

n

d−r

(3)

n

where A, C ∈ L(R , R ), B1 , D1 ∈ L(R , R ), B2 ∈ L(R , R ) and rank D1 = r . For this equation, we establish the equivalence between approximate controllability, approximate null controllability and some conditional viability criterion. Finally, an easily computable condition is provided. 2. Preliminaries Let us recall the following definition of approximate controllability Definition 2.1. We say that the equation (1) is approximately controllable if, for all x ∈ Rn , all T > 0, all η ∈ L2 (Ω;FT ; P ;Rn ), and all ε > 0, there

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exists an admissible control u such that h i 2 E |y(T, x, u) − η| ≤ ε. Moreover, we say that the equation (1) is approximately null controllable if the above condition holds for the particular case η = 0. In order to obtain computable algebraic criteria for approximate controllability in the case where the control is not allowed to act on the noise (i.e. D = 0), the authors of Ref. 3 use the notion of strict invariance. This concept (slightly modified) had already been used in Ref. 4 to obtain, by algebraic Riccati equation methods, a characterization of stochastic linear equations admitting a feedback that stabilizes the system for all noise intensities. Definition 2.2. Given m + 1 linear operators L; M1 ; ...; Mm ∈ L(Rn , Rn ), a linear subspace V ⊂ Rn is said to be (L; M1 ; ...; Mm )-strictly invariant if LV ⊂ Span{V ; M1 V ; ...; Mm V }. We notice that a linear subspace V ⊂ Rn is (L; M1 ; ...; Mm )-strictly invariant if and only if there exist operators K1 , ..., Km ∈ L(Rn , Rn ) such that Ki V ⊂ V and (L + M1 K1 + ... + Mm Km )V ⊂ V or, equivalently, if and only if for all v ∈ V there exist w1 , ..., wm ∈ V such that Lv + M1 w1 + ... + Mm wm ∈ V. Since we allow the control to act on the noise, we have to require a stronger property than strict invariance. Namely, we give the following Definition, which extends the notion of strict invariance used in Ref. 3 to establish a characterization of l.i.t.v. Definition 2.3. Given the linear operators L, M, N ∈ L(Rn , Rn ) and two linear subspaces V ⊂ Rn and U ⊂ Rn , we say that V is (L; M )-strictly invariant conditioned to (N, U ) if for all v ∈ V there exists w ∈ V such that w − N v ∈ U and Lv + M w ∈ V. Remark 2.1. It is easy to see that V is (L; M )-strictly invariant conditioned to (N, U ) if and only if there exists a linear operator K ∈ L(Rn , Rn ) such that KV ⊂ V , (K − N )V ⊂ U and (L + M K)V ⊂ V . Remark 2.2. For all N it can be easily observed that V is (L; M )-strictly invariant conditioned to (N, Rn ) if and only if it is (L; M )-strictly invariant (i.e., if U = Rn is the full space we find the notion of strict invariance).

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Remark 2.3. For arbitrary linear subspaces V, U ⊂ Rn , the largest subspace of V which is (L; M )-strictly invariant conditioned to (N, U ) can be obtained in at most n iterations by considering the following schema V0 = V ; Vi+1 = {v ∈ Vi : M ((U + N v) ∩ Vi ) ∩ (Vi − Lv) 6= ϕ} , i ∈ N. 3. The Dual Equation We consider the following backward stochastic differential equation ½ dp(t) = − (A∗ p(t) + C ∗ q(t)) dt + q(t)dW (t), 0 ≤ t ≤ T, p(T ) = η ∈ L2 (Ω, FT , P ; Rn ).

(4)

It has been established in Ref. 5 that for all T > 0 and all η ∈ L2 (Ω, FT , P ; Rn ) Eq. (4) admits a unique solution (p, q) ∈ M 2 (0, T ; Rn ) × M 2 (0, T ; Rn ) (where M 2 (0, T ; Rn ) stands for the set of Rn -valued processes which are Ft -progressively measurable and square integrable over Ω × [0,hT ] with respect to i P ⊗dt). Moreover, p has continuous trajectories 2 and E sups∈[0,T ] |p(s)| < ∞. We are able at this point to prove the connection between approximate controllability for Eq. (1) and the backward stochastic differential equation Eq. (4): Proposition 3.1. The equation (1) is approximately-controllable if and only if, for all T > 0, every solution of (4) such that B ∗ p(s) + D∗ q(s) = 0, P − a.s., for all s ∈ [0, T ] is trivial. Moreover, the equation (1) is approximately null controllable if and only if, for all T > 0, every solution of (4) such that B ∗ p(s) + D∗ q(s) = 0, P − a.s., for all s ∈ [0, T ] satisfies p(0) = 0. Proof. Let us fix T ≥ 0. Using Itˆo’s formula for hp(T ), y(T, x, u)i we have "Z # T E [hp(T ), y(T, x, u)i]−E[hp(0), xi] = E hB ∗ p(s)+D∗ q(s), u(s)i ds (5) 0

We denote by L2P ([0, T ], Rd ) the hR space of iall predictable processes T u : Ω × [0, ∞[−→ Rd satisfying E 0 |u(s)|2 ds < ∞ and consider the linear operator MT given by MT : L2P ([0, T ], Rd ) −→ L2 (Ω, FT , P ; Rn ), MT u = y(T, 0, u).

(6)

It is straightforward that if and only ¡ 2Eq. (1) is dapproximately-controllable ¢ 2 if, for all T > 0, MT LP ([0, T ], R ) is dense in L (Ω, FT , P ; Rn ). The equality (5) implies MT∗ η = B ∗ p + D∗ q. We use the fact that the image of a

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linear operator is dense if and only if the kernel of its adjoint is trivial and the uniqueness and the continuity of the solution of Eq. (4) to get η = 0 if and only if p(s) = 0, P − a.s., for all s ∈ [0, T ] and q(s) = 0 dsdP -almost everywhere on [0, T ] × Ω. In order to prove the second assertion in our statement, we introduce the linear operator LT : Rn −→ L2 (Ω, FT , P ; Rn ), LT x = y(T, x, 0).

(7)

It is obvious that approximate null controllability is equivalent to the fact that for all T > 0, LT [Rn ] ⊂ MT [L2P ([0, T ], Rd )] (or Ker (MT∗ ) ⊂ Ker (L∗T )). To conclude the proof of this second part, we use (5) for u = 0 and obtain L∗T η = p(0). The conclusion follows. ¤ In Ref. 1, for the case where D is a full rank matrix, the author was able to transform the equation into the equivalent form dy(t) = (Ay(t) + A1 u0 (t) + Bu00 (t))dt + u0 (t)dW (t), 0 ≤ t ≤ T.

(8)

Using this idea, for the case where 0 ≤ rank D = r ≤ n, we get the following linear stochastic differential equation, which is in fact equivalent to Eq. (1): dy(t) = (Ay(t) + B1 u0 (t) + B2 u00 (t))dt + (Cy(t) + D1 u0 (t))dW (t), where A, C ∈ L(Rn , Rn ), B1 , D1 ∈ L(Rr , Rn ), B2 ∈ L(Rd−r , Rn ) and rank D1 = r. Since rank D1 = r we establish the existence of F ∈ L(Rn , Rn ) solution of D1∗ F + B1∗ = 0. In this case, the dual equation becomes ½ dp(t) = [−(A∗ + C ∗ F )p(t) − C ∗ q(t)] dt + (F p(t) + q(t))dW (t), (9) p(T ) = η. From Ref. 5, Theorem 4.1, we have that for all T > 0 and all η ∈ L2 (Ω, FT , P ; Rn ), the backward stochastic differential equation (9) admits an unique solution (p, q) ∈ M 2 (0, T ; Rn ) × M 2 (0, T ; Rn ) such that p has continuous trajectories and " # sup |p(s)|2 < ∞.

E

s∈[0,T ]

As before, using Itˆo’s formula we show that E hp(T ), y(T )i = E hp(0), y(0)i "Z # T +E (hB2∗ p(t), u00 (t)i + hD1∗ q(t), u0 (t)i) dt , 0

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and, with the same arguments as for the previous proposition we can prove Proposition 3.2. The equation (3) is approximately-controllable if and only if, for all T > 0, every solution of (9) such that B2∗ p(s) = 0 and D1∗ q(s) = 0, P − a.s., for all s ∈ [0, T ] is trivially reduced to 0. Moreover, the equation (3) is approximately null controllable if and only if, for all T > 0, every solution of (9) such that B2∗ p(s) = 0 and D1∗ q(s) = 0, P − a.s., for all s ∈ [0, T ] satisfies p(0) = 0. Approximate controllability property for (3) can be expressed with the help of the following quadratic cost function (Z T hD E J(θ, q(·)) = E Π(Ker B2∗ )⊥ p(t, q, θ), p(t, q, θ) 0

D +

)

Ei Π(Ker

D1∗ )⊥ q(t), q(t)

dt .

Indeed, let us consider the Stochastic Linear Quadratic optimal control problem (SLQ) For each θ ∈ Rn find an admissible control q(·) such that not

J(θ, q(·)) = inf J(θ, q(·)) = V (θ) Definition 3.1. The (SLQ) problem is said to be (1) solvable at θ ∈ Rn if there exists an admissible control q(·) such that J(θ, q(·)) = V (θ). In this case, q(·) is called an optimal control. (2) pathwise uniquely solvable at θ ∈ Rn if it is solvable at θ and, whenever q1 (·) and q2 (·) are two optimal controls on the same space, it holds P ({q1 (t) = q2 (t), for almost every t ∈ [0, T ]}) = 1. The following proposition gives the connection between approximate controllability, approximate null controllability and the (SLQ) problem. Proposition 3.3. We have equivalence between the following assertions: (i) The equation (3) is approximately-controllable. (ii) For all T > 0, all θ ∈ Rn , and all q ∈ L2P ([0, T ], Ker D1∗ ) such that B2∗ p(s, q, θ) = 0, P − a.s.., for all s ∈ [0, T ] it must hold that q(s) = 0 dsdP -almost everywhere on [0, T ] × Ω and θ = 0; ½ a) The (SLQ) problem is pathwise uniquely solvable at θ=0; (iii) b) The equation (3) is approximately null controllable.

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Proof. We only have to prove the equivalence between the last two assertions. Let us suppose that (ii) holds true. It is obvious that (SLQ) is solvable at θ = 0 and that q(·) ≡ 0 is an optimal control process. If we consider q(·) to be another optimal control at θ = 0, we must have J(0, q(·)) = 0. Therefore, we get q ∈ L2P ([0, T ], Ker D1∗ ) and B2∗ p(s, q, θ) = 0, P − a.s., for all s ∈ [0, T ]. We now use (ii) to obtain that q(s) = 0 dsdP -almost everywhere on [0, T ] × Ω and the condition (iii) in the statement is proved. For the converse, let us fix T > 0, θ ∈ Rn , and q ∈ L2P ([0, T ], Ker D1∗ ) such that B2∗ p(s, q, θ) = 0, P − a.s., for all s ∈ [0, T ]. Since (3) is approximately null controllable we get θ = 0 and J(0, q(·)) = 0. Thus, q(·) is an optimal control for (SLQ) at θ = 0 and the pathwise uniqueness yields q(s) = 0 dsdP -almost everywhere on [0, T ] × Ω. The proof is now complete.

¤

Remark 3.1. The previous proposition shows that, in order for Eq. (3) to be approximately -controllable, (SLQ) must be pathwise uniquely solvable at 0. The backward stochastic differential equation (9) may be interpreted as the following forward differential equation ½ dp(t) = [−(A∗ + C ∗ F )p(t) − C ∗ q(t)] dt + (F p(t) + q(t))dW (t); (10) p(0) = θ ∈ Rn . Therefore, for all θ ∈ Rn , all linear subspace V ⊂ Rn , and all q ∈ L2P ([0, T ], V ) there exists an unique predictable solution p(·, q, h i θ) of (10)

with continuous trajectories such that E sups∈[0,T ] |p(s, q, θ)|2 < ∞. The approximate controllability conditions given by Proposition 3.2 become Proposition 3.4. The equation (3) is approximately-controllable if and only if, for all T > 0, all θ ∈ Rn , and all q ∈ L2P ([0, T ], Ker D1∗ ) such that B2∗ p(s, q, θ) = 0, P − a.s., for all s ∈ [0, T ], it holds q(s) = 0, dsdP -almost everywhere on [0, T ] × Ω and θ = 0 . The equation (3) is approximately null controllable if and only if, for all T > 0, all θ ∈ Rn and all q ∈ L2P ([0, T ], Ker D1∗ ) such that B2∗ p(s, q, θ) = 0, P − a.s., for all s ∈ [0, T ], it holds θ = 0.

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4. Conditional Local in Time Viability Motivated by Proposition 3.4, we extend the notion of ”local in time viability” to ”conditional local in time viability”. Definition 4.1. Let U, V ⊂ Rn be two linear subspaces of Rn . The family of all θ ∈ V for which there exists a T > 0 and q ∈ L2P ([0, T ], U ) such that p(s, q, θ) ∈ V, P − a.s., for all s ∈ [0, T ] is called the viability kernel of V conditioned to U with respect to Eq. (10) (we denote this set by V iab(V /U )). Moreover, we say that V is local in time viable conditioned to U with respect to Eq. (10) if V iab(V /U ) = V. If U and V are two linear subspaces of Rn , we denote by ΠU ⊥ (respectively ΠV ⊥ ) the orthogonal projections on U ⊥ (respectively V ⊥ ). For all N ≥ 1, let us consider the following Riccati equations with values in S n (the family of symmetric, non-negative matrix of n × n type):  0 PN (s) = −PN (s)(A∗ + C ∗ F ) − (A + F ∗ C)PN (s) + F ∗ PN (s)F    −(F ∗ P (s)−P (s)C ∗ )(I+N Π ⊥ +P (s))−1 (P (s)F −CP (s))  N N N N N U (11) +N Π , ⊥ V     PN (T ) = 0; Ito’s formula applied to hPN (T − t)p(t), p(t)i yields E[hPN (T − t)p(t), p(t)i] "Z # T¡ ¢ 2 2 2 =E N (|ΠV ⊥ p(s)| +|ΠU ⊥ q(s)| )+|q(s)| ds "Z

t

−E

(12)

# ¤¯¯2 1£ ¯ −1 2 ¯[fN (PN (s))] [fN (PN (s))] (PN (s)F −CPN (s))p(s)−q(s) ¯ ds

T¯

t

where fN (PN (s)) = I + N ΠU ⊥ + PN (T − s). Since I + N ΠU ⊥ À 0 (that is I + N ΠU ⊥ is positive) and N ΠV ⊥ ≥ 0 (that is N ΠV ⊥ is nonnegative), the Riccati equation (11) admits a unique solution with values in S n (see Ref. 6, condition (4.23) and Theorem 7.2). Proposition 4.1. The viability kernel of V conditioned to U with respect to Eq. (10) has the following representation: n o V iab(V |U ) = θ ∈ V : ∃T > 0 s.t. lim hPN (T )θ, θi < ∞ . N →∞

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Proof. Let us consider θ ∈ V iab(V |U ). Then, there exist T > 0 and q ∈ L2P ([0, T ], U ) such that p(s, q, θ) ∈ V, P − a.s., for all s ∈ [0, T ]. We RT recall that (12) holds true and get E [hPN (T )θ, θi] ≤ E 0 |q(s)|2 ds. For the converse, we may choose, for all N, the optimal control process q N ∈ L2P ([0, T ], Rn ) and, again by (12), we have ∞ > E [hPN (T )θ, θi] "Z # T ¡ ¢ =E N (|ΠV ⊥ p(s, q N , θ)|2 + |ΠU ⊥ q N (s)|2 ) + |q N (s)|2 ds . 0

The sequence (q N )N ≥0 is bounded in L2P ([0, T ], Rn ), and there exists a suitable subsequence (still denoted by (q N )N ≥0 ) such that q N converges to q weakly in L2P ([0, T ], Rn ). Since (10) is affine in q, we get the convergence p(s, q N , θ) → p(s, q, θ). Therefore, we have ΠV ⊥ p(s, q, θ) = 0 P − a.s., for all s ∈ [0, T ]. On the other hand, it can be easily seen that "Z # T

E 0

N |ΠU ⊥ q N (s)|2 ds ≤ lim hPN (T )θ, θi , N

hR i T which leads to E 0 |ΠU ⊥ q N (s)|2 )ds → 0 when N → ∞. Let us notice the fact that "Z # "Z # T T E |ΠU ⊥ q(s)|2 ds = E hΠU ⊥ q(s), q(s)i 0

0

"Z

= lim E N →∞

hΠU ⊥ q(s), q N (s)i

0

"Z

# 21

T

≤ lim E N →∞

#

T

0

|ΠU ⊥ q N (s)|

2

"Z E

# 21

T

|q(s)|

2

t

=0 We deduce that q ∈ L2P ([0, T ], U ). The proof of our proposition is now complete. ¤ One can show that Proposition 4.2. The viability kernel of the linear subspace V ⊂ Rn conditioned to the linear subspace U ⊂ Rn with respect to (10) is conditional locally in time viable. In particular, the conditional viability kernel is locally in time viable.

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Proof. Let us consider θ ∈ V iab (V |U ). Then, there exist T > 0 and q ∈ L2P ([0, T ], U ) such that p(s, q, θ) ∈ V, P − a.s., for all s ∈ [0, T ]. Therefore, using (12), we have "Z # T

E [hPN (T − s)p(s, q, θ), p(s, q, θ)i] ≤ E

|q(r)|2 dr .

s

Thus, by a monotone convergence argument and Proposition 4.1, we infer that p(s, q, θ) ∈ V iab(V |U ) P − a.s. ¤ 5. The Main Result We are now able to prove our main result: Theorem 5.1. We have equivalence between the following assertions: (1) The equation (3) is approximately-controllable. (2) The equation (3) is approximately null controllable. (3) The viability kernel of Ker B2∗ conditioned to Ker D1∗ is trivial. Proof. We only have to prove (3) =⇒ (1). In order to establish the result, suppose that V iab(Ker B2∗ |Ker D1∗ ) is trivial and let q ∈ L2P ([0, T ], KerD1∗ ) such that p(s, q, θ) ∈ Ker B2∗ , P − a.s., for all s ∈ [0, T ]. We get θ = 0, and p(s, q, θ) ∈ V iab(Ker B2∗ |Ker D1∗ ). Therefore, p(s, q, θ) = 0, P − a.s., for all s ∈ [0, T ]. Recall that p(·, q, θ) is the solution of Eq. (10) to conclude q(s) = 0, dsdP -almost everywhere on [0, T ] × Ω. ¤ The following theorem gives a method to obtain conditional local in time viability using conditional strict invariance, thus providing an algebraic criterion for approximate controllability. Theorem 5.2. The linear subspace V ⊂ Rn is local in time viable conditioned to the linear subspace U ⊂ Rn with respect to (10) if and only if V is (A∗ ; C ∗ )-strictly invariant conditioned to (F, U ). Proof. Let us first suppose that V is (A∗ ; C ∗ )-strictly invariant conditioned to (F, U ). We wish to prove that V is local in time viable conditioned to U. In order to prove this, it suffices to notice that there exists a linear operator K ∈ L(Rn , Rn ) such that KV ⊂ V , (K − F )V ⊂ U and (A∗ + C ∗ K)V ⊂ V. Indeed, for any θ ∈ V, we consider the linear stochastic differential equation ½ dp(t) = −(A∗ + C ∗ K)p(t)dt + Kp(t)dW (t), p(0) = θ.

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Obviously, the solution of this equation is in V. Moreover, if we set q(t) = (K − F )p(t) ∈ U , we notice that p(t, q(t), θ) = p(t) ∈ V for all t > 0. For the converse let us fix θ ∈ V. We suppose that p(s, q, θ) ∈ V, P −a.s., for all s ∈ [0, T ], for some T > 0 and q ∈ L2P ([0, T ], U ). We multiply (10) with (I − ΠV ) to obtain  ∗ ∗ ∗  d(I − ΠV )p(t) = (I − ΠV ) [−(A + C F )p(t) − C q(t)] dt + (I − ΠV )(F p(t) + q(t))dW (t),  p(0) = θ. Since we have supposed that p(s, q, θ) ∈ V, P − a.s., the quadratic variation of (I − ΠV )p is zero (i.e. F p(t) + q(t) ∈ V, dtdP -almost everywhere on [0, T ] × Ω). Moreover, we have (I − ΠV ) [−(A∗ + C ∗ F )p(t) − C ∗ q(t)] = 0, dtdP -almost everywhere on [0, T ] × Ω. At this point, we consider the linear subspace W ⊂ V W = {θ ∈ V s.t. ∃α ∈ V : α − F θ ∈ U, A∗ θ + C ∗ α ∈ V } . It is straightforward that p(t, q, θ)∈W, dtdP -almost everywhere on [0, T ]×Ω. We use the continuity of the trajectories of p to finally get θ ∈ W. This implies that V = W (i.e. V is (A∗ ; C ∗ )-strictly invariant conditioned to (F, U )). ¤ Since the viability kernel of Ker B2∗ conditioned to Ker D1∗ is locally in time viable conditioned to Ker D1∗ , we deduce that V iab(KerB2∗ |KerD1∗ ) is the largest space which is (A∗ ; C ∗ )-strictly invariant conditioned to (F, KerD1∗ ) for some F solution for the equation D1∗ F + B1∗ = 0. Thus, we can state our final result Theorem 5.3. We have equivalence between the following assertions: (1) The equation (3) is approximately-controllable. (2) The equation (3) is approximately null controllable. (3) The largest linear subspace of KerB2∗ which is (A∗ ; C ∗ )-strictly invariant conditioned to (F, KerD1∗ ) is the trivial subspace {0}. The latter assertion is easily computable as seen in the Remark 2.3. References 1. S.G. Peng, Backward stochastic differential equation and exact controllability of stochastic control systems, Progr. Natur. Sci., 4, 3 (1994), 274–284. 2. Y. Liu, S. Peng, Infinite horizon backward stochastic differential equation and exponential convergence index assignment of stochastic control systems, Automatica, 38 (2002), 1417–1423.

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3. R. Buckdahn, M. Quincampoix, G. Tessitore, A Characterization of Approximately Controllable Linear Stochastic Differential Equations, Stochastic Partial Differential Equations and Applications, G. Da Prato and L. Tubaro, Eds., Series of Lecture Notes in pure and appl. Math., Vol. 245 (Chapman & Hall, 2006), 253–260. 4. J.L. Willems, J.C. Willems, Robust stabilization of uncertain systems, SIAM J. Contr. Optim., 21 (1983), 342–372. 5. E. Pardoux, S.G. Peng, Adapted solutions of a backward stochastic differential equation, Systems and Control Letters, 14 (1990), 55–61. 6. J. Yong, X.Y. Zhou, Stochastic Controls (Hamiltonian Systems and HJB Equations), (Springer Verlag, Berlin, 1999). 7. R. Buckdahn, S. Peng, M. Quincampoix, C. Rainer, Existence of stochastic control under state constraints, C.R. Acad. Sci., S´erie I, 327 (1998), 17–22. 8. R. Buckdahn, M. Quincampoix, A. Rascanu, Viability property for a backward stochastic differential equation and applications to partial differential equations, Probab. Theory Rel. Fields, 116, 4 (2000), 485–504. 9. M. Quincampoix, C. Rainer, Stochastic control and compatible subsets of constraints, Bull. Sci. Math., 129 (2005), 39–55. 10. R. Buckdahn, M. Quincampoix, C. Rainer, A. Rascanu, Stochastic control with exit time and constraints. Application to small time attainability of sets, Appl. Math. Optim., 49 (2004), 99–112.

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NUMERICAL SOLUTIONS OF TWO-POINT BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS USING PARTICULAR NEWTON INTERPOLATING SERIES GHIOCEL GROZA1 and NICOLAE POP2 1 Department

of Mathematics and Informatics, Technical University of Civil Engineering, 124 Lacul Tei, Sec.2, 020396, Bucharest, Romania, E-mail: [email protected] 2 North University of Baia Mare, Dept. of Math. and Comp. Science, Victoriei 76, 43012, Baia Mare, Romania E-mail: nic [email protected] Approximate solutions of two-point boundary value problems for linear differential equations are studied. An approximative solution is a partial sum of a series constructed by means of Newton interpolating polynomials which verify the boundary conditions. Keywords: Newton interpolating series; Two-point boundary value problem.

1. Introduction Consider a sequence of real numbers {xk }k≥1 . In this paper we study the socalled Newton interpolating series defined in Section 2 by means of Newton interpolating polynomials Nn (x), n≥1 at x1 , x2 , ..., xn+1 . These series are useful generalization of power series which, in particular forms, were used in number theory to prove the transcendence of some values of exponential series. Taking into account the importance of power series in the theory of initial value problems for differential equations, it seems to be very useful to study Newton interpolating series in order to find the solution of the multipoint boundary value problem for differential equations. Section 3 deals with functions which are representable into Newton interpolating series. Theorem 1 gives a criterion for a function to be representable into a Newton interpolating series. The representation of solutions of linear differential equations into Newton interpolating series are considered in Section 4, Theorem 2. An appli-

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cation of this theorem is given in the last section. 2. Newton Interpolating Series Let {xk }k≥1 be a sequence of real numbers. We construct the polynomials ui (x) =

i Y

(x − xk ), i = 1, 2, ..., u0 (x) = 1,

(1)

k=0

where x is a real variable. We call an infinite series of the form ∞ X

ai ui (x),

(2)

i=0

where ai ∈ R, a Newton interpolating series with real coefficients ai at {xk }k≥1 . In this paper we study two particular cases. First we suppose that ( 0, if k ≡ 0(mod2), xk = (3) 1, if k ≡ 1(mod2). In second case we consider x1 = 0, x2 = 1, x3 = xk =

1 2

and for k ≥ 4

2s + 1 , where 2m + 1 < k ≤ 2m+1 + 1, s = k − 2m − 2. 2m+1

(4)

Given a particular convergent Newton interpolating series on [0,1] of the form (3) we consider the series ∞ X

(1)

ai ui (x),

(5)

i=0

where, when the series

∞ P

ak bi,k converge,

k=i+1 (1)

ai

=

∞ P

ak bi,k , bk−1,k = k, b0,1 = 1,

k=i+1

(6)

bi,k = bi,k−1 (xi+1 − xk ) + bi−1,k−1 , (1)

with b−1,k = 0. If the coefficients ai derivative series of (3).

are well defined (5) is called the

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3. Functions Represented into Newton Interpolating Series Consider a function f : [0, 1] → R and xk ∈ [0, 1], k = 1, 2, ... . We say that f can be represented into Newton interpolating series at {xk }k≥1 if there exists a series of the form (3) which converges uniformly to f on [0,1]. In order to give a sufficient condition for a function f to be represented into Newton interpolating series we need the following lemma. Lemma 3.1. If the real numbers xk ∈ [0, 1], k = 1, 2, ... are given by (5), then for every x ∈ [0, 1]: a) |u2n +1 (x)| ≤

(2n − 1)! . (2n )2n

(7)

b) For every positive integer m ∈ (2n + 1, 2n+1 ], n ≥ 1, |um (x)| ≤

Proof. a) Since u2n +1 (x) =

2n +1 Q i=1

(m − 3)! (m 2)

m−2

.

(8)

µ ¶ i−1 j x − n , a) is obviously for x = n , 2 2

j+1 j < x < n with j ≤ 2n − 1, then 2n 2 j+1 j 1 1 2 2n + 1 − (j + 1) (2n − 1)! |u2n +1 (x)| ≤ n n ... n n n ... ≤ . n 2 2 2 2 2 2 (2n )2n

j = 0, 1, ..., 2n . If

b) If m ∈ (2n + 1, 2n+1 ], then m = 2n + 2 + t, t = 0, 1, ..., 2n − 2. Hence, for

2j − 1 2j + 1 ≤ x < n+1 with j ≤ 2n − 1, n+1 2 2 ¯¡ ¢¡ 1 |um (x)| = |u2n +1 (x)| ¯ x − 2n+1 x−

3 2n+1

¢

¡ ... x −

2t+1 2n+1

¢¯ ¯

j−1 t (2n − 1)! Y 2j + 1 − (2k + 1) Y 2k + 1 − (2j − 1) ≤ (2n )2n 2n+1 2n+1 k=0 j−1 Q

(2n − 1)! = (2n )2n

k=0

k=j

2(j − k)

t Q

2(k − j + 1)

k=j t+1

(2n+1 )

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(2n − 1)!

j−1 Q

(j − k)

t Q

k=j 2n +t+1 n (2 )

=

t+1

≤

(k − j + 1)

k=0

(2n − 1)! (2n ) 2n +t · 2n (2n )

≤

(2n + t − 1)! 2n +t (2n )

≤

(m − 3)! m−2

(m 2)

.

¤

Theorem 3.1. Suppose f : [0, 1] → R is a C ∞ function and Ms = sup |f (s) (x)|. x∈[0,1]

If xk , k = 1, 2, ... are given either by (4) or by (5) and lim

s→∞

Ms = 0, s!

(9)

then f can be represented uniquely into a Newton interpolating series at {xk }k≥1 . Proof. If xk is given by (4) the result follows from Ref. 1, Theorem 3.2. Let xk be given by (5). By Newton’s interpolating formula there exists ξ = ξ(x, n) ∈ [0, 1] such that f (x) = Nn (x) +

f (n+1) (ξ) un+1 (x), (n + 1)!

(10)

where Nn (x) = f1 + f1,2 (x − x1 ) + ... + f1,2,...,n+1 (x − x1 )...(x − xn )

(11)

is the Newton interpolating polynomial with respect to x1 , x2 , ..., xn+1 and f1,2,...,k+1 is the divided difference of order k. By (7), (7) and (10) we obtain that there exists a positive integer n0 ≥ 3 such that for every n ≥ n0 and for every x ∈ [0, 1] |f (x) − Nn (x)| ≤

Mn+1 (n − 2)! · . (n + 1)! ( n+1 )n−1 2

(12)

Now by Stirling’s formula (n − 2)! =

¶n−2 θ µ p n−2 n−2 e 12(n−2) , θn−2 ∈ (0, 1). 2π(n − 2) e

(13)

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By (12) and (13) it follows that, for every n ≥ n0 , p ¡ ¢n−2 Mn+1 2 2π(n − 2) n−2 e |f (x) − Nn (x)| ≤ · ¡ n+1 ¢n−1 (n + 1)! 2 p Mn+1 4 2π(n − 2) · . ≤ (n + 1)! n+1

169

(14)

Hence Nn (x) converges uniformly to f and the function is represented into a Newton interpolating series at {xk }k≥1 . By taking x = xk , k = 1, 2, ... it follows the uniqueness. ¤ Corollary 3.1. Suppose f : [0, 1] → R is a C ∞ function and there exists C > 0 such that for every s f verifies the condition Ms ≤ C(s − 1)!

(15)

and xk , k = 1, 2, ... are given by (4) or (5). Then f can be represented uniquely into a Newton interpolating series, the derivative Newton interpolating series of f is well defined and f 0 can be represented uniquely into a Newton interpolating series given by the derivative series of f which is obtained by termwise differentiation of Newton interpolating series of f . Proof. If xk are given by (4) the corollary follows from Ref. 1, Corollary 2.2. We remak that u0k (x) =

u0k (x) =

k X uk (x) , x − xj j=1

k−1 X

(16)

bj,k uj (x),

(17)

j=0

where bj,k are given by (6). If xk are given by (5), we can prove using (15) and the method of proof of Lemma 4.1 that |u02n +1 (x)| ≤

(2n − 1)!(2n + 1) (2n )2n −1

(18)

and for every positive integer m ∈ (2n + 1, 2n+1 ], n ≥ 1, |u0m (x)| ≤

(m − 3)!m (m 2)

m−3

.

(19)

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By Theorem 3.1 f =

∞ P i=0

ai ui , where

ai = f1,2,...,i+1 =

f (i) (ξi ) , ξi ∈ [0, 1]. i!

(20)

Thus by (19) and Stirling’s formula |am u0m (x)| ≤ C

(m − 1)! (m − 3)!m · ¡ ¢m−3 ≤ bm , m m!

(21)

2

´m−3 ³ ∞ √ √ P with bm = 4C 2π m − 3 · 2(m−3) . Since the series bi converges, em i=0

by Weierstrass criterion the series ∞ X

ai u0i (x)

(22)

i=0

is uniformly convergent on [0, 1]. Then it follows that the sum of series (22) is equal to f 0 and the corollary follows by using (17). ¤ 4. Solutions of Differential Equations which are Representable into Newton Interpolating Series Consider a linear differential equation with analytic function coefficients y (n) (x) = c(x) + b0 (x)y(x) + b1 (x)y 0 (x) + ... + bn−1 (x)y (n−1) (x),

(23)

with x ∈ [0, 1], where c, bi ∈ C ∞ ([0, 1]), i ∈ {0, 1, ..., n − 1} and there exists C0 > 0 such that n o (j) max kc(j) k∞ , kbi k∞ , i ∈ {0, 1, ..., n − 1} < C0j+1 , j = 0, 1, ... . (24) It is well known that the solutions of this equation are also analytic functions (e.g. Ref. 2, Ch. IV, or Ref. 3, Sec. 3.4). We shall prove that these solutions can be represented into Newton interpolating series at {xk }k≥1 where xk is given by (4) or (5). Firstly we prove the following result. Lemma 4.1. For D > 0 and a, b, c positive integers with b < c we define b Q

S(D, T, a, b, c) =

T X i=0

j=1 c Q j=1

(T + a + j) · (T − i + j)

Di · i!

(25)

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Then for every ² > 0, there exists T0 = T0 (²) such that S(D, T, a, b, c) ≤ ²

(26)

for all T ≥ T0 . Proof. For T ≥ 2c we can write b Q

S(D, T, a, b, c) ≤

(T + a + j)

[ T2 ] i X D

j=1 c Q

(T − [T /2] + j)

i=0

i! (27)

j=1 T X

+Dc

i=[ T2 ]+1

Di−c (T + a + 1)...(T + a + b) · (i − c)! (i − c + 1)...i(T − i + 1)...(T − i + c)

where [x] is the integral part of x (i.e. the largest integer less equal with x). Since k(T − k + 1) ≥ T for k = i − c + 1, ..., i, then there exists T1 such that for every T ≥ T1 T X i=[ T2 ]+1

Di−c (T + a + 1)...(T + a + b) · (i − c)! (i − c + 1)...i(T − i + 1)...(T − i + c) T X

≤

i=[ T2 ]+1

Di−c . (i − c)!

(28)

Thus for every ² ≥ 0 we can choose T0 = T0 (²) ≥ T1 such that b Q

(T + a + j)

j=1 c Q

≤

(T − [T /2] + j)

² , 2eD

(29)

j=1

Dc

T X i=[

T 2

]+1

Di−c ² ≤ . (i − c)! 2

(30)

Now the lemma follows by (19)–(9). ¤ Theorem 4.1. If {xk }k≥1 is a sequence of real numbers from [0, 1] of the form either (4) or (5), then every solution y ∈ C n ([0, 1]) and its derivatives y (k) , k = 1, 2, ..., n of the equation (16) are represented into Newton interpolating series at {xk }k≥1 .

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Proof. For any s ≥ n we can write y (s) (x) = c(s−n) (x) +

n−1 X X s−n k=0 i=0

(s − n)! (i) b (x)y (s−n−i+k) (x) i!(s − n − i)! k

and thus, by (17), ky (s) k∞ ≤ kc(s−n) k∞ +

n−1 X s−n X k=0 i=0

(s − n)! C i+1 ky (s−n−i+k) k∞ . i!(s − n − i)! 0

(31)

If xk is given by (4) the theorem follows from Ref. 1, Theorem 4.1. Let xk be given by (5). We choose a positive integer u ≥ n + 1. Then by Lemma 5.1 there exists s0 ≥ u such that for every s ≥ s0 n−1 X

S(C0 , s − n + k − u, n − k, u − n, u − k) ≤

k=0

1 , 3C0

1 C0s−n ≤ (s − u)! 3C0

(32)

(33)

and, since, for k ≤ n − 1, (s − n − i + k − u)! <1 (s − n − i)! and lim

s→∞

(s − n)...(s − u + 1) i C0 = 0, i!

i = s − n + k − u + 1, ..., s − u, n−1 X

s−n X

1 (s−n−i+k−u)! (s−n)(s−n−1)...(s−u+1)C0i ≤ i!(s−n−i)! 3C0

(34)

k=0 i=s−n+k−u+1

Now we choose C1 > 1 such that for all s ≤ s0 ky (s) k∞ ≤ C1 (s − u)!,

(35)

where here m! means 1 if m < 0. Suppose (35) holds for s ≤ t, where t ≥ s0 and we shall prove that it is also true for s = t + 1 and thus for all s = 0, 1, . . .. Indeed by (17), (10)–(35)

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° (t+1) ° °y ° ∞ Ã ≤ C1

C0t+2−n +

n−1 X X t+1−n k=0

Ã ≤ C0 C1 C0t+1−n +

n−1 X k=0

i=0

(t + 1 − n)!(t + 1 − n − i + k − u)! i+1 C0 i!(t + 1 − n − i)!

!

Ãt+1−n+k−u X

(t + 1 − n)!C0i i!(t+1−n−i+k−u+1)...(t+1−n−i)

i=0

!! (t + 1 − n)!(t + 1 − n − i + k − u)!C0i + i!(t + 1 − n − i)! i=t+1−n+k−u+1 Ã n−1 X C0t+1−n ≤ C0 C1 (t+1−u)! · + S(C0 , t+1−n+k−u, n−k, u−n, u−k) (t+1−u)! t+1−n X

k=0

+

n−1 X

t+1−n X

(t+1−n−i+k−u)! (t+1−n)...(t+1−u+1)C0i i!(t+1−n−i)!

!

k=0 i=t+1−n+k−u+1

≤ C1 (t+1−u)!. (k)

Consequently, if we denote by Ms

¯ ¯ = sup ¯y (s+k) (x)¯, k = 0, 1, ..., n, we x∈[0,1]

obtain Ms(k) < (s − k − (u − k))!C1

(36)

and by Theorem 3.1 it follows that y, y 0 , ..., y (n) can be represented into Newton interpolating series at {xk }k≥1 . ¤ 5. Application In this section we present a method based on Newton interpolating series to approximate the solution of a two-point value problem for equation (16). For simplicity we take n = 2, V = C ∞ ([0, 1]) and we define the operator L : V → V such that Ly(x) = y 00 (x) − b1 (x)y 0 (x) − b0 (x)y(x),

(37)

with the coefficients bi ∈ V, i = 0, 1, satisfying the conditions (17). Let αi , i = 0, 1 be linear functionals on V of the form αi (y) = y(i)

(38)

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such that the system {α0 , α1 } is linearly independent over ker L. We want to approximate the solution of the two-point boundary value problem Ly = c, α0 (y) = α1 (y) = 0,

(39)

where c ∈ V satisfies (17). Denote by V1 = {y ∈ V : α0 (y) = α1 (y) = 0} and suppose that the restriction of L to V1 denoted also by L is an isomorphisms between the vector spaces V1 and V. By putting Y = {f ∈ V, ∃ C(f ) > 0, kf (s) k∞ < C(f )s+1 , s = 0, 1, . . .} and X = L−1 (Y ) we find two linear subspaces of V and V1 , respectively. For a fixed t ≥ 2 we endow Y and X with the norms kf kY =

t−2 X

kf (k) k∞

and respectively

kykX := kLykY .

(40)

k=0

For every s ≥ 2, consider the finite-dimensional subspaces ( ) s X ˜ Xs−2 = ys , ys (x) = ai ui (x), α0 (ys ) = α1 (ys ) = 0, ai ∈ R i=0

˜ s−2 ) ⊂ Y , which are Banach spaces with respect to the and Y˜s−2 := L(X ˜ s−2 be the linear projector with norms (40). For any s ≥ t let Ps−2 : X → X ˜ s−2 . Choose the Ps−2 (y) := Ns defined by (10). We remark that Ns ∈ X distinct collocation points (see Ref. 4, Ch. XIV, or Ref. 5, p.52) zi ∈ [0, 1], i ∈ {1, 2, . . . , N (s)}, with N (s) ≥ s − 1, such that 0, 1 ∈ {z1 , ..., zN (s) }. t P Since L : (X, k · kt ) → (Y, k · kY ) is bounded, where kykt = ky (k) k∞ , then for every ² > 0 we can find s ≥ 2 and an element ys (x) =

s X

˜s, ai ui (x) ∈ X

k=0

(41)

i=0

where the coefficients ai are determined by least-square method such that ¯µ µ s ¯ ¶¶(k) X ¯ ¯ ² (k) ¯ L aj uj (zi ) − c (zi )¯¯ < , ¯ 4(t − 1) (42) j=0 k ∈ {0, 1, . . . , t − 2} and α0 (ys ) = α1 (ys ) = 0. As (Lys )(k) and c(k) are uniformly continuous on [0, 1], we can choose N (s) and |∆N (s) | (the maximum mesh length of partition ∆N (s) determined by zi ) such that for each x ∈ [0, 1] there exists

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zj and ¯ ¯ ¯(Lys − c)(k) (x) − (Lys − c)(k) (zj )¯ <

² , 4(t − 1)

(43)

k ∈ {0, 1, . . . , t − 2}. ² Thus kLys − ckY < and this implies that 2 kLy − Lys kY ≤ kLy − ckY + kLys − ckY < ² and ky − ys kX < ². Hence lim ky − ys kt = 0

s→∞

and the approximate solution of our problem is given by ys with the coefficients ai determined by least-square method from the system (k)

(Lys − c)

(zi ) = 0,

k ∈ {0, 1, ..., t − 2},

i ∈ {1, 2, ..., N (s)}.

(44)

Example. Consider the two-point boundary value problem (Ref. 6, p. 140) y 00 (x) − 2500y(x) = 2500 cos2 (πx), x ∈ [0, 1], y(0) = y(1) = 0.

(45)

We know that this two-point boundary value problem has a solution y which can be represented by a Newton interpolating series of each of the forms (4) or (5). Moreover, by Corollary of Theorem 3.1, the derivatives y 0 and y 00 of y can be represented by a Newton interpolating series which are the derivative series of Newton interpolating series of y. We approximate the solution by taking the partial sums S33 (x) =

33 X

ai ui (x).

i=0

The boundary conditions imply a0 = a1 = 0. Using, for example, 32 equij distant collocation nodes zj+1 = , j = 0, 1, ..., 31 we obtain the coeffi32 cients ai . Table 1 lists the absolute errors in y. The computations were performed on a computer with a 14-hexadecimal-digit mantissa. Note that the errors are clearly unacceptable in simple shooting method.

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Table 1

x

simple shooting (Ref. 6, p. 141)

Newton series (3) with n = 33

Newton series (4) with n = 33

0.1

.19 · 10−7

.25 · 10−6

.13 · 10−10

0.2

.28 · 10−5

.17 · 10−8

.25 · 10−12

0.3

.41 · 10−3

.70 · 10−10

.14 · 10−11

0.4

.61 · 10−1

.15 · 10−9

.11 · 10−11

0.5

.90 · 10

.72 · 10−10

.64 · 10−9

0.6

.13 · 104

.25 · 10−9

.48 · 10−8

0.7

.20 · 106

.49 · 10−9

.41 · 10−7

0.8

.29 · 108

.81 · 10−7

.22 · 10−5

0.9

.44 · 1010

.12 · 10−4

.27 · 10−5

1.0

.65 · 1012

0

0

Errors associated with the example References 1. G. Groza and N. Pop, Approximate solutions of multipoint boundary value problems for linear differential equations by polynomial functions (to appear). 2. P. Hartman, Ordinary Differential Equations, (Wiley, New York, 1964). 3. V. Barbu, Differential Equations (in Romanian), (Junimea Press, Ia¸si, 1985). 4. L.V. Kantorovich and G.P. Akilov, Functional analysis (in Russian), (Nauka, Moscow, 1977). 5. S.G. Mikhlin, Error Analysis in Numerical Processes, (John Wiley & Sons, Berlin, 1991). 6. U. Ascher, R. Mattheij and R. Russell, Numerical Solution of Boundary Value Problem for Ordinary Differential Equations, (Prentice-Hall Inc., New Jersey, 1988). 7. R.A. DeVore and G.G. Lorentz, Constructive Approximation, (SpringerVerlag, Berlin, 1993).

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ON THE INTEGRAL AND ASYMPTOTIC REPRESENTATION OF SINGULAR SOLUTIONS OF ELLIPTIC BOUNDARY VALUE PROBLEMS NEAR COMPONENTS OF SMALL DIMENSIONS OF BOUNDARY N. JITARAS ¸U State University of Moldova, Chi¸sin˘ au E-mail: [email protected]; [email protected] In the bounded domain G ⊂ Rn with smooth boundary Γ composed from varieties Γk of dimensions 0 ≤ nk ≤ n − 1 the elliptic boundary value problem (BVP) is considered. Using the Green’s function of usual elliptic BVP the integral representations of singular solutions near intern components of small dimensions are obtained. In the case of homogeneous operator with constant coefficients the asymptotic representations of solutions with power singularity on Γk is obtained too. Keywords: Elliptic equation; Boundary problem; Integral and asymptotic representation.

1. Statement of the Problem Let G0 ⊂ Rn be a bounded domain with n − 1-dimensional boundary Γ0 ∈ C ∞ and Γk ∈ C ∞ are the nk -dimensional manifolds without boundary which lie inside of G0 , 0 ≤ nk < n − 1. Denote by κk = n − nk the codimension of Γk . Suppose that Γk ∩ Γj = ∅ for k 6= j. Denote by G the κ κ [ [ domain G = G0 \ Γj and by Γ = Γj the boundary of G. Moreover, j=1

j=0

we suppose that the every manifold Γk is oriented. It means that the reper (τ, ν)sliding along Γk on any way preserves orientation when it comes back in the initial point. Here we consider the problem of integral representation of singular solutions of boundary value problem (BVP) in the domain G and of their asymptotic representation near the manifold Γk . For simplicity we consider the case of one elliptic equation of arbitrary order 2m. Denote by H s (G0 ) (H s (G)), H s (Γk ), s ∈ R1 , the usual Sobolev’s spaces

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in G0 and by C s (G0 ), C s (G) and C s (Γk )(s ≥ 0) denote the usual Holder’s spaces on G0 , G and Γk respectively. ¯ 0 and Let L(x, Dx ) be an uniformly elliptic operator of order 2m in G Bj (x, Dx )(j = 1, . . . , m) be the system of operators which covers L(x, Dx ) on Γ0 Refs. 1,2. Consider the elliptic boundary value problem in G0 L(x, Dx )u(x) = f (x), Bj (x, Dx )u(x) = ϕj (x),

x ∈ Γ0 ,

x ∈ G0 ,

(1)

j = 1, . . . , m,

(2)

and the following boundary value problem in G (the Sobolev’s problem): to find the solution of the equation (1) in G, which satisfies the boundary conditions (3) on Γ0 and boundary conditions on Γk : Bjk (x, Dx )u(x) = ψjk (x),

j = 1, . . . , σk ,

k = 1, . . . , χ.

(3)

1 Here and in that follows Dx = (∂/∂x1 , ..., ∂/∂xn ), Dxα = ∂ α1 /∂xα 1 ... αn αn s ∂ /∂xn . These problems has been investigated in various spaces H , C s of functions Refs. 1,2. In particular, in Ref. 2 the theorems of full collection of homomorphisms, generated by the operators of these problems in the scales of Sobolev’s spaces H s , s ∈ R1 , are proved. In the theory mentioned above the boundary value problems (1), (3) and ¯ 0 or G, ¯ respectively, since the operator of the (1)–(4) were investigated in G BVP in the domain has nontrivial kernel. The dim ker A0 (orA) = Ns 6= ∅ and depend on the degrees of smoothness of the solution u(x) near and on the variety Γk . The BV conditions (3) or (4) have sense as traces on Γk and for smooth solutions overdetermine the BVP. The conditions (4) which lose sense as traces on Γk , by no means are connected with the behavior of the solution u(x) near Γk . However, in many problems from applied science the necessity for studying of solutions, having power singularities on manifolds Γk of various dimensions arises Refs. 3,4. In this case it is necessary to give on Γk other natural boundary conditions. These conditions are concerned either (with the asymptotical behavior) of the solution u(x) near the manifold Γk or with giving on Γk of a regularized solution. For this it is necessary to know the asymptotic behavior of the solution u(x) near boundary. In the following the problem of finding the solution u(x) to the equation (1) in G0 or G, satisfying BV conditions (3) on Γ0 we are going to name BVP (1), (3) in G0 or G respectively. We denote by A0 and A the operators, generated by these BVP respectively.

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2. Formulation of the Problem, Integral Representations of Singular Solutions Now we intend to study the BVP in G. Under the supposition that ker A0 =∅ we are looking for the solutions u(x) ∈ H s (G0 ) of the BVP (1), (3) in G from the space H s (G). It is natural to seek solution u(x) of BVP in G as a restriction u(x) = U (x)|G , where U ∈ H s (G0 ) is the solution of BVP (1), (3) in G0 . It is evident, that if u1 (x) and u2 (x) are two solutions of the problem (1), (3) in G0 with boundary function ϕ(x) and right hand sides f1 (x) and f2 (x) respectively, and f1 (x) = f2 (x) in G, then u(x) = u1 (x) − u2 (x) is the solution to problem (1), (3) in G0 with f 0 (x) = f1 (x) − f2 (x) and ˜ suppf 0 (x) ⊂ ∪χk=1 Γk = Γ. ˜ Denote by Ns = kerA = {u0 ∈ H s (G), Lu0 (x) = f 0 (x), suppf 0 (x) ⊂ Γ}. Then every solution u(x) of the BVP (1), (3) in G has the form u(x) = u0 (x) + u0 (x) where u0 (x) is a solution of the BVP (1), (3) with fixed ˜ = γ ∈ C ∞, κ = n − q f0 (x) ∈ H s (G) ), u0 (x) ∈ Ns . For simplicity let Γ be codimension of γ. Locally γ is diffeomorphic to Rq ; x = (x0 , x00 ) are the local coordinates near γ, where x0 ∈ Rq are the tangential coordinates and x00 = (xq+1 , . . . , xn ) are the ”normal” coordinates. For obtaining the representation of u0 (x) ∈ H s (G) near γ it is necessary to know the structure of the distribution f 0 (x) ∈ H s−2m (G0 ), with support on γ. It is true the following Lemma. Lemma 2.1. (Ref. 5) The non-zero element f 0 (x) ∈ H s−2m (G0 ) is concentrated at γ if and only if s < −κ/2 + 2m and there exist the elements fσ (x0 ) ∈ H s−2m+|σ|+κ/2 , such that f 0 (x) =

X

|σ| ≤ χs = [2m − s − κ/2]

³ ´ Dνσ fσ (x0 ) × δ(x00 ) ,

(4)

|σ|≤χs

where, by definition of the Dirac measure δ(γ), X ¯ (f, v)G = hfσ (x0 ), Dνσ v(x)iγ , ∀v ∈ C ∞ (G), |σ|≤χs

(·, ·)G and h·, ·iγ are the ”scalar products” on G and γ respectively, [α] is the integer part of the number α. It is known (Refs. 6,7), that if the problem (1), (3) in G0 is elliptic and kerA0 = ∅, then there exists the Green’s function G(x, y) of BVP and for

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a sufficiently smooth function f (x) the solution u(x) of the homogeneous BVP (1), (3) in G admits the integral representation Z u(x) = G(x, y)f (y)dy. (5) G0 0

(G0 ) and suppf 0 (x) ⊂ γ. Consider ³ ´ X Dνσ fσ (x0 ) × δε (x00 ) , fε0 (x) =

Let now f (x) = f (x) ∈ H

s−2m

|σ|≤χs 00

C0∞ (Rκ ), δε (x00 )

where δε (x ) ∈ → δ(x00 ), as ε → 0 in the sense of distri0 butions in D (G0 ), χs = [−s − κ/2 + 2m]. Replacing fε0 (x) instead of f (x) in formula (5) for u(x), after integrating by parts with respect to variables x00 , and passing to the limit as ε → 0, we obtain the integral representation of solution u0 (x): X Z ¯ νσ G(x, y)fσ (y 0 )dy 0 , D ¯ ν = −Dν . u0 (x) = D (6) |σ|≤χs

γ

3. The Asymptotic Representation of Singular Solutions The behavior of the function u0 (x) near γ is very complicated, but the same expression of the Green’s function G(x, y) is very complicated and depends on the local structure of the operator L(x, ∂x ) near γ. Nevertheless, in some cases we can obtain the asymptotic representation of the singular part of u0 (x) near γ or here the principal part of this representation. For this suppose that near γ L is a homogeneous operator of order 2m with constant coefficients: L(x, Dx ) = L0 (Dx ). Then the Green function G(x, y) has the form G(x, y) = E(x, y) + g(x, y),

(7)

where E(x, y) is the fundamental solution (Ref. 8) of the operator L0 (Dx ) and g(x, y) is a regular, sufficiently smooth function near γ. For simplicity we suppose that γ locally coincides with Rq , x = (x0 , x00 ) is an arbitrary point near γ and suppfσ (y 0 ), fσ ∈ C0∞ , is concentrated in the neighborhood U (x00 ) of the point x00 ∈ γ. Then X Z ¯ xσ00 E(x0 − y 0 , x00 )fσ (y 0 )dy 0 + u u0 (x) = D ˜0 (x), (8) |σ|≤χs

Rq

where u ˜0 (x)is a smooth function. For fσ (y 0 ) ∈ C0∞ (Rq ) we consider the functions Z ¯ xσ00 E(x0 − y 0 , x00 )fσ (y 0 )dy 0 , vσ (x) = D (9) Rq

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¯ ν = −Dν . It is known (Ref. 8) that where D |Dxσ00 E(x0 − y 0 , x00 )| < C|x − y|2m−n−|σ| ln |x − y|,

(10)

where ln |x−y| is dropped for 2m−n−|σ| < 0. Moreover, if 2m−n−|σ| < 0, then E (σ) (z 0 , x00 ) are homogeneous functions of degrees 2m − n − |σ| and if −2m + n − |σ| ≥ q, i.e. n − q − 2m − |σ| = κ − 2m − |σ| ≡ αs ≥ 0, then the integrals Z E (σ) (z 0 , x00 )fσ (x0 − z 0 )dz 0 (11) vσ (x) = Rq

are hypersingular integrals with homogeneous kernels. To study the integrals vσ we use the known procedure of regularization of divergent integrals (regularization in Hadamar’s sense), by separating the singular and regular parts. Let n ≥ 3, ρ = |x0 |, d = |x00 |. Denote by Pα (x0 , z 0 )f (x0 ) =

α X (k) 0 α X X f (x ) (−z 0 )k ≡ Pλ (x0 , z 0 )f k!

λ=0 k=λ

λ=0

the segment of the Taylor expansion of the function f (x0 − z 0 ) near the point z 0 = 0 Z ³ ´ vσ0 = E (σ) (z 0 , x00 ) fσ (x0 −z 0 )−Pασ −1 (x0 , z 0 )fσ −θPασ (x0 , z 0 ) fσ dz 0 (12) Rq

is the regularization of the integral vσ (x) at the point z 0 = 0, θ(z 0 ) = 1 for |z 0 | < 1 and θ(z 0 ) = 0 for |z 0 | > 1. Evidently that the integrals vσ (x) could be presented in the form Z vσ (x) = vσ0 (x) − E (σ) (z 0 , x00 )Pασ −1 (z 0 , Dx0 )fσ dz 0 − Z Rq − Pασ (z 0 , Dx0 )fσ dz 0 ≡ |z 0 |<1

αX σ −1 Z ≡ vσ0 (x) + E (σ) (z 0 , x00 )Pλ (z 0 , Dx0 )fσ dz 0 + q R λ=0 Z (σ) 0 + E (z , x00 )Pσ (z 0 , Dx0 )fσ dz 0 ≡ |z 0 |<1

≡ vσ0 (x) +

αX σ −1

(13)

Iλ [fσ ] + Iασ [fσ ].

λ=0

Firstly we consider the integrals Iλ [fσ ]: (k) X fσ(k) (x0 ) Z X fσ (x0 ) Iλ [fσ ] = E (σ) (z 0 , x0 )(−z 0 )k dz 0 ≡ (−1)λ Ikσ . k! k! Rq |k|=λ

|k|=λ

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Since E (σ) (z 0 , x00 ) is a homogeneous function of degree 2m − n − |σ|, after the change of variables z 0 = d · ξ 0 , we obtain Z Ikσ = E (σ) (z 0 , x00 )z 0k dz 0 = βkσ (w00 )d−ασ +λ , Rq

where

Z E (σ) (ξ 0 , w00 )ξ 0k dξ 0 ,

00

βkσ (w ) =

w00 = x00 /|x00 |.

Rq

Hence Iλ [fσ ]=(−1)λ

X

(k)

βkσ (w00 )

|k|=λ

Now we consider

fσ (x0 ) −ασ +λ d ≡Qλ (Dx0 )fσ (x0 )d−ασ +λ . (14) k!

Z

Iασ [fσ ] =

|z 0 |<1

E (σ) (z 0 , x00 )Pασ (z 0 , Dx0 )fσ (x0 )dz 0 = X

= (−1)ασ

|k|=ασ

Z Aσk (x00 ) =

(k)

Aσk (x00 )

fσ (x0 ) , k!

(15)

E (σ) (z 0 , x00 )z 0k dz 0 . |z 0 |<1

After the substitution of variables z 0 = d · ξ 0 and passing to spherical coordinates (ω 0 , ρ), we obtain Z Aσk (x00 ) = E (σ) (ξ 0 , ω 00 )ξ 0k dξ 0 = |ξ 0 |
0

Since E (σ) (ρω 0 , ω 00 ) is homogeneous of the degree 2m − n − |σ| = −|k| − q, then ³ρ ω 00 ´³ ρ ´|k|+q . E (σ) (ρω 0 , ω 00 )ρ|k|+q = E (σ) ω 0 , λ λ λ For λ = (1 + ρ2 )1/2 we obtain lim E (σ) (ρω 0 , ω 00 ) = lim E (σ)

ρ→∞

ρ→∞

³ρ λ

ω0 ,

ω 00 ´³ ρ ´|k|+q = E (σ) (ω 0 , 0). λ λ

Hence, the integral on ρ in (16) can be written in the form

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Z 1 I= E (σ) (ρω 0 , ω 00 )ρ|k|+q−1 dρ+ 0 Z ∞ Z d−1 dρ dρ (17) (σ) 0 00 |k|+q (σ) 0 + (E (ρω , ω )ρ −E (ω , 0)) + E (σ) (ω 0 , 0) +o(d)= ρ ρ 1 1 R ∞ (σ) 0 00 |k|+q−1 (σ) 0 = regρ=∞ 0 E (ρω , ω )ρ dρ − E (ω , 0) ln d + o(d), where o(d) → 0 as d → 0, and the sum of the first two terms of the right hand side of (17) is the regularized integral in the point ρ = ∞. Then for Aσk (x00 ) we obtain Aσk (x00 ) = −aσk ln d + bσk (x00 ) = o(d), where Z bσk (ω 00 ) =

|ω 0 |=1

Z aσk = E (σ) (ω 0 , 0)ω 0k dω 0 , |ω 0 |=1 Z ∞ ³ ´ ω 0k regρ=∞ E (σ) (ρω 0 , ω 00 )ρ|k|+q−1 dρ dω 0 + o(d). 0

Hence, for Iασ [fσ ] we have Iασ [fσ ] = −Aσ (Dx0 )fσ (x0 ) ln d + Bσ (Dx0 )fσ (x0 ) + o(d),

(18)

where Aσ (Dx0 )fσ (x0 ) = (−1)ασ

X |k|=ασ

Bσ (Dx0 )fσ (x0 ) = (−1)ασ

X |k|=ασ

(k)

aσk

fσ (x0 ) , k!

(19)

(k)

bσk (ω 00 )

fσ (x0 ) . k!

(20)

Now we can formulate the following result. Theorem 3.1. Let fσ (x0 ) ∈ C0ασ +1 (Rq ). The function vσ near manifold γ is represented by vσ (x) = vσ0 (x) +

αX σ −1 λ=0

Qλ (Dx0 )fσ (x0 )d−ασ +λ −

(21)

−Aσ (Dx0 )fσ (x0 ) ln d + Bσ (Dx0 )fσ (x0 ) + o(d), where the function vσ0 (x) and the operators Qλ (Dx0 ), Aσ (Dx0 ) and Bσ (Dx0 ) are defined by (12), (14), (19) and (8), and o(d) tends to 0 as d tends to 0.

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Summing the equalities (21) with respect to σ for all σ such that ασ ≥ 0 we obtain the asymptotic representation for the singular part of the solution u0 (x) near γ. The representation (21) are quite complicate but they represent vσ (x) in the general case. These representations can be simplified in some particular cases (Ref. 9). References 1. S. Agmon, A. Douglis, I. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm.Pure Appl. Math., 12 (1959), 623–727. 2. Y. Roitberg, Elliptic Boundary Value Problems in the Spaces of Distributions, (Kluwer Acad. Publ., Vol. 384, 1996). 3. L.P. Nizhnik, On point interaction in quantum mechanics, Ukr. Math. Zh. 49, 11 (1997), 1557–1560. 4. L.P. Nizhnik, Boundary value problems with singular conditions on boundary components of small dimensions, Meth. Func. Anal. Topology, 7, 1 (2001), 76–81. 5. Y. Roitberg, Boundary Value Problems in the Spaces of Distributions, (Kluwer Acad. Publ., Vol. 498, 1999). 6. Yu.P. Krasovsky, Separation of singularities for Green’s function, Izv. Akad. Nauk SSSR, ser. mat., 31, 5 (1967), 997–1010. 7. V.A. Solonnikov, On Green’s matrices for elliptic boundary value problens, Trudy Mat. Inst. Akad. Nauk SSSR, 110 (1970), 107–145. 8. F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, (Intersciens Publishers, 1955). 9. N. Jitara¸su, On the Sobolev Boundary Value Problem with Singular and Regularized Boundary Conditions for Elliptic Equations. Anal. and Optimiz. of differential systems. (Kluwer Acad. Publ., 2003), 219–226. 10. I.M. Ghelfand, G.E. Shilov, Generalizid Functions and Operations over These Functions, (Fizmatgiz, Moscow, 1959). 11. S.G. Samco, A.A. Kilbas, O.I. Marichev, Integrals and Derivatives of Fractional Order and Their Applications, (Science and Tekhnik, Minsk, 1987).

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AN EXISTENCE RESULT FOR A CLASS OF NONLINEAR DIFFERENTIAL SYSTEMS RODICA LUCA-TUDORACHE Department of Mathematics, ”Gh. Asachi” Technical University, 11 Bd. Carol I, Iasi 700506, Romania E-mail: [email protected] In a real Hilbert space we study the existence, uniqueness and asymptotic behaviour of the strong and weak solutions to a class of nonlinear differential systems subject to some extreme conditions and initial data. For the proofs of our theorems we use several results from the theory of monotone operators and of nonlinear evolution equations of monotone type. Keywords: Nonlinear differential system; Extreme condition; Maximal monotone operator; Cauchy problem; Strong solution; Weak solution.

1. Introduction We shall investigate the nonlinear differential system  dun vn (t) − vn−1 (t)   (t) + + cn A(un (t)) 3 fn (t), dt h (S)   dvn (t) + un+1 (t) − un (t) + d B(v (t)) 3 g (t), n n n dt h for n = 1, 2, ..., and 0 < t < T, in the space H, with the extreme condition (EC)

0 (v0 (t), s1 w10 (t), ..., sm wm (t))T ∈ −Λ((u1 (t), w1 (t), ..., wm (t))T ),

for 0 < t < T , and the initial data ½ un (0) = un0 , vn (0) = vn0 , n = 1, 2, ..., (ID) wi (0) = wi0 , i = 1, m. Here H is a real Hilbert space with the scalar product h·, ·i and the associated norm k · k, T > 0, m ∈ IN , h > 0, cn > 0, dn > 0, for all n ∈ IN and si > 0, for all i = 1, m. The unknown functions are un , vn : (0, T ) → H, n ∈ IN from (S) and wi : (0, T ) → H, i = 1, m from (EC). A and B are

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multivalued operators in H and Λ is a multivalued operator in H m+1 which satisfy some assumptions. This problem is a discrete version with respect to x (with H = IR) of some problems which have applications in the theory of integrated circuits (see Refs. 1, 2, 3 and 4). In the particular case Ã ! Λ11 (v0 ) T Λ((v0 , w1 , ..., wm ) ) = , Λ22 ((w1 , ..., wm )T ) where Λ11 : H → H, Λ22 : H m → H m , the condition (EC) becomes (EC)1

v0 (t) ∈ −Λ11 (u1 (t)),

(EC)2

0 (t))T ∈ −Λ22 ((w1 (t), ..., wm (t))T ). (s1 w10 (t), ..., sm wm

The condition (EC)2 with (ID)2 , that is w0 (t) ∈ −S −1 Λ22 (w(t)), w(0) = w0 , where S=diag(s1 , ..., sm ), w(t)=(w1 (t), ..., wm (t))T , w0 =(w10 , ..., wm0 )T , e e give us by integrating the function w(t)=S(t)w 0 , where {S(t); t ≥ 0} is −1 the semigroup generated by −S Λ22 . The problem (S)+(EC)1 +(ID)1 has been investigated in Refs. 5 and 6. We shall prove the existence, uniqueness and asymptotic behaviour of the strong and weak solutions of our problem (S)+(EC)+(ID), by applying some results from the theory of monotone operators and of nonlinear evolution equations of monotone type in Hilbert spaces (see Refs. 7, 8, 9, 10). We present the assumptions that we shall use in the paper (H1) i) The operators A : D(A) ⊂ H → H, B : D(B) ⊂ H → H are maximal monotone, possibly multivalued, 0 ∈ A(0), 0 ∈ B(0). ii) There exist a0 > 0, b0 > 0 such that a) for all u1 , u2 ∈ D(A), γ1 ∈ A(u1 ), γ2 ∈ A(u2 ) we have hγ1 − γ2 , u1 − u2 i ≥ a0 ku1 − u2 k2 . b) for all v1 , v2 ∈ D(B), δ1 ∈ B(v1 ), δ2 ∈ B(v2 ) we have hδ1 − δ2 , v1 − v2 i ≥ b0 kv1 − v2 k2 . m+1 m+1 (H2) i) The operator Λ : D(Λ) ⊂ H →H is maximal monotone, µ ¶ Λ11 Λ12 possibly multivalued. Moreover Λ = with Λ21 Λ22

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Λ11 : D(Λ11 ) ⊂ H → H, Λ12 : D(Λ12 ) ⊂ H m → H, Λ21 : D(Λ21 ) ⊂ H → H m , Λ22 : D(Λ22 ) ⊂ H m → H m , Ã ! Λ11 (u1 ) + Λ12 ((u2 , ..., um+1 )T ) T Λ((u1 , u2 , u3 , ..., um+1 ) ) = . Λ21 (u1 ) + Λ22 ((u2 , ..., um+1 )T ) ii) There exists c0 > 0 such that for all x, y ∈ D(Λ), x = (x1 , x2 )T ∈ H ×H m , y = (y 1 , y 2 )T ∈ H ×H m , with x2 6= y 2 and w1 ∈ Λ(x), w2 ∈ Λ(y), then hw1 − w2 , x − yiH m+1 ≥ c0 kx2 − y 2 k2H m . (H3) (D(A) × H m ) ∩ D(Λ) 6= ∅. (H4) i) The operators Λ11 and Λ12 are bounded on bounded sets. ii) (D(A) × H m ) ∩ (intD(Λ)) 6= ∅. (H5) The constant h > 0. (H6) i) The constants cn > 0, dn > 0, for all n ∈ IN . ii) There exist the constants χ1 > 0, χ2 > 0 such that cn ≥ χ1 > 0, dn ≥ χ2 > 0, ∀ n ∈ IN . (H7) S = diag(s1 , ..., sm ) with sj > 0, ∀ j = 1, m. Remark 1.1. The assumption (H2) i) is a technical one and it is automatically satisfied if Λ is a matrix with elements from H. 2. The Results We shall express our problem (S)+(EC)+(ID) as a Cauchy problem in a certain Hilbert space. For, we consider the Hilbert space X = lh2 (H) × lh2 (H) × H m , where

( lh2 (H) =

(un )n ⊂ H,

∞ X n=1

) kun k2 < ∞

(= l2 (H)),

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with the scalar product h((un )n , (vn )n , w), ((un )n , (v n )n , w)iX = h(un )n , (un )n il2 (H) + h(vn )n , (v n )n il2 (H) + hw, wis h

=

∞ X n=1

h (hun , un i + hvn , v n i) +

h

m X

si hwi , wi i ,

i=1

where w = (w1 , ..., wm )T , w = (w1 , ..., wm )T ∈ H m . We define the operator A : D(A) ⊂ X → X with D(A) = {((un )n , (vn )n , w) ∈ X, (u1 , wT )T ∈ D(Λ)}, ½µµ ¶ µ ¶ ¶ vn − vn−1 un+1 − un A((un )n , (vn )n , w) = , , z ∈ X, h h n n ¾ v0 ∈ −Λ11 (u1 ) − Λ12 (w), z ∈ S −1 Λ21 (u1 ) + S −1 Λ22 (w) . We also define the operator B : D(B) ⊂ X → X with D(B) = {((un )n , (vn )n , w) ∈ X, un ∈ D(A), vn ∈ D(B), n ≥ 1, {(cn A(un ))n } ⊂ l2 (H), {(dn B(vn ))n } ⊂ l2 (H)}, B((un )n ,(vn )n ,w) = {((cn γn )n , (dn δn )n , 0), γn ∈ A(un ), δn ∈ B(vn ), n ≥ 1}. Theorem 2.1. If the assumptions (H2) i), (H5) and (H7) hold, then the operator A is maximal monotone in X. Theorem 2.2. If the assumptions (H1) i), (H5), (H6) i) and (H7) hold, then the operator B is maximal monotone in X. For the proof of Theorem 1.2, see Ref. 6. Theorem 2.3. If the assumptions (H1) i), (H2) i), (H3), [(H4) i) or (H4) ii)], (H5), (H6) i) and (H7) hold, then the operator A + B is maximal monotone. Using the operators A and B the problem (S)+(EC)+(ID) can be equivalently expressed as the following Cauchy problem in the space X   dU (t) + A(U (t)) + B(U (t)) 3 F (t), 0 < t < T dt (P)  U (0) = U0 , where U =((un )n , (vn )n , w), U0 =((un0 )n , (vn0 )n , w0 ), F =((fn )n , (gn )n , 0), w = (w1 , ..., wm )T , w0 = (w10 , ..., wm0 )T . The main result for our problem (S)+(EC)+(ID) is

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Theorem 2.4. Assume that the assumptions (H1) i), (H2) i), (H3), [(H4) i) or (H4) ii)], (H5), (H6) i) and (H7) hold. If (u10 , w10 , ..., wm0 )T ∈ D(Λ), un0 ∈ D(A), vn0 ∈ D(B), for all n ≥ 1, with (un0 )n , (vn0 )n ∈ l2 (H), {(cn A(un0 ))n }, {(dn B(vn0 ))n }⊂l2 (H), and (fn )n , (gn )n ∈W 1,1 (0, T ; l2 (H)), then there exist unique functions un , vn , n ≥ 1, wi , i = 1, m, (un )n , (vn )n ∈ W 1,∞ (0, T ; l2 (H)), wi ∈ W 1,∞ (0, T ; H), i = 1, m, (u1 (t), w1 (t), ..., wm (t))T ∈ D(Λ), un (t) ∈ D(A), vn (t) ∈ D(B), for all n ≥ 1 and for all t ∈ [0, T ], that verify the system (S) and the extreme condition (EC) for all t ∈ [0, T ) and the initial data (ID). Moreover un , vn , n ≥ 1 and wi , i = 1, m are everywhere differentiable from right in the topology of H and µ ¶0 d+ un vn − vn−1 = fn − cn A(un ) − , n≥2 dt h µ ¶0 d + vn un+1 − un = gn − dn B(vn ) − , n≥1 dt h (1)  +   0 d u1 v1 − v0  dt  f1 − c1 A(u1 ) −     , h  d+ w  = −S −1 Λ21 (u1 ) − S −1 Λ22 (w) dt with −v0 (t) ∈ Λ11 (u1 (t)) + Λ12 (w(t)), for all t ∈ [0, T ), where w = (w1 , ..., wm )T . If U0 ∈ D(A) ∩ D(B) and F ∈ L1 (0, T ; X) then, by Corollary 2.2, Chapter III in Ref. 7 we deduce that the problem (P) has a unique weak solution U ∈ C([0, T ]; X), that is there exist (Fek )k ⊂ W 1,1 (0, T ; (l2 (H))2 ), Fek → Fe, as k → ∞, strongly in L1 (0, T ; (l2 (H))2 ), Fe = ((fn )n , (gn )n ) and (Uk )k ⊂ W 1,∞ (0, T ; X), Uk (0) = U0 , Uk → U as k → ∞, strongly in C([0, T ]; X), strong solutions for the problems dUk (t) + (A + B)(Uk (t)) 3 Fk (t), for a.a. t ∈ (0, T ), k = 1, 2, ..., dt where Fk = (Fek , 0). We present in what follows an existence result for the stationary problem associated to (P). Theorem 2.5. Assume that the assumptions (H1), (H2), (H3), [(H4) i) or (H4) ii)], (H5), (H6) and (H7) hold. Then the equation (A + B)(U ) 3 0 e ∈ D(A) ∩ D(B). has a unique solution U

(2)

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Using now Theorem 3.1 and Theorem 3.9 in Ref. 8 we obtain Theorem 2.6. Assume that the assumptions (H1), (H2), (H3), [(H4) i) or (H4) ii)], (H5), (H6) and (H7) hold, Fe = ((fn )n , (gn )n ) ∈ L1loc (IR+ ; (l2 (H))2 ) verifies the condition Fe(t) → Fe0 , as t → ∞, e = ((pn )n , (qn )n , r), strongly in (l2 (H))2 , Fe0 = ((fn0 )n , (gn0 )n ), and U T r = (r1 , ..., rm ) , is the unique solution of the equation (2). Then e , strongly in X, where U (t) = ((un (t))n , (vn (t))n , w(t)), w(t) = U (t) → U (w1 (t), ..., wm (t))T , t ≥ 0, is an arbitrary weak solution of the equation (P)1 (for t > 0), that is ∞ X

m X

(kun (t) − pn k2 + kvn (t) − qn k2 ) +

n=1

kwi (t) − ri k2 → 0, as t → ∞,

i=1

strongly in H. More precisely, Z e kX ≤ e−d0 t kU (0) − U e kX + kU (t) − U

t

ed0 (s−t) kF (s) − F0 kX ds, ∀ t ≥ 0,

0

where F = (Fe, 0), F0 = (Fe0 , 0). If

dFe ∈ L1 (IR+ ; (l2 (H))2 ) and U (0) ∈ D(A + B) then dt ° + ° °d U ° ° lim (t)° ° =0 t→∞ ° dt X

and ° Z Z ∞° + °d U ° ° dt ≤ 1 k((A+B)(U (0))−F (0))0 kX + 1 ° (t) ° ° dt d0 d0 0

X

0

∞° °

° ° ° dF (t)° dt. ° dt ° X

3. Proofs Proof of Theorem 3.1. Because D(Λ) 6= ∅ (by (H2) i)), we have D(A) 6= ∅. Besides, it is well defined in X; if ((un )n , (vn )n , w) ∈ D(A), then A((un )n , (vn )n , w) ∈ X. We suppose without loss of generality (for an easy writing) that Λ is single-valued. First we shall prove that the operator A is monotone; indeed, for U = ((un )n , (vn )n , w), U =((un )n , (v n )n , w)∈D(A), w = (w1 , ..., wm )T , w = (w1 , ..., wm )T we obtain after some computations

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191

® A(U ) − A(U ), U − U X ¶ µ ¶ ¿µ À v n − v n−1 vn − vn−1 − , (un − un )n = h h n n l2 (H) Àh ¿µ ¶ µ ¶ un+1 − un un+1 − un + − , (vn − v n )n h h 2 (H) n n lh −1 ® + S Λ21 (u1 ) + S −1 Λ22 (w) − S −1 Λ21 (u1 ) − S −1 Λ22 (w), w − w s µ ¶ µ ¿ µ ¶ ¶À u1 u1 − u1 u1 −Λ ≥ 0, = Λ , w w w−w H m+1 because Λ is monotone (by (H2) i)). Next we shall prove that A is maximal monotone. By Proposition 2.2 in Ref. 8 it is sufficient (and necessary) to show that for any λ > 0 (equivalently there exists a λ > 0 such that) R(I + λA) = X. We consider λ = h and we shall show that for any Y = ((xn )n , (yn )n , z) ∈ X, z = (z1 , ..., zm )T ∈ H m , the equation (I + hA)(U ) = Y

(3)

has a solution U = ((un )n , (vn )n , w) ∈ D(A). The equation (3) is equivalent to  un + vn − vn−1 = xn      vn + un+1 − un = yn , n = 1, 2...,  v0 = −Λ11 (u1 ) − Λ12 (w),     w + h[S −1 Λ21 (u1 ) + S −1 Λ22 (w)] = z, The system (4) can be written as  un + vn − vn−1 = xn      vn + un+1 − un = yn , n = 1, 2, ...,    Ã ! −v0  u1   µ ¶  .   z−w =Λ   w  S h

(4)

(5)

We shall use a similar idea from Ref. 6. We look for a solution of (5) in the form un = u1n + u2n , vn = vn1 + vn2 , n = 1, 2, ..., and w ∈ H m , where ((u1n )n , (vn1 )n ) is a solution to

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 1  u1 + vn1 − vn−1 = xn   n 1 1 1 vn + un+1 − un = yn , n = 1, 2, ...,    v 1 = 0,

(6)

0

((u2n )n , (vn2 )n )

and = a((pn )n , (qn )n ), where a ∈ H will be determined below and ((pn )n , (qn )n ) ∈ (l2 (IR))2 is solution of the system   p + q1 = p   1 (7) pn + qn − qn−1 = 0, n = 2, 3, ...   q + p − p = 0, n = 1, 2, ..., n

n+1

n

where p > 0. The problem (6) has a solution. To prove this, we consider the operator A0 : D(A0 ) = (l2 (IR))2 → (l2 (IR))2 , A0 ((un )n , (vn )n ) = ((vn − vn−1 )n , (un+1 − un )n ), v0 = 0. Then the equation (6) is equivalent to U + A0 (U ) = Y.

(8)

The operator A0 is monotone, single-valued, everywhere defined and continuous. Then by Proposition 2.4 in Ref. 8 we deduce that A0 is maximal monotone and so the equation (8) (and also the problem (6)) has a unique solution. Using the same argument used before, we deduce that the problem (7) (here H = IR) has a unique solution ((pn )n , (qn )n ) ∈ (l2 (IR))2 . By a direct computation (see also Ref. 6) the solution of (7) is Ã Ã √ !n−1 √ √ !n 3− 5 5−1 3− 5 pn = p, ∀ n ≥ 1. p ; qn = 2 2 2 Evidently un = u1n + u2n = u1n + apn and vn = vn1 + vn2 = vn1 + aqn verify the relations (5)1 for n = 2, 3, ... and (5)2 for n = 1, 2, ... We shall determine a ∈ H and w ∈ H m such that  u1 + v1 − v0 = x1      ! Ã 1  −u21 − v12   u1 + u21 Ã !  µ ¶ −v0 . ⇔ u1 z−w =Λ  µ ¶  S w = Λ    z − w  h  w  S h By definition of u21 and v12 the above relation is equivalent to     Ã 1 ! Ã 1 ! −ap1 − aq1 −ap u1 + ap1  µ ¶ ¶  = Λ u1 + ap1 ⇔ µ , (9)   z − w =Λ z−w  S w w S h h

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where u11 is the solution of (6). We denote

√

ζ = ap1 = √

5−1 ap. 2

193

(10)

5+1 ζ and (9) becomes 2 √  Ã 1 ! Ã ! 5+1 ζ 0 u1 + ζ  2 z . = (11)  +Λ S w w h S h The equation (11) can be written as Ã ! Ã ! Ã ! 0 ζ ζ z , in H m+1 , Λ1 + Λ2 = (12) S w w h m+1 where the operators Λ1 :ÃH m+1 → ,) Λ2 : D(Λ2 ) ⊂ H m+1 → H m+1 !H (µ ¶ 1 u1 + ζ ζ ∈ D(Λ) are defined by with D(Λ2 ) = ; w w √  ! Ã ! Ã ! Ã 1 5+1 ζ ζ ζ u1 + ζ   2 = . Λ1 =Λ  , Λ2 w w w w S h The operator Λ2 is maximal monotone in H m+1 and Λ1 is single-valued, everywhere defined, continuous and strongly monotone, because * Ã ! Ã !+ √ m X ζ ζ si 5+1 Λ1 , = kζk2 + kwi k2 2 h w w m+1 i=1 So ap =

H

Ã ≥ θ0 (√ where θ0 = min

2

kζk +

m X i=1

! kwi k

2

°µ ¶°2 ° ζ ° ° = θ0 ° , ° w ° H m+1 )

5 + 1 si , , i = 1, m . 2 h

Therefore, by Corollary 1.3, Chapter II in Ref. 7 we deduce that Λ1 +Λ2 is maximal monotone and coercive, then the equation (12) has a unique solution (ζ, wT )T ∈ D(Λ2 ). By (10) we obtain a ∈ H, which together with w satisfy the equation (9). So we proved the existence of solution U = ((un )n , (vn )n , w) ∈ D(A) of the system (4). Therefore the operator A is maximal monotone in X.

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Proof of Theorem 2.1. The operator A + B : D(A) ∩ D(B) ⊂ X → X has D(A) ∩ D(B) = {((un )n , (vn )n , w) ∈ X, un ∈ D(A), vn ∈ D(B), n ≥ 1, (u1 , wT )T ∈ D(Λ), {(cn A(un ))n }, {(dn B(vn ))n } ⊂ l2 (H)} 6= ∅, by (H1) i), (H3). We suppose again, for an easy writing that A, B and Λ are single-valued. We firstly suppose that (H4) i) holds. The operator A + B is monotone, because A and B are monotone. To prove that A + B is maximal monotone, we shall prove that for any F0 = ((fn0 )n , (gn0 )n , ζ 0 ) ∈ X the equation U + A(U ) + B(U ) = F0 has a solution U ∈ D(A) ∩ D(B). For, let F0 ∈ X be done. The equation (13) is equivalent to  vn − vn−1   un + + cn A(un ) = fn0 h   vn + un+1 − un + dn B(vn ) = g 0 , n = 1, 2, ... n h with     −v0 u1 µ ¶ µ ¶  s1 (ζ10 − w1 )   w1  −v0 u1     = Λ ⇔ = Λ ,    .  .. 0 . S(ζ − w) w     . . 0 sm (ζm − wm ) wm 0 T ) , w = (w1 , ..., wm )T . where ζ 0 = (ζ10 , ..., ζm We shall approximate the above problem by the following one ( λ U + A(U λ ) + Bλ (U λ ) = F0

U λ ∈ D(A), λ > 0,

(13)

(14)

(15)

(16)

where Bλ ((un )n , (vn )n , w) = ((cn Aλ (un ))n , (dn Bλ (vn ))n , 0) with Aλ , Bλ the Yosida approximations of A, respectively B, (Aλ = λ1 (I − JλA ), Bλ = λ1 (I − JλB )). By the properties of Aλ and Bλ we deduce that Bλ is everywhere defined in X, single-valued, monotone and continuous. Using also Theorem 3.1 (A is maximal monotone), it follows that A + Bλ is maximal monotone, for each λ > 0. Hence, for each λ > 0 the problem (16) has a solution U λ = ((uλn )n , (vnλ )n , wλ ) ∈ D(A). The problem (16) is equivalent to  λ λ   uλ + vn − vn−1 + cn Aλ (uλ ) = f 0  n n n h (17) λ λ    v λ + un+1 − un + dn Bλ (v λ ) = g 0 , n = 1, 2, ... n n n h

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with 



−v0λ



uλ1

195



Ã λ! ! Ã  λ   u1 −v0λ  w1   s1 (ζ10 − w1λ )    = Λ  =Λ , (18)  ..  ⇔   .. 0 λ wλ S(ζ − w )  .    . λ λ 0 wm ) − wm sm (ζm λ T where wλ = (w1λ , ..., wm ) .

Let U 0 = ((u0n )n , (vn0 )n , w0 ) ∈ D(A), v10 = −Λ11 (u01 ) − Λ12 (w0 ). We denote by Fλ = ((fnλ )n , (gnλ )n , ζ λ ) := U 0 + A(U 0 ) + Bλ (U 0 ), λ > 0.

(19)

The set {Bλ (U 0 ); λ > 0} is bounded in the space X, because kBλ (U 0 )k2X =

∞ X

h(c2n kAλ (u0n )k2 + d2n kBλ (vn0 )k2 )

n=1

≤

∞ X

h(c2n kA(u0n )k2 + d2n kB(vn0 )k2 ) = kB(U 0 )k2X , ∀ λ > 0.

n=1

We deduce by the above inequality and (19) that kFλ kX ≤ const., for all λ > 0. We denote by const. some different positive constants which are independent of λ. Using (16) and (19) we obtain kU λ kX ≤ const., for all λ > 0, and therefore ∞ X n=1

h(kuλn k2 + kvnλ k2 ) +

m X

si kwiλ k2 ≤ const., ∀ λ > 0.

i=1

Because {uλ1 ; λ > 0} and {wiλ ; λ > 0}, i = 1, m are bounded in H, by (H4) i) we deduce that {v0λ ; λ > 0} is also bounded in H. So we obtain that {A(U λ ); λ > 0} is bounded in X. By (16) we get {Bλ (U λ ); λ > 0} is bounded in X, that is kBλ (U λ )kX ≤ const., for all λ > 0, which is equivalent to ∞ X

h(kcn Aλ (uλn )k2 + kdn Bλ (vnλ )k2 ) ≤ const., ∀ λ > 0.

(20)

n=1

In what follows we shall prove that the sets {(uλn )n ; λ > 0}, λ > 0}, {wiλ ; λ > 0}, i = 1, m are Cauchy sequences. For this, let U = ((uλn )n , (vnλ )n , wλ ), U µ = ((uµn )n , (vnµ )n , wµ ) ∈ D(A) be solutions for the problem (16), (λ, µ > 0). By (16) we obtain {(vnλ )n ; λ

U λ − U µ + A(U λ ) − A(U µ ) + Bλ (U λ ) − Bµ (U µ ) = 0.

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We multiply the above relation by U λ − U µ in X and after some computations involving (20), we deduce ® kU λ − U µ k2X ≤ − Bλ (U λ ) − Bµ (U µ ), U λ − U µ X . Therefore ∞ X

h(kuλn − uµn k2 + kvnλ − vnµ k2 ) +

n=1

m X

si kwiλ − wiµ k2

i=1

∞ X ® ® ≤ − [cn h Aλ (uλn )−Aµ (uµn ), uλn −uµn +dn h Bλ (vnλ )−Bµ (vnµ ), vnλ −vnµ ], n=1

λ, µ > 0, and finally we get ∞ X

(kuλn − uµn k2 + kvnλ − vnµ k2 ) +

n=1

m X

kwiλ − wiµ k2 ≤ const.(λ + µ), ∀ λ, µ > 0.

i=1

By the last inequality we deduce that {(uλn )n ; λ > 0}, {(vnλ )n ; λ > 0} are Cauchy sequences in l2 (H), and {wiλ ; λ > 0}, i = 1, m are Cauchy sequences in H. Therefore there exist lim (uλn )n = (un )n , lim (vnλ )n = λ→0

λ→0

(vn )n , strongly in l2 (H), (evidently uλn → un , vnλ → vn , as λ → 0, strongly in H, for all n ≥ 1), and lim wiλ = wi , i = 1, m, strongly in H. Then we λ→0

λ T ) → (u1 , w1 , ..., wm )T , as λ → 0, strongly in H m+1 have (uλ1 , w1λ , ..., wm 0 λ T λ − wm )) ; λ > 0} is bounded in and, because {(−v0 , s1 (ζ10 − w1λ ), ..., sm (ζm m+1 H , on a subsequence, denoted in the same way, we have 0 0 λ T −wm ))T , −wm )) → (−v0 , s1 (ζ10 −w1 ), ..., sm (ζm (−v0λ , s1 (ζ10 −w1λ ), ..., sm (ζm

as λ → 0, weakly in H m+1 , with v0 ∈ H. Because Λ is demicontinuous, by (18) and the above relations, we obtain (u1 , w1 , ..., wm )T ∈ D(Λ) and 0 − wm ))T = Λ((u1 , w1 , ..., wm )T ), (−v0 , s1 (ζ10 − w1 ), ..., sm (ζm

that is we have (15). Next, by uλn → un and vnλ → vn , as λ → 0, strongly in H, we deduce that JλA uλn → un , JλB vnλ → vn , as λ → 0, strongly in H, for all n ≥ 1. Because {Aλ (uλn ); λ > 0}, {Bλ (vnλ ); λ > 0}, n ≥ 1 are bounded (from (20), for any n fixed we have kcn Aλ (uλn )k ≤ const., so kAλ (uλn )k ≤ c1n const., for all λ > 0), and A, B are demiclosed, we deduce that un ∈ D(A), vn ∈ D(B), for all n ≥ 1 and Aλ (uλn ) → pn , Bλ (vnλ ) → qn , as λ → 0, weakly in H, (eventually on some subsequences) and pn = A(un ), qn = B(vn ), for all n ≥ 1. By passing to λ → 0 in (17)+(18) we deduce that U = ((un )n , (vn )n , w), w = (w1 , ..., wm )T , is a solution for (14)+(15).

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By (14) we obtain (cn A(un ))n , (dn B(vn ))n ∈ l2 (H), so U ∈ D(A) ∩ D(B) and A + B is maximal monotone. If (H4) ii) holds, by Theorems 3.1 and 1.2 we have that A and B are maximal monotone with intD(A) = {((un )n , (vn )n , w) ∈ X, (u1 , w1 , ..., wm )T ∈ intD(Λ)}, and so intD(A)∩D(B) 6= ∅. Therefore, by Corollaire 2.7 in Ref. 8 we deduce that A + B is maximal monotone. Proof of Theorem 2.2. By Theorem 2.1 the operator A + B : D(A) ∩ D(B) ⊂ X → X is maximal monotone in X. Using Theorem 2.2 and Corollary 2.1, Chapter III in Ref. 7, we deduce that for U0 = ((un0 )n , (vn0 )n , w0 ) ∈ D(A) ∩ D(B), w0 = (w10 , ..., wm0 )T and Fe = ((fn )n , (gn )n ) ∈ W 1,1 (0, T ; (l2 (H))2 ), the problem (P) has an unique strong solution U = ((un )n , (vn )n , w) ∈ W 1,∞ (0, T ; X), w = (w1 , ..., wm )T , U (t) ∈ D(A) ∩ D(B), ∀ t ∈ [0, T ). By extending correspondingly the functions fn , gn , n≥1, in the interval [0, T +ε], with ε > 0, we obtain U (T ) ∈ D(A) ∩ D(B). The solution U is everywhere differentiable from right and d+ U (t) = (F (t) − A(U (t)) − B(U (t)))0 , ∀ t ∈ [0, T ), dt that is the relations (1) are verified. In addition we have ° + ° ° Z t° °d U ° ° ° dF 0 ° ° ° ° ° dt (t)° ≤ k(F (0)−A(U0 )−B(U0 )) kX + ° ds (s)° ds, ∀ t ∈ [0, T ). 0 X X Proof of Theorem 3.1. Under the assumptions (H1) ii) and (H2) ii) the operator A + B is strongly monotone. Indeed, for all U = ((un )n , (vn )n , w), U =((un )n , (v n )n , w) ∈ D(A) ∩ D(B), w=(w1 , ..., wm )T , w=(w1 , ..., wm )T , Z ∈ (A + B)(U ), Z ∈ (A + B)(U ) we have ∞ X ® Z − Z, U − U X ≥ h(a0 cn kun − un k2 + b0 dn kvn − v n k2 ) +

m X

n=1

c0 kwi − wi k2 ≥ a0 χ1

i=1 m X

+c0

∞ X

hkun − un k2 + b0 χ2

n=1 2

∞ X

hkvn − v n k2

n=1 2

kwi − wi k ≥ d0 kU − U k ,

i=1

where d0 = min{a0 χ1 , b0 χ2 , c0 /si , i = 1, m} > 0. Therefore the operator A + B is coercive, and by Corollaire 2.4 in Ref. 8 we deduce that R(A + B) = X. Hence the equation (2), which can be written as

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 vn − vn−1   + cn A(un ) 3 0   h  un+1 − un + dn B(vn ) 3 0, n = 1, 2...,   h    (−v0 , 0, ..., 0)T ∈ Λ((u1 , w1 , ..., wm )T ), e =((pn )n , (qn )n , r) ∈ D(A) ∩ D(B), r=(r1 , ..., rm )T , has a unique solution U 2 (pn )n , (qn )n ∈ l (H), (p1 , r1 , ..., rm )T ∈ D(Λ), pn ∈ D(A), qn ∈ D(B), for all n ≥ 1. References 1. C.A. Marinov, P. Neittaanmaki, Mathematical Models in Electrical Circuits: Theory and Applications, (Kluwer Academic Publishers, Dordrecht, 1991). 2. G. Moro¸sanu, C.A. Marinov, P. Neittaanmaki, Well-posed nonlinear problems in integrated circuits modelling, Circ. Syst. Sign. Proc., 10 (1991), 53–69. 3. R. Luca, An existence result for a nonlinear hyperbolic system, Differ. Integral Equ., 8 (1995), 887–900. 4. R. Luca, Monotone boundary conditions for a class of nonlinear hyperbolic systems, Int. J. Pure Appl. Math. (2006), in press. 5. R. Luca, Existence for a class of discrete hyperbolic problems, Adv. Difference Equ., (2006), (2006), doi:10.1155/ADE/2006/89260, 14p. 6. R. Luca, Existence and asymptotic behaviour for a discrete hyperbolic system, J. Math. Anal. Appl., (2006), doi:10.1016/j.jmaa.2006.06.048, 15p. 7. V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, (Noordhoff, Leyden, 1976). 8. H. Brezis, Op´erateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, (North-Holland, Amsterdam, 1973). 9. V. Lakshmikantham, S. Leela, Nonlinear Differential Equations in Abstract Spaces, International Series in Nonlinear Mathematics: Theory, Methods and Applications, Vol. 2, (Pergamon Press, 1981). 10. G. Moro¸sanu, Nonlinear Evolution Equations and Applications, (Editura Academiei-D. Reidel, Bucure¸sti-Dordrecht, 1988).

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WELL-POSEDNESS FOR A NON-AUTONOMOUS MODEL OF FAST DIFFUSION GABRIELA MARINOSCHI Institute of Mathematical Statistics and Applied Mathematics Calea 13 Septembrie 13, 050711 Bucharest, Romania, E-mail: [email protected] The paper deals with the study of the well-posedness of a non-autonomous nonlinear model of fast diffusion involving a blowing-up diffusivity. In particular it can describe the water infiltration in a porous medium characterized by the time and space variability of the pore structure. The model turns into an abstract Cauchy problem with a time dependent nonlinear multivalued operator in a Hilbert space. Results concerning the existence and uniqueness of the solution are proved. Numerical results intended to show the influence of the variable porosity upon the flow are provided. Keywords: Fast-diffusion; Non-autonomous evolution equations; m-accretive operators; Infiltration in porous media.

1. Introduction Let Ω be an open bounded subset of RN , (N = 1, 2, 3), with a piecewise smooth boundary Γ := ∂Ω. We assume that Γ = Γu ∪ Γα , where Γu and Γα are disjoint smooth subsets of the boundary. We denote the space variable by x = (x1 , x2 , x3 ) ∈ Ω and the time by t ∈ (0, T ), T > 0. Also, we shall denote by i = (i1 , i2 , i3 ) the unit vector in the reference system with Ox3 downwards directed and by ν the outward normal to the boundary Γ. We are concerned with the boundary value problem with initial data ∂(m(x, t)v) − ∆β ∗ (v) + ∇ · K(v) 3 f, in Q = Ω × (0, T ), ∂t v(x, 0) = v0 (x), in Ω, K(v) · ν − ∇β ∗ (v) 3 u(x, t), on Σu = Γu × (0, T ), v(x, t) = 0, on Σα = Γα × (0, T ).

(1)

The meaning of these equations will be explained a little farther by the definition of the weak solution.

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Problem Hypotheses Let vs ∈ (0, +∞) be a constant. Let β : (−∞, vs ) → [ρ, +∞) be a positive, continuous, monotonically increasing function on (0, vs ), with β(r) ≡ ρ on (−∞, 0]. We assume that β satisfies the hypotheses: (iβ ) β(r) ≥ ρ > 0, ∀r < vs , (iiβ ) lim β(r) = +∞, r%vs

(iiiβ ) there exists

Ks∗

∈ (0, +∞),

Z Ks∗

r

:= lim

r%vs

β(ξ)dξ. 0

In our problem β ∗ is the multivalued function defined as ½ Rr β(ξ)dξ, if r < vs , ∗ 0 β (r) := [Ks∗ , +∞), if r = vs ,

(2)

having the properties (i) (β ∗ (r) − β ∗ (r))(r − r) ≥ ρ(r − r)2 , ∀r, r ≤ vs , (ii) lim β ∗ (r) = −∞, r→−∞

(iii) lim β ∗ (r) = Ks∗ . r%vs

We also notice that β ∗ is convex. The vector K(r) has the components Kj : (−∞, vs ]→[0, Ksj ], j=1, 2, 3, Kj (r) = 0 for r < 0 and Kj (vs ) = Ksj ∈ (0, ∞). The functions Kj are monotonically increasing and Lipschitz continuous for r ≤ vs . Hence, there exists M > 0, such that (iiK ) |K(r) − K(r)| ≤ M |r − r| , ∀r, r ≤ vs . Finally, we assume that ∂m ∈ C 1 (Q), ∂t m(x, t) ∈ [m1 , m2 ], ∀(x, t) ∈ Q, where m1 , m2 ∈ R, m1 > 0.

m ∈ C 1 (Q), mt :=

(3)

By the hypotheses (iiβ ) and (iiiβ ) the first equation in (2) is associated to a fast diffusion (see e.g., Ref. 1). Moreover, this equation is assigned to the diffusion with transport in porous media and, in particular, to the water infiltration in soils. In this context v is the water saturation, β is the water diffusivity and K is the water conductivity vector. The presence of the function m can describe either a porous medium with a time and space dependent porosity or a sorption-desorption process in which water may be sorbed and kept for a while by the solid and released totally or partially into the fluid after some time. We have to add that in the case of water

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infiltration in soils, θ(x, t) = m(x, t)v(x, t) is the volumetric water content, or the moisture of the soil. Obviously, some other boundary conditions can be considered, but we have chosen these ones being motivated by a practical interest in water infiltration into soils. The flux boundary condition on the upper boundary Σu represents the inflow due to a water supply (rain) and the condition on Σα expresses the fact that the domain is large enough and moisture does not reach this boundary. Previous existence and uniqueness studies for solutions to the ellipticparabolic problems leading to equations of type (2) with m constant have been presented in the literature especially using a technique inspired by the method of entropy solutions introduced by S.N. Krushkov in Ref. 2. Other studies following Krushkov’s method are due to J. Carillo (see Refs. 3–5), F. Otto (see Ref. 6), H.W. Alt and S. Luckhaus (see Ref. 7). We mention also the paper of E. Chasseigne and J.L. Vazquez (see Ref. 8) and the papers of J.L. Vazquez (see Ref. 9,10) related to fast diffusion equations. In the paper Ref. 11 a model of the saturated-unsaturated flow lying on a special definition of the boundary conditions that changes during the phenomenon evolution, has been developed for a finite value of the diffusivity at saturation. The model studied in this paper was introduced in Ref. 1. For m constant the well-posedness of such models of diffusion was approached in the papers Ref. 12–14 (the last for a superdiffusion model), using techniques from the theory of evolution equations with m-accretive operators in Hilbert spaces. In this paper we shall investigate the existence of the solution to the model with m time and space variable, in relation with the results of Kato, proved in Ref. 15 and extended by Crandall and Pazy in Ref. 16 to general cases involving quasi m-accretive operators.

Functional Framework We consider the Hilbert space V := {v ∈ H 1 (Ω); v = 0 on Γα } ³R ´1/2 2 with the norm kvkV = Ω |∇v(x)| dx and its dual, V 0 . We endow V 0 with the scalar product (v, v)V 0 := v(ψ)

(4)

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where ψ is the solution to the problem ∂ψ = 0 on Γu , ψ = 0 on Γα . (5) ∂ν For the sake of simplicity we shall denote the norm and the scalar product in L2 (Ω) with no subscript and if any confusion is avoided we shall not indicate in the integrands those function arguments that represent the integration variables. −∆ψ = v,

Definition 1. Let m ∈ C 1 (Q), mt ∈ C 1 (Q), v0 ∈ L2 (Ω), v0 ≤ vs , a.e. x ∈ Ω, f ∈ L2 (0, T ; V 0 ), u ∈ L2 (0, T ; L2 (Γu )). We call a solution to (2) a function v that satisfies v∈ mv ∈ β ∗ (v) ∈ v≤ v(0) = and

¿

C([0, T ]; L2 (Ω)) ∩ L2 (0, T ; V ), C([0, T ]; L2 (Ω)) ∩ L2 (0, T ; V ) ∩ W 1,2 (0, T ; V 0 ), L2 (0, T ; V ), vs , a.e. (x, t) ∈ Q, v0 ,

d(mv) (t), ψ dt

À

(6)

Z + V 0 ,V

(∇ζ · ∇ψ − K(v) · ∇ψ) dx ΩZ

= hf (t), ψiV 0 ,V −

u(t)ψdx, a.e. t ∈ (0, T ), ∀ψ ∈ V,

(7)

Γu

for ζ(x, t) ∈ β ∗ (v(x, t)), a.e. (x, t) ∈ Q. We replace in (2) θ(x, t) := m(x, t)v(x, t), θ0 := m(x, 0)v0 ,

(8)

and for each t∈[0, T ] we introduce the operator A(t) : D(A(t)) ⊂ V 0 →V 0 , on ½ µ ¶ ¾ z D(A(t)) := z ∈ L2 (Ω); ∃η ∈ V, η(x, t) ∈ β ∗ a.e. x ∈ Ω , m(x, t) (9) by the relation µ ¶ ¶ Z µ z hA(t)z, ψiV 0 ,V := ∇η · ∇ψ − K · ∇ψ dx, ∀ψ ∈ V, m(x, t) Ω µ ¶ (10) z ∗ for η(x, t) ∈ β . m(x, t) Also, if u ∈ L2 (0, T ; L2 (Γu )) we define fΓu ∈ L2 (0, T ; V 0 ) by

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Z hfΓu (t), ψiV 0 ,V := −

u(t)ψdσ, ∀ψ ∈ V,

(11)

Γu

and so we are led to the Cauchy problem dθ (t) + A(t)θ(t) = f (t) + fΓu (t), a.e. t ∈ (0, T ), dt θ(0) = θ0 .

(12)

We notice that if the solution to (12) exists, then it is a solution in the generalized sense to (2). Equivalently to (12) we can still write µ ¶ À ¶ Z T¿ Z µ dθ θ ∇η · ∇ϕ − K (t), ϕ(t) dt + · ∇ϕ dxdt dt m 0 Q V 0 ,V Z T Z (13) uϕdσdt, = hf (t), ϕ(t)iV 0 ,V dt − Σu 0 µ ¶ θ ∀ϕ ∈ L2 (0, T ; V ), for η(x, t) ∈ β ∗ a.e. (x, t) ∈ Q, m(x, t) and this relation is equivalent to (7), too. For a later use we still define ½Rr ∗ β (ξ)dξ, if r ≤ vs , 0 j(r) := (14) +∞, if r > vs , where the left limit of β ∗ at v = vs was specified in (iii). The function j is proper, convex and lower semicontinuous (see Ref. 1). The purpose to our paper is to study the existence and uniqueness of the solution to (2), as introduced by Definition 1, or equivalently, of the solution to the non-autonomous Cauchy problem (12). 2. The Approximating Problem Since the operator A is multivalued we shall first investigate an approximating problem obtained by replacing the multivalued function β ∗ by the continuous approximation βε∗ defined for each ε > 0 by ( β ∗ (r), if r < vs − ε βε∗ (r) := (15) K ∗ −β ∗ (v −ε) β ∗ (vs − ε) + s ε s (r − vs ), if r ≥ vs − ε. This function is differentiable on R, except at r = vs (it has bounded lateral derivatives at vs but they do not coincide) and we denote ( β(r), if r < vs − ε βε (r) := Ks∗ −β ∗ (vs −ε) (16) , if r ≥ vs − ε. ε Moreover, βε (r)≥ρ, ∀r∈R, βε∗ satisfies (i) all over on R, βε∗ (vs )=Ks∗ and (iv) lim βε∗ (r) = +∞. r→∞

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We extend the functions Kj by Kj (vs ) at the right of vs , but since they do not depend on ε and for the sake of simplicity we shall denote this extension by K, too. Therefore, we get the approximating problem dθ (t) + Aε (t)θ(t) = f (t) + fΓu (t), a.e. t ∈ (0, T ), (17) dt θ(0) = θ0 , where Aε (t) : D(Aε (t)) ⊂ V 0 → V 0 is defined by ¶ µ ¶ ¶ µ Z µ z(x) z(x) ∗ hAε (t)z, ψiV 0 ,V := ∇βε ·∇ψ−K ·∇ψ dx, ∀ψ∈V (18) m(x, t) m(x, t) Ω and

½ D(Aε (t)) :=

µ 2

z ∈ L (Ω);

βε∗

z m(x, t)

¶

¾ ∈V

.

Equivalently to (17) we can write À µ ¶ µ ¶ ¶ Z T¿ Z µ θ θ dθ ∗ (t), ϕ(t) dt + ∇βε ·∇ϕ − K ·∇ϕ dxdt dt m m Q 0 V 0 ,V (19) Z T Z = hf (t), ϕ(t)iV 0 ,V dt − uϕdσdt, ∀ϕ ∈ L2 (0, T ; V ). 0

Σu

We also define

Z jε (r) = 0

r

βε∗ (ξ)dξ, ∀r ∈ R

(20)

and notice that ∂jε (r)=βε∗ (r) (this is in fact the Gˆateaux differential of jε ). We remark that we cannot apply Lions’ theorem for the nonautonomous case in order to prove the existence of the solution to (17) because the operator Aε (t) if were defined from V to V 0 would be not monotone. That is why we introduced it as an unbounded operator from V 0 to V 0 and we shall approach the problem in the framework of the evolution equations with m-accretive operators in Hilbert spaces. To this end we prove first a lemma that gathers some important properties of Aε (t). Lemma 2. Let Aε (t) be the operator defined by (18). (a) The domain of the operator Aε (t) is independent of t and D(Aε (t)) = D(Aε (0)) = V. (b) For each ε > 0 and t ∈ [0, T ] fixed, the operator Aε (t) is quasi m-accretive on V 0 . (c) For θ ∈ V and 0 ≤ s, t ≤ T we have kAε (t)θ − Aε (s)θkV 0 ≤ |t − s| g (kθkV 0 ) (kA(t)θkV 0 + 1), where g : [0, ∞) → [0, ∞) is a nondecreasing function.

(21)

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θ Proof. (a) If θ ∈ V then m ∈ V, since m ∈¡ C 1¡(Q)¢¢and m ¡is in¢ the bounded ∂ θ θ ∂θ 2 interval [m1 , m2 ] with m1 > 0. Next, ∂x βε∗ m = βε m ∂xj ∈ L (Ω) j because βε (r) is bounded on R for each ε > 0. Thus, V ⊂ D(Aε (t)). ∗ Conversely, we notice ¡ ¢ that by (i) theθ inverse of βε is Lipschitz with the 1 ∗ θ constant ρ , so βε m ∈ V implies m ∈ V and finally θ ∈ V, therefore D(Aε (t)) ⊂ V. (b) Let λ > 0, ε > 0 and t ∈ [0, T ] be fixed. We compute ° °2 ¡ ¢ ( I + λAε (t))θ − (I + λAε (t))θ, θ − θ V 0 = °θ − θ°V 0 ·Z µ µ ¶ µ ¶¶ µ µ ¶ µ ¶¶ ¸ θ θ θ θ ∗ ∗ + ∇ βε − βε · ∇ψ − K −K · ∇ψ dx, m m m m Ω

where −∆ψ = θ − θ, ∂ψ ∂ν on Γu and ψ = 0 on Γα . By a few computations we get ¡ ¢ (I + λAε (t)) θ − (I + λAε (t)) θ, θ − θ V 0 ° °2 ° λρ ° (22) °θ − θ°2 , ≥ (1 − λω) °θ − θ°V 0 + 2m2 2

m2 with ω = M . This shows that Aε (t) is quasi monotone for λ = λ0 2ρm21 sufficiently small, λ0 ω < 1. Next we have to prove that R(I + λAε (t)) = V 0 , i.e., to show that

vε + Aε (t)vε = g

(23)

0 has a solution vε ∈¡D(A ¢ ε (t)) for any g ∈ V . ∗ vε If we denote βε m =ζ∈V, due to the fact that βε∗ is continuous and monotonically increasing on R and R(βε∗ )=(−∞, ∞) it follows that its inverse

G(ζ) := m(βε∗ )−1 (ζ) is continuous from V to L2 (Ω) because ° ° ° ¡ ¢° °G(ζ) − G(ζ)° = °m (βε∗ )−1 (ζ) − (βε∗ )−1 (ζ) ° ° ° ° m2 ° °ζ − ζ ° ≤ m2 cΩ °ζ − ζ ° , ∀ζ, ζ ∈ V. ≤ V ρ ρ

(24)

(25)

Here we used Poincar´e inequality with the constant cΩ . So, (23) can be rewritten as G(ζ) + B0 ζ = g

(26)

0

with B0 : V → V defined by µ ¶ ¶ Z µ 1 hB0 ζ, ψiV 0 ,V := ∇ζ · ∇ψ − K G(ζ) · ∇ψ dx, ∀ψ ∈ V. m Ω

(27)

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By some computations we deduce that the operator G + B0 is continuous, monotone and coercive, so it is surjective. Detailed calculations for m constant are given in Ref. 1. In particular Aε (t) is closed. To prove (c) we calculate 2

kAε (t)θ − Aε (s)θkV 0 = hAε (t)θ − Aε (s)θ, ψiV 0 ,V , for θ ∈ V, where −∆ψ = Aε (t)θ − Aε (s)θ, ∂ψ ∂ν = 0 on Γu , ψ = 0 on Γα . We have ¶ µ ¶¸ Z ½ · µ θ θ 2 kAε (t)θ − Aε (s)θkV 0 = ∇ βε∗ − βε∗ · ∇ψ m(x, t) m(x, s) Ω · µ ¶ µ ¶¸ ¾ θ θ − K −K · ∇ψ dx m(x, t) m(x, s) ¯Z ¶ µ ¶¸ ¯ · µ ¯ ¯ θ θ ≤ ¯¯ (Aε (t)θ − Aε (s)θ) βε∗ − βε∗ dx¯¯ m(x, t) m(x, s) Ω ° ° ° θ ° θ ° +M ° ° m(x, t) − m(x, s) ° kψkV ° ° ° θ θ ° K ∗ − β ∗ (vs − ε) ° kAε (t)θ − Aε (s)θkV 0 ° − ≤ s ° m(x, t) m(x, s) ° ε V ° µ ¶° ° ° 1 1 ° +M cΩ ° °θ m(x, t) − m(x, s) ° kAε (t)θ − Aε (s)θkV 0 ° µ V ¶° ° ° 1 1 ° . − = C(ε) kAε (t)θ − Aε (s)θkV 0 ° θ ° m(x, t) m(x, s) °V 1 and Now, let us notice that by (3) it follows that the functions m(x,t) ´ ³ 1 ∇ m(x,t) are Lipschitz continuous with respect to t, uniformly with respect to x, i.e., there exist M1 and M2 positive, such that ¯ ¯ ¯ 1 1 ¯¯ ¯ (28) ¯ m(x, t) − m(x, s) ¯ ≤ M1 |t − s| , ∀x ∈ Ω,

¯ µ ¶ µ ¶¯ ¯ ¯ 1 1 ¯∇ ¯ ≤ M2 |t − s| , ∀x ∈ Ω. −∇ ¯ m(x, t) m(x, s) ¯

(29)

Taking into account (28) and (29) we deduce the estimate ° µ ¶° q ° ° 1 1 ° ≤ C0 |t − s| kθk , C0 = 2(M 2 + M 2 c2 ), °θ − 1 2 Ω V ° m(x, t) m(x, s) °V so that we get that kAε (t)θ − Aε (s)θkV 0 ≤ C0

µ

¶ Ks∗ − β ∗ (vs − ε) + M cΩ kθkV . ε

(30)

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We write θ =

207

θ mm

and still have that r ³ ° ° ´ °θ° 2 ° ° kθkV ≤ Cm ° ° , Cm = 2 m22 + cΩ kmkC 1 (Q) . m V

(31)

Also, from the inequality µ ¶ À µ ¶ µ ¶ µ ¶¶ ¿ Z µ θ θ θ θ θ ∇βε∗ = ·∇ −K ·∇ dx Aε (t)θ, m V 0 ,V m m m m Ω ° °2 ° ° ° ° µ ¶ ¯ µ ¶¯2 Z √ √ °θ° °θ° °θ° θ ¯¯ θ ¯¯ ° ° ° ° ° ° , ≥ ρ − ≥ βε ∇ dx − 3K 3K s° s° ° ° ° ¯ ¯ m m m V m V m °V Ω where Ks = max {Ksj }, we deduce that j=1,2,3

° °2 ¿ À °θ° θ ° ° ρ ° ° ≤ Aε (t)θ, m m

V 0 ,V

V

and still

° ° ° ° °θ° °θ° ° ° ° + Ks ° ° ≤ (kAε (t)θkV 0 + Ks ) ° °m° m V

√ ° ° °θ° ° ° ≤ max{1, 3Ks } (kAε (t)θk 0 + 1) . V °m° ρ V

V

(32)

Combining (30), (31) and (32) we finally obtain that kAε (t)θ − Aε (s)θkV 0 ≤ CA |t − s| (kAε (t)θkV 0 + 1), √ 3Ks }

with CA = C0 C(ε)Cm max{1,ρ

.¤

Now we can pass to the proof of the existence and uniqueness results for problem (12). 3. Main Results Theorem 3. Let m ∈ C 1 (Q), mt ∈ C 1 (Q), v0 ∈ L2 (Ω), v0 (x) ≤ vs a.e. x ∈ Ω, f ∈ L2 (0, T ; V 0 ), u ∈ L2 (0, T ; L2 (Γu )).

(33)

Then, the Cauchy problem (12) has a unique solution v ∈ C([0, T ]; L2 (Ω)) ∩ L2 (0, T ; V ), v(x, t) ≤ vs a.e. (x, t) ∈ Q mv ∈ C([0, T ]; L2 (Ω)) ∩ L2 (0, T ; V ) ∩ W 1,2 (0, T ; V 0 ), β ∗ (v) ∈ L2 (0, T ; V ).

(34)

Proof. The proof will be done in three steps and is essentially based on the quasi m-accretivity of the operator Aε (t).

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Step 1. In this part we shall transform the Cauchy problem (17) into an equivalent homogeneous problem, according to a technique developed in Ref. 17. This transformation is necessary in order to prove the existence of the solution to the approximating non-autonomous problem (17) using a result given in Ref. 16. First, let us assume that f ∈ W 1,1 (0, ∞; V 0 ), fΓu ∈ W 1,1 (0, ∞; V 0 ), v0 =

θ0 ∈ V, m0

(35)

where m0 = m(x, 0). We denote F = f + fΓu ∈ W 1,1 (0, ∞; V 0 ) and introduce the space X := V 0 × L1 (0, ∞; V 0 ) with the norm µ ¶ Z ∞ θ 2 2 ∈ X. kΘkX = kθkV 0 + kγ(s)kV 0 ds, for Θ = γ 0 Then we define the operator Aε (t) : D(Aε (t)) = D(Aε (t)) × W 1,1 (0, ∞; V 0 ) ⊂ X → X by

µ Aε (t)(w, γ) =

Aε (t)w − γ(0) −γ 0

¶ ,

(36)

for all (w, γ) ∈ D(Aε (t)) × W 1,1 (0, ∞; V 0 ). If we denote µ ¶ θ(t) P(t) = , Γ(t)

(37)

where Γ ∈ W 1,∞ (0, T ; W 1,1 (0, ∞; V 0 )) is defined by Γ(t)(s) := F (t + s), ∀s ∈ (0, ∞),

(38)

we can write the problem dP (t) + Aε (t)P(t) = 0, a.e. t ∈ (0, T ), dt µ ¶ θ0 P(0) = . F (s)

(39)

It is easily seen that the problem corresponding to the first component in (39) is exactly (17). The operator Aε (t) satisfies the same properties proved for Aε (t) in Lemma 2. Indeed, its domain is time-independent and according to the results given in Refs. 17,18 it turns out that the operator Aε (t) is quasi m-accretive on X . Finally, the property (c) follows because 2

2

kAε (t)Θ − Aε (τ )ΘkX = kAε (t)θ − Aε (τ )θkV 0 .

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Consequently, we can apply the main result in Ref. 16 and assert that under the hypotheses (35) the problem (39) has a unique strong solution in the domain of the operator Aε (t), implying that problem (17) has a unique strong solution θε ∈ C([0, T ]; V 0 )∩W 1,2 (0, T ; V 0 ) with βε∗ (θε ) ∈ L2 (0, T ; V ). The latter implies also that θε ∈ L2 (0, T ; V ). The solution satisfies the estimate ¶ ¶°2 µ Z t° µ Z ° ° ∗ θε θε ° m(x, t)jε (t) dx + βε (τ ) ° ° ° dτ mµ Ω µZ 0 V ¶ ¶ m (40) θ0 ≤4 m0 (x)jε dx + C1 + β1 exp(CL t), for t ∈ [0, T ]. m0 Ω Moreover, if we consider two solutions θε and θε corresponding to the pairs of data (v0 , u) and (v 0 , u) we have the estimate Z t ° ° ° ° °θε (t) − θε (t)°2 0 + °θε (τ ) − θε (τ )°2 dτ ≤ β2 C2 . (41) V 0

Here C1 , CL , β1 , C2 and β2 are constants independent of ε, namely ) ( Z T 2 2 2 2 kf (t)kV 0 + ctr ku(t)kL2 (Γu ) dt , C1 := 3 3Ks meas(Ω)T + 0 Z T Z T ° °2 ° °2 2 2 ° ° ° ° C2 := θ0 −θ0 V 0 + f (t)−f (t) V 0 dt+ctr ku(t)−u(t)kL2 (Γu ) dt, (42) CL :=

kmt kC 1 (Q) m1

0

0

, β1 :=

4c2Ω CL1 , ½

CL1 :=

m2 β2 := γ2 exp(α2 T ), γ2 := max 1, ρ

¾

2

kmt kC 1 (Q) m1

.

¶ µ M 2 m2 γ2 , , α2 := 1 + ρm21

(43)

and cΩ and ctr are the constant occurring in the Poincar´e inequality and trace theorem, respectively. To prove estimate (40) we need some intermediate results. Let z ∈ V 0 and m ∈ W 1,∞ (Ω). Then we have mz ∈ V 0 . Indeed, we set (mz)(ψ) := z(mψ), ∀ψ ∈ V

(44)

and this is well defined. Moreover, we can easily see that if m, mt ∈ C 1 (Q) and θε ∈ L2 (0, T ; V 0 ) then we have (in the sense of distributions) that µ ¶ d θε θε dθε =m + mt . (45) dt dt m m

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Also, we can deduce the identity µ ¶ µ ¶À Z t¿ d θε θε m dτ (τ ), βε∗ (τ ) dτ m µ 0Z V 0 ,V µ ¶ m Z ¶ θε θ0 = m(x, t)jε (t) dx − m0 (x)jε dx. m m0 Ω Ω Indeed, let us denote y = (−∞, ∞),

θε m

(46)

and define the function ϕ : L2 (Ω) × (0, T ) → Z

ϕ(y, t) :=

m(x, t)jε (y)dx.

(47)

Ω

This function is continuous with respect to y and t and ∂y ϕ(y, t) = m(x, t)∂jε (y) = m(x, t)βε∗ (y). Moreover, we have À Z t¿ dy dτ m (τ ), βε∗ (y) dτ 0 V 0 ,V À À Z t¿ Z t¿ dy dy = (τ ), m(τ )βε∗ (y) dτ = (τ ), ∂ϕ(y, τ ) dτ dτ dτ 0 0 V 0 ,V V 0 ,V Z t Z Z ∂ϕ(y, τ ) = dτ = m(x, t)jε (y) dx − m0 (x)jε (y0 ) dx. ∂τ Ω Ω 0 This proves (46). Next, we apply (19) for ϕ = θmε and obtain À µ ¶ µ ¶ Z tZ Z t¿ θε θε dθε θε ∗ , dτ + ∇βε ·∇ dxdτ dτ m m m 0 ,V 0 µ V¶ µ 0¶ Ω Z tZ θε θε ≤ K ·∇ dτ dx m 0 Ω ° °m ° ° Z t Z t ° θε ° ° θε ° ° dτ + ° ° + kf (τ )kV 0 ° (τ ) (τ ) dτ. ku(τ )k L2 (Γu ) ° °m ° m °L2 (Γu ) 0 0 V We write θε = m √t 2 m

1

θε √ √ m m

and apply (45), because

√

m ∈ C 1 (Q) and

√ d m dt

∈ C (Q). We recall that m ≥ m1 > 0. We have µ ¶ À Z t ¿p Z tZ d θε θε m θ θ √ √ ε √ t ε dxdτ m(τ ) (τ ), (τ ) dτ + dτ m m m 2 mm 0 0 Ω V 0 ,V °2 ° ¶° ° Z t° µ Z t° ° ° °° ° θε ° °K θε (τ ) ° ° θε (τ )° dτ ° (τ )° dτ ≤ +ρ ° ° °m ° ° m m °V 0 V ° °0 ° ° Z t Z t ° θε ° ° θε ° ° dτ + ctr ° dτ, ° + kf (τ )kV 0 ° (τ ) ku(τ )k (τ ) L2 (Γu ) ° °m ° m °V 0 0 V

=

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where ctr √ is the constant in the trace theorem. We recall that K is bounded |K(r)| ≤ 3Ks and after a few calculations involving also (44) we get ° ¶ À Z t¿ µ Z t° ° θε ° 2 d θε θε ° ° √ (τ ), √ (τ ) dτ + ρ ° m (τ )° dτ dτ m m 0 0 V V 0 ,V °2 ¶2 Z t° Z tZ µ ° ° ρ θ |mτ | ° θε (τ )° dτ + C1 + 1 √ε ≤ dτ dx. 2 0 ° m °V 2ρ 2 0 Ω m m Integrating the first term on the left-hand side with respect to τ we obtain ° ° °2 °2 °2 Z t° ¶2 Z µ ° ° θε ° ° ° ° θε ° √ (t)° +ρ ° θε (τ )° dτ ≤ ° √θ0 ° +C1 +CL √ dxdτ, (48) ° m0 ° ° m ° ° ° m Q 0 m V ´ ³ km k t C 1 (Q) because, due to (3) there exists a positive constant CL = such m1 that |mt | ≤ CL . (49) m By Gronwall lemma we obtain the ε-independent estimate Ã° ! °2 Z t° ° °2 °2 ° ° θε ° ° ° ° ° √ (t)° +ρ ° θε (τ )° dτ ≤ ° √θ0 ° +C1 exp(CL t), for t ∈ [0, T ]. (50) ° m ° ° ° ° m0 ° 0 m V Moreover, we notice that

so that

θ0 m0

° ° ° ° ° θ0 ° √ ° θ0 ° ° ° √ ° ≤ m2 ° ° m0 ° ° m0 °

∈ L2 (Ω) implies that

√θ0 m0

∈ L2 (Ω).

(51)

¡ ¢ To prove estimate (40) we apply (19) for ϕ=βε∗ θmε and write θε =m θmε . Using (45) and (46) we obtain after some computations that µ ¶ µ ¶ ¶°2 Z Z tZ Z ° µ ° θε θε ∗ θε 1 t° °βε∗ θε (τ ) ° dτ mjε (t) dx + βε mτ dxdτ + ° ° m m 2 0 m Ω 0 Ωm V µ ¶ Z θ0 C1 ≤ m0 jε dx + . m0 2 Ω Further we have that µ ¶ ¶°2 Z Z ° µ ° θε 1 t° °βε∗ θε (τ ) ° dτ mjε (t) dx + ° ° m 2 0 m Ω µ ¶ ¶° Z Z t ° Vµ ° ∗ θε ° θ0 1 °βε ° dτ ≤ m0 jε dx + C1 + (τ ) ° ° m0 4 0 m Ω V ¶2 2 Z tZ µ mτ θε 2 √ dxdτ, +cΩ m m 0 Ω

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where we recall that cΩ is the constant in Poincar´e inequality. Since m2τ m ≤ CL1 (defined in (42)) we still have µ ¶ ¶°2 Z t° µ Z ° ° ∗ θε θε ° ° m(x, t)jε (t) dx + °βε m (τ ) ° dτ m Ω 0 V °2 µZ µ ¶ ¶ Z t° ° ° θε θ0 ° √ (t)° dτ. ≤4 m0 (x)jε dx + C1 + 4c2Ω CL1 ° m0 m ° Ω 0 Using (50) we obtain (40) as claimed. Now, we write (19) for two pairs of data, subtract the equations and multiply the difference by θε (t)−θε (t) scalarly in V 0 . We obtain the estimate on the basis of the quasi m-accretivity of Aε (t) (see (22)) and via Gronwall lemma. Step 2. In the second step we assume that v0 ∈ L2 (Ω), v0 ≤ vs a.e. x ∈ Ω and f, fΓu ∈ L2 (0, T ; V 0 ). We can extend these functions by 0 for t ∈ (T, +∞) and we consider the sequences {fn }n≥1 ∈ W 1,1 (0, ∞; V 0 ), {fΓnu }n≥1 ∈ W 1,1 (0, ∞; V 0 ) and {v0n }n≥1 ⊂ V such that Fn = fn +fΓnu → F = f + fΓu in L2 (0, T ; V 0 ) and v0n → v0 in L2 (Ω), as n → ∞, and apply the previous step. The result follows by passing to limit as n → ∞. Moreover, we can write µ ¶ Z θ0 /m Z vs Z vs θ0 ∗ ∗ jε = βε (r)dr ≤ βε (r)dr ≤ β ∗ (r)dr = j(vs ) ≤ Ks∗ vs . m0 0 0 0 Plugging this result in (40) we obtain µ ¶ ¶°2 Z Z t° µ ° ∗ θε ° θε °βε ° dτ m(x, t)jε (t) dx + (τ ) ° ° mε m Ω 0 V ∗ ≤ 4(m2 Ks vs meas(Ω) + C1 ) + β1 exp(CL t), ∀t ∈ [0, T ],

(52)

in which the right-hand constant does not depend on ε. Using (18) we see that Aε (t)θε is bounded in L2 (0, T ; V 0 ) if θε is bounded in L2 (0, T ; V ). 2 0 ε Then, by (17) we get that dθ dt is bounded in L (0, T ; V ), Step 3. In this part we shall obtain the solution to (12) by passing to limit as ε → 0. The estimates (50) and (52) entitles us to deduce that on successive subsequences we have µ ¶ θε βε∗ → ζ weakly in L2 (0, T ; V ), as ε → 0, (53) m θε → v weakly in L2 (0, T ; V ), as ε → 0, (54) m θε → θ weakly in L2 (0, T ; V ), as ε → 0. (55)

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Since the sequence

© dθε ª dt

ε

213

is bounded in L2 (0, T ; V 0 ), we get that

dθ dθε → weakly in L2 (0, T ; V 0 ), as ε → 0. dt dt

(56)

By (54) and (55) we deduce that θ = mv.

(57)

It still follows that θ(0) = θ0 . We conclude also that {θε }ε is compact in L2 (0, T ; L2 (Ω)), i.e., θε → θ strongly in L2 (0, T ; L2 (Ω)) as ε → 0.

(58)

The weakly convergence of βε∗ (vε ) to η ∈ β ∗ (v) may be proved like in Ref. 1 using the fact that ∂j(v(x, t) = β ∗ (v(x, t)) a.e. (x, t) ∈ Q. Finally we can pass to limit as ε → 0 in (19) and obtain (13). For the uniqueness we pass to the limit as ε → 0 in (41), using the weakly l.s.c. property and we get thus the uniqueness of θ. Since v = mθ, the result is obvious. ¤ Remark. If m is space dependent only we can treat the degenerate case for m(x) vanishing in Ω0 , an open bounded subset of Ω, with K a function of the form ½ K0 (x), if (x, t) ∈ Q0 = Ω0 × (0, T ), K(x, v) = Km (v), if (x, t) = Qm , Q = Qm ∪ Q0 . In the approximating problem previously introduced we replace m by mε (x) = ε + m(x) and all results remain true for the solution to problem (17). We notice that estimate (52) does not depend on ε (because CL = 0) and we can pass to the limit as ε → 0. However, a separate analysis in the domains Q0 and Qm has to be done and the uniqueness proof requires a few other intermediate results. The degenerate case with m(x) was studied in Ref. 19 (without transport) and in Ref. 1 for other boundary conditions. The degenerate case with m function of x and t can be handled in t| an analogous way as done for that with m(x). Because in this case |m m may blow up (and CL is no longer that given by (42)) we have to assume t| expressly that there exists a positive constant CL , such that |m m ≤ CL .This m2

t| implies that mt = |mt | |m m ≤ kmt kC 1 (Q) CL := CL1 and therefore estimate (52) does not depend on ε.

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4. Numerical Results We illustrate the results previously developed by an example related to water infiltration into an isotropic, nonhomogeneous deformable soil. In this situation, water conductivity has only the vertical component K(v)i3 , so that the transport term in equation (2) reduces to ∂K(v) ∂x3 . The numerical computations have been performed with the Comsol Multiphysics 3.2 (see Ref. 20) for the dimensionless 1-D model using the following data: the domain is Ω = (0, 5), the time runs in (0, T ) with T = 1.5, c(c − 1) , for v ∈ [0, 1); β ∗ (v) = (c−1)v c−v , for v ∈ [0, 1], (c − v)2 (c − 1)v 2 K(v) = , for v ∈ [0, 1], v0 (x) = (25 − x2 )/50, u(x, t) = 0.5. c−v

β(v) =

The parameter c ∈ (1, ∞) shows the degree of nonlinearity of the soil, i.e., if c is close to 1 then the soil behaves strongly nonlinear, and if c >> 1 the soil is weakly nonlinear. In the computations we used the value c = 1.02.

Fig. 1a (contour plot m = 0.5)

Fig. 1b (contour plot m(x))

Fig. 2a (breakthrough curves)

Fig. 2b (breakthrough curves)

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Fig. 1c (contour plot m(x, t))

215

Fig. 2c (breakthrough curves)

The simulations are intended to put into evidence the way in which the variable porosity of a strongly nonlinear soil influences the flow. To enhance the comparisons of the solutions we present the results of three simulations for 2

a) m = 0.5; b) m(x) = 0.5e−(x−2.5) + 0.1; 2 2 c) m(x, t) = 0.5e−[(x−2.5) +(t−0.75) ] + 0.1. The contour plots of the solution v and the breakthrough curves at some moments of time are illustrated for m = constant = 0.5, in Fig. 1a and Fig. 2a, respectively. The contour plots are the maps of the solution values in the plane (x, t) (with the space coordinate in the horizontal axis and the time in the vertical one) and the darker areas correspond to the highest values (close to saturation) of v. The breakthrough curves represent the graphics of the solution (in the vertical axis) along Ox (in the horizontal axis) at the times t = 0 (initial distribution of v), 0.5, 1.0 and 1.5. Figs. 1b and 2b and Figs. 1c and 2c show the contour plots and the breakthrough curves at the same times as before for the cases corresponding to m function of x and m function of both x and t, respectively. We can see how the extent of the saturation subdomains enlarges with the porosity increasing. Acknowledgment. This paper was elaborated under the Contract CEEX–05-D11-36/2005, 3-14730/2005, financed by the Romanian Ministry of Education and Research. References 1. G. Marinoschi, Functional Approach to Nonlinear Models of Water Flow in Soils, (Springer, 2006). 2. S.N. Krushkov, Generalized solutions of the Cauchy problem in the large for first-order nonlinear equations, Soviet Math. Dokl., 10 (1969), 785–788.

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3. J. Carillo, On the uniqueness of the solution of the evolution dam problem, Nonlinear Analysis, 22 (1994), 573–607. 4. J. Carillo, Entropy solutions for nonlinear degenerate problems, Arch. Rational Mech. Anal., 147 (1999), 269–361. 5. J. Carillo, P. Wittbold, Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems, J. Differential Equations, 156 (1999), 93–121. 6. F. Otto, L1 -contraction and uniqueness for unstationary saturatedunsaturated porous media flow, Adv. Math. Sci. Appl., 7 (1997), 537–553. 7. H.W. Alt, S. Luckhaus, Quasi-linear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311–341. 8. E. Chasseigne, J.L. Vazquez, Theory of extended solutions for fast diffusion equations in optimal classes of data. Radiation from singularities, Archive Rat. Mech. Anal., 164 (2002), 133–187. 9. J.L. Vazquez, Darcy’s law and the theory of shrinking solutions of fast diffusion equations, SIAM J. Math. Anal., 35, 4 (2003), 1005–1028. 10. J.L. Vazquez, Symmetrization and mass comparison for degenerate nonlinear parabolic and related elliptic equations, Advanced Nonlinear Studies, 5 (2005), 87–131. 11. I. Borsi, A. Farina, A. Fasano, On the infiltration of rain water through the soil with runoff of the excess water, Nonlinear Analysis Real World Applications, 5 (2004), 763–800. 12. G. Marinoschi, A free boundary problem describing the saturated unsaturated flow in a porous medium, Abstr. Appl. Anal., 9 (2004), 729–755. 13. G. Marinoschi, A free boundary problem describing the saturated unsaturated flow in a porous medium. II. Existence of the free boundary in the 3-D case, Abstr. Appl. Anal., 8 (2005), 813–854. 14. V. Barbu, G. Marinoschi, Existence for a time dependent rainfall infiltration model with a blowing up diffusivity, Nonlinear Analysis Real World Applications, 5, 2 (2004), 231–245. 15. T. Kato, Nonlinear semi-groups and evolution equations, J. Math. Soc. Japan, 19 (1967), 508–520. 16. M.G. Crandall, A. Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math., 11 (1971), 57–94. 17. C.M Dafermos, M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, J. Functional Anal., 13 (1973), 97–106. 18. M.G. Crandall, A. Pazy, An approximation of integrable functions by step functions with an application, Proc. Amer. Math. Soc., 76, 1 (1979), 74–80. 19. A. Favini, G. Marinoschi, Existence for a degenerate diffusion problem with a potential-type nonlinear operator (preprint). 20. Comsol Multiphysics 3.2, license no. 1025226, 2005. 21. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, (Noordhoff International Publishing, Leyden, 1976). 22. P. Benilan, S.N. Krushkov: Quasilinear first-order equations with continuous nonlinearities, Russian Acad. Sci. Dokl. Math., 50, 3 (1995), 391-396.

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BACKWARD STOCHASTIC GENERALIZED VARIATIONAL INEQUALITY ˘ ¸ CANU(∗∗) LUCIAN MATICIUC(∗) and AUREL RAS (∗) Department of Mathematics, “Gheorghe Asachi” Technical University of Iasi, Bd. Carol I, no. 11, Iasi - 700506, Romania, E-mail: [email protected] (∗∗) Faculty

of Mathematics, “Alexandru Ioan Cuza” University of Iasi, Bd. Carol I, no. 9, and “Octav Mayer” Mathematics Institute of the Romanian Academy, Bd. Carol I, no. 8, Iasi - 700506, Romania, E-mail: [email protected] We prove the existence and uniqueness of a solution for the (generalized) backward stochastic variational inequality (BSVI) with random terminal time    dYt + F (t, Yt , Zt ) dt + G (t, Yt ) dAt ∈ ∂ϕ (Yt ) dt (1) (BSV I) : +∂ψ (Yt ) dAt + Zt dWt , 0 ≤ t ≤ τ,   Y = ξ, τ where (At )t≥0 is a continuous one-dimensional increasing Ft -progressively measurable process satisfying A0 = 0 and ∂ϕ and ∂ψ are subdifferentials operators. Keywords: Backward stochastic differential equations; Variational inequalities.

1. Assumptions and Results Let {(Ω, F, P) ; Ft , Wt }t≥0 be a d-dimensional standard Brownian motion (Ft is the natural filtration generated by Wt ). For λ, µ ≥ 0, τ : Ω → [0, ∞) a stopping time and {At : t ≥ 0} , A0 = 0, a continuous one-dimensional increasing progressively measurable stochastic process (p.m.s.p.), we introduce the notations: Skλ,µ [0, τ ] is the Banach space of continuous p.m.s.p. f : Ω×[0, ∞) → Rk such that · ³ ¸ ¯ ¯2 ´ 1/2 kf kS = E sup eλt+µAt ¯f (t) ¯ <∞. 0≤t≤τ

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Mkλ,µ [0, τ ] is the Hilbert space of p.m.s.p. f : Ω × [0, ∞) → Rk such that ¸ · ³Z τ ¯2 ´ 1/2 λs+µAs ¯ ¯ ¯ <∞. e f (s) ds kf kM = E 0

and that

˜ λ,µ M k

[0, τ ] is the Hilbert space of p.m.s.p. f : Ω × [0, ∞) → Rk such · ³Z kf kM˜ = E

τ

λs+µAs

e

´ ¯ ¯ ¯f (s) ¯2 dAs

¸1/2 <∞.

0

We formulate the assumptions on the BSVI (1). • Assume there exist α, β ∈ R, L ≥ 0 and η, γ : [0, ∞) × Ω → [0, ∞) p.m.s.p. such that the functions F : Ω × [0, ∞) × Rk × Rk×d → Rk and G : Ω × [0, ∞) × Rk → Rk satisfy for all t ≥ 0, y, y 0 ∈ Rk , z, z 0 ∈ Rk×d : (i)

F (·, ·, y, z) is p.m.s.p.

(ii)

y −→ F (ω, t, y, z) : Rk → Rk is continuous, a.s. ® 2 y − y 0 , F (t, y, z) − F (t, y 0 , z) ≤ α |y − y 0 | , a.s. ¯ ¯ ¯F (t, y, z) − F (t, y, z 0 ) ¯ ≤ L kz − z 0 k , a.s. ¡ ¢ |F (t, y, z)| ≤ ηt + L |y| + kzk , a.s.

(iii) (iv) (v)

(2)

and (i)

G (·, ·, y) is p.m.s.p.

(ii)

y −→ G (ω, t, y) : Rk → Rk is continuous, a.s. ® 2 y − y 0 , G (t, y) − G (t, y 0 ) ≤ β |y − y 0 | , a.s. ¯ ¯ ¯G (t, y) ¯ ≤ γt + L |y| , a.s.

(iii) (iv)

(3)

• The terminal date ξ is an Rk -valued Fτ -measurable random variable such that there exists λ > 2α + 2L2 + 1, µ > 2β + 1 : ³ ´ def 2 Γ (τ ) = Eeλτ +µAτ |ξ| + ϕ (ξ) + ψ (ξ) Z τ h i (4) 2 2 +E eλs+µAs |ηs | ds + |γs | dAs < ∞. 0 k

• The functions ϕ, ψ : R → (−∞, +∞] satisfy (i) (ii)

ϕ, ψ are proper convex l.s.c. functions, ϕ (y) ≥ ϕ (0) = 0, ψ (y) ≥ ψ (0) = 0,

The subdifferentials are defined by © ª ∂ϕ (x) = v ∈ Rk : hv, y − xi + ϕ (x) ≤ ϕ (y) , ∀y ∈ Rk

(5)

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and similar for ψ. The existence result for (1) will be obtained via Yosida approximations. Define for ε > 0 the convex C 1 -function ϕε by ½ ¾ 1 2 k ϕε (y) = inf |y − v| + ϕ (v) : v ∈ R 2ε (and similar for ψε ).

y−Jε y , then y −→ ∇ϕε (y) ε 1 2 is an montone Lipschitz function and ϕε (y) = |y − Jε y| + ϕ (Jε y) 2ε (analog for ψε ). Denoting Jε y=(I+ε∂ϕ)−1 (y) and ∇ϕε (y) =

• We introduce now the compatibility assumptions: for all ε > 0, t ∈ [0, T ], y ∈ Rk and z ∈ Rk×d ® (i) ∇ϕε (y) , ∇ψε (y) ≥ 0, ® ®+ (6) (ii) ∇ϕ (y) , G (t, y) ≤ ∇ψε (y) , G (t, y) , ε ® ®+ (iii) ∇ψε (y) , F (t, y, z) ≤ ∇ϕε (y) , F (t, y, z) , where a+ = max {0, a} . Definition 1.1. (Y, Z, U, V ) will be called a solution of BSVI (1) if satisfies ˜ λ,µ [0, τ ] , Z ∈ M λ,µ [0, τ ] , (a) Y ∈ Skλ,µ [0, τ ] ∩ Mkλ,µ [0, τ ] ∩ M k k×d ˜ λ,µ [0, τ ] , (b) U Z∈ Mkλ,µ [0, τ ] , V ∈ M k τ λs+µAs (c) E e [ϕ (Ys ) ds + ψ (Ys ) dAs ] < ∞, 0

(d) (Yt , Ut ) ∈ ∂ϕ, P (dω) ⊗ dt , (Yt , Vt ) ∈ ∂ψ, P (dω) ⊗ A (ω, dt) a.e. on Ω × [0, τ ] , Z Z Z τ

τ

(e) Yt +

Us ds + t∧τ

Z

Vs dAs = ξ + t∧τ

τ

+

Z

F (s, Ys , Zs ) ds t∧τ

τ

G (s, Ys ) dAs − t∧τ

(7)

τ

Zs dWs , for all t ≥ 0 a.s. t∧τ

In all that follows, C denotes a constant, which may depend only on µ, α, β and L, which may vary from line to line. Proposition 1.1. Let the assumptions (2), (3), (5) and (6). If (Y, Z, U, V ) ˜ U ˜ , V˜ ) are corresponding solutions to ξ and ξ˜ which satisfy (4), and (Y˜ , Z, then Z τ h i E eλs+µAs |Ys − Y˜s |2 (ds + dAs ) + ||Zs − Z˜s ||2 ds 0 i h (8) ˜2 . +E sup eλt+µAt |Yt − Y˜t |2 ≤ E eλτ +µAτ |ξ − ξ| 0≤t≤τ

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Proof. From Itˆo’s formula we have Z τ eλ(t∧τ )+µAt∧τ |Yt∧τ − Y˜t∧τ |2 + eλs+µAs |Ys − Y˜s |2 (λds + µdAs ) t∧τ Z τ Z τ ˜s ids+2 +2 eλs+µAs hYs −Y˜s , Us −U eλs+µAs hYs −Y˜s , Vs −V˜s idAs t∧τ t∧τ Z τ + eλs+µAs ||Zs − Z˜s ||2 ds t∧τ Z τ ® ˜ 2 +2 = eλτ +µAτ |ξ−ξ| eλs+µAs Ys −Y˜s , F (s, Ys , Zs ) −F (s, Y˜s , Z˜s ) ds t∧τ Z τ ® +2 eλs+µAs Ys − Y˜s , G (s, Ys ) − G(s, Y˜s ) dAs Zt∧τ τ ® −2 eλs+µAs Ys − Y˜s , (Zs − Z˜s )dWs t∧τ

˜s ids ≥ 0, hYs − Y˜s , Vs − V˜s idAs ≥ 0, Since hYs − Y˜s , Us − U ® ˜ 2 +1)|Ys −Y˜s |2 + 1 ||Zs −Z˜s ||2 2 Ys −Y˜s , F (s, Ys , Zs )−F (s, Y˜s , Z˜s ) ≤ (2˜ α+2L 2 ® and 2 Ys − Y˜s , G (s, Ys ) − G(s, Y˜s ) ≤ (2β˜ + 1)|Ys − Y˜s |2 , then (using also the Burkholder–Davis–Gundy’s inequality) the inequality (8) follows. ¤ The main result of the paper is Theorem 1.1. Let the assumptions (2)–(4), (6) be satisfied. Then there exists a unique solution (Y, Z, U, V ) for (1). 2. A Priori Estimates Consider the approximating equation Z τ Z τ Z τ Ytε + ∇ϕε (Ysε ) ds + ∇ψε (Ysε ) dAs = ξ + F (s, Ysε , Zsε ) ds t∧τZ τ t∧τ t∧τ Z τ + G (s, Ysε ) dAs − Zsε dWs , ∀t ≥ 0, P − a.s. t∧τ

(9)

t∧τ

Since ∇ϕε , ∇ψε : Rk → Rk are Lipschitz functions then, by a standard argument (Banach fixed point theorem when y → F (t, y, z) and y → G (t, y) are uniformly Lipschitz functions and Lipschitz approximations when y → αy − F (t, y, z) and y → βy − G (t, y) are continuous monotone functions) (see also Ref. 1 and 2) the equation (9) has a unique solution ¡ ¢ ˜ λ,µ [0, τ ] × M 2,µ,λ [0, τ ] . (Y ε , Z ε ) ∈ Skλ,µ [0, τ ] ∩ Mkλ,µ [0, τ ] ∩ M k k×d

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Proposition 2.1. Let the assumptions (2)–(4), (6) be satisfied. Then Z τ ¢ λs+µAs¡ λt+µAt ε 2 E sup e |Yt | +E e |Ysε |2(ds+dAs )+kZsε k2 ds ≤CΓ(τ ). (10) 0≤t≤τ

0

2

Proof. Ito’s formula for eλt+µAt |Ytε | yields Z τ Z τ 2 λ(t∧τ )+µAt∧τ ε 2 λs+µAs ε 2 e |Yt∧τ | + e |Ys | (λds+µdAs )+ eλs+µAs kZsε k ds t∧τ Z τ h t∧τ i ® ® +2 eλs+µAs Ysε , ∇ϕε (Ysε ) λds + Ysε , ∇ψε (Ysε ) µdAs t∧τ Z τ ® 2 = eλτ +µAτ |ξ| + 2 eλs+µAs Ysε , F (s, Ysε , Zsε ) ds t∧τ Z τ Z τ λs+µAs e eλs+µAs hYsε , Zsε dWs i . +2 hYsε , G(s, Ysε )idAs − 2 t∧τ

t∧τ

From Schwartz’s inequality and assumptions (2)-(4) ¯ ¯ ® 2 2 Ys , F (s, Ys , Zs ) ≤ 2α |Ys | + 2L |Ys | kZs k + 2 |Ys | ¯F (s, 0, 0) ¯ ¯ ¯2 ¡ ¢ 1 2 2 ≤ 2α + 2L2 + 1 |Ys | + kZs k + ¯F (s, 0, 0) ¯ 2 and ¯ ¯ ¯2 ® 2 2 ¯ 2 Ys , G(s, Ys , Zs ) ≤ 2β |Ys | +2 |Ys | ¯G(s, 0)¯ ≤ (2β+1) |Ys | +¯G(s, 0)¯ Hence

Z λ(t∧τ )+µAt∧τ

ε 2 |Yt∧τ |

τ

2

eλs+µAs |Ysε | (λ − 2α − 2L2 − 1)ds t∧τ Z Z τ 1 τ λs+µAs 2 2 e kZsε k ds + eλs+µAs |Ysε | (µ − 2β − 1) dAs + 2 t∧τ t∧τ Z τ ³¯ ´ ¯2 ¯ ¯2 2 ≤ eλτ +µAτ |ξ| + eλs+µAs ¯F (s, 0, 0) ¯ ds + ¯G (s, 0) ¯ dAs t∧τ Z τ λs+µAs −2 e hYsε , Zsε dWs i e

+

t∧τ

that clearly yields (for λ > 2α + 2L2 + 1 and µ > 2β + 1): Z τ ¤ λs+µAs £ 2 2 E e |Ysε | (ds + dAs ) + kZsε k ds ≤ C Γ (τ ) 0

Since, by Burkholder–Davis–Gundy’s inequality, ¯Z τ ¯ µZ τ ¶1/2 ¯ ¯ E sup ¯¯ eλs+µAs hYsε , Zsε dWs i¯¯ ≤3E e2(λs+µAs ) | hYsε , Zsε i |2 ds 0≤t≤τ

t∧τ

0

1 2 ≤ E sup eλt+µAt |Ytε | + C E 4 0≤t≤τ λt+µAt

then it follows E sup e 0≤t≤τ

Z

|Ytε |2 ≤CΓ(τ ).

τ

e 0

λs+µAs

2

kZsε k ds,

The proof is complete. ¤

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Proposition 2.2. Let the assumptions (2)–(4), (6) be satisfied. Then there exists a positive constant C such that for all stopping time θ ∈ [0, τ ]: ·Z τ ³¯ ´¸ ¯ ¯ ¯ λs+µAs ¯ ε ¯2 ε ¯2 ¯ e (a) E ∇ϕε (Ys ) ds + ∇ψε (Ys ) dAs ≤ C Γ (τ ) ·Z

0

τ

(b) E 0

´¸ ³ ¡ ¢ ¡ ¢ eλs+µAs ϕ Jε (Ysε ) ds + ψ Jˆε (Ysε ) dAs ≤ C Γ (τ )

³¯ ¯2 ¯ ¯2 ´ (c) Eeλθ+µAθ ¯Yθε − Jε (Yθε ) ¯ + ¯Yθε − Jˆε (Yθε ) ¯ ≤ ε C Γ (τ ) ,

(11)

³ ¡ ¢ ¡ ¢´ (d) Eeλθ+µAθ ϕ Jε (Yθε ) + ψ Jˆε (Yθε ) ≤ C Γ (τ ) . Proof. Essential for the proof is the stochastic subdifferential inequality introduced by Pardoux and R˘a¸scanu in Ref. 2. We will use this inequality for our purpose. First we write the subdifferential inequality eλs+µAs ϕε (Ysε ) ≥ (eλs+µAs − eλr+µAr )ϕε (Ysε ) + eλr+µAr ϕε (Yrε ) ® +eλr+µAr ∇ϕε (Yrε ) , Ysε − Yrε for s = ti+1 ∧ τ ,r = ti ∧ τ , where t = t0 < t1 < t2 < ... < t ∧ τ and 1 ti+1 − ti = , then summing up over i, and passing to the limit as n → ∞, n we deduce Z τ ® ε eλτ +µAτ ϕε (ξ) ≥ eλ(t∧τ )+µAt∧τ ϕε (Yt∧τ )+ eλs+µAs ∇ϕε (Ysε ) , dYsε t∧τ Z τ + ϕε (Ysε ) d(eλs+µAs ) t∧τ

We have the similar inequalities for the function ψε . Hence by the equation (9), we have Z τ ¯ ¯2 ¡ ¢ λ(t∧τ )+µAt∧τ ε ε e ϕε (Yt∧τ ) + ψε (Yt∧τ ) + eλs+µAs ¯∇ψε (Ysε ) ¯ dAs t∧τ Z τ Z τ ¯ ¯ ¡ ¢ λs+µAs ¯ ε ¯2 λs+µAs ϕε (Ysε ) +ψε (Ysε ) (λds+µdAs ) ∇ϕε (Ys ) ds+ e + e t∧τ t∧τ Z τ ® + eλs+µAs ∇ϕε (Ysε ) , ∇ψε (Ysε ) (ds + dAs ) t∧τ Z τ ¡ ¢ ® λτ +µAτ ≤e ϕε (ξ) + ψε (ξ) + eλs+µAs ∇ϕε (Ysε ) , F (s, Ysε , Zsε ) ds t∧τ Z τ ® λs+µAs ε + e ∇ϕε (Ys ), G(s, Ysε ) dAs t∧τ

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τ

+ Zt∧τ τ + Zt∧τ τ − t∧τ

223

® eλs+µAs ∇ψε (Ysε ) , F (s, Ysε , Zsε ) ds ® eλs+µAs ∇ψε (Ysε ) , G (s, Ysε ) dAs ® eλs+µAs ∇ϕε (Ysε ) + ∇ψε (Ysε ) , Zsε dWs

The result follows by combining this with (10), assumptions (6) and the following inequalities ¯2 ¯2 1 ¯¯ 1 ¯¯ y − Jε (y) ¯ ≤ ϕε (y) , y − Jˆε (y) ¯ ≤ ψε (y) , 2ε¡ 2ε ¢ ¡ ¢ ϕ Jε (y) ≤ ϕε (y) ≤ ϕ (y) , ψ Jˆε (y) ≤ ψε (y) ≤ ψ (y) , ¯2 ¡ ® 1¯ 2 2¢ ∇ϕε (y) , F (s, y, z) ≤ ¯∇ϕε (y) ¯ + 3 ηs2 + L2 |y| + L2 ||z|| , 4 ® ¡ 1 2 2¢ ∇ψε (y) , G (s, y) ≤ |∇ψε (y)| + 2 γs2 + L2 |y| , 4 ® ®+ ∇ψε (y) , F (s, y, z) ≤ ∇ϕε (y) , F (s, y, z) ¯2 ¡ 1¯ 2 2¢ ≤ ¯∇ϕε (y) ¯ + 3 ηs2 + L2 |y| + L2 ||z|| 4 ® ®+ ∇ϕε (y) , G (s, y) ≤ ∇ψε (y) , G (s, y) ¯2 ¡ 1¯ 2¢ ≤ ¯∇ψε (y) ¯ + 2 γs2 + L2 |y| . ¤ 4 Proposition 2.3. Let assumptions (2)–(4), (6) be satisfied. Then Z τ ¯ ° ° ¢ λt+µAt ¡ ¯ ¯Ysε − Ysδ ¯2 (ds + dAs ) + °Zsε − Zsδ °2 ds E e 0 ¯ ¯2 +E sup eλt+µAt ¯Ytε − Ytδ ¯ ≤ C (ε + δ) Γ (τ ) 0≤t≤τ

Proof. By Itˆo’s formula λ(t∧τ )+µAt∧τ

e

Z

τ

¯ ε ¯ δ ¯2 ¯Yt∧τ − Yt∧τ +

λs+µAs

Z

τ

t∧τ

¯ ¯2 eλs+µAs ¯Ysε − Ysδ ¯ (λds + µdAs )

® ε Ys − Ysδ , ∇ϕε (Ysε ) − ∇ϕδ (Ysδ ) ds

+2 e Z τ t∧τ ® +2 eλs+µAs Ysε − Ysδ , ∇ψε (Ysε ) − ∇ψδ (Ysδ ) dAs t∧τ Z τ ® =2 eλs+µAs Ysε − Ysδ , F (s, Ysε , Zsε ) − F (s, Ysδ , Zsδ ) ds Z τ t∧τ ® +2 eλs+µAs Ysε − Ysδ , G (s, Ysε ) − G(s, Ysδ ) dAs t∧τ

(12)

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Z

τ

−

e

Z ° ε ° °Zs −Zsδ °2 ds−2

λs+µAs

t∧τ

τ

® eλs+µAs Ysε −Ysδ , (Zsε −Zsδ )dWs

t∧τ

We have ® 1 2 Ysε −Ysδ, F (s,Ysε, Zsε )−F (s,Ysδ,Zsδ ) ≤(2α+2L2 )|Ysε −Ysδ |2 + kZsε −Zsδ k2 2 and ¯ ¯2 ® 2 Ysε − Ysδ , G (s, Ysε ) − G(s, Ysδ ) ≤ 2β ¯Ysε − Ysδ ¯ . Since ∇ϕε (y) ∈ ∂ϕ (Jε y) and ∂ϕ is a monotone operator then ® 0 ≤ ∇ϕε (Ysε ) − ∇ϕδ (Ysδ ), Jε (Ysε ) − Jδ (Ysδ ) ¯ ¯2 ¯ ¯2 ® = ∇ϕε (Ysε ) − ∇ϕδ (Ysδ ), Ysε − Ysδ − ε¯∇ϕε (Ysε ) ¯ − δ ¯∇ϕδ (Ysδ )¯ ® + (ε + δ) ∇ϕε (Ysε ) , ∇ϕδ (Ysδ ) . Hence ® ® ∇ϕε (Ysε ) − ∇ϕδ (Ysδ ), Ysε − Ysδ ≥ − (ε + δ) ∇ϕε (Ysε ) , ∇ϕδ (Ysδ ) and in the same manner ® ® ∇ψε (Ysε ) − ∇ψδ (Ysδ ), Ysε − Ysδ ≥ − (ε + δ) ∇ψε (Ysε ) , ∇ψδ (Ysδ ) . Consequently, Z τ ε δ 2 eλ(t∧τ )+µAt∧τ |Yt∧τ − Yt∧τ | + eλs+µAs |Ysε − Ysδ |2 (λ − 2α − 2L2 )ds t∧τ Z τ Z 1 τ λs+µAs ε + eλs+µAs |Ysε − Ysδ |2 (µ − 2β) dAs + e ||Zs − Zsδ ||2 ds 2 t∧τ t∧τ Z τ h ® ≤ 2 (ε + δ) eλs+µAs ∇ϕε (Ysε ) , ∇ϕδ (Ysδ ) ds (13) t∧τ Z τ i ® ® + ∇ψε (Ysε ) , ∇ψδ (Ysδ ) dAs − 2 eλs+µAs Ysε − Ysδ , (Zsε − Zsδ )dWs . t∧τ

Now, from (11-a) Z

τ

2 (ε + δ) E t∧τ

h ® ∇ϕε (Ysε ) , ∇ϕδ (Ysδ ) ds i ® + ∇ψε (Ysε ) , ∇ψδ (Ysδ ) dAs ≤ C (ε + δ) Γ (τ )

eλs+µAs

and clearly by standard calculus the inequality (12) follows.

¤

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3. Proof of the Main Result We give now the proof of the Theorem 3.1. Proof. Uniqueness is a consequence of Proposition 1.1. The existence of the solution (Y, Z, U, V ) is obtained as limit of the (Ysε , Zsε , ∇ϕε (Ysε ) , ∇ψε (Ysε )) . From Proposition 2.3 we have ˜ λ,µ [0, τ ], ∃ Z ∈ M λ,µ ∃ Y ∈ Skλ,µ [0, τ ] ∩ Mkλ,µ [0, τ ] ∩ M k k×d ˜ λ,µ [0, τ ], lim Z ε = Z in M λ,µ . lim Y ε = Y in Skλ,µ [0, τ ] ∩ Mkλ,µ [0, τ ] ∩ M k k×d

ε&0

ε&0

From (11-a) and (11-c) we have ˜ λ,µ [0, τ ] lim Jε (Y ε ) = Y in Mkλ,µ [0, τ ], lim Jˆε (Y ε ) = Y in M k

ε&0

and

ε&0

h¯ ¯2 ¯ ¯2 i lim E eλθ+µAθ ¯Jε (Yθε ) − Yθ ¯ + ¯Jˆε (Yθε ) − Yθ ¯ = 0,

ε&0

for any stopping time θ, 0 ≤ θ ≤ τ. Using Fatou’s Lemma, from (11-b), (11-d) and the fact that ϕ is l.s.c. we obtained (7-c). Denoting U ε = ∇ϕε (Y ε ), V ε = ∇ψε (Y ε ), from (11-a) it follows: ·Z τ ¸ ¡ ε2 ¢ λs+µAs ε 2 E e |U | ds + |V | dAs ≤ C Γ (τ ) 0

˜ λ,µ [0, τ ] such that for a Hence there exists U ∈ Mkλ,µ [0, τ ] and V ∈ M k subsequence εn & 0 U εn * U, weakly in the Hilbert space Mkλ,µ [0, τ ] ˜ λ,µ [0, τ ] V εn * V, weakly in the Hilbert space M k and then ·Z E

³ ´¸ 2 2 eλs+µAs |U | ds + |V | dAs θ ·Z τ ¸ ¡ εn 2 ¢ λs+µAs εn 2 ≤ lim inf E e |U | ds + |V | dAs ≤ C Γ (τ ) . τ

n→∞

θ

Passing now to lim in (9) we obtain (7-e). ¡ ¢ ˜ λ,µ [0, τ ] . Since ∇ϕε (Ytε ) ∈ ∂ϕ Jε (Ytε ) and Let u ∈ Mkλ,µ [0, τ ],¢ v ∈ M k ¡ ∇ψε (Ytε ) ∈ ∂ψ Jˆε (Ytε ) , ∀t ≥ 0, then as signed measures on Ω × [0, τ ] ® ¡ ¢ eλs+µAs Usε , us − Jε (Ysε ) P (dω) ⊗ ds + eλs+µAs ϕ Jε (Ysε ) P (dω) ⊗ ds ≤ eλs+µAs ϕ(us ) P (dω) ⊗ ds

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and

® eλs+µAs Vsε , vs − Jˆε (Ysε ) P (dω) ⊗ A (ω, ds) ¡ ¢ +eλs+µAs ψ Jˆε (Ysε ) P (dω) ⊗ A (ω, ds) ≤ eλs+µAs ψ(vs ) P (dω) ⊗ A (ω, ds) .

Passing to lim inf in these last two inequalities we obtain (7-d). The proof is complete. ¤ Remark 3.1. The study of the BSVI (1) is motivate, via a nonlinear Feynman-Ka¸c representation formula (see Ref. 3, by the existence problem of the viscosity solution (in the sense of Ref. 4) of parabolic variational inequality with nonlinear multivalued Neumann-Dirichlet boundary conditions of the form  d d X  ∂2u ∂u ∂u(t, x) 1 X  ∗  + (σσ )ij (t, x) + bi (t, x)   ∂t 2 ∂x ∂x ∂x  i j i  i,j=1 i=1     + f (t, x, u(t, x), (∇uσ)(t, x)) ∈ ∂ϕ(u(t, x)), (t, x) ∈ (0, T )×D,   ∂u(t, x)    + g(t, x, u(t, x)) ∈ ∂ψ(u(t, x), (t, x) ∈ (0, T ) × Bd (D) ,   ∂n     u(T, x) = h(x), x ∈ D. This applications is in our study. References 1. E. Pardoux, S. Zhang, Generalized BSDE and nonlinear Neumann boundary value problems, Probab. Theory Relat. Fields, 110 (1998), 535–558. 2. E. Pardoux, A. R˘ a¸scanu, Backward stochastic differential equations with subdifferential operator and related variational inequalities, Stochastic Processes and their Applications, 76 (1998), 191–215. 3. E. Pardoux, S. Peng, Backward SDE’s and quasilinear PDE’s, Stochastic PDE and Their Applications, B.L. Rozovski, R.B. Sowers (Eds.), Lecture Notes in Computer Science, Vol. 176, (Springer Verlag, 1992). 4. M. Crandall, H. Ishii, P.L. Lions, User’s guide to the viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1–67.

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TRICHOTOMY FOR LINEAR SKEW-PRODUCT SEMIFLOWS MIHAIL MEGAN∗

CODRUT ¸ A STOICA∗∗

LARISA BULIGA∗

∗ Department

of Mathematics, West University of Timi¸soara, Timi¸soara, Romania E-mails: [email protected], [email protected] www.uvt.ro ∗∗ Department

of Mathematics and Computer Science, ”Aurel Vlaicu” University of Arad, Arad, Romania E-mail: [email protected] www.uav.ro

In this paper we present the property of uniform trichotomy for skew-product semiflows in Banach spaces. We give several characterizations, the obtained theorems and propositions being generalizations of some well-known results on asymptotic behaviors of linear differential equations. There are also presented several examples of semiflows, cocycles and linear skew-product semiflows in Banach spaces. Keywords: Semiflow; Cocycle; Linear skew-product semiflow; Uniform exponential dichotomy; Uniform exponential trichotomy.

1. Preliminaries and Definitions In recent years, the fact that some nonlinear equations have been modelled by means of linear skew-product flows led to an important development of the respective theory, as in Ref. 1. Some questions concerning their asymptotic behavior have been answered by extending classical theorems for stability, expansiveness and dichotomy for equations on Banach sequence or function spaces in the case of cocycles in Banach spaces, as in Ref. 2 and 3. Let X be a metric space and F(X) the set of all functions f : X → X. Definition 1.1. A mapping ϕ : R+ × X → X is called a semiflow on X if it satisfies the following properties (s1 ) ϕ(0, x) = x for all x ∈ X (s2 ) ϕ(t + s, x) = ϕ(t, ϕ(s, x)) for all (t, s, x) ∈ R2+ × X.

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If ϕ is a semiflow on X then we shall denote by Sϕ the function Sϕ : R+ → F(X) defined by Sϕ (t)x = ϕ(t, x) for all (t, x) ∈ R+ × X. Example 1.1. Let V be a Banach space and let X = C(R+ , V ) be the set of all continuous functions x : R+ → V . For each x ∈ X and t ∈ R+ we define the translate xt by xt (s) = x(t + s) for all (t, s) ∈ R2+ . We will assume that X is given with the topology of uniform convergence on bounded sets. In this case X is a metric space and the mapping ϕ : R+ × X → X, ϕ(t, x) = xt is a semiflow on X = C(R+ , V ) and Sϕ is the semigroup of translations. Let X be a metric space, V a Banach space and B(V ) the space of all bounded linear operators on V . Definition 1.2. A mapping Φ : R+ × X → B(V ) is called a cocycle over the semiflow ϕ : R+ × X → X on E = R+ × X if it satisfies the following properties (c1 ) Φ(0, x) = x for all x ∈ X (c2 ) Φ(t + s, x) = Φ(t, ϕ(s, x))Φ(s, x) for all (t, s, x) ∈ R2+ × X (c3 ) there exist M ≥ 1 and ω > 0 such that kΦ(t, x)vk ≤ M eωt kvk for all (t, x, v) ∈ E × V . Definition 1.3. If Φ is a cocycle over the semiflow ϕ on E = R+ × X then the function π : R+ × X × V → X × V defined by π(t, x, v) = (ϕ(t, x), Φ(t, x)v) for all (t, x, v) ∈ R+ × X × V is called a linear skew-product semiflow on X ×V. Note that there is a one-to-one correspondence between the linear skewproduct semiflows π and the pairs (ϕ, Φ) consisting of a semiflow ϕ and a cocycle Φ over ϕ. Linear skew-product semiflows arise rather naturally in the study of differential equations as the following example shows. Example 1.2. Let ϕ : R+ ×X → X be a continuous semiflow on a compact metric space X, V a Banach space and A : X → B(V ) a continuous mapping. For all x ∈ X the mapping t 7→ A(ϕ(t, x)) defines a continuous

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and bounded B(V )-valued function. If Φ denotes the fundamental solution of the linear differential equation v(t) ˙ = A(ϕ(t, x))v(t) then the function π(t, x, v) = (ϕ(t, x), Φ(t, x)v) is a linear skew-product semiflow on X × V . 2. On Uniform Exponential Trichotomy for Linear Skew-product Semiflows Let V be a given Banach space and let X be a metric space. We define the product space E = X × V. Definition 2.1. A mapping P : E → E is said to be a projector if P is continuous and has the form P (x, v) = (x, P (x)v) where P (x) is a linear projection on Ex = {x} × V . This means that P (x) : Ex → Ex is a bounded linear mapping that satisfies P (x)P (x) = P 2 (x) = P (x) for all x ∈ X. If P is a projector on E then the mapping Q : E → E given by Q(x, v) = (x, v − P (x)v) is also a projector on E, called the complementary projector to P . Definition 2.2. A projector P on E is said to be invariant if one has P (ϕ(t, x))Φ(t, x) = Φ(t, x)P (x) for all t ≥ 0 and x ∈ X. Remark 2.1. If the projector P is invariant, then the projector Q is invariant as well. Let π = (ϕ, Φ) be a linear skew-product semiflow on E = X × V . If λ ∈ R then we denote by Φλ the mapping Φλ : R+ × X → X defined by Φλ (t, x) = e−λt Φ(t, x). One refers to πλ = (ϕ, Φλ ) as the shifted linear skew-product semiflow.

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The property of uniform exponential dichotomy is defined on similar basis as in Refs. 2,4 and 5. Definition 2.3. The linear skew-product semiflow π = (ϕ, Φ) has uniform exponential dichotomy on E with characteristics N and α, β, where N ≥ 1 and α < 0 < β, if there exist two projectors P and Q defined over E such that (ued1 ) the projectors P and Q are invariant on E (ued2 ) for each x ∈ X the projections P (x) and Q(x) commute and P (x) + Q(x) = I and P (x)Q(x) = 0 for all x ∈ X (ued3 ) the following inequalities hold for all t ≥ 0, x ∈ X and v ∈ V kΦ(t, x)P (x)vk ≤ N eαt kP (x)vk

(1)

kQ(x)vk ≤ N e−βt kΦ(t, x)Q(x)vk .

(2)

As a natural generalization we can consider the following definition. Definition 2.4. We say that π = (ϕ, Φ) has uniform exponential trichotomy on E with characteristics N and ν1 , ν2 , ν3 , ν4 , where N ≥ 1 and ν1 < ν2 ≤ 0 ≤ ν3 < ν4 , if there exist three projectors P , Q and R defined over E such that the following properties hold (uet1 ) each of the projectors P , Q and R is invariant on E (uet2 ) for each x ∈ X the projections P (x), Q(x) and R(x) commute and one has P (x) + Q(x) + R(x) = I and P (x)Q(x) = P (x)R(x) = Q(x)R(x) = 0 for all x ∈ X (uet3 ) the following four inequalities hold for all t ≥ 0, x ∈ X and v ∈ V kΦ(t, x)P (x)vk ≤ N kP (x)vk eν1 t

(3)

kQ(x)vk ≤ N kΦ(t, x)Q(x)vk e−ν4 t

(4)

kΦ(t, x)R(x)vk ≤ N kR(x)vk eν3 t

(5)

kR(x)vk ≤ N kΦ(t, x)R(x)vk e−ν2 t

(6)

Remark 2.2. For R = 0 we obtain the property of uniform exponential dichotomy.

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Example 2.1. Let us consider V = R3 with the norm k(v1 , v2 , v3 )k = |v1 | + |v2 | + |v3 |. We denote C = C(R+ , R+ ) the set of all continuous functions x : R+ → R+ . We consider that C is given with the uniform convergence topology on compact subsets of R+ . C is metrizable with respect to the metric d(x, y) =

∞ X 1 dn (x, y) 2n 1 + dn (x, y) n=1

where dn (x, y) = sup |x(t) − y(t)|. t∈[0,n]

If x ∈ C then for every t ∈ R+ we denote by xt ∈ C the function xt (s) = x(t + s). Let us consider X = {ft , t ∈ R+ }, where f : R+ → R∗+ is a decreasing function with lim f (t) = α > 0. t→∞

Then (X, d) is a metric space and ϕ : R+ × X → X, ϕ(t, x)(s) = x(t + s) is a semiflow on X. Then Φ : R+ × X → B(V ), given by Φ(t, x)(v1 , v2 , v3 ) = (e−2tf (0)+

Rt 0

x(s)ds

v1 , e

Rt 0

x(s)ds

v2 , e−tf (0)+2

Rt 0

x(s)ds

v3 )

is a cocycle on X × V . We consider the projections P (x)(v1 , v2 , v3 ) = (v1 , 0, 0), Q(x)(v1 , v2 , v3 ) = (0, v2 , 0), R(x)(v1 , v2 , v3 ) = (0, 0, v3 ). Following relations hold kΦ(t, x)P (x)vk ≤ e−tf (0) kP (x)vk kΦ(t, x)Q(x)vk ≥ eαt kQ(x)vk e−tf (0) kR(x)vk ≤ kΦ(t, x)R(x)vk ≤ etf (0) kR(x)vk which proves that the linear skew-product semiflow π = (ϕ, Φ) is uniformly exponentially trichotomic.

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Proposition 2.1. A linear skew-product semiflow π is uniformly exponentially trichotomic if and only if there exist three projectors P1 , P2 and P3 over E with the properties (i) each of the projectors P1 , P2 and P3 is invariant on E (ii) for each x ∈ X the projections P1 (x), P2 (x) and P3 (x) commute and one has P1 (x) + P2 (x) + P3 (x) = I and Pi (x)Pj (x) = 0 for all x ∈ X and all i, j ∈ {1, 2, 3}, i 6= j, two nondecreasing functions f, g : [0, ∞) → [1, ∞) with lim f (t) = lim g(t) = ∞

t→∞

t→∞

such that (uet01 ) f (t) kΦ(t, x)P1 (x)vk ≤ kP1 (x)vk (uet02 ) f (t) kP2 (x)vk ≤ kΦ(t, x)P2 (x)vk (uet03 ) kP3 (x)vk ≤ g(t) kΦ(t, x)P3 (x)vk ≤ g 2 (t) kP3 (x)vk for all (t, x) ∈ R+ × X and all v ∈ V. Proof. Necessity. As π is uniformly exponentially trichotomic with characteristics given by Definition 2.4, we can consider function f (t) = N −1 eνt , where ν = min{−ν1 , ν4 } and function g(t) = N eµt , where µ = ν3 − ν2 . Sufficiency. Let us define P = P1 , Q = P2 and R = P3 . To prove the first inequality of (uet3 ) let us denote n = [t] , t ≥ 0. Then there exist s ∈ N and r ∈ [0, s) such that t = ns + r. For all x ∈ X and v ∈ V , we obtain successively kΦ(t, x)P (x)vk = kΦ(ns + r, x)P (x)vk ≤ kΦ(r, ϕ(ns, x))Φ(ns, x)P (x)vk ≤ ≤ (f (r))−1 kΦ(ns, x)P (x)vk ≤ ... ≤ N −1 e−ν1 t kP (x)vk where we have denoted N = f (s)(f (r))−1 > 1, ν1 = ln f (s) > 0. Similarly can be proved the other three inequalities.

¤

Definition 2.5. If the application (t, x) → Φ(t, x)v is continuous for every v ∈ V then π = (ϕ, Φ) is called a strongly continuous linear skew-product semiflow on E. We emphasize some connections between the asymptotic behaviors of a linear skew-product semiflow and the shifted semiflow (see Ref. 5 and 6).

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Lemma 2.1. A linear skew-product semiflow π = (ϕ, Φ) has uniform exponential growth if and only if there exists α > 0 such that the shifted linear skew-product semiflow πα = (ϕ, Φα ) is uniformly exponentially stable. Proof. Necessity. If π has uniform exponential growth then there exist M ≥ 1, ν > 0 such that kΦ(t, x)vk ≤ M eν(t−s) kΦ(s, x)vk for all t ≥ s ≥ 0 and x ∈ X, v ∈ V . If we consider α = 2ν > 0 we obtain kΦα (t, x)vk ≤ M e−ν(t−s) kΦα (s, x)vk for all t ≥ s ≥ 0, x ∈ X, v ∈ V and hence πα is uniformly exponentially stable. Sufficiency. If there exists α > 0 such that πα = (ϕ, Φα ) is uniformly exponentially stable then there exist N > 1 and ν > 0 such that kΦ(t, x)vk = eαt kΦα (t, x)vk ≤ N e(α−ν)(t−s) kΦ(s, x)vk for all t ≥ s ≥ 0, x ∈ X and v ∈ V where we have denoted ½ α − ν, if α > ν ω= 1, if α ≤ ν We obtain that π has uniform exponential growth.

¤

Lemma 2.2. A linear skew-product semiflow π = (ϕ, Φ) has uniform exponential decay if and only if there exists α > 0 such that the shifted linear skew-product semiflow π−α = (ϕ, Φ−α ) is uniformly exponentially instable. Proof. The proof is similar with the one of Lemma 2.1.

¤

The next result gives an integral characterization for the property of uniform exponential trichotomy for linear skew-product semiflows. Theorem 2.1. Let π = (ϕ, Φ) be a strongly continuous linear skew-product semiflow on E and let P1 , P2 and P3 be three projectors with the properties (i) each of the projectors P1 , P2 and P3 is invariant on E (ii) for each x ∈ X the projections P1 (x), P2 (x) and P3 (x) commute and one has P1 (x) + P2 (x) + P3 (x) = I and Pi (x)Pj (x) = 0

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for all x ∈ X and all i, j ∈ {1, 2, 3}, i 6= j and with the property that there exist t0 > 0 and c ∈ (0, 1) such that kΦ(t0 , x)P1 (x)vk ≤ c kP1 (x)vk

(7)

kP2 (x)vk ≤ c kΦ(t0 , x)P2 (x)vk

(8)

for all x ∈ X and all v ∈ V. Then the linear skew-product semiflow is uniformly exponentially trichotomic if and only if (uet001 ) there exists M1 > 0 such that Z ∞ kΦ(τ, x)P1 (x)vk dτ ≤ M1 kΦ(t, x)P1 (x)vk t

for all (t, x) ∈ R+ × X and all v ∈ V (uet002 ) there exists M2 > 0 such that Z t kΦ(τ, x)P2 (x)vk dτ ≤ M2 kΦ(t, x)P2 (x)vk 0

for all (t, x) ∈ R+ × X and all v ∈ V (uet003 ) there exist M > 0 and α > 0 such that Z t e−α(τ −s) kΦ(τ, x)P3 (x)vk dτ ≤ M kΦ(s, x)P3 (x)vk s

for all (t, s, x) ∈ R+ 2 × X, t ≥ s and all v ∈ V (uet004 ) there exist M > 0 and α > 0 such that Z t eα(t−τ ) kΦ(τ, x)P3 (x)vk dτ ≤ M kΦ(t, x)P3 (x)vk s

for all (t, s, x) ∈ R+ 2 × X, t ≥ s and all v ∈ V. Proof. Necessity. Let us define P1 = P , P2 = Q and P3 = R, where projectors P , Q and R are given by Definition 2.4. We also denote by N and ν1 , ν2 , ν3 , ν4 the characteristics of the uniformly exponentially trichotomic linear skew-product semiflow π. In order to prove (uet001 ), it is obvious that the integral inequality hold for M1 = −N ν1−1 . For (uet002 ) we consider M2 = N ν4−1 . It is immediate to prove (uet003 ) and (uet004 ) if we define the constants α = min{2ν3 , −2ν2 } and M = max{N ν3−1 , −N ν2−1 }.

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Sufficiency. As in Ref. 3, from the property that there exist t0 > 0 and c ∈ (0, 1) such that relation (7) hold it follows that kΦ(t, x)P1 (x)vk ≤ N1 eν1 kP1 (x)vk for all (t, x, v) ∈ E × V , where ν1 = lnt0c , N1 = M e(ω+ν1 )t0 and constants M and ω are given by statement (c3 ) of Definition 1.2. From the fact that there exist t0 > 0 and c ∈ (0, 1) such that inequality (8) hold it follows that kP2 (x)vk ≤ N2 e−ν4 t kΦ(t, x)P2 (x)vk for all (t, x, v) ∈ E × V , where ν4 = − lnt0c and N2 = M eωt0 . According to Ref. 5, statements (uet003 ) and (uet004 ) are integral characterizations of the properties of uniform exponential stability respectively instability for shifted semiflows through projector P3 , which leads, by means of Lemma 2.1 and Lemma 2.2, to the properties of uniform exponential growth and decay, as following relations show kP3 (x)vk ≤ N e−ν2 t kΦ(t, x)P3 (x)vk kΦ(t, x)P3 (x)vk ≤ N eν3 t kP3 (x)vk for all (t, x, v) ∈ E × V , where ν2 ≤ 0 ≤ ν3 and N > 1. Hence, the uniform exponential trichotomy for π is proved.

¤

In order to characterize the trichotomy by means of two dichotomies, we consider three invariant projectors P , Q and R such that P (x) + Q(x) + R(x) = I, P (x)Q(x) = Q(x)R(x) = P (x)R(x) = 0 and 2

2

2

2

2

2

kP (x)v + R(x)vk = kP (x)vk + kR(x)vk , kQ(x)v + R(x)vk = kQ(x)vk + kR(x)vk for all x ∈ X and all v ∈ V . Proposition 2.2. The linear skew-product semiflow π = (ϕ, Φ) has uniform exponential trichotomy with projectors P , Q, R and characteristics ν1 , ν2 , ν3 , ν4 and N if and only if (uet000 1 ) for all ν ∈ (ν1 , ν2 ), πν has uniform exponential dichotomy with characteristics (N, αν ) and associated projectors (Pν , Qν ) where αν = min{ν − ν1 , ν2 − ν}, Pν = P and Qν = Q + R

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(uet000 2 ) for all µ ∈ (ν3 , ν4 ), πµ has uniform exponential dichotomy with characteristics (N, αµ ) and associated projectors (Pµ , Qµ ) where αµ = min{µ − ν3 , ν4 − µ}, Pµ = P + R and Qµ = Q Proof. Necessity. To prove (uet000 1 ) we consider relations (3), (4) and (6) of Definition 2.4 as well as Lemma 2.1, with the remark that, without any loss of generality, we can consider ν4 = −ν1 . For ν ∈ (ν1 , ν2 ) we have kΦν (t, x)P (x)vk = e−νt kΦ(t, x)P (x)vk ≤ N e−(ν−ν1 )t kP (x)vk for all (t, x, v) ∈ E × V . Also 2

2

k[Q(x) + R(x)]vk ≤ N 2 e(2ν1 +2ν)t kΦν (t, x)Q(x)vk + 2

+N 2 e(−2ν2 +2ν)t kΦν (t, x)R(x)vk ≤ N 2 e−2(ν2 +ν)t kΦν [Q(x) + R(x)]vk . We have proved the uniform exponential dichotomy for πν with characteristics (N, αν ) where αν = min{ν − ν1 , ν2 − ν}. The proof of (uet000 2 ) is similar. Sufficiency. It follows by means of Definition 2.3, Lemma 2.1 and Lemma 2.2. ¤ References 1. S.N. Chow, H. Leiva, Existence and roughness of the exponential dichotomy for linear skew-product semiflows in Banach spaces, J. Diff. Equations, 120 (1995), 429–477. 2. M. Megan, A.L. Sasu, B. Sasu, Uniform exponential dichotomy and admissibility for linear skew-product semiflows, Operator Theory, Advances and Applications, 153 (2004), 185–195. 3. M. Megan, A.L. Sasu, B. Sasu, Exponential stability and exponential instability for linear skew-product flows, Mathematica Bohemica, 129, 3 (2004), 225–243. 4. M. Megan, C. Stoica, On Uniform Exponential Trichotomy of Evolution Operators in Banach Spaces, Preprint Series in Mathematics, no. 2, (West University of Timi¸soara, 2006), 1–13. 5. M. Megan, C. Stoica, L. Buliga, On Asymptotic Behaviors of Evolution Operators in Banach Spaces, Preprint Series in Mathematics, no. 5, (West University of Timi¸soara, 2006), 1–22. 6. C. Stoica, M. Megan, Uniform Exponential Instability of Evolution Operators in Banach Spaces, Preprint Series in Mathematics, no. 7, (West University of Timi¸soara, 2006), 1–6.

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FEEDBACK CONTROL OF CONSTRAINED PARABOLIC SYSTEMS IN UNCERTAINTY CONDITIONS VIA ASYMMETRIC GAMES BORIS S. MORDUKHOVICH∗ Department of Mathematics, Wayne State University Detroit, Michigan 48202, USA; [email protected] THOMAS I. SEIDMAN Department of Mathematics and Statistics, University of Maryland Baltimore County Baltimore, MD 21250, USA; [email protected] This paper concerns control problems for multidimensional linear parabolic equations subject to hard/pointwise constraints on both boundary controls and state dynamic/output functions in the presence of uncertain perturbations within given regions. Such problems are formalized as minimax problems of optimal control, where the control strategy is sought as a feedback law depending on the current state position. Problems of this type are among the most important while the most challenging and difficult in control theory and applications. Based on the Maximum Principle in the theory of parabolic equations and on time convolutions in the theory of Fourier transforms, we reduce the problems under consideration to certain asymmetric games. This allows us to discover significant properties of feasible and optimal feedback controls for constrained parabolic systems. Keywords: Feedback control; Parabolic systems; Minimax synthesis; Pointwise constraints; Convolutions; Asymmetric games.

1. Introduction This paper concerns feedback control design of state-constrained linear parabolic systems functioning under uncertain disturbances/perturbations. The original motivation came from some practical applications to automatic control of the groundwater regime in irrigation networks, where the ∗ Research

of this author was partially supported by the USA National Science Foundation under grants DMS-0304989 and DMS-0603846 and by the Australian Research Council under grant DP-0451168.

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main objective was to neutralize the adverse effect of uncertain weather and environmental conditions; see Ref. 1. Problems of this type are formalized as the minimize design of parabolic systems, which unavoidably requires the usage of feedback control inputs acting through boundary conditions; see Refs. 1,2 for the initial descriptions via the one-dimensional heat equation. Among the most important specific features of control systems modeled in Refs. 1,2 in order to meet practical requirements we mention the following: • distributed uncertain perturbations taking values within given closed areas with only the bounds assumed to be known; • pointwise/hard constraints of the magnitude type on control functions acting through the Dirichlet boundary conditions, which offer the least regularity properties for the linear parabolic dynamics; • pointwise state constraints (of the magnitude type) on the observed motions; • infinite horizon of dynamic processes. It has been well recognized that problems of this type are among the most challenging and difficult in control theory being at the same time among the most important for applications. To the best of our knowledge, a variety of approaches and results developed in the theories of differential games, H∞ -control, and Riccati’s feedback synthesis are not applicable to such problems; see, e.g., Refs. 3–6 and also Refs. 2,7,8 with the discussions and references therein. The approach developed in Ref. 2 for the case of one-dimensional heat/diffusion equations and then partly extended in Refs. 7,9,10 to multidimensional settings mainly concerns the system reaction to the extreme perturbations, which are well justified to be the worst in the original environmental situation Refs. 1,2. In this way the structure and parameters of control functions are computed by using the Pontryagin maximum principle (Ref. 11) for ODE approximating systems of optimal control, with the further adjustment to the parabolic dynamics and the exclusion of unstable vibrations. In this paper we suggest another approach to the minimax synthesis of (hard) constrained parabolic systems, which is based on their reduction to asymmetric games by using time convolutions. This approach is based on certain fundamental properties of the linear parabolic dynamics (notably on the Maximum Principle for parabolic equations) and allows us to clarify important characteristics of feasible and optimal feedback controls and

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perturbations in the problems under consideration. The rest of the paper is organized as follows. In Section 2 we formulate and discuss the basic minimax problem of feedback control for constrained parabolic systems, with imposing the standing assumptions. Section 3 concerns some new underlying properties of linear parabolic equations with irregular Dirichlet boundary controls and distributed perturbations; these properties are important for understanding the subsequent results while are certainly of a broader independent interest. In Section 4 we describe the convolution approach to the original minimax problem and discuss its possible extensions to a broader class of parabolic control systems. Finally, Section 5 is devoted to the study of the asymmetric game appearing in this approach, which is again of an independent interest while allowing us to clarify some important structural properties of feedback controls and find efficient conditions for controllability and non-controllability in the original problem — the latter means that feasible feedback control regulators may not exist in some situations.

2. Problem Formulation and Discussion Let Ω ⊂ IRn be an open bounded domain with the closure cl Ω and the boundary ∂Ω, which is assumed to be a sufficiently smooth (n − 1)dimensional manifold. Define a self-adjoint and uniformly strongly elliptic operator A on L2 (Ω) by

A := −

n X i,j=1

aij (x)

∂2 − c, ∂xi ∂xj

(1)

where c ∈ IR and where the functions aij : cl Ω → IR satisfy the properties: aij ∈ C ∞ (clΩ), aij (x) = aji (x) for all x ∈ Ω, n X i,j=1

aij (x)ξi ξj ≥ ν

n X

ξi2

with some ν > 0

i=1

whenever x ∈ Ω and ξ = (ξ1 , ..., ξn ) ∈ IRn .

i, j = 1, ..., n, (2)

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Fix the final time T > 0 and consider the parabolic control system  ∂y   + Ay = w(t) a.e. in Q := [0, T ] × Ω,    ∂t (3) y(0, x) = 0, x ∈ Ω,      y(t, x) = u(t), (t, x) ∈ Σ := [0, T ] × ∂Ω, with controls u(·) acting in the Dirichlet boundary conditions and distributed perturbations w(·) on the right-hand side of the parabolic equation. Define the sets of admissible controls u(·) and admissible perturbations w(·) by, respectively, ¯ © ª Uad := u ∈ L2 (0, T )¯ u(t) ∈ [−¯ α, α] a.e. t ∈ [0, T ] , (4) ¯ © ª ¯ a.e. t ∈ [0, T ] , Wad := w ∈ L2 (0, T )¯ w(t) ∈ [−β, β]

(5)

¯ β are given. Note that control where the positive numbers α ¯ , α and β, and perturbation functions look similarly via (4) and (5) — except they are situated in the different parts of the parabolic system (3) — while their roles in the minimax control problem formulated below are completely opposite. To formulate this problem, we first observe that for every (u, w) ∈ L2 (0, T )×L2 (0, T ) the parabolic system (3) admits a unique generalized solution y ∈ L2 (Q); see the next section for more details and representations. Then fix a point x0 ∈ Ω at which we are able to collect information about the system performance, and let η > 0 be an assigned number to bound feasible system motions. The basic minimax feedback control problem (P ) under consideration in this paper is as follows: nZ T ¯ ¯ o ¯u(y(t, x))¯ dt minimize the cost J(u) := max (6) w(·)∈Wad

0

over feasible controls u ∈ Uad formed via the feedback law ¡ ¢ u(t) = u y(t, x0 )

(7)

depending on feasible trajectories to the parabolic system (3) that are generated by u(t) through the Dirichlet boundary conditions and satisfy the pointwise state constraints |y(t, x0 )| ≤ η for all t ∈ [0, T ]

(8)

whatever perturbation w ∈ Wad is taken in (3). Let us emphasize that perturbations w(·) in (3) are uncertain: the only information available for them is the bounds β and β in (5). The main objective of feasible feedback controls u = u(y) in (7) constrained by (4) is to

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keep the system motion y(t) = y(t, x0 ) at the assigned point of observation x = x0 ∈ Ω within the prescribed area (8) for any feasible perturbations w ∈ Wad . Furthermore, the optimization in (P ) is understood in the minimax sense: to give the minimum value to the cost functional (6) in the case of the worst perturbations w(·) maximizing the integral functional in (6) subject to all the constraints imposed. Note that the optimal solution of the minimax problem (P ) is naturally to understand as a saddle point (¯ u, w) ¯ ∈ Uad × Wad , e.g., as a couple of the interrelated worst perturbation and optimal control; see Ref. 8. A major complication in the case of problem (P ) under consideration comes from the feedback structure of the control action (7) depending on the current position of the observed dynamics y(t) = y(t, x0 ). To proceed further, we study in the next section certain fundamental properties of the linear parabolic dynamics important in what follows as well as for their own sake. 3. Monotonicity of the Parabolic Dynamics In this section we give an explicit representation of the appropriate solutions (transients) y = y(t, x) of the parabolic system (3) corresponding to (u, w) ∈ L2 (0, T ) × L2 (0, T ) and on this base establish the underlying monotonicity property of transients with respect to both controls and perturbations, which is a crucial manifestation of the parabolic dynamics under the assumptions made. First observe that the input data (u, w) ∈ Uad × Wad are just measurable, i.e., they are irregular to ensure the existence of the classical smooth solutions y to (3). Nevertheless, for all (u, w) ∈ Uad × Wad the parabolic system (3) admits a unique generalized solution y = y(t, x) ∈ L2 (Q); this is proved, e.g., in Ref. 12. Let us establish an efficient representation of such generalized solutions to (3) via the Fourier series, which is important in what follows. Given the operator A in (1), consider the homogeneous boundary value problem ½ −Aϕ + λϕ = 0, (9) ϕ|∂Ω = 0 and recall that the number component λ in the nontrivial pair (λ, ϕ) satisfying (9) is an eigenvalue, while ϕ is the corresponding eigenfunction for the operator A under the Dirichlet boundary condition. According to Ref. 13, Theorems 8.37, 8.38, the assumptions in (2) imposed on the operator A ensure the fulfillment of the following properties:

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(a) The eigenvalues λi , i = 1, 2, ..., are real and form a nondecreasing sequence, which accumulates only at ∞; (b) The corresponding orthonormal system {ϕi (x)} ⊂ C ∞ (Ω) of eigenfunctions is complete in L2 (Ω). (c) The first eigenvalue λ1 is simple and has the positive eigenfunction ϕ1 (x). The next representation of transients to system (3) is essentially related to given setting, where both controls u = u(t) and perturbations w = w(t) in (3) are spatially constant, i.e., independent of the space variable. Theorem 3.1. (transient representation for linear parabolic systems with Dirichlet boundary conditions). Let (u, w) ∈ L2 (0, T ) × L2 (0, T ) in (3) under the assumptions (2) on the elliptic operator A, and let (λi , ϕi ) be the corresponding eigenvalues and eigenfunctions of A with the weights Z µi := ϕi (x) dx, i = 1, 2, .... Ω

Then the unique solution y ∈ L2 (Q) to (3) admits the representation y(t, x) =

∞ X i=1

³Z

t

Z

w(θ)eλi θ dθ + (c + λi )

µi 0

´ u(θ)eλi θ dθ e−λi t ϕi (x), (10)

t

0

where the series in (10) strongly converges in the space L2 (Q). Proof. Let D(0, T ) be the space of C ∞ -functions on (0, T ) with compact supports. Since D(0, T ) is dense in L2 (0, T ), we pick sequences {uk } and {wk } from D(0, T ) such that (uk , wk ) → (u, w) strongly in L2 (0, T ) × L2 (0, T ) as k → ∞.

(11)

By Ref. 14, system (2.2) has a unique classical solution yk for each (uk , wk ) ∈ D(0, T ) × D(0, T ). As mentioned, for each (u, w) ∈ L2 (0, T ) × L2 (0, T ) there is a unique generalized solution y ∈ L2 (Q). Moreover, the linear operator (u, w) → y from L2 (0, T ) × L2 (0, T ) into L2 (Q) is continuous by Ref. 12. Combining this with the strong convergence in (11), we get yk → y strongly in L2 (Q) as k → ∞.

(12)

Consider now the sequence {(uk , wk )} from (11) and denote hk :=yk −uk for each k = 1, 2, .... We can easily conclude that hk is the unique classical

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solution to the system  ∂h  0    ∂t + Ah = wk (t) − uk (t) + cuk (t), h(0, x) = 0, x ∈ Ω,     h(t, x) = 0, (t, x) ∈ Σ, with the homogeneous Dirichlet boundary conditions. It is well known (see, e.g., Ref. 14) that hk admits the representation Z t ∞ ³Z t ´ X λi θ hk (t, x)= µi wk (θ)e dθ+(c+λi ) uk (θ)eλi θ dθ e−λi t ϕi (x) i=1

−uk (t)

0 ∞ X

0

(13)

µi ϕi (x) for each k = 1, 2....

i=1

Since one obviously has hk + uk = yk for each k ∈ IN and since ∞ X

µi ϕi (x) = 1

i=1

with the strong convergence in L2 (Ω) (by the orthonormality property of {ϕi (x)} in L2 (Ω) and the construction of µi in the theorem), we conclude that the triple (uk , wk , yk ) satisfies (10) for all k = 1, 2, .... Passing there to the limit as k → ∞ with taking (11) and (12) into account, we arrive at the limiting representation (10) for the reference triple (u, w, y) ∈ L2 (0, T ) × L2 (0, T ) × L2 (Q) and thus conclude the proof of the theorem. ¤ It is well known (see, e.g., Ref. 14) that classical solutions to the parabolic system (3) satisfy the Maximum Principle, which is a fundamental result of the parabolic dynamics. Based on this result and on a certain smooth approximation technique, we establish next the principal monotonicity property of generalized solutions (in the sense above) to (3) with respect to both controls and perturbations (u, w) ∈ L2 (0, T ) × L2 (0, T ). Theorem 3.2. (monotonicity property of transients). Let y1 and y2 be the generalized solutions to the parabolic system (3) corresponding to the pairs (u1 , w1 ) ∈ L2 (0, T ) × L2 (0, T ) and (u2 , w2 ) ∈ L2 (0, T ) × L2 (0, T ), respectively. If u1 (t) ≥ u2 (t) and w1 (t) ≥ w2 (t) a.e. in (0, T ), then we have y1 (t, x) ≥ y2 (t, x) a.e. in Q.

(14)

Proof. Given the triples (u1 , w1 , y1 ) and (u2 , w2 , y2 ) in the theorem, denote u(t) := u1 (t) − u2 (t), w(t) := w1 (t) − w2 (t), y(t, x) := y1 (t, x) − y2 (t, x).

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By linearity of (3), y ∈ L2 (Q) is the generalized solution to (3) corresponding to the nonnegative L2 -inputs u(t) ≥ 0 and w(t) ≥ 0 for a.e. t ∈ [0, T ].

(15)

Furthermore, the triple (u, w, y) satisfies the Fourier representation (10) by Theorem 3.1. To justify (14), we need to show that y(t, x) ≥ 0 for a.e. t ∈ [0, T ]. We are going to do it by using the Maximum Principle for classical solutions to parabolic equations and the following smooth approximation procedure. Let ρ(r) be a C ∞ -function on IR having the properties: (a)

ρ(r) = 0

if |r| ≥ 1,

(b)

ρ(r) ≥ 0 Z ρ(r)dr = 1.

if |r| ≤ 1,

(c)

IR

For each ε > 0 and v ∈ L2 (0, T ), define the function Z ³ 1 t − r´ vε (t) = ρ v(r) dr. ε IR ε

(16)

Observe by Ref. 14, pp. 84–85, that vε is a C ∞ -function whose support belongs to an ε−neighborhood of the support for v and that Z T Z T 2 kvε (t)k dt ≤ kv(t)k2 dt for all ε ≥ 0, 0 0 (17) Z T

kvε (t) − v(t)k2 dt → 0 as ε ↓ 0.

0

Furthermore, vε (t) ≥ 0 for all t ∈ [0, T ] whenever v(t) ≥ 0 for a.e. t ∈ [0, T ]. Consider the initial-boundary value problem  ∂y     ∂t + Ay = wε (t) in Q, (18) y(0, x) = 0, x ∈ Ω,     y(t, x) = u (t), (t, x) ∈ Σ, ε

where uε and wε are the smooth approximations built by (16) upon the given functions u(t) and w(t), respectively, satisfying (15). As mentioned, relationships (15) and (16) imply that uε (t) ≥ 0 and wε (t) ≥ 0 for a.e. t ∈ [0, T ] and all ε > 0, and that the corresponding ¡ ¢problem (18) admits the unique classical solution yε ∈ C 1,2 [0, T ] × cl Ω generated by the

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smooth input (uε , wε ). Applying now the Maximum Principle for parabolic equations (see, e.g., Ref. 14, Theorem 2.1), we get yε (t, x) ≥ 0 for all (t, x) ∈ cl Q.

(19)

By passing to the limit in the solution representation (10) of Theorem 3.1 and taking into account the uniqueness of L2 (Q)-solutions to the parabolic system (3), we conclude that the strong convergence of (uε , wε ) → (u, w) is L2 (0, T ) implies the strong convergence of yε → y in L2 (Q) as ε ↓ 0. Thus the convergence property from (17) applied to (uε , wε ) and the classical result of real analysis yield the pointwise convergence yε (t, x) → y(t, x) a.e. in Q as ε ↓ 0 along a subsequence of {ε}. Combining the latter with (19) and with the corresponding result from Ref. 8, Lemma 4.2, we conclude that y(t, x) ≥ 0 for a.e. (t, x) ∈ Q, which ends the proof of the theorem.

¤

4. Convolution Approach In this section we describe a new approach to the formulation and study of the minimax feedback control problem (P ) defined in Section 2. This approach (see also the next section) is based on reducing (P ) to an asymetric game via time convolutions well developed and largely applied, e.g., in the theory of Fourier transforms. We refer the reader to Ref. 15 for a variety of results on convolutions and their applications. Recall that the time convolution of two real-valued functions v1 (t) and v2 (t) is defined by the following integral over the corresponding domain Z (v1 ∗ v2 )(t) := v1 (s)v2 (t − s) ds. We start our consideration with the one-dimensional heat equation in (3) written as  ∂y ∂2y   − a 2 = w(t) for x ∈ (−1, 1), a.e. t ∈ [0, T ],   ∂t ∂x (20) y(0, x) = 0, −1 < x < 1,     y(t, −1) = y(t, 1) = u(t) a.e. t ∈ [0, T ]. Here the perturbation w = w(t) is spatially constant and is subject to the assumed bounds |w(t)| ≤ β a.e. t ∈ [0, T ],

(21)

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while the control u = u(t) is subject to the imposed pointwise constraints |u(t)| ≤ α a.e. t ∈ [0, T ]

(22)

with some given positive constants α and β. Both control and perturbations (u, w) are supposed to be (Lebesgue) measurable on [0, T ], i.e., they belong to the admissible regions Uad in (4) and to Wad in (5), respectively, where the symmetricity is assumed for simplicity. For definiteness, choose the middle point x0 of the domain Ω = (−1, 1) as the point of observation ξ(t) := y(t, 0), where we collect information about the system performance. By the system linearity, we can represent the (generalized) solution y ∈ L2 (Q), with Q = [0, T ]×(−1, 1), to (20) generated by (u, w) ∈ Uad ×Wad as y = B1 w + B2 u, where B1 w is the solution to (20) with homogeneous boundary conditions (i.e., with u ≡ 0), and where B2 u is the solution to (20) as a homogeneous equation (i.e., with w ≡ 0); cf. representation (10) in Theorem 3.1. Setting further Z t W (t) := w(τ ) dτ, 0

observe that the the spatially constant function W satisfies the original system (20) except the boundary conditions, and thus B1 w = W − B2 W. On the other hand, if z is the solution to the homogeneous system ¯ ¯ ∂z ∂2z ¯ ¯ − a 2 = 0, z¯ = 1, z¯ = 0, ∂t ∂x x=±1 t=0 then B2 u is the time convolution of u(·) with the time derivative zt (·, ·). Letting now ψ(·) := zt (·, 0), we get [B2 u](t, 0) = [ψ ∗ u](t), and so our observation ξ has the following convolution representation in terms of u and w: ϕ ∗ w + ψ ∗ u with ϕ := 1 − ψ.

(23)

Note that ϕ and ψ in (23) have their supports in IR+ — actually in [0, T ], since we are restricting attention to the given time interval [0, T ]. Applying the Maximum Principle for the heat equation as in the proof of Theorem 3.2, we conclude that both ϕ and ψ in (23) are positive on (0, T ]. Finally, we observe that we could have fixed α = 1 in (22) and β = 1

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in (21) without any loss of generality if we were to have correspondingly rescaled ϕ and ψ. Thus α = β = 1 in what follows. The crucial requirement (8) on the system performance is now written as |ξ(t)| ≤ η on [0, T ] for any w as in (21).

(24)

The primary goal of feasible feedback controls ¡ ¢ u = u(ξ) with u(t) = u ξ(t) , 0 ≤ t ≤ T,

(25)

such that u(t) satisfies (22), is to keep the motion ξ in (23) under observation within the constraint region (24) for any feasible perturbation w(t). The optimization problem over all the feasible feedback controls and feasible perturbations consists of minimizing the maximum functional J given by nZ T ¡ ¢ o J(u) := max |u ξ(t) | dt , (26) w

0

where the maximum in (26) is taken with respect to feasible perturbations w = w(·), while feasible controls u = u(ξ) are formed by a feedback law, i.e., in closed loop u = u(ξ) in (25) Our phrasing is significant: there is no suggestion that w is already known on [0, T ], so u(·) cannot be prescribed in open loop as u = u(t), but must be constructed progressively as the observation history provides more information. Thus such a construction must be a map C : ξ 7→ u subject to (22), (24), and the causality condition that £ ¤ £ ¤ ξ1 (·)=ξ2 (·) on [0, t] =⇒ C(ξ1 ) (t)= C(ξ2 ) (t) for each t ∈ [0, T ]. (27) The cost functional appearing in (26) is then really J = J(C) to be minimized over feedback laws C satisfying relationships (22), (24), and (27) with ξ(t) and u(t) given in (23) and (25), respectively. Of course, a preliminary requirement to this optimization problem is imposing conditions on the initial data α, β, η, and T ensuring the nonemptiness of the admissible set of feedback laws written as © ª © ª C ad := C satisfying (22), (24), (27) 6= ∅. (28) It is easy to observe that the device above readily applies to the case of multidimensional linear parabolic systems (3) considered in Sections 2 and 3. In this case, we fix some point x0 ∈ Ω from the open bounded domain Ω ⊂ IRn and evaluate the system performance at ξ(t) = y(t, x0 ) as in Section 2 representing it further in the convolution form (23). This device leads us to the following result summarizing the above derivations and discussions.

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Theorem 4.1. (minimax feedback control problem via convolutions). The minimax feedback control problem (P ) formulated in Section 2 is equivalent to minimizing the cost functional of the maximum type J = J(C) in (26) over feedback laws C : ξ 7→ u subject to (22), (24), and (27), where the observation ξ is given by (23) via the time convolutions involving positive functions ϕ and ψ. Furthermore, the above convolution approach can be extended to a more general control problem (Pe) for multidimensional parabolic systems formulated as follows. Given an open bounded domain Ω ⊂ IRn with the sufficiently smooth boundary ∂Ω, consider the linear parabolic system  on QT := (0, T ] × Ω,  yt − ∆ · A∆y = w(t)ω ¯ ¯ (29) ¯  y ¯¯ = u(t)e ω, y¯ ≡ 0 on Ω, ∂Ω

t=0

subject to appropriate regularity requirements on the entries of the positive definite n × n matrix A = A(x), where ω and ω e are given positive functions on Ω and ∂Ω, respectively. Let Γ be an arbitrary linear functional on states y = y(t, ·) specified before as ® Γ, y = y(t, x0 ), t ∈ [0, T ], with some x0 ∈ Ω. In the general case, define the system observation ξ by ® ξ(t) := Γ, y(t, ·) on [0, T ].

(30)

Then we have the convolution representation of (14) along the parabolic system (13): ® ® ϕ ∗ w + ψ ∗ u with ϕ = Γ, zt and ψ = Γ, zet , (31) where z is the solution to the system  on QT ,  zt − ∆ · A∆z = ω ¯ ¯  z ¯¯ = 0, z ¯¯ ≡ 0 on Ω ∂Ω

t=0

with the homogeneous boundary conditions, while ze is the solution to the homogeneous parabolic equation  z=0 on QT ,  zet − ∆ · A∆e ¯ ¯ ¯  ze¯¯ = ω e , ze¯ ≡ 0 on Ω. ∂Ω

t=0

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The generalized feedback control problem (Pe) is that of minimizing the functional J =©J(C) in (26) over feedback laws C : ξ 7→ u belonging to the admissible set C}ad from (12) with observation ξ given by the convolutions in (15). Again we note that the constants α and β in (22) and (21) can be fixed as α = β = 1 without loss of generality by absorbing this in the specification of ϕ and ψ in the convolution representation (15) of the observation ξ from (14) satisfying the requirements in (24) and (27). 5. Asymmetric Games In this section we formulate and study a certain asymmetric game, which incorporates the most essential features of both minimax control problems (P ) and (Pe) and allows us to establish important properties of feasible and optimal feedback controls and worst perturbations via the corresponding game strategies. Given two positive functions ψ, ϕ ∈ L1 (IR+ ), a number η > 0, and a time bound T > 0, formulate the asymmetric game (G) as follows: A fox and a hound are considered as moving points f (·) and h(·) in IR given by the convolutions f = ϕ ∗ w and h = ψ ∗ u in IR,

(32)

i.e., controlled by providing measurable inputs u and w, respectively, subject to the pointwise constraints |u(t)| ≤ 1 and |w(t)| ≤ 1 for a.e. t ≥ 0.

(33)

The fox wins if she can ever escape, i.e., can get f (t) farther than η from h(t) at some time t < T for any hound’s feasible strategy u from (33). Conversely, the hound wins if he can track successfully, i.e., can keep h(t) no farther than η from f (t) during the entire interval [0, T ] for any fox’s feasible strategy w from (33). There are no ties in the game. As the information structure for this game, we are to assume that the hound knows (and remembers) his own input u and the resulting motion h and observes (and remembers) the relative position ξ(t) = f (t) − h(t). We may attribute to the fox perfect information about both inputs, perhaps even knowing the hound’s strategy for causally determining his input u in response to each ξ(·).

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Note that in view of (33), the parameters α and β in the constraints of (22) and (21) appear in the present formulation as Z T Z T 1 1 α = kψkL := ψ(t) dt and β = kϕkL := ϕ(t) dt (34) 0

0

for the positive functions ψ and ϕ from (32); so β is the effective range (in time T ) of the fox and, correspondingly, α is the effective range of the hound. We are primarily concerned with the hound’s tracking strategy. The first question is whether the hound does have a winning strategy; and if so, the next question — corresponding to the minimax problems (P ) and (Pe) from Section 4 — would be selecting among winning strategies to minimize the cost max-functional J in (26); we call such hound’s strategy an absolutely winning one. From the control formulation/interpretation of (G) in terms of convolutions, it is clear that at any time τ the future evolutions of f and h in (32) include some history dependence. Thus it would seem desirable for the hound to know also at least the past history of the fox’s input w — which has not been provided directly — for better predictions of the fox’s future motion in constructing u(τ ). The next theorem justifies such a well-posedness of the game (and of the minimax control problems) under consideration. Theorem 5.1. (well-posedness of the game). Let ψ, ϕ ∈ L1 [0, T ] be given functions with ϕ 6≡ 0. Then the histories of u(·) and of ξ := ϕ ∗ w − ψ ∗ u on [0, τ ]

(35)

uniquely determine the restriction of w to [0, τ ]. Proof. Take ψ, ϕ to vanish outside [0, T ] (since anything else is irrelevant), and take uτ , wτ as the restrictions of u, w extended to vanish outside [0, τ ] (since anything else is irrelevant at time τ ). Then ξτ (·) defined by ξτ := ϕ ∗ wτ − ψ ∗ uτ coincides with ξ defined by (35) on [0, τ ]. Employing the Fourier transforms (Ref. 15) (denoted by fb as usual) of the functions under consideration, the convolutions become simply products. Thus, rearranging slightly, we have ϕ bτ w bτ = ψbτ u bτ + ξbτ .

(36)

Note that ψ, uτ , and ξτ are known at time τ (by prescription, memory, and observation), and so the product ϕ bτ w bτ is known by (36). Since ψ (restricted to [0, T ]) and wτ are bounded with compact support, each of

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the factors ϕ bτ and w bτ is entire analytic by the Paley-Wiener Theorem; see, e.g., Ref. 15, Chapter 7. Thus w bτ is uniquely determined, and so is wτ by inverting the Fourier transform, which completes the proof of the theorem. ¤ Remark 5.1. (infinite horizon). If game (G) is considered with infinite horizon (i.e., with T = ∞), then Theorem 5.1 still holds true. Indeed, in this case we take some T0 > τ , and the same arguments apply. Thus, despite the nominal asymmetry of the specified information structure, we can assume that at each τ the hound has perfect causal information: both the input functions u and w on [0, τ ] as well as knowing ψ and ϕ. Note in this connection that we are here assuming exact observation and computation ignoring, e.g., the question of continuity of the map uτ , ξτ 7→ wτ whose existence has been assured by Theorem 5.1. Next we consider the question of whether game (G) would be a win for the fox or for the hound — obviously depending on ψ, ϕ and T, η. It is convenient for this to define Ψ := ψ ∗ 1 and Φ := ϕ ∗ 1, i.e., to let Z t Z t Ψ(t) := ψ(s) ds and Φ(t) := ϕ(s) ds. (37) 0

0

In accordance with the monotonicity property of the parabolic dynamics as in Theorem 3.2 and in Section 4, we always suppose that the input functions ψ and ϕ are positive, which implies by (37) that the integral images Ψ and Φ are increasing — to α and β, respectively. The next statement indicates two cases when an absolutely winning strategy for the hound and a winning strategy for the fox in game (G) can be found directly. Note that we can see this easily in the abstract game framework, while in fact it is essentially based on the fundamental monotonicity of transients with respect to both controls and perturbations (eventually on the parabolic Maximum Principle) in the original context of parabolic dynamics developed in Section 3. Note also that, due the symmetricity of game (G) and the original control problem (P ), assertion (ii) of the next proposition admits the symmetric extreme version w ≡ −1 corresponding to the other bound of the fox’s constraints in (33). Proposition 5.1. (winning strategies). Consider game (G) under the assumptions made. The following assertions hold: (i) Let Φ(t) ≤ η on [0, T ]. Then standing still u ≡ 0 is an absolutely winning strategy for the hound.

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(ii) Suppose that Φ(t) > Ψ(t) + η for some t ∈ (0, T )

(38)

(this is certainly the case if β > α + η in (34) before rescaling). Then taking the extreme action w ≡ 1 is a winning strategy for the fox. Proof. To justify (i), observe that the condition (0 ≤)Φ(t) ≤ η on [0, T ] ensures the fulfillment of the state constraint |ξ(t)| = |f (t) − h(t)| = |f (t)| ≤ η on [0, T ] for any fox’s strategy w from (33) by putting u ≡ 0 into the motion equation (35). It is obvious at the same time that the hound’s strategy u ≡ 0 minimizes the cost functional J in (26), i.e., it is an absolutely winning one. To prove (ii), observe from (35) and the positivity of ψ and ϕ that the bigger fox’s action w is, the bigger hounds’s input u should be applied to keep the game motion ξ in (35) within the prescribed area |ξ(t)| ≤ η for all t ∈ [0, T ].

(39)

It is easy to conclude from (35) that the fox’s extreme strategy w ≡ 1 under condition (38) does not allow the hound to keep ξ within (39) whatever his strategy u constrained by (33) applies. This means that the fox wins. ¤ The above analysis seems to be inadequate to treat game (G) when assumption (38) is violated. To see this, we introduce a notion of agility for the motion characteristics as meaning that a change in the input would more rapidly maneuver (by convolution) the resulting motion. Leaving this notion a bit vague, it is natural to say, e.g., that ϕ(t) e = 2ϕ(2t) would be more agile than ϕ, with the same norm. We are now concerned with the effect of having the fox more agile than the hound. Example 5.1. (agility). Let η = 1, T = 2, and ( ( 2 for 0 ≤ t ≤ 2, 3 for 0 ≤ t ≤ 1, ψ(t) = ϕ(t) = 0 else. 0 else; Then we compute the integrals ( 3t for 0 ≤ t ≤ 1, Φ(t) = 3 for t ≥ 1;

( Ψ(t) =

2t for 0 ≤ t ≤ 2, 4 for t ≥ 2.

With η = 1 this gives Φ(t) < Ψ(t) + η except for equality at t = 1: the scenario of (38) cannot happen — and even continuing indefinitely with

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u ≡ 1 (while the fox also continues her extreme strategy w ≡ 1) would not put the hound too far from the fox. On the other hand, suppose the fox were to use the input function ( ( 1 for 0 ≤ t ≤ 1, 3t for 0 ≤ t ≤ 1. w(t) = so f (t) = −1 for t ≥ 1; 6 − 3t for t ≥ 1. Even knowing this in advance, what could the hound do? Unless u ≡ 1 on [0, 1], one would have h(1) < Ψ(1) = 1, and thus f (1) = Φ(1) = 3 would give |ξ(t)| > η = 1, i.e., a win for the fox. The choice ( ( 1 for 0 ≤ t ≤ 1, 2t for 0 ≤ t ≤ 1, e u e(t) = gives h(t) = −1 for 1 ≤ t ≤ 2 2 for 1 ≤ t ≤ 2. e Thus, as f (2) = 0, we would have |ξ(2)| = 2 > η, and a win for the fox. Any other input choice u(·) avoiding the fox’s win at t = 1 would necessarily give u ≥ u e; so h ≥ e h, and again a win for the fox before t = 2 = T .

References 1. B.S. Mordukhovich, Optimal control of the groundwater regime on engineering reclamation systems, Water Resources, 12 (1986), 244–253. 2. B.S. Mordukhovich, Minimax design for a class of distributed parameter systems, Autom. Remote Control, 50 (1990), 262–283. 3. T. Ba¸sar, P. Bernhard, H∞ -Optimal Control and Related Minimax Design Problems, (Birkh¨ auser, Boston, MA, 1995). 4. N.N. Krasovskii, A.I. Subbotin, Game-Theoretical Control Problems, (Springer, New York, 1988). 5. B. van Keulen, H∞ -Control for Distributed Parameter Systems: A StateSpace Approach, (Birkh¨ auser, Boston, MA, 1993). 6. I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theory, published in two volumes, (Cambridge University Press, Cambridge, UK, 2000). 7. B.S. Mordukhovich, Minimax design of constrained parabolic systems, in: S. Chen et al. (Eds.), Control of Distributed Parameter and Stochastic Systems, (Kluwer, Boston, MA, 1999), 111–118. 8. B.S. Mordukhovich, K. Zhang, Minimax control of parabolic equations with Dirichlet boundary conditions and state constraints, Appl. Math. Optim., 36 (1997), 323–360. 9. B.S. Mordukhovich, I. Shvartsman, Optimization and feedback control of constrained parabolic systems under uncertain perturbations, in: M. de Queiroz et al. (Eds.), Optimal Control, Stabilization and Nonsmooth Analysis, Lecture Notes Cont. Inf. Sci., Vol. 301, (Springer, New York, 2004), 121–132.

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10. B.S. Mordukhovich, K. Zhang, Robust suboptimal control of constrained parabolic systems under uncertaintly conditions, in: G. Leitmann et al. (Eds.), Dynamic and Control, (Gordon and Breach, Amsterdam, 1999), pp. 81–92. 11. L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko, The Mathematical Theory of Optimal Processes, (Wiley, New York.) 12. J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, (Springer, Berlin, 1971). 13. D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, (Springer, Berlin, 1983). 14. O.A. Ladyzhenskaya, A.I. Solonnikov, N.N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, (American Mathematical Society, Providence, RI, 1968). 15. W. Rudin, Functional Analysis, (McGraw-Hill, New York, 1991).

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IMPLICIT PARALLEL SOLVERS IN COMPUTATIONAL ELECTROCARDIOLOGY MARILENA MUNTEANU∗ and LUCA FRANCO PAVARINO Department of Mathematics, Universit` a di Milano, Via Saldini 50, 20133 Milano, Italy E-mail: [email protected], [email protected]; www.mat.unimi.it This paper presents a parallel solver for the Monodomain and Bidomain systems modeling the bioelectrical activity of cardiac tissue in three dimensions. These systems consists of nonlinear parabolic reaction-diffusion equations coupled with a stiff system of several ordinary differential equations, known as Luo-Rudy I model. Instead of the more common explicit or semi-implicit time discretizations, we consider here the fully implicit Backward Euler and linearly implicit Rosenbrock methods, that allow a time step-size selection without stability constraints. Trilinear finite elements are employed in the space discretization. The nonlinear systems originated at each time step of Backward Euler are solved by a Newton-Krylov method, where the outer iteration of an inexact Newton method solves the linear systems of the Jacobian update with a Krylov method such as GMRES with a block Jacobi preconditioner. The same linear solver is applied to the linear systems of the Rosenbrock method. Parallel numerical results on a Linux Cluster show that the proposed parallel solver is scalable for the Monodomain system, with both linear and nonlinear iteration counts bounded from above by a constant independent of the number of processors. This does not hold for the Bidomain system, since the linear iteration counts increase with the number of processors. Keywords: Monodomain and Bidomain models; Parallel solver; Finite elements; Implicit time discretization; Newton-Krylov methods.

1. Introduction The bioelectrical activity of the heart is here described by the multiscale Monodomain or Bidomain models, consisting of nonlinear parabolic ∗ This

work was supported by Istituto Nazionale di Alta Matematica Francesco Severi, Roma. The participation at ICAADE was supported by the Ministero degli Affari Esteri, under the grant 9 — Technological and scientific cooperation between Italy and Romania 2006–2008.

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reaction-diffusion equations describing the macroscopic evolution of the electrical potentials in the cardiac tissue, coupled with a stiff system of ordinary differential equations, known as the Luo-Rudy I (LR1) model, see Ref. 1, describing the microscopic ionic currents through the cellular membrane. The numerical resolution of these reaction-diffusion models is computationally intensive, because of the interaction of the different scales in space and time, the nonlinearities involved and the very severe ill-conditioning of the discrete systems arising at each time step. The treatment of the nonlinearities of these models is a challenging issue occurring in the discretization process. Most numerical studies employ explicit or semi-implicit time discretizations where the nonlinear reaction term is treated explicitly and the diffusion term implicitly, see e.g. Refs. 2 and 3. Often splitting schemes are also employed in order to decouple the ODE system from the nonlinear parabolic system, see e.g. Ref. 4. Instead, we consider here the fully implicit Backward Euler and linearly implicit Rosenbrock methods, see Lang and Verwer5 , that take into account the complete nonlinearities of the model and allow a time step-size selection without stability constraints. The nonlinear systems originated at each time step of Backward Euler are solved by a Newton-Krylov method, with an inexact Newton outer iteration and a Krylov space inner iteration for the Jacobian update, such as GMRES with a block Jacobi preconditioner. The same linear solver is applied to the linear systems of the Rosenbrock method. Our three dimensional parallel codes have evolved from the parallel solver presented in Colli Franzone and Pavarino4 which takes into account the orthotropic anisotropy of the cardiac tissue. A previous implicit parallel solver for the Bidomain system has been studied by Murillo and Cai6 for two dimensional domains and for the simplified FitzHugh-Nagumo membrane model. We present the results of several parallel simulations on a Linux Cluster that show that the proposed parallel solver is scalable for the Monodomain system, with both linear and nonlinear iteration counts bounded from above by a constant independent of the number of processors. This does not hold for the Bidomain system, since the linear iteration counts increase with the number of processors, indicating that the simple one-level block Jacobi preconditioner must be replaced with a more powerful multilevel preconditioner.

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2. Mathematical Models The Bidomain model captures the most important features of the excitation and repolarization on the myocardial tissue at the macroscopic level, see Ref. 7. The intra and extracellular potentials ui , ue and the gating variables w are the solutions of the following system of reaction-diffusion equations (see Ref. 4):  ∂v i  χCm − div(Di ∇ui ) + χIion (v, w) = Iapp in Ω × (0, T )    ∂t   ∂v e −χCm − div(De ∇ue ) + χIion (v, w) = Iapp in Ω × (0, T ) (1)  ∂t      ∂w − R(v, w) = 0, v(x, t) = ui (x, t) − ue (x, t) in Ω × (0, T ), ∂t together with the Neumann boundary conditions nT Di ∇ui = 0,

nT De ∇ue = 0, on ∂Ω × (0, T ),

initial conditions v(x, 0) = ui (x, 0) − ue (x, 0) = v0 (x),

w(x, 0) = w0 (x),

and compatibility conditions: Z Z i e Iapp dx = Iapp dx. Ω

(2)

Ω

The last equation of (1) is a system of ordinary equations describing the ionic currents through the cellular membrane. We consider here the LuoRudy I (LR1) membrane model that uses 6 gating variables and 1 ionic concentration, w = (w1 , ..., w7 ); see Ref. 1 for details. A rigorous mathematical derivation of the macroscopic model and existence, uniqueness and regularity results can be found in Refs. 7 and 8. λDi Assuming that Di = λDe with λ constant and setting D = and 1+λ i e λIapp Iapp Iapp = + the Bidomain system reduces to the simplified 1+λ 1+λ Monodomain model for the transmembrane potential v  ∂v   χCm − div(D(x)∇v) + χIion (v, w) = Iapp in Ω × (0, T ) ∂t   ∂w − R(v, w) = 0 in Ω × (0, T ) ∂t with Neumann boundary condition for v and initial condition for v, w.

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3. Space and Time Discretization We use hexahedral isoparametric Q1 finite elements and denote by M, A, Ai,e the mass matrix and the stiffness matrices arising from the finite element discretization, see Colli Franzone and Pavarino4 . Consequently we obtain the semidiscrete Galerkin approximation of the Monodomain system χCm M

∂vh + Avh + χMIhion (vh , wh ) = MIhapp ∂t

and Bidomain system ∂ χCm M ∂t

Ã

ui,h ue,h

!

Ã +A

ui,h

!

ue,h

Ã +χ

MIhion (vh , wh ) −MIhion (vh , wh )

!

Ã =

MIi,h app −MIe,h app

! ,

where · M=

M −M −M M

¸

· A=

Ai 0 0 Ae

¸ .

In both cases, these equations are coupled with the semidiscrete approximations of the LR1 gating and concentration system ∂wh = R(vh , wh ). ∂t Stability and convergence estimates of the semidiscrete Galerkin approximation of the Bidomain system coupled with the FitzHugh-Nagumo model have been studied by Sanfelici9 . 3.1. Backward Euler One of the simplest unconditionally stable methods for advancing the solution in time is the Backward Euler (BE) scheme. A general theory for a posteriori error estimates for nonlinear, abstract evolution equations in Hilbert spaces was carried out by Nochetto, Savar´e and Verdi10 . In the special case of the Bidomain system Colli Franzone and Savar´e7 established an a priori error estimate. At each time-step we must solve the nonlinear system F (u) = 0, where in the Monodomain case

(3)

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¶  χCm M + A vh n+1 + χMIhion (vh n+1 , wh n+1 )−  ∆t µ ¶      χCm  − Mvh n + MIhapp F (u) =    ∆t     n+1 w − wn n+1 n+1 − R(v ,w ) ∆t µ

and in the Bidomain case Ã !  µ ¶ Ã un+1 ! MIhion (vhn+1 , whn+1 ) χCm i M+A +χ −   ∆t  un+1 −MIhion (vhn+1 , whn+1 ) e     Ã n! Ã   i,h ! MI u   app χCm i F (u) =  . − − M   ∆t une −MIe,h   app      n+1  w − wn n+1 n+1 − R(v ,w ) ∆t The nonlinear problem (3) is solved by the Newton-Krylov-Schwarz method (NKS), see Cai and al.11 , where a Newton scheme is used as outer iteration while each Jacobian linear system is solved by a Krylov method with a Schwarz-type preconditioner. Here we consider the simplest block Jacobi preconditioner. We refer to Knoll and Keyes12 for a review of NKS methods and to Deuflhard13 , sections 2.2.4 and 3.2.3 for a inexact Newton-GMRES convergence theory. 3.2. A 3o order Rosenbrock method (Ros3P) The Monodomain and the Bidomain systems can be written in the abstract form B

∂u = f (u(t)), ∂t

where, in the Monodomain case u = (v, w1 , . . . , w7 ), µ B=

χCm M 0 0 I7×7

¶

 , f (u(t)) = 

−Avh − χMIhion (vh , wh ) + MIhapp R(vh , wh )

and in the Bidomain case u = (ui , ue , w1 , . . . , w7 )

 

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µ B=

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χCm M 0 0 I7×7 

 −A f (u(t)) =  

µ

¶ ,

ui,h ue,h

Ã

¶ −χ

MIhion (vh , wh )

!

−MIhion (vh , wh )

Ã +

MIi,h app −MIe,h app

!  . 

R(vh , wh ) Rosenbrock methods belong to a large class of methods which try to avoid nonlinear systems and replace them by a sequence of linear systems, see Ref. 14. The convergence and order conditions of Rosenbrock-type methods ∂u applied to differential-algebraic systems of the form B(u) = f (u) with ∂t nonsingular matrix B have been studied by Lubich and Roche15 . For nonlinear parabolic problems, Lang and Verwer5 proposed a 3-stage, thirdorder, A-stable method called Ros3P. In order to avoid matrix-vector operations we use the transformed form of a Rosenbrock scheme (cf. Ref. 5); in the special case when B doesn’t depend on time and solution, the Ros3P method has the following form: ¯ 1 ∂f (u) ¯¯ step 0: compute E = B − J(un ), where J(un ) = ; γ∆t ∂u ¯u=un step 1: define U1 = un and compute Un1 , solution of the linear system EUn1 = f (U1 ); step 2: define U2 = un + a21 Un1 and compute Un2 , solution of the linear system EUn2 = f (U2 ) + B

c21 Un1 ; ∆t

step 3: define U3 = un + a31 Un1 + a32 Un2 and compute Un3 , solution of the linear system i hc c32 31 Un1 + Un3 ; EUn3 = f (U3 ) + B ∆t ∆t step 4: advance the numerical solution un+1 = un +

3 X i=1

mi Uni .

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4. Parallel Implementation and Numerical Results Our Fortran code is based on the Portable Extensible Toolkit for Scientific computing (PETSc)16 library available as an open source software from Argone National Laboratory. All tests were performed on the Linux cluster Ulisse at the University of Milan (http://cluster.mat.unimi.it/) that has 92 processors Xeon. We refer to Colli Franzone and Pavarino4 for the parameters calibration of the Monodomain and Bidomain models. We report both scalability results for a few time steps and simulations of complete cardiac cycle. 4.1. Scalability We perform 10 times steps with constant stepsize dt = 0.05 msec. The linear systems are solved (in parallel) by GMRES method with Block-Jacobi preconditioning. We denote by it. the number of linear iteration at time t=0.5 msec, av.it. the average number of linear iteration over the first 10 time-steps, nit. the number of nonlinear iteration at time t=0.5 msec and av.nit. the average number of nonlinear iteration over the 10 time-steps. In Tables 1–4, we report these quantities for a test of scaled speedup up to 64 processors, where the mesh sizes and processor counts are increased proportionally so that the local mesh size per processor remains constant. Both Backward Euler and Rosenbrock methods are scalable for the MonodomainLR1 model, since both linear and nonlinear iterations remain constant, while they are not scalable for the Bidomain-LR1 model since the linear iterations increase with the number of processors (even if the nonlinear iterations remain constant). 4.2. Complete cardiac cycle We performed simulations on a three-dimensional domain representing a slab of cardiac tissue of size 0.5 × 0.5 × 0.05 cm3 discretized with a mesh 50 × 50 × 5. The linear systems are solved (in parallel) by GMRES method with Block-Jacobi preconditioning. In Tables 5 and 6, we report the time spent computing the Jacobian matrix (TimeComputJac), the time for the solution of the nonlinear system (TimeSNES) and the total CPU time (Time), for both the MonodomainLR1 and Bidomain-LR1 models, both the Backward Euler and Ros3P parallel solvers and both fixed and adaptive time step strategies (see Ref. 17). The timings clearly show that the Jacobian assembling takes up only a small percentage of the total CPU time. Therefore future effort to improve

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efficiency should focus on improving the linear iteration counts within the Newton Jacobian update and the adaptive time step strategy (see Ref. 18). Table 1.

Monodomain-LR1 model; time integrator-BE

nr.proc

mesh

unknowns

lit

av.lit.

nit

av.nit

8=2·2·2 16 = 4 · 2 · 2 32 = 4 · 4 · 2 64 = 8 · 4 · 2

40 × 40 × 40 80 × 40 × 40 80 × 80 × 40 160 × 80 × 40

512.000 1.024.000 2.048.000 4.096.000

18 18 18 13

29,3 29,3 28,6 28,0

3 3 3 2

4,8 4,8 4,6 4,4

Table 2.

Bidomain-LR1 model; time integrator-BE

nr.proc

mesh

unknowns

it

av.it.

nit

av.nit

8=2·2·2 16 = 4 · 2 · 2 32 = 4 · 4 · 2 64 = 8 · 4 · 2

30 × 30 × 30 60 × 30 × 30 60 × 60 × 30 120 × 60 × 30

243.000 486.000 972.000 1.944.000

104 110 130 132

194,2 192,8 214,6 217,9

3 3 3 3

4,7 4,6 4,4 4,1

Table 3.

Monodomain-LR1 model; time integrator-Ros3P

nr.proc

mesh

unknowns

lit

av.lit.

8=2·2·2 16 = 4 · 2 · 2 32 = 4 · 4 · 2 64 = 8 · 4 · 2

40 × 40 × 40 80 × 40 × 40 80 × 80 × 40 160 × 80 × 40

512.000 1.024.000 2.048.000 4.096.000

21 21 21 21

19,7 19,7 19,7 19,7

Table 4.

Bidomain-LR1 model; time integrator-Ros3P

nr.proc

mesh

unknowns

it

av.it.

8=2·2·2 16 = 4 · 2 · 2 32 = 4 · 4 · 2 64 = 8 · 4 · 2

30 × 30 × 30 60 × 30 × 30 60 × 60 × 30 120 × 60 × 30

243.000 486.000 972.000 1.944.000

168 174 204 221

157,2 159,6 181,7 208,6

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Table 5. Adaptive time step selection mesh = 50 × 50 × 5, Tmax = 400, nr. proc = 10. TimeComputeJac = time to assemble the Jacobian, TimeSNES = total time for solving the nonlinear system, TimeRos3P = total time for the Rosenbrock solver, Time = total CPU time model

Backward Euler

Ros3P

mono LR

TimeComputeJac = 80,379 TimeSNES = 2159,281 Time = 2170,955

TimeComputeJac = 46,072 TimeRos3P = 3398,792 Time = 3414,86

TimeComputeJac = 3,7% TimeSNES TimeComputeJac = 921,419 TimeSNES = 7149,694 Time = 7178

TimeComputeJac = 1,35%TimeRos3P TimeComputeJac = 508,076 TimeRos3P = 12972,79 Time = 12986,683

TimeComputeJac = 12,88 % TimeSNES

TimeComputeJac = 3,91% TimeRos3P

bi LR

Table 6. Fixed time-step mesh = 50 × 50 × 5, Tmax = 400, nr. proc = 10 TimeComputeJac = total time to assemble the Jacobian, TimeSNES = total time for solving the nonlinear system, TimeRos3P = total time for the Rosenbrock solver, Time = total CPU time model

Backward Euler

Ros3P

mono LR

TimeComputeJac = 562,165 TimeSNES = 13752,169 Time = 13816,83

TimeComputeJac = 364,237 TimeRos3P = 21766,812 Time = 21849,27

TimeComputeJac = 4,08% TimeSNES TimeComputeJac = 7527,959 TimeSNES = 39373,615 Time = 39552,15

TimeComputeJac = 1,67% TimeRos3P TimeComputeJac = 4952,747 TimeRos3P = 59629,155 Time = 59771,49

TimeComputeJac = 9,11% TimeSNES

TimeComputeJac = 8,3% TimeRos3P

bi LR

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The number of linear iterations as function of time for Backward Euler and Ros3P time integrators with adaptive time-step strategy are reported in Fig. 1 and with fixed time-step, in Fig. 2.

60

250

BE Ros3P

BE Ros3P

50 200

linear iterations

linear iterations

40

30

150

100

20

50

10

0

0

50

100

150

200

250

300

350

400

50

100

150

200

250

300

350

400

time

time

Fig. 1. Adaptive time-step strategy, number of linear iterations for a full heartbeat on a slab 50 × 50 × 5, with 10 processors; Monodomain-LRI model on the left side and Bidomain-LRI model on the right side. Solver: GMRES+Block Jacobi.

25

200

BE Ros3P

BE Ros3P 180

20

160

linear iterations

linear iterations

140

15

10

120

100

80

60

40

5

20

0

0

50

100

150

200

time

250

300

350

400

50

100

150

200

250

300

350

400

time

Fig. 2. Fixed time-step strategy, number of linear iterations for a full heartbeat on a slab 50 × 50 × 5, with 10 processors; Monodomain-LRI model on the left side and Bidomain-LRI model on the right side. Solver: GMRES+Block Jacobi.

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References 1. C. Luo, Y. Rudy, A model of the ventricular cardiac action potential: depolarization, repolarization, and their interaction, Circ. Res., 68 (1991), 1501– 1526. 2. G. Akrivis, M. Crouzeix, C. Makridakis, Implicit-explicit multistep finite element methods for nonlinear parabolic problems, Math. Comp., 67, 2 (1998) 457–477. 3. J. Pormann, A simulation system for the Bidomain equations, PhD thesis, Duke University, Dep. of Electr. Comp. Eng., (1999). 4. P. Colli Franzone, L.F. Pavarino, A parallel solver for reaction-diffusion equations in computational electrocardiology, Math. Models Methods Appl. Sci., 14, 6 (2004), 883–911. 5. J. Lang, J. Verwer, Ros3P-An accurate third-order Rosenbrock solver designed for parabolic problems, BIT, 41, 4 (2001), 731–738. 6. M. Murillo, X.-C. Cai, A fully implicit parallel algorithm for simulating the non-linear electrical activity of the heart, Numer. Linear Algebra Appl., 11, 2–4 (2004), 261–277. 7. P. Colli Franzone, G. Savare, Degenerate Evolution Systems Modeling the Cardiac Electric Field at Micro- and Macroscopic Level, Evolution equations, semigroups and functional analysis (Milano, 2000), 49–78, Progr. Nonlinear Differential Equations Appl., 50, (Birkh¨ auser, Basel, 2002). 8. M. Pennacchio, G. Savar´e, P. Colli Franzone, Multiscale modeling for the bioelectric activity of the Heart, SIAM J. Math. Anal., 37, 4 (2006), 1333– 1370. 9. S. Sanfelici, Convergence of the Galerkin approximation of a degenerate evolution problem in electrocardiology, Numer. Meth. Part. Differ. Equ., 18, 2 (2002), 218–240. 10. R.H. Nochetto, G. Savar`e, C. Verdi, A posteriori error estimates for variable time-step discretizations for nonlinear evolution equations, Comm. Pure Appl. Math., 53, 5 (2000), 525–589. 11. X.-C. Cai, W.D. Gropp, D.E. Keyes, M.D. Tidriri, Newton-Krylov-Schwarz methods in CFD, in Proceedings of the International Workshop on Numerical Methods for the Navier-Stokes equations, (Vieweg, Braunschweig, 1995), 183–200. 12. D.A. Knoll, D.E. Keyes, Jacobian-free Newton-Krylov methods: a survey of approches and applications, J. Comput. Phys., 193, 2 (2004), 357–397. 13. P. Deuflhard, Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics, 35, (Springer-Verlag, Berlin, 2004). 14. E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics, 14, (Springer-Verlag, Berlin, 1996). 15. Ch. Lubich, M. Roche, Rosenbrock methods for differential-algebraic systems with solution-dependent singular matrix multiplying the derivative, Computing, 43, 4 (1990), 325–342.

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16. S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M. Knepley, L.Cufman McInnes, B.F. Smith and H.Zhang, PETSc user manual, Tech. Rep ANL95/11-Revision 2.1.5, Argonne National Laboratory (2002). 17. N. Hooke, Efficient simulation of action potential propagation in a Bidomain, PhD thesis, Duke Univ., Dept. of Comput. Sci., (1992). 18. P.Colli Franzone, P. Deuflhard, B. Erdmann, J. Lang, L.F. Pavarino, Adaptivity in space and time for reaction-diffusion systems in electrocardiology, SIAM J. Sci. Comput., 28, 3 (2006), 942–962.

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FAN’S INEQUALITY IN THE CONTEXT OF MP -CONVEXITY∗ (a) CONSTANTIN

P. NICULESCU and

(b) IONEL

ROVENT ¸A

Department of Mathematics, University of Craiova, Craiova 200585, Romania (a) E-mail: [email protected] (b) E-mail: [email protected] Several analogies of Fan’s inequality are proved in the context of Mp -convexity. As a consequence, a Nash equilibrium theorem is obtained. Keywords: Mp -convexity; Min-max inequality; Nash equilibrium.

1. Introduction In what follows we are interested in a class of functions having a nice behavior under the action of means. The weighted Mp -mean is defined for pairs of positive numbers x, y by the formula  ((1 − λ)xp + λy p )1/p , if p ∈ R\{0}    x1−λ y λ , if p = 0 Mp (x, y, 1 − λ, λ) =  min{x, y}, if p = −∞   max{x, y}, if p = ∞, where λ ∈ [0, 1]. If p is an odd number, we can extend Mp to pairs of real numbers. Let E be a linear topological space and assume that C is a nonempty compact and convex subset of E. Definition 1.1. We say that a function f : C −→ R is Mp -concave if f ((1 − λ)x + λy) ≥ Mp (f (x), f (y), 1 − λ, λ) for all x, y ∈ C and λ ∈ (0, 1). ∗ Supported

by Grant CNCSIS 80/2006.

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Thus the M1 -concave functions are the usual concave functions, while the M∞ -concave functions, are precisely the quasi-concave functions. A celebrated result due to Ky Fan asserts that any function f :C×C→R+ which is quasi-concave in the first variable and lower semicontinuous in the second variable verifies the inequality min sup f (x, y) ≤ sup f (z, z). y∈C x∈C

(1)

z∈C

The aim of this paper is to prove a complementary result, precisely: Theorem 1.1. Suppose that f : C × C → R+ is a function which is Mp concave and lower-semicontinuous in each variable. Then min sup Mpp (f (x, y), f (y, x), 1 − λ, λ) ≤ sup f p (z, z), y∈C x∈C

z∈C

for all λ ∈ (0, 1) and p ∈ R. Our technics yields also the following fact: Theorem 1.2. Let C be a nonempty compact and convex subset of E, and let f : C × C → R+ be a function which is Mp -concave in the first variable and lower-semicontinuous in the second variable. Then min max Mpp (f (y, y), f (x, x), 1 − λ, λ) ≤ sup f p (z, z) y∈C x∈C

z∈C

for all λ ∈ (0, 1) and p ∈ R. For p an odd number f is allowed to take negative values. 2. Proof of the Main Result We actually prove a much more general result: Theorem 2.1. Assume f : C × C → R+ is a function which is Mp concave and lower-semicontinuous in each variable and let g : C → C be a continuous onto function. Then min sup Mpp (f (x, y), f (y, x), 1−λ, λ)≤ sup Mpp (f (z, g(z)), f (g(z), z), 1−λ, λ), y∈C x∈C

z∈C

for all λ ∈ (0, 1) and p ∈ R. Theorem 3.1 represents the particular case where g is the identity of C. The proof of Theorem 2.1 is based on the KKM-Theorem, whose statement is recalled here for the convenience of the reader:

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Theorem 2.2. (Knaster-Kuratowski-Mazurkievicz). Suppose that for every point x in a nonempty set X ⊂ E there is an associated closed subset M (x) ⊂ X such that [ conv F ⊂ M (x) x∈F

holds for all finite subsets F ⊂ X. Then for any finite subset F ⊂ X we have \ M (x) 6= ∅. x∈F

Hence if some subset M (z) is compact, we have \ M (x) 6= ∅. x∈X

Theorem 2.2 is one of the many results known to be equivalent to Brouwer’s fixed point theorem. See Ref. 1. Proof of Theorem 2.1. We attach to g : C → C and λ ∈ [0, 1] the family of sets M (g(x))x∈C , where M (g(x)) consists of all y ∈ C such that Mpp (f (x, y), f (y, x), 1 − λ, λ) ≤ sup Mpp (f (z, g(z)), f (g(z), z), 1 − λ, λ). z∈C

We will show that this family satisfies the hypothesis of the KKMTheorem. In fact, g(x) ∈ M (g(x)) for every x ∈ C and [ conv F ⊂ M (g(x)) x∈F

for every finite subset F ⊂ C. For example, if F consists of two elements g(x1 ) and g(x2 ), we have to show that u = (1 − α)g(x1 ) + αg(x2 ) ∈ M (g(x1 )) ∪ M (g(x2 ))

(2)

for every α ∈ (0, 1). Our argument is by reductio ad absurdum. If (2) fails, then for some α ∈ (0, 1) we have Mpp (f (x1 , u), f (u, x1 ), 1−λ, λ)> sup Mpp (f (z, g(z)), f (g(z), z), 1−λ, λ), (3) z∈C

and Mpp (f (x2 , u), f (u, x2 ), 1−λ, λ)> sup Mpp (f (z, g(z)), f (g(z), z), 1−λ, λ). (4) z∈C

The Intermediate Value Theorem yields a β ∈ [0, 1] such that xβ = βx1 + (1 − β)x2 ∈ C verifies g(xβ ) = u = (1 − α)g(x1 ) + αg(x2 ).

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Since f is Mp -concave in each variable it follows that Mpp (f (xβ , g(xβ )), f (g(xβ ), xβ ), 1 − λ, λ) = (1 − λ)f p ((1 − β)x1 + βx2 , u) + λf p (u, (1 − β)x1 + βx2 ) is not less than (1 − λ)((1 − β)f p (x1 , u) + βf p (x2 , u)) +λ((1 − β)f p (u, x1 ) + βf p (u, x2 )) = (1 − β)((1 − λ)f p (x1 , u) + λf p (u, x1 )) +β((1 − λ)f p (x2 , u) + λf p (u, x2 )) = (1 − β)Mpp (f (x1 , u), f (u, x1 ), 1 − λ, λ) +βMpp (f (x2 , u), f (u, x2 ), 1 − λ, λ) > (1 − β) sup Mpp (f (z, g(z)), f (g(z), z), 1 − λ, λ) z∈C

+β sup Mpp (f (z, g(z)), f (g(z), z), 1 − λ, λ) z∈C

= sup Mpp (f (z, g(z)), f (g(z), z), 1 − λ, λ), z∈C

a contradiction. Thus (2) follows. By the KKM-Theorem we infer that \

M (g(x)) 6= ∅,

x∈C

which means the existence of y ∈ C such that Mpp (f (x, y), f (y, x), 1 − λ, λ) ≤ sup Mpp (f (z, g(z)), f (g(z), z), 1 − λ, λ), z∈C

for every x ∈ C, equivalently, sup Mpp (f (x, y), f (y, x), 1 − λ, λ)

x∈C

≤ sup Mpp (f (z, g(z)), f (g(z), z), 1 − λ, λ). z∈C

In conclusion, min sup Mpp (f (x, y), f (y, x), 1 − λ, λ) y∈C x∈C

≤ sup Mpp (f (z, g(z)), f (g(z), z), 1 − λ, λ). z∈C

¤

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3. A Nonsymmetric Extension of Fan’s Theorem The aim of this section is to prove the following result: Theorem 3.1. Let D and F be two two nonempty, compact and convex subsets of E and let g be a continuous onto function g : D → F . Then for every function f : D × F → R+ which is quasi-concave in the first variable and lower-semicontinuous in the second variable the following inequality holds: min sup f (x, y) ≤ sup f (z, g(z)). y∈F x∈D

z∈D

This result extends Fan’s Theorem (which corresponds to the case where D = F and g(z) = z for all z ∈ D). Proof. Consider the family of sets ½ ¾ M (g(x)) = y ∈ F : f (x, y) ≤ sup f (z, g(z)) ,

for x ∈ D.

z∈D

We will show that this family verifies the assumptions of the KKMTheorem. In fact, it is easy to see that g(x) ∈ M (g(x)), for x ∈ D. Let A be a nonempty finite subset of F, say A = {g(x1 ), g(x2 )} for simplicity. We have to prove that [ conv A ⊂ M (g(x)), (5) g(x)∈A

that is, (1 − λ)g(x1 ) + λg(x2 ) ∈ M (g(x1 )) ∪ M (g(x2 )), for all λ ∈ (0, 1). Indeed, if the contrary is true, then for a suitable λ ∈ (0, 1) we have f (x1 , (1 − λ)g(x1 ) + λg(x2 )) > sup f (z, g(z)) z∈D

and f (x2 , (1 − λ)g(x1 ) + λg(x2 )) > sup f (z, g(z)). z∈D

The intermediate Value Theorem yields an α ∈ [0, 1] such that xλ = αx1 + (1 − α)x2 ∈ D verifies (1 − λ)g(x1 ) + λg(x2 ) = g(xλ ).

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Then f (xλ , g(xλ )) = f (αx1 + (1 − α)x2 , (1 − λ)g(x1 ) + λg(x2 )) ≥ min {f (x1 , (1 − λ)g(x1 ) + λg(x2 )), f (x2 , (1 − λ)g(x1 ) + λg(x2 ))} > sup f (z, g(z)), z∈D

a\contradiction that shows that (5) works. By the KKM-Theorem, M (g(x)) 6= ∅, and this fact assures the existence of a y0 ∈ F such x∈D

that f (x, y0 ) ≤ sup f (z, g(z)) for every x ∈ D. z∈D

Consequently, sup f (x, y0 ) ≤ sup f (z, g(z)), which yields x∈D

z∈D

min sup f (x, y0 ) ≤ sup f (z, g(z)). y∈F x∈D

¤

z∈D

A similar argument yields the following theorem in the case of Mp convex functions: Theorem 3.2. Let C be a nonempty, compact and convex subset of E and let g be a continuous onto function g : C → C. Then for every Mp -convex function f : C × C → R+ which is upper-semicontinuous with respect to each variable we have max inf Mpp (f (x, y), f (y, x), 1−λ, λ)≥ inf Mpp (f (z, g(z)), f (g(z), z), 1−λ, λ), x∈C y∈C

z∈C

for all λ ∈ (0, 1). In the same manner we can prove the following result for the quasiconvex functions: Theorem 3.3. Let D and F be two nonempty, compact and convex subsets and let g be a continuous onto function g : D → F . Let f : D × F → R+ be a quasi-convex function in the second variable and upper-semicontinuous in the first variable. Then we have max inf f (x, y) ≥ inf f (z, g(z)). x∈D y∈F

z∈D

Remark 3.1. We impose the condition that g is an onto function in order to assure the applicability of KKM Theorem.

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4. Further Results An important application of Theorem 3.1 is the existence of a g-equilibrium, a fact that generalizes the existence of a Nash equilibrium. Theorem 4.1. Let C = C1 × C2 × ... × Cn , where Ci , i = 1, ..., n are nonempty, compact and convex subsets of E, let g = (g1 , g2 , ..., gn ) : C → C be a continuous onto function and let fi : C → C be a function which is lowersemicontinuous in the second variable and xi → (fi )(y1 , ..., g(xi ), ...yn ) is quasi concave for every i = 1, ..., n. Then there exists an y ∈ C such that fi (y) ≤ fi (y1 , ..., g(xi ), ..., yn )), for every xi ∈ Ci , i = 1, ..., n.

Proof. Let f (x, y) =

n X

(fi (y) − fi (y1 , ..., g(xi ), ..., yn ))). It is easy to see

i=1

that f satisfies the assumptions of Theorem 3.1. This yields an y ∈ C such that sup f (x, y) ≤ sup f (z, g(z)) = 0. x∈C

z∈C

Letting xi = (y1 , y2 , ..., xi , ..., yn ) (i = 1, ..., n) in the last inequality we conclude that fi (y) − fi (y1 , ..., g(xi ), ..., yn )) ≤ 0 for every xi ∈ Ci , i = 1, ..., n. ¤ The following result due to M. Sion2 has important applications in convex analysis and games theory: Theorem 4.2. Let D and F be two nonempty, compact and convex subsets and let f : D × F → R+ be a function which is upper-semicontinuous and quasi-concave function in the first variable, and lower-semicontinuous and quasi-convex in the second variable. Then min max f (x, y) = max min f (x, y). y∈F x∈D

x∈D y∈F

This result can be derived via Theorems 3.1 and 3.3 above provided that inf f (z, g1 (z)) = sup f (z, g2 (z)).

z∈D

z∈D

(6)

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for suitably chosen continuous onto functions g1 and g2 . In fact, the hypothesis of Sion’s theorem make possible to apply Theorems 3.1 and 3.3 and thus inf f (z, g1 (z)) ≤ max min f (x, y) ≤ min max f (x, y) ≤ sup f (z, g2 (z)),

z∈D

x∈D y∈F

y∈F x∈D

z∈D

for all continuous onto functions g1 , g2 : D → F . While the topological condition (6) can be easily verified in a number of particular cases, we do not know how general is it. Results of this type, concerning the existence of continuous onto maps relating compact convex sets, may be found in Ref. 3. References 1. J.M. Borwein and A.S. Lewis, Convex Analysis and Nonlinear Optimization, (Springer-Verlag, Berlin, 2000). 2. M. Sion, On general minimax theorems, Pacific J. Math, 8 (1958), 171–176. 3. J.G. Hocking and G.S. Young, Topology, (Dover, 1988). 4. J. Kindler, Uber Spiele auf konvexen Mengen, in Operations Res. Vehrfahren, (Hain, Meisenheim/Glan, 1977), 695-704. 5. J. Kindler, A simple proof of Sion’s Minimax Theorem, The Mathematical Association of America, 112 (2005), 356–358.

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RANGE CONDITION IN OPTIMIZATION AND OPTIMAL CONTROL N.H. PAVEL Department of Mathematics, Ohio University, Athens OH 45701, USA E-mail: [email protected] This paper deals with maximum principles for locally Lipschitz cost functionals, under constraints given by functional equations associated with pairs of closed range unbounded operators. The so called ”range condition” introduced by the author, plays an essential role. Such abstract maximum principles have both a unifying effect in this area and applications to optimal control of some partial differential equations. Keywords: Cost functionals; Fr´ echet derivatives; Clarke’s generalized gradients; Range conditions; Maximum principles; Optimal control.

1. Introduction In the early 1995 the author has started the investigation of the problem ½ Minimize the cost functional L(y, u) = L(y o , uo ), (y o , uo ) ∈ M (P) subject to the constraints: (y, u) ∈ M = {(y, u), Ay = Bu + f }. He observed that if the linear(unbounded) operators A and B are densely defined on a Hilbert space H, with closed ranges R(A) and R(B) and if the range condition (RC) (the relationship between R(A) and R(B)) below holds: (RC)

R(A) ⊆ R(B), or vice versa: R(B) ⊆ R(A)

then one can obtain maximum principles of the form: If (y o , uo ) is an optimal pair, then there is p such that: A∗ p ∈ ∂y L(y o , uo ), B ∗ p ∈ −∂u L(y o , uo ). Here ∂L could mean Fr´echet derivative, or subdifferential or Clarke’s generalized gradients, depending on the conditions on L, A and B. Such maximum principles have applications to optimal control of some differential systems including some PDE (see Refs. 1–6). Moreover, this

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abstract scheme includes the essentials of many existing results on optimal control, such as those in Barbu7 (i.e. it has an unifying effect, too). This idea was extensively extended to more general cases by the author in some joint papers with Aizicovici and Motreanu, and by Voisei8,9 . Voisei has proved that actually it suffices to assume that only one of A and B has a closed range and that in continuous case, the range condition (RC) (the relationship between R(A) and R(B)) above, is not necessary. We can also call the (RC) above — the strong range condition (SRC) between R(A) and R(B). A significant particular case is: L(y, u) = g(y) + h(u), with g and h Fr´echet differentiable, or locally Lipschitz continuous, or subdifferentiable in the sense of convex analysis. In this paper we show that the range condition (RC) above can still be relaxed to the weak range condition (WRC) as in Hypothesis (H2) below. Moreover, we give a simple proof (via Proposition 2.1) of the fact that at least in the case in which A and B are continuous, the range condition (WRC) in (H2) or (RC) above are not necessary. So the following interesting open problems remain to be investigated: (OP1) Are there cases in which the range conditions of type (H2) in Theorem 2.1 are necessary? More precisely, are there pairs of operators A, B — not both continuous, for which the range condition is not satisfied and the conclusion of Theorem 2.1 fails? (OP2) Are there pairs A, B not both continuous for which the range condition (RC) above is not satisfied but the conclusion of Theorem 2.1 holds? (OP3) Are there pairs A, B satisfying the weak range condition (WRC) in (H2) below, but not satisfying the range condition (RC) above? Let us recall the definition of the generalized directional derivative in the sense of Clarke. Definition 1.1. Let f : U → R be a locally Lipschitz function on an open subset U of a Banach space X. The generalized directional derivative of f at a point x ∈ U in the direction v ∈ X is the number f o (x; v) := lim sup y→x t↓0

1 (f (y + tv) − f (y)). t

The generalized gradient of f at the point x ∈ U is the nonempty subset of X ∗ defined as follows: ∂f (x) := {z ∈ X ∗ : hz, vi ≤ f o (x; v), ∀v ∈ X}. The basic hypotheses are the following:

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(H1) A : D(A) ⊂ H → H and B : D(B) ⊂ H → H are linear densely defined operators. (H2) The range R(B) of B is closed and the weak range condition (WRC) below holds. (WRC) S AB = {v ∈ D(A); Av ∈ R(B)}, is dense in H. Here one can interchange the role of A and B. (H3) L : H × H → R is Fr´echet differentiable, or locally Lipschitz continuous. The case of g and h lower semicontinuous convex proper functions, requires additional hypotheses discussed in Refs. 1,8. In this case, both A and B must be also closed with closed range. 2. Maximum Principles under Minimal Conditions ½ (P)

Locally minimize the cost functional L(y, u) = g(y) + h(u) subject to the constraints Ay = Bu + f .

Set

M = {(y, u), Ay = Bu + f }

(1)

We say that (y o , uo ) ∈ M is a local optimal pair for the problem P above, if there is a neighborhood U of (y o , uo ) such that L(y, u) ≥ L(y o , uo ) for all (y, u) ∈ U ∩ M. Theorem 2.1. Assume that the Hypotheses (H1), H(2) and (H3) hold and that (y o , uo ) is an optimal pair for problem (P) above. Then there is p such that: A∗ p ∈ −∂y L(y o , uo ), B ∗ p ∈ ∂u L(y o , uo ). Remark 2.1. If the strong range condition (RC) holds (RC) R(A)⊆R(B), then S AB =D(A) so the weak range condition (WRC) above holds too. Obviously, in Theorem 2.1 we can interchange the role of A and B. Proof of Theorem 2.1. Part I. The case of Fr´echet differentiability of g and h. Clearly for every t∈R and Av=Bw, we have (y o +tv, uo +tw)∈M , so L(y o + tv, uo + tw) − L(y o , uo ) ≥ 0, for allt ∈ R

(2)

which implies

D E ˙ o ), w = 0, for all Av = Bw hg(y ˙ o ), vi + h(u For v = 0, (3) yields D E ˙ o ), w = 0, for all w, with Bw = 0 h(u

(3) (4)

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˙ o ) is orthogonal to the kernel K(B) of B, so h(u ˙ o) This means that h(u ∗ ∗ ˙ belongs to the range R(B ) of B , i.e. there is p such that h(uo ) = B ∗ p. Substituting into (3) we have: hg(y ˙ o ), vi + hB ∗ p, wi = 0, for all Av = Bw

(5)

As w ∈ D(B), this implies hg(y ˙ o ), vi + hp, Bwi = 0, for all Av = Bw i.e. for all v ∈ S AB

(6)

hg(y ˙ o ), vi + hp, Avi = 0, for all v ∈ S AB

(7)

so which is supposed to be dense in H. This implies p ∈ D(A∗ ) and hg(y ˙ o ), vi + hA∗ p, vi = 0, for all v ∈ S AB

(8)

which implies: A∗ p = −g(y ˙ o ). Part II. The case: g and h — locally Lipschitz. In this case (2) implies the existence of ξ ∈ ∂g(yo ) and η ∈ ∂h(uo ) satisfying hξ, vi + hη, wi = 0, for all Av = Bw

(9)

which implies as above B ∗ p = η ∈ ∂h(uo ) and A∗ p = −ξ ∈ ∂g(yo ) which completes the proof. Remark 2.2. For the conclusion of Theorem 2.1, the range conditions of type (H2) in Theorem 2.1 may not necessary, as shown below. Indeed, in the case in which A and B are linear bounded operators, the range condition (H2) in Theorem 2.1 is not necessary. To prove this, let us recall a classical result: Proposition 2.1. Assume A : H → H is linear bounded and J is Fr´echet differentiable or locally Lipschitz continuous. 1. Let : J(y o ) = minJ(y) subject to: Ay = f . Then necessarily there is p ˙ o ) = A∗ p , with Ay o = f . such that J(y ˙ o ) = A∗ p 2. If J is convex then the necessary condition J(y (with Ay o = f ) is sufficient, too. ˙ o ) = A∗ p 3. If J is not convex, then the necessary condition J(y o (with Ay = f ) may not be sufficient. Proof. The proof of Part 1 is standard. o 0 D Part 2. E Let f (t) = J(y + tz), with Az = 0 and t ∈ R. Then f (0) = ˙ o ), z = hA∗ p, zi = hp, Azi = 0 so f (t) ≥ f (0) for all t ∈ R, as f is J(y convex, too. Therefore: J(y o + tz) ≥ J(y o ) for all t ∈ R and Az = 0. Now

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let Aw = f . We do have Ay o = f , so z = w − y o ∈ N (A) , i.e. Az = 0, so J(y o + z) ≥ J(y o ), i.e. J(w) ≥ J(y o ), for all w with Aw = f , q.e.d. Part 3. Let us consider the example: ¶ µ 01 (10) A= 01 2 ˙ f =(5, 5), J(y1 , y2 )=y13 + y2 , p=(1, 0) and y o =(0, 5). Clearly J(y)=(3y 1 , 1) o ˙ )=A∗ p with y=(y1 , y2 ). It is easy to verify that the necessary conditions J(y with Ay o =f are satisfied, but obviously y o is not a minimum point of J(y o )= min J(y) subject to: Ay=f , as such a minimum does not exist. Now let us consider the problem (P) with A and B linear bounded. Define the operator A from H ×H into H ×H by A(y, u) = (Ay−Bu, 0).

Proposition 2.2. Let A and B be linear bounded operators from H into itself, L Fr´echet differentiable, and let (yo , uo ) be a minimum point (a solution of Problem (P)). Then there is (p, q) such that: ˙ o )) = A∗ (p, q) = (A∗ p, −B ∗ q). (g(y ˙ o ), h(u Proof. Clearly, Ay = Bu + f iff A (y, u)= (f, 0). It is easy to check that A∗ (p, q) = (A∗ p, −B ∗ q). According to Proposition 2.1, with A and H × H ˙ o )) = in place of A and H, respectively, there is (p, q) such that: (g(y ˙ o ), h(u ∗ ∗ ∗ A (p, q) = (A p, −B q), which completes the proof. Remark 2.3. Therefore, the conclusion of Proposition 2.2 holds without any assumption on the range conditions of type (H2) in Theorem 2.1. Thus the range condition in Theorem 2.1 is not necessary in the case in which A and B are both linear bounded operators on H. See also the open problems (OP1)–(OP3) in Section 1. References 1. J.K. Kim and N.H. Pavel, Optimal control of the periodic wave equation, in Proceedings of Dynamical Systems and Applications. Vol. 2, (Atlanta, Georgia, 1995), 309–314. 2. N.H. Pavel, Periodic solutions to nonlinear 2-D equations, in: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Kartsatos, Ed., Lecture Notes in Pure and Appl. Math., Vol. 178, (Marcel Dekker, New York, 1996), 243–249. 3. J.K. Kim and N.H. Pavel, T -periodic solutions of the wave equation in the resonant case, Proc. of Nonlin. Funct. Anal. and Appl., Vol. 1, (Kyungnam Univ. Masan, Korea, 1996), 139–145. 4. J.K. Kim and N.H. Pavel, Optimal control of the periodic wave equation, in Proc. Dynamical Systems and Applications, Dynamic Publishing Inc., Vol. 2, (P.O. Box 48654 Atlanta GA (USA), 1996), 309–314.

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5. S. Aizicovici, D. Motreanu and N.H. Pavel, Fully nonlinear programming problems with closed range operators, Lecture Notes in Pure and Appl. Math., Vol. 225, (Marcel Dekker Inc., New York, 2002), 19–30. 6. S. Aizicovici, D. Motreanu and N.H. Pavel, Nonlinear mathematical programming and optimal control, Dynamics of Continuous, Discrete and Impulsive Systems, Serie A: Mathematical Analysis, 11 (2004), 503–524. 7. V. Barbu, Optimal control of the one dimensional wave equation, Appl. Math. Optimiz., 35 (1997), 77–90. 8. M.D. Voisei, First-order necessary conditions of optimality for strongly monotone nonlinear control problems, JOTA, 116, 2 (2003) 421–436. 9. M.D. Voisei, First-order necessary conditions of optimality for nonlinear programming problems associated with linear operators, Adv. Math. Sci. Appl., 13, 2 (2003). 10. S. Aizicovici, D. Motreanu and N.H. Pavel, Nonlinear programming problems associated with closed range operators, Applied Mathematics and Optimization, 40 (1999), 211–228. 11. D. Motreanu and N.H. Pavel, Tangency, Flow Invariance for Differential Equations and Optimization Problems, Monographs and Textbooks in Pure and Appl. Mathematics, Vol. 219, (Marcel Dekker, New York-Basel, 1999). 12. N.H. Pavel, Periodicity and stability of semigroups via ”λk ∈ ρ(C k ) =⇒ λ ∈ ρ(C)”, in: Advances in Nonlinear Dynamics, Martynyuk and Sivasundaran, Eds., (Gordon and Breach Publ., Amsterdam 1997), 117–127. 13. J.K. Kim and N.H. Pavel, Existence and regularity of weak periodic solutions of the 2–D wave equation, Nonlinear Analysis, 32 (1998), 861–870. 14. S.C. Gao, Pavel, N.H., Optimal control of a functional equation associated with closed range self-adjoint operators, Proc. Amer. Math. Soc., 126 (1998), 2979–2986. 15. S. Gao, N.H. Pavel, N. Schirmeister, Optimal control of a functional equation associated with closed range operators, Comm. Appl. Anal., 2 (1998), 383– 392. 16. I. Hrinca, J.K. Kim and N.H. Pavel, The kernel of some hyperbolic differential operators and theorems of numbers theory, Communications in Applied Analysis, 4, (2000), 527–532. 17. N.H. Pavel, Closed range weak or mild solution operators in the resonant case and applications to optimal control, Advances in Mathematical Sciences and Applications, 10, (2000), 851–871. 18. N.H. Pavel, Closed range mild solution operators and applications to optimal control of PDE, in Proceedings of the 25th Congress of the American Romanian Academy of Arts and Sciences (ARA), Case Western Reserve University, (Cleveland Ohio, USA, Polytechnic International Press, 2000), 294–299. 19. J.K. Kim, G. Moro¸sanu and N.H. Pavel, Closed range mild solution operators and nonconvex optimal control via orthogonality and generalized gradients, Journal of Nonlinear Analysis: Series A Theory and Methods, 49, (2002), 247–264.

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LIMITS OF SOLUTIONS TO THE INITIAL BOUNDARY DIRICHLET PROBLEM FOR SEMILINEAR HYPERBOLIC EQUATION WITH SMALL PARAMETER A. PERJAN State University of Moldova, Chi¸sin˘ au E-mail: [email protected] We study the existence of the limits of solution to singularly perturbed initial boundary value problem of hyperbolic-parabolic type with boundary Dirichlet condition for the semilinear wave equation. We prove the convergence of solutions and also the convergence of gradients of solutions to perturbed problem to the corresponding solutions to the unperturbed problem as the small parameter tends to zero. We show that the derivatives of solution relative to time-variable possess the boundary layer function of the exponential type in the neighborhood of t = 0. Keywords: Semiliniar wave equation; Singular perturbation; Boundary layer function.

1. Statement of the Problem Let Ω ∈ Rn be an open and bounded set with the smooth boundary ∂Ω. Consider the following initial boundary value problem for the semilinear wave equation, which will be called (Pε ):  00 εu + u0ε − ∆uε + |uε |p uε = f (x, t), x ∈ Ω, t > 0,    ε uε |t=0 = u0 (x), u0ε |t=0 = u1 (x), x ∈ Ω,    uε |x∈∂Ω = 0, t ≥ 0, where ε is a small positive parameter and 0 denote the derivative relative to variable t. We will study the behavior of the solutions to the problem (Pε ) as ε → 0. Our aim is to establish the relationship between the solutions to the problem (Pε ) and the corresponding solutions to the unperturbed problem (P0 ):

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 0 v − ∆v + |v|p v = f (x, t), x ∈ Ω, t > 0,    v|x=0 = u0 (x), x ∈ Ω,    v|x∈∂Ω = 0, t ≥ 0, Singularly perturbed problems of hyperbolic-parabolic type was studied by many authors. Here we mention only some works which are dedicated to such kind of problems and which contained a large list of references (see Refs. 1–8). 2. Solvability of Problems (Pε ) and (P0 ) Denote by (·, ·) the scalar product in the real Hilbert space L2 (Ω) and by h·, ·i denote the pairing between H −1 (Ω) and H01 (Ω). Definition 2.1. We say a function uε ∈ L2 (0, T ; H01 (Ω)) with u0ε ∈ L2 (0, T ; L2 (Ω)), u00ε ∈ L2 (0, T : H −1 (Ω)) is a solution to the problem (Pε ) provided ( εhu00ε (t), ηi + (u0ε (t), η) + (∇uε (t), ∇η) + (|uε (t)|p uε (t), η) = (f (t), η), ∀η ∈ H01 (Ω),

a.e.

t ∈ (0, T ),

u(0) = u0 ,

u0 (0) = u1 .

Definition 2.2. We say a function v ∈ L2 (0, T ; H01 (Ω)) with v 0 ∈ L2 (0, T ; H −1 (Ω)) is a solution to the problem (P0 ) provided ( 0 hv (t), ηi + (∇v(t), ∇η) + (|v(t)|p v(t), η) = (f (t), η), ∀η ∈ H01 (Ω),

a.e.

t ∈ (0, T ),

v(0) = u0 .

Theorem 2.1. (Ref. 9) Let T > 0 and p verifies condition " p ∈ [0, 2/(n − 2)], if n ≥ 3, p ∈ [0, ∞),

if

n = 1, 2.

(1)

If f ∈ W 1,1 (0, T ; L2 (Ω)), u0 ∈ H01 (Ω) ∩ H 2 (Ω), u1 ∈ H01 (Ω), then there exists a unique solution to the problem (Pε ) such that uε ∈ W 1,∞ (0, T ; H01 (Ω)) ∩ L∞ (0, T ; H 2 (Ω)), ∞ −1 u00ε ∈ L∞ (0, T ; L2 (Ω)), u000 (Ω)). ε ∈ L (0, T ; H

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Theorem 2.2. (Ref. 10) Let T > 0. If f ∈ W 1,1 (0, T ; L2 (Ω)), u0 ∈ H01 (Ω)∩H 2 (Ω), then there exists a unique solution v ∈ W 1,∞ (0, T ; L2 (Ω))∩ W 1,2 (0, T ; H01 (Ω)) to the problem (P0 ) and the estimate kvkW 1,∞ (0,t;L2 (Ω)) + kvkW 1,2 (0,t;H01 (Ω)) + ³Z t Z ´1/2 + |v(x, τ )|p+2 dxdτ + 0

Ω

0

Ω

³Z t Z ´1/2 + |v(x, τ )|p |v 0 (x, τ )|2 dxdτ ≤ ³

(2)

´

≤ C ku0 kH 2 (Ω) + kf kW 1,1 (0,t;L2 (Ω)) is true for t ∈ [0, T ]. 3. Main Results Theorem 3.1. Suppose that p verifies condition (1), f ∈W 2,1 (0, T ; L2 (Ω)), u0 , u1 , α = f (0) − u1 + ∆u0 − |u0 |p u0 ∈ H01 (Ω) ∩ H 2 (Ω), then there exist constants C = C(n, p, Ω) and ε0 = ε0 (Ω, n, p) such that for 0 < ε ≤ ε0 the following estimates kuε − vkC([0,T ];L2 (Ω)) ≤ M (T )ε1/2 ,

(3)

kuε − vkL∞ (0,T ;H01 (Ω)) ≤ M (T )ε1/4

(4)

are fulfilled, where uε is the solution to the problem (Pε ), v is the solution to the problem (P0 ), and M (T ) = M (T, kf kW 2,1 (0,T ;L2 (Ω)) , ku1 kH 2 (Ω) , kαkH 2 (Ω) ). If in addition p ∈ [1, 2/(n − 2)] and f ∈ W 2,∞ (0, T ; L2 (Ω)), then for 0 < ε ≤ ε0 the estimate ku0ε − v 0 − αe−t/ε kL∞ (0,T ;L2 (Ω)) ≤ M1 (T )ε1/4

(5)

is true, where M1 (T ) = M (T ) + kf 00 kL∞ (0,T ;L2 (Ω)) . The relationship (5) shows that the derivative u0ε has the singular behavior relative to the small values of the parameter ε in neighborhood of the set {(x, t)|x ∈ Ω, t = 0}. It means that this neighborhood is the boundary layer for u0ε and the function αe−t/ε is the boundary layer function for u0ε . The proofs of relations (3), (4) and (5) are based on two key points. The first one is the relationship between the solutions to the problem (P0 )

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and (Pε ) in the linear case. The second key point represents an apriori estimates of solutions to the problem (Pε ), which are uniform relative to ε for the small values of parameter ε. 4. Apriori Estimates for the Solutions to Problem (Pε ) In what follows, we will use the following notations. For k ∈ N and p ∈ [1, ∞] we denote | · |0 =: k · kL2 (Ω) ;

k · kk =: k · kH k (Ω) ;

| · |W k,p =: k · kW 1,p (0,T ;L2 (Ω)) ; T

k · k =: k · kH01 (Ω) ;

k · kW k,p =: k · kW k,p (0,T ;H01 (Ω)) . T

Also, in what follows, we agree that all constants which depend only on n, p and Ω we will denote by C. Denote by E(u; t) = ku(t)k + kukW 1,2 + |u0 (t)|0 + t

√

εku0 (t)k + ε|u00 (t)|0 +

√

ε|u00 |W 0,2 . t

Lemma 4.1. Let f ∈ W 1,1 (0, T ; L2 (Ω)), u0 ∈ H01 (Ω) ∩ H 2 (Ω), u1 ∈ H01 (Ω) and p verifies condition (1). Then there exists positive constant C such that for any solution u to the problem (Pε ) the following estimate E(uε ; t) ≤ CM0 (t),

t ∈ [0, T ],

0 < ε < 1/2,

(6)

hold, where M0 (t) = M0 (ku0 k2 , ku1 k, |f |W 1,1 ). t If in addition f ∈ W 2,1 (0, T ; L2 (Ω)), u1 , α ∈ H01 (Ω) ∩ H 2 (Ω), then there exists ε0 = ε0 (Ω, M0 ) ∈ (0, 1) such that the function zε (t) = u0ε (t) + αe−t/ε satisfies the estimate kzε kW 0,∞ + |zε0 |W 0,∞ + kzε kW 1,2 ≤ M (t), t

t

t

t ∈ [0, T ],

0 < ε ≤ ε0 ,

(7)

where α = f (0) − u1 + ∆u0 − |u0 |p u0 , and M (t) = (kαk2 , ku1 k2 , |f |W 1,2 ). t

This Lemma can be proved as Lemma 2 from Ref. 8. 5. Relationship Between the Solutions to Problems (Pε) and (P0) in the Linear Case The relationship between the solutions to problems (Pε ) and (P0 ) in the linear case, i. e. in the case when the terms |uε |p uε and |v|p v in problems (Pε ) and (P0 ) are missing was inspired by the work Ref. 11.

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At first we shall give some properties of the kernel K(t, τ, ε) of the transformation which realizes this connection. For ε > 0 denote ´ 1 ³ K1 (t, τ, ε) + 3K2 (t, τ, ε) − 2K3 (t, τ, ε) , K(t, τ, ε) = √ 2 πε where

n 3t − 2τ o ³ 2t − τ ´ λ √ , 4ε 2 εt n 3t + 6τ o ³ 2t + τ ´ K2 (t, τ, ε) = exp λ √ , 4ε 2 εt nτ o ³t + τ ´ λ √ , K3 (t, τ, ε) = exp ε 2 εt Z ∞ 2 λ(s) = e−η dη.

K1 (t, τ, ε) = exp

s

Lemma 5.1. (Ref. 7) The function K(t, τ, ε) possesses the following properties: (i) For any fixed ε > 0 K ∈ C({t ≥ 0} × {τ ≥ 0}) ∩ C ∞ (R+ × R+ ); (ii) Kt (t, τ, ε) = εKτ τ (t, τ, ε) − Kτ (t, τ, ε),

t > 0, τ > 0;

(iii) εKτ (t, 0, ε) − K(t, 0, ε) = 0, t ≥ 0; n τ o 1 exp − , τ ≥ 0; (iv) K(0, τ, ε) = 2ε 2ε (v) For each fixed t > 0, s, q ∈ N there exist constants C1 (s, q, t, ε) > 0 and C2 (s, q, t) > 0 such that |∂ts ∂τq K(t, τ, ε)| ≤ C1 (s, q, t, ε) exp{−C2 (s, q, t)τ /ε}, (vi) K(t, τ, ε) > 0,

t ≥ 0,

τ > 0;

τ ≥ 0;

(vii) Let ε be fixed, 0 < ε ¿ 1 and H be a Hilbert space. For any ϕ : [0, ∞)→H continuous on [0, ∞) such that |ϕ(t)|≤M exp{Ct}, t ≥ 0, the relationship Z ∞ ¯¯ ¯¯ Z ∞ ¯¯ ¯¯ e−τ ϕ(2ετ )dτ ¯¯ = 0; K(t, τ, ε)ϕ(τ )dτ − lim ¯¯ Z

t→0

H

0

0

∞

(viii)

K(t, τ, ε)dτ = 1,

t ≥ 0;

0

Z (ix)

0

∞

¡ ¢ K(t, τ, ε)|t − τ |q dτ ≤ Cεq/2 1 + tq/2 ,

q ∈ [0, 1].

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(x) Let f ∈ W 1,∞ (0, ∞; H). Then there exists positive constant C such that Z ∞ ¯¯ ¯¯ √ √ ¯¯ ¯¯ K(t, τ, ε)f (τ )dτ ¯¯ ≤ C ε(1 + t)kf 0 kL∞ (0,∞;H) , ¯¯f (t) − H

0

t ≥ 0. (xi) There exists C > 0 such that Z tZ

n

∞

K(τ, θ, ε) exp 0

−

0

θo dθdτ ≤ Cε, t ≥ 0, ε > 0. ε

Lemma 5.2. Suppose that f ∈ L∞ (0, ∞; L2 (Ω)) and u ∈ W 2,∞ (0, ∞; L2 (Ω)) ∩ L∞ (0, ∞; H01 (Ω)) is a solution to the problem:  00 0   ε(u (t), η) + (u (t), η) + (∇u(t), ∇η) = (f (t), η),  ∀η ∈ H01 (Ω), a.e.t ∈ (0, ∞),    u(0) = u0 , u0 (0) = u1 . Then the function v0 which is defined by Z ∞ v0 (t) = K(t, τ, ε)u(τ )dτ 0

is the solution to the problem   (v00 (t), η) + (∇v0 (t), ∇η) = (f0 (t, ε), η), ∀η ∈ H01 (Ω), ∀t > 0,  v (0) = R ∞ e−τ u(2ετ )dτ, 0 0 where Z

∞

f0 (t, ε) = F0 (t, ε) +

K(t, τ, ε)f (τ )dτ, 0

n 3t o ³ 1 h F0 (t, ε) = √ 2 exp λ 4ε π

r ´ ³ 1 r t ´i t −λ u1 , ε 2 ε

Moreover, v0 ∈ W 2,∞ (0, ∞; L2 (Ω)) ∩ L∞ (0, ∞; H01 (Ω)).

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6. Proof of Theorem 3.1 Without loss of generality, in what follows we may consider that f ∈ W 2,l (0, ∞; L2 (Ω)). Indeed, if f ∈ W 2,l (0, T ; L2 (Ω)) for l ∈ [1, ∞], then there exits the extension f˜ ∈ W 2,l (0, ∞; L2 (Ω)) such that kf˜kW 2,l (0,∞;L2 (Ω)) ≤ C(T )kf˜kW 2,l (0,T ;L2 (Ω)) . If u ˜ is a solution to the problem (Pε ) with the same initial conditions u0 , u1 and the right-hand side f˜, then according to Theorem 2.1 we have that u(t) = u ˜(t) for t ∈ [0, T ]. Similarly, if v˜ is a solution to the problem (P0 ) with the same initial condition u0 and the right-hand side f˜, then according to Theorem 2.2 we have that v(t) = v˜(t) for t ∈ [0, T ]. Proof of estimates (3). If uε is the solution to the problem (Pε ), then according to Lemma 5.2 the function Z ∞ wε (t) = K(t, τ, ε)uε (τ )dτ 0

is the solution to the problem ´ ³ ´ ³  wε0 (t), η + (∇wε (t), ∇η) = F (t, ε), η , ∀η ∈ H01 (Ω), t > 0, R∞  wε (0) = 0 e−τ uε (2ετ )dτ, where

Z

Z

∞

F (t, ε) = F0 (t, ε) +

∞

K(t, τ, ε)f (τ )dτ − 0

(8)

K(t, τ, ε)|uε (τ )|p uε (τ )dτ.

0

Using the estimate for u0ε from (6), the estimate for zε from (7) and properties (viii) and (x) of kernel K(t, τ, ε) from Lemma 5.1 we obtain the following estimates √ kuε −wε kC([0,T ];L2 (Ω)) ≤ C(T ) ε|u0ε |W 0,∞ ≤ M0 (T )ε1/2 , 0 < ε ≤ 1/2, (9) T

kuε − wε kW 0,∞ T

√ ≤ C(T ) εku0ε kW 0,∞ ≤ M (T )ε1/2 , 0 < ε ≤ ε0 . T

(10)

As |wε (t)|0 ≤ |uε (t)|0 , then using the H¨older inequality and Sobolev’s embedding theorem we get ° ¯ ¯ ¯ ° ¯ ¯ ¯ °uε (t)|p uε (t) − |wε (t)|p wε (t)¯ ≤ C ¯uε (t) − wε (t)|uε (t)|p ¯ ≤ 0

≤ Ckuε (t) − wε (t)kL2n/(n−2) (Ω) kuε (t)kpLpn (Ω) ≤ ≤ Ckuε (t) − wε (t)kkuε (t)kp ≤ M (T )ε1/2 ,

0 < ε ≤ ε0 ,

0

(11) t ∈ [0, T ].

Denote by y(t) = v(t) − wε (t), where v is the solution to problem (P0 ) and wε is the solution to problem (8). Then the function y is the solution to the following problem:

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³ ´ ³ ´  y 0 (t), η + (∇y(t), ∇η) + (|v(t)|p v(t) − |wε (t)|p wε (t), η) = F1 (t, ε), η ,  ∀η ∈ H 1 (Ω), 0

t > 0,

where

y(0) = u0 − wε (0), Z

F1 (t, ε)=f (t)−F0 (t, ε)−

³ ´ K(t, τ, ε) f (τ )−|uε (τ )|p uε (τ ) dτ −|wε (t)|p wε (t).

∞

0

Due to the estimate (2) (in the linear case) for the function y we get the estimate Z t |F1 (τ, ε)|0 dτ, t ≥ 0. (12) |y(t)|0 ≤ |y(0)|0 + 0

From the estimate (6) it follows that Z ∞ |y(0)|0 ≤ e−τ |u0 − uε (2ετ )|0 dτ ≤ Z ∞ Z 2ετ 0 ≤ e−τ |u0ε (s)|0 dsdτ ≤ Cε|u0ε |W 0,∞ ≤ εM0 (T ). 0

(13)

T

0

2

As q(s) = es λ(s) ≤ C for s ∈ [0, ∞), then Z t Z th n 3τ o ³r τ ´ ³ 1 r τ ´i |F0 (τ, ε)|0 dτ ≤ C|u1 |0 exp λ +λ dτ = 4ε ε 2 ε 0 0 n oh ³p ´ ³ p ´i Rt τ τ 1 τ = C|u1 |0 0exp − 4ε q dτ ≤Cε|u1 |0 , t∈[0, T ]. ε +q 2 ε Using the properties (viii) and (x) from Lemma 5.1, we have Z t¯ Z ∞ ¯ √ ¯ ¯ K(τ, s, ε)f (s)ds¯dτ ≤ M (T ) ε, t ∈ [0, T ]. ¯f (τ ) − 0

(14)

(15)

0

Let us evaluate the difference

Z

∞

p

I(t) = |wε (t)| wε (t) −

K(t, τ, ε)|uε (τ )|p uε (τ )dτ.

0

Due to Sobolev’s embedding theorem and the estimates (6), (7), we get ¯³ ¯ ¯ ´0 ¯ ¯ ¯ ¯ ¯ ¯ |uε (s)|p uε (s) ¯ = (p + 1)¯u0ε (s)|uε (s)|p ¯ ≤ 0

≤ ku0ε (s)kL2n/(n−2) (Ω) ku(s)kpLpn (Ω) ≤ ≤ Cku0ε (s)kkuε (s)kp ≤ M (T ),

s ≥ 0,

0

0 < ε ≤ ε0 ,

and, consequently, for 0 < ε ≤ ε0 we have Z ∞ ¯ ¯ √ ¯ ¯ p |u (t)| u (t) − K(t, τ, ε)|uε (τ )|p uε (τ )dτ ¯ ≤ M (T ) ε, ¯ ε ε 0

0

t ∈ [0, T ].

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Hence, from the last estimate and (11), we get Z ∞ ¯ ° ° ¯ p K(t, τ, ε)|uε |p uε (τ )dτ ¯ + |I(t)|0 ≤ °uε (t)| uε − 0

0

° ¯ √ ° ¯ +°wε (t)|p wε − |uε (t)|p uε (t)¯ ≤ M (T ) ε, 0

Therefore

Z

t

√ |I(τ )|0 dτ ≤ M (T ) ε,

t ∈ [0, T ],

t ∈ [0, T ],

0 < ε ≤ ε0 .

0 < ε ≤ ε0 .

(16)

0

Gathering the estimates (14), (15) and (16), we have Z t √ |F1 (τ, ε)|0 dτ ≤ M (T ) ε, t ∈ [0, T ], 0 < ε ≤ ε0 . 0

Using the last estimate and (13), from (12) follows the estimate √ |y(t)|0 ≤ M (T ) ε, t ∈ [0, T ], 0 < ε ≤ ε0 .

(17)

Finally, the estimates (9) and (17) involve (3). Proof of estimate (4). To prove the estimate (4) we have to evaluate kykW 0,∞ . To this end we observe that due to (2) for y 0 is true the estimate T Z t |y 0 (t)|0 ≤ |Y0 |0 + |F10 (τ, ε) − a0 (τ )|0 dτ, t ∈ [0, T ], (18) 0

p

where a(t) = |v(t)| v(t) − |wε (t)|p wε , Y0 = ∆y(0) + F1 (0, ε) − a(0). Using the estimate (6), we get |Y0 |0 ≤ M (T ). As εKτ (t, τ, ε) − K(t, τ, ε) = − and Z ∞³ 0

(19)

´ 3 ³ √ K1 (t, τ, ε) − K2 (t, τ, ε) 4ε π

´ ³ ¡ 1p ¢ ¡p ¢´ K1 (t, τ, ε)+K2 (t, τ, ε) dτ ≤ Cε λ − t/ε +e3t/4ε λ t/ε ≤ Cε, 2

then for f ∈ W 1,∞ (0, ∞; L2 (Ω)), due to properties (ii), (iii) and (v) from Lemma 5.1, we obtain the estimate ¯Z ∞ ¯ ¯ ¯ Kt (t, τ, ε)f (τ )dτ ¯ = ¯ 0 0 Z ¯ ∞³ ¯ ´ ¯ ¯ =¯ εKτ (t, τ, ε) − K(t, τ, ε) f 0 (τ )dτ ¯ ≤ (20) 0 Z 0∞ ³ ´ K(t, τ, ε) + K2 (t, τ, ε) |f 0 (τ )|0 dτ ≤ M (T ). ≤ Cε−1 0

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Similarly, using the Sobolev’s embedding theorem, estimate (6) and properties (ii), (iii), (v) and (viii) from Lemma 5.1, we obtain the estimates ¯Z ∞ ¯ ¯ ¯ Kt (t, τ, ε)|uε (τ )|p uε (τ )dτ ¯ ≤ Ckuε kW 1,∞ ≤ M (T ), 0 < ε ≤ ε0 , (21) ¯ 0

0

T

¯³ ´0 ¯ ¯ ¯ ¯ |wε (t)|p wε (t) ¯ ≤ Ckwε0 (t)kkwε (t)kp ≤ 0 ¯¯ Z ∞ ¯¯ ¯¯ ¯¯ ≤ M0 (T )¯¯ Kt (t, τ, ε)uε (τ )dτ ¯¯ ≤ M (T ) 0 < ε ≤ ε0 , t ∈ [0, T ].

(22)

0

By direct computation we can show that Z t |F00 (τ, ε)|0 dτ ≤ C|u1 |0 ≤ M (T ).

(23)

The estimates (20), (21), (22) and (23) involve the estimate Z t |F10 (τ, ε)|0 dτ ≤ M (T ), 0 < ε ≤ ε0 , t ∈ [0, T ].

(24)

0

0

As

then

³ 1/n |a0 (t)|0 ≤ C kv(t)kLpn (Ω) kv 0 (t)kL2n/(n−2) (Ω) + ´ 1/n +kwε (t)kLpn (Ω) kwε0 (t)kL2n/(n−2) (Ω) ≤ ³ ´ ≤ C kv 0 (t)k(kv(t)k + 1) + kwε0 (t)kkwε (t)k , Z

t

|a0 (τ )|0 dτ ≤ M (T ),

0 < ε ≤ ε0 ,

t ∈ [0, T ].

(25)

0

Using the estimates (19), (24) and (25), from (18) we obtain |y 0 (t)|0 ≤ M (T ),

0 < ε ≤ ε0 ,

t ∈ [0, T ].

(26)

As (y 0 (t), y(t)) + ky(t)k2 + (a(t), y(t)) = (F1 (t, ε), y(t)) |F1 (t, ε)|0 ≤ M (T ),

0 < ε ≤ ε0 ,

t ∈ [0, T ],

then due to (26) we get ³ ´ √ ky(t)k2 ≤ |y(t)|0 |F1 (t)|0 + |y 0 (t)|0 ≤ M (T ) ε, 0 < ε ≤ ε0 , t ∈ [0, T ]. From the last estimates and (10) the estimate (4) follows.

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Proof of estimate (5). According to Lemma 5.2 the function w1ε (t), which is defined as Z ∞ w1ε (t) = K(t, τ, ε)zε (τ )dτ 0

is a solution to the problem ( 0 (w1ε (t), η) + (∇w1ε (t), ∇η) = (F(t, ε), η), R∞ w1ε (0) = 0 e−τ zε (2ετ )dτ, where

Z

Z

∞

F(t, ε) =

∞

K(t, τ, ε)f1 (τ, ε)dτ − (p + 1) 0

∀η ∈ H01 (Ω),

t > 0,

K(t, τ, ε)|uε (τ )|p zε (τ )dτ.

0

Using the estimates (6), (7) and properties (viii), (ix) and (x) from Lemma 5.1, similarly as the estimates (9) and (10) were obtained, we obtain the following estimates √ |zε (t) − w1ε (t)|0 ≤ M (T ) ε, t ∈ [0, T ], 0 < ε ≤ ε0 , (27) and kzε (t) − w1ε (t)k ≤ M (T )ε1/4 ,

t ∈ [0, T ],

0 < ε ≤ ε0 .

(28)

Under the conditions on f, u0 and u1 , if v is a solution to problem (P0 ), then the function v1 (t) = v 0 (t) is a solution to the problem ³ ´ ³ ´ ³ ´  v 0 (t), η + (∇v1 (t), ∇η) + (p + 1) |v(t)|p v1 (t), η = f 0 (t), η , 1  ∀η ∈ H 1 (Ω), a.e. t ∈ (0, ∞), v (0) = z (0). 1

0

ε

Denote by R(t) = w1ε (t) − v1 (t). Then the function R is a solution to the problem ³ ´ ³ ´ ³ ´  R0 (t), η + (∇R(t), ∇η) + (p + 1) |v(t)|p R(t), η = F1 (t, ε), η ,  ∀η ∈ H 1 (Ω), a.e. t ∈ (0, ∞), R(0) = w (0) − z (0), 1ε

0

ε

where F1 (t, ε) = F(t, ε) + (p + 1)|v(t)|p w1ε (t) − f 0 (t). For function R the estimate

Z

|R(t)|0 ≤ |R(0)|0 +

t

|F1 (τ, ε)|0 dτ,

t ≥ 0,

(29)

0

holds. Using the estimate (7) we get Z ∞ |R(0)|0 ≤ e−τ |zε (0) − zε (2ετ )|0 dτ ≤ εM (T ). 0

(30)

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To estimate the second term of the right-hand side of (29) we will present F1 (t, ε) in the following form: F1 (t, ε) = I1 (t, ε) + I2 (t, ε) + (p + 1)I3 (t, ε), where

Z

∞

I1 (t, ε) =

K(t, τ, ε)f 0 (τ )dτ − f 0 (t),

0

Z

³ ´ K(t, τ, ε) (p + 1)|uε (τ )|p α − ∆α e−τ /ε dτ, 0 Z ∞ I3 (t, ε) = |v(t)|p w1ε (t) − K(t, τ, ε)|uε (τ )|p zε (τ )dτ. ∞

I2 (t, ε) =

0

Using properties (viii) and (x) from Lemma 5.1, we obtain the estimate Z t √ (31) |I1 (τ, ε)|0 dτ ≤ M1 (T ) ε, t ∈ [0, T ], ε > 0. 0

As

° ¯ ° ¯ °uε (t)|p α¯ ≤ Ckuε (t)k2 kαk ≤ M (T ), 0

t ∈ [0, T ],

then property (xi) from Lemma 5.1 permits to estimate I2 (t, ε) Z t Z tZ ∞ |I2 (τ, ε)|0 dτ ≤ M (T ) K(τ, s, ε)e−s/ε dsdτ ≤ M (T )ε, 0

0

t ∈ [0, T ],

0

(32)

0 < ε ≤ ε0 .

Further, using Sobolev’s embedding theorem and estimates (4) and (28), we get ° ¯ ° ¯ °v(t)|p w1ε (t) − |uε (t)|p zε (t)¯ ≤ 0 ¯ ¯ ¯ ³¯ ¯ ¯ ¯ p p ¯ ≤ ¯w1ε (t)(|uε (t)| − |v(t)| )¯ + ¯(w1ε (t) − zε (t))|uε (t)|p ¯ ≤ 0 0 ³ ´ 1/n 1/n ≤ kw1ε (t)kL2n/(n−2) (Ω) kuε (t)kLpn (Ω) + kv(t)kLpn (Ω) + ´ 1/n (33) +kuε (t)kLpn (Ω) kw1 (t) − z(t)kL2n/(n−2) (Ω) ≤ ³ ´ ≤ Ckw1ε (t)k kuε (t)k + kv(t)k + 1 + +Ckuε (t)kp kw1ε (t) − zε (t)k ≤ M (T )ε1/4 , t ∈ [0, T ] 0 < ε ≤ ε0 .

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If p ∈ [1, 2/(n − 2)], then using the estimates (6), (7) and property (ix) from Lemma 5.1, we obtain the following estimate Z ∞ ° ¯ ° ¯ K(t, τ, ε)°uε (τ )|p zε (τ ) − |uε (t)|p zε (t)¯ dτ ≤ 0 0 Z ∞ ¯ Z t° ¯ ¯ ¯ ° ¯ ¯ ≤ C K(t, τ, ε)¯ °uε (s)|p−1 |u0ε (s)||zε (s)|+|uε (s)|p |zε0 (s)|¯ ds¯dτ ≤ 0 0 τ Z ∞ ¯Z t ³ ´ ¯ (34) ¯ ¯ K(t, τ, ε)¯ ≤ M (T ) ku0ε k + kzε0 (s)k ds¯dτ ≤ 0 τ Z ∞ ≤ M (T ) K(t, τ, ε)|t − τ |1/2 dτ ≤ M (T )ε1/4 , 0

t ∈ [0, T ]

0 < ε ≤ ε0 .

The estimates (33), (34) imply Z t |I3 (τ )|0 dτ ≤ M (T )ε1/4 ,

t ∈ [0, T ],

0 < ε ≤ ε0 .

0

From the last estimate and (31), (32) it follows that Z t |F1 (τ, ε)|0 dτ ≤ M1 (T )ε1/4 , t ∈ [0, T ], 0 < ε ≤ ε0 .

(35)

0

From (29), due to (30) and (35) it follows that |R(t)|0 ≤ M1 (T )ε1/4 ,

t ∈ [0, T ],

0 < ε ≤ ε0 .

The last estimate and (27) imply the estimate (5). Theorem 3.1 is proved. ¤ References 1. C.G. Hsiao , R.J. Weinacht, Singular perturbations for semilinear hyperbolic equation, SIAM J. Math. Anal., 14 (1983), 1168–1179. 2. A. Benaouda, M. Madaune-Tort, Singular perturbations for nonlinear hyperbolic-parabolic problems, SIAM J. Math. Anal., 18 (1987), 137–148. 3. A.J. Milani, Long time existence and singular perturbation results for quasilinear hyperbolic equations with small parameter and dissipation term. II, Nonlinear Analysis. Theory. Methods and Applications, 11 (1987), 1371– 1381. 4. B.F. Esham, R.J. Weinacht, Hyperbolic-parabolic singular perturbations for quasilinear problems, SIAM J. Math. Anal., 20 (1989), 1344–1365. 5. X. Mora, J. Sola-Morales, The singular limit dynamics of semilinear damped wave equations, J. Differ. Eq., 78 (1989), 262–307. 6. B. Najman, Time singular limit of semilinear wave equation with damping, J. Math. Anal. Appl., 174 (1993), 95–117.

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7. A. Perjan, Linear singular perturbations of hyperbolic-parabolic type, Buletinul Acad. S ¸ . R.M., ser. Matematica, (42) (2003), 95–112. 8. A. Perjan, Limits of solutions to the semilinear wave equation with small parameter, Buletinul Acad. S ¸ . R. M., ser. Matematica, (50) (2006), 65–84. 9. J.L. Lions, Quelques M´ethodes de Resolution des Probl`emes aux Limites Non Lin´eares, (Dunod Gauthier-Villars, Paris, 1969). 10. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, (Nordhoff, Leyden, 1976). 11. M.M. Lavrenitiev, K.G. Reznitskaia, B.G. Yahno, The Inverse OneDimentional Problems from Mathematical Physics, (Nauka, Novosibirsk, 1982).

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WELL-POSEDNESS OF THE FIXED POINT PROBLEM FOR MULTIVALUED OPERATORS ADRIAN PETRUS ¸ EL and IOAN A. RUS Babe¸s-Bolyai University Cluj-Napoca, Department of Applied Mathematics, Kog˘ alniceanu 1, 400084, Cluj-Napoca, Romania. E-mail: [email protected] E-mail: [email protected] The purpose of this paper is to define two concepts of well-posed fixed point problem for multivalued operators and to give some metric properties of these notions. Several examples of well-posed fixed point problem are given. Some open problems are pointed out. Keywords: Pompeiu-Hausdorff functional; The gap functional; Fixed point; Strict fixed point; Multivalued Picard operator; Nonself multivalued operator.

1. Introduction Throughout this paper, the standard notations and terminologies in nonlinear analysis are used. For the convenience of the reader we recall some of them (see Refs. 1–7, etc.). Let (X, d) be a metric space. We will use the following symbols: P (X) = {Y ⊂ X| Y 6= ∅}, Pb (X) := {Y ∈ P (X)| Y is bounded } Pcl (X) := {Y ∈ P (X)| Y is closed}, Pb,cl (X) := Pb (X) ∩ Pcl (X). If T :[ X → P (X) is a multivalued operator, then for Y ∈ P (X), T (Y ) := T (x) will denote the image of the set Y , the set of all nonempty x∈Y

invariant subsets of T will be denoted by I(T ) := {Y ∈ P (X)|T (Y ) ⊂ Y }, while the graph of the multivalued operator T is denoted by Graf T := {(x, y) ∈ X × X | y ∈ T (x)}. Also T 1 (x) := T (x), . . . , T n+1 (x) = T (T n (x)), n ∈ N, x ∈ X denote the iterate operators of T . By definition, the orbit of T starting from x ∈ X is a sequence (xn )n∈N ⊂ X such that x0 := x, xn+1 ∈ T (xn ), for each n ∈ N. Throughout the paper FT := {x ∈ X| x ∈ T (x)} (respectively (SF )T :=

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{x ∈ X| {x} = T (x)}) denotes the fixed point set, (respectively the strict fixed point set) of the multivalued operator T . The following (generalized) functionals are used in the main sections of the paper. The gap functional (1) Dd : P (X)×P (X) → R+ ∪{+∞}, Dd (A, B) := inf{d(a, b)| a∈A, b∈B}. The δ generalized functional (2) δd : P (X) × P (X) → R+ ∪ {+∞}, δd (A, B) := sup{d(a, b)| a∈A, b∈B}. The excess generalized functional (3) ρd : P (X) × P (X) → R+ ∪ {+∞}, ρd (A, B) := sup{Dd (a, B)| a∈A}. The Pompeiu-Hausdorff generalized functional (4) Hd :P (X)×P (X)→R+ ∪ {+∞}, Hd (A, B) := max{ρd (A, B), ρd (B, A)}. It is well-known that (Pb,cl (X), Hd ) is a complete metric space provided (X, d) is a complete metric space. Let (X, d), (Y, d0 ) be metric spaces and T : X → Pcl (Y ) be a multivalued operator. Then T is said to be an a-contraction if a ∈ ]0, 1[ and Hd0 (T (x1 ), T (x2 )) ≤ ad(x1 , x2 ), for all x1 , x2 ∈ X. Also, T is called contractive if Hd0 (T (x1 ), T (x2 )) < d(x1 , x2 ), for all x1 , x2 ∈ X, with x1 6= x2 . The notion of well-posed fixed point problem for singlevalued operators was defined by F.S. De Blasi, J. Myjak8 and S. Reich, A.J. Zaslavski9 and also studied by I.A. Rus in Ref. 10. The aim of this paper is to define and study the well-posedness of the fixed point problem for multivalued operators. Some open problems are pointed out. 2. Well-posedness of Fixed Point Problems We give first two definitions for a well-posed fixed point problem. Definition 2.1. Let (X, d) be a metric space, Y ∈P (X) and T : Y →Pcl (X) be a multivalued operator. The fixed point problem is well-posed for T with respect to Dd iff: (a1 ) FT = {x∗ } (b1 ) If xn ∈ Y , n ∈ N and Dd (xn , T (xn )) → 0, as n → +∞ then xn → x∗ , as n → +∞.

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Definition 2.2. Let (X, d) be a metric space, Y ∈P (X) and T : Y →Pcl (X) be a multivalued operator. The fixed point problem is well-posed for T with respect to Hd iff: (a2 ) (SF )T = {x∗ } (b2 ) If xn ∈ Y , n ∈ N and Hd (xn , T (xn )) → 0, as n → +∞ then xn → x∗ , as n → +∞. Remark 2.1. It’s easy to see that Hd (u, T (u)) = δd (u, T (u)) and hence (b2 ) implies (b1 ). Moreover, FT = (SF )T = {x∗ } implies that: if the fixed point problem is well-posed for T with respect to Dd then the fixed point problem is well-posed for T with respect to Hd . Example 2.1. Let (X, d) be a complete metric space and T : X → Pcl (X) be a multivalued a-contraction. Suppose that (SF )T 6= ∅. Then the fixed point problem is well-posed for T with respect to Dd and with respect to Hd too. Proof. Since (SF )T 6= ∅ and T is an a-contraction we have (see Ref. 5, 87) that FT =(SF )T ={x∗ }. Suppose Dd (xn , T (xn )) → 0, as n → +∞. Then d(xn , x∗ )≤Dd (xn , T (xn ))+H(T (xn ), T (x∗ )) ≤ Dd (xn , T (xn ))+a·d(xn , x∗ ). 1 Hence d(xn , x∗ ) ≤ 1−a ·Dd (xn , T (xn )) and the conclusion follows. The second conclusion follows from Remark 2.1. ¤ Example 2.2. Let X = (0, +∞), d(x, y) := |x − y| and T : X → Pcl (X) defined by T (x) := { 12 x, 12 x + 1}. Then: (a) FT = {2} (b) (SF )T = ∅ (c) T is 12 -contraction (d) the fixed point problem is not well-posed for T with respect to Dd . Indeed, if xn < 1, then D(xn , T (xn )) = 12 xn → 0, as n → +∞ implies xn 9 2, as n → +∞. Example 2.3. Let X = R, d(x, y) := |x − y| and T : X → Pcl (X) defined by ½ [−2, −1], for x < 0 T (x) := [ 13 x, 12 x], for x ≥ 0 Then: (a) FT = [−2, −1] ∪ {0} (b) (SF )T = {0} (c) T |R+ : R+ → R+ is 12 -contraction (d) the fixed point problem is well-posed for T with respect to Hd .

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Let us prove (d). First we may remark that H(x, T (x)) ≥ 12 , for each x ∈ R∗− . If we suppose that H(xn , T (xn )) → 0, as n → +∞, then there exists n0 ∈ N∗ such that xn ∈ R+ , for each n ≥ n0 . Now the conclusion follows from (c). ¤ Some abstract results are given now. Lemma 2.1. Let X be a nonempty set and d, d0 two metrics on X. Suppose that d, d0 are metric equivalent. Let T : X → Pcl (X) be a multivalued operator. Then: (i) The fixed point problem for T is well-posed with respect to Dd if and only if it is well-posed with respect to Dd0 . (ii) The fixed point problem for T is well-posed with respect to Hd if and only if it is well-posed with respect to Hd0 . Proof. (i) Let c1 , c2 > 0 such that d ≤ c1 d0 and d0 ≤ c2 d. Then Dd ≤ c1 Dd0 and Dd0 ≤ c2 Dd . Let x∗ ∈ X be the unique fixed point of T . Let xn ∈ X, n ∈ N be such that Dd0 (xn , T (xn )) → 0, as n → +∞. Then Dd (xn , T (xn )) ≤ c1 Dd0 (xn , T (xn )) → 0, as n → +∞. d

Since the fixed point problem is well-posed for Dd we get that xn → x∗ , as n → +∞. As consequence we have d0 (xn , x∗ ) ≤ c2 d(xn , x∗ ) → 0, as n → +∞. In a similar way, interchanging the roles of d and d0 we get the conclusion. (ii) The second conclusion can be established in a similar way, by taking into account that if d ≤ c1 d0 and d0 ≤ c2 d then δd ≤ c1 δd0 and δd0 ≤ c2 δd . ¤ In a similar way, we have: Lemma 2.2. Let X be a nonempty set and d, d0 two metrics on X. Suppose that d, d0 are topologically equivalent and there exists c > 0 such that d ≤ cd0 . Let T : X → Pcl (X) be a multivalued operator. Then: (i) If the fixed point problem for T is well-posed with respect to Dd then it is well-posed with respect to Dd0 . (ii) If the fixed point problem for T is well-posed with respect to Hd then it is well-posed with respect to Hd0 . An abstract result is: Lemma 2.3. Let (X, d) be a compact metric space and T : X → P (X) be a closed multivalued operator (i.e. the graph of T is a closed set). (i) If FT = {x∗ } then the fixed point problem is well-posed with respect to Dd .

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(ii) If (SF )T = {x∗ } and T is lower semicontinuous, then the fixed point problem is well-posed with respect to Hd . Proof. (i) Let xn ∈ X, n ∈ N be such that Dd (xn , T (xn )) → 0, as n → +∞. d

Let (xni )i∈N be a convergent subsequence of (xn )n∈N . Suppose xni → x e, d

as i → +∞. Then there exists yni ∈ T (xni ), i ∈ N, such that yni → x e, as i → +∞. Since T is closed we get that x e = x∗ . Since the space X is d compact, we obtain the desired conclusion: xn → x∗ , as n → +∞. (ii) Let xn ∈ X, n ∈ N be such that Hd (xn , T (xn )) → 0, as n → +∞. d

Let (xni )i∈N be a convergent subsequence of (xn )n∈N . Suppose xni → x e, as i → +∞. Since T is continuous and Hd (xni , T (xni )) → 0, as i → +∞, we immediately get that Hd (e x, T (e x)) = 0. Hence x e ∈ (SF )T and so x e = x∗ . d

Taking into account that the space X is compact we have that xn → x∗ , as n → +∞. ¤ As consequence, we have the following result for contractive multivalued operators. Theorem 2.1. Let (X, d) be a compact metric space and T : X → Pcl (X) be a multivalued contractive operator. Then: (i) If cardFT ≤ 1, then the fixed point problem for T is well posed with respect to Dd . (ii) If (SF )T 6= ∅ and T is lower semicontinuous, then the fixed point problem for T is well posed with respect to Hd . Proof. (i) By a theorem of Smithson11 , we have that FT = {x∗ }. Since T is contractive, we get that it is upper semicontinuous. The proof follows now from Lemma 2.3 (i). (ii) If (SF )T 6= ∅, then, from the contractive condition, it follows (SF )T = {x∗ }. Moreover FT = (SF )T = {x∗ }. Indeed, if y ∈ FT , with y 6= x∗ , then H(T (x∗ ), T (y)) = δ(x∗ , T (y)) < d(x∗ , y). On the other hand, if cardT (y) > 1, we get δ(x∗ , T (y)) ≥ d(x∗ , y) and hence y = x∗ . The proof follows from Lemma 2.3 (ii). ¤ 3. Multivalued Picard Operators The following concept was introduced by A. Petru¸sel and I.A. Rus in Ref. 12 (see also Ref. 4). Definition 3.1. Let (X, d) be a complete metric space and T : X→Pcl (X). By definition, T is called a multivalued Picard operator (briefly MP operator) if:

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(i) (SF )T = FT = {x∗ } H

(ii) T n (x) →d {x∗ }, as n → ∞, for each x ∈ X. A natural question is the following: Problem 3.1. For which MP operators the fixed point problem is wellposed? Since many examples of MP operators come from generalized multivalued contractions, the main purpose of this section is to study the following open problem: Problem (3.1)’ For which generalized multivalued contractions the fixed point problem is well-posed? The first result of this section is: Theorem 3.1. Let (X, d) be a complete metric space and T : X → Pcl (X) ´ c type multivalued operator, i. e. there exists q ∈]0, 1[ such that for be a Ciri´ each x, y ∈ X we have H(T (x), T (y)) ≤ ≤ q · max{d(x, y), D(x, T (x)), D(y, T (y)), 21 (D(x, T (y)) + D(y, T (x)))}. Suppose card(FT ) ≤ 1. Then the fixed point problem is well-posed with respect to D. ´ c (see Ref. 13) we have that FT 6= ∅. Hence FT = {x∗ }. Proof. From Ciri´ Consider (xn )n∈N ⊂ X, such that D(xn , T (xn )) → 0, as n → +∞. Then d(xn , x∗ ) ≤ D(xn , T (xn )) + H(T (xn ), T (x∗ )) ≤ D(xn , T (xn )) 1 +q max{d(xn , x∗),D(xn , T (xn )), (D(xn , T (x∗))+D(x∗, T (xn )))} 2 1 ∗ ≤D(xn , T (xn ))+q max{d(xn , x ),D(xn , T (xn )), (d(xn , x∗)+D(x∗, T (xn )))} 2 1 ∗ ≤ D(xn , T (xn ))+q max{d(xn , x ), D(xn , T (xn )), d(xn , x∗ )+ D(xn , T (xn ))} 2 1 = D(xn , T (xn )) + q max{D(xn , T (xn )), d(xn , x∗ ) + D(xn , T (xn ))}. 2 q+2 Hence d(xn , x∗ )≤ max{ 2(1−q) , 1+q}·D(xn , T (xn )). Then we immediately ∗ get d(xn , x )≤(1+q)·D(xn , T (xn ))→0 as n→+∞. The proof is complete. ¤

Remark 3.1. In particular, under the same conditions as before, the fixed point problem for the so-called Reich type operators (i.e. there exist a, b, c ∈

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R+ with a + b + c < 1 such that H(T (x), T (y)) ≤ ad(x, y) + bD(x, T (x)) + cD(y, T (y)), for all x, y ∈ X, see Ref. 14) is well-posed with respect to D. ´ c type operators is the following. Another well-possedness result for Ciri´ Theorem 3.2. Let (X, d) be a complete metric space and T : X → Pcl (X) ´ c type multivalued operator such that (SF )T 6= ∅. be a Ciri´ Then the fixed point problem is well-posed with respect to H. Proof. Let us show that (SF )T = {x∗ }. Let x∗ , y ∗ ∈ (SF )T be arbitary. Then d(x∗ , y ∗ )=H(T (x∗ ), T (y ∗ ))≤q max{d(x∗ , y ∗ ), D(x∗ , T (x∗ )), D(y ∗ , T (y ∗ )), 1 1 (D(x∗ , T (y ∗ ))+D(y ∗ , T (x∗ )))}=q max{d(x∗ , y ∗ ), (d(x∗ , y ∗ ) + d(y ∗ , x∗ ))} 2 2 = qd(x∗ , y ∗ ). Hence (SF )T = {x∗ }. The proof follows now from Theorem 3.1, via Remark 2.1. ¤ Recall now the notion of strict comparison function. A function ϕ : R+ → R+ is said to be a strict comparison function (see Ref. 5) if it is ∞ X strict increasing and ϕn (t) < +∞, for each t > 0. As consequence, we n=1

also have ϕ(t) < t, for each t > 0, ϕ(0) = 0 and ϕ is continuous in 0. Let (X, d), (Y, d0 ) be metric spaces and T : X → Pcl (Y ) be a multivalued operator. Then T is called a ϕ-contraction if ϕ : R+ → R+ is a strict comparison function and for all x1 , x2 ∈ X we have that Hd0 (F (x1 ), F (x2 )) ≤ ϕ(d(x1 , x2 )). Some well-possedness results for multivalued ϕ-contractions are given now. Theorem 3.3. Let (X, d) be a complete metric space and T : X → Pcl (X) be a multivalued ϕ-contraction such that card(FT ) ≤ 1. Suppose that the function ψ : R+ → R+ , given by ψ(t) := t − ϕ(t) is strict increasing and lim ψ(t) = +∞.

t→+∞

Then the fixed point problem is well-posed with respect to D.

Proof. From Wegrzyk’s theorem (see Ref. 15) we have that FT 6= ∅. Hence FT = {x∗ }. Let (xn )n∈N ⊂ X be such that D(xn , T (xn )) → 0, as n → +∞. Then d(xn , x∗ ) ≤ D(xn , T (xn ))+Hd (T (xn ), T (x∗ )) ≤ D(xn , T (xn ))+ ϕ(d(xn , x∗ )). Hence ψ(d(xn , x∗ )) ≤ D(xn , T (xn )). Since ψ −1 is continuous in 0, we have d(xn , x∗ ) ≤ ψ −1 (D(xn , T (xn ))) → 0, as n → +∞. The proof is complete. ¤

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Theorem 3.4. Let (X, d) be a complete metric space and T : X → Pcl (X) be a multivalued ϕ-contraction such that (SF )T 6= ∅. Suppose that the function ψ : R+ → R+ , given by ψ(t) := t − ϕ(t) is strict increasing and lim ψ(t) = +∞. t→+∞

Then the fixed point problem is well-posed with respect to H.

Proof. As before, we will show that (SF )T = {x∗ }. Let x∗ , y ∗ ∈ (SF )T . Then d(x∗ , y ∗ ) = H(T (x∗ ), T (y ∗ )) ≤ ϕ(d(x∗ , y ∗ )). Since ϕ is a strict comparison function we get that y ∗ = x∗ and hence (SF )T = {x∗ }. The conclusion follows from Theorem 3.3, via Remark 2.1. ¤ Remark 3.2. It is easy to see that, if the hypothesis of Theorem 3.2 or Theorem 3.4 are satisfied, then FT = (SF )T = {x∗ }. S. Reich in Ref. 14 introduced the following class of multivalued operators: Definition 3.2. Let (X, d) be a metric space and T : X → Pb (X). Then T is called a multivalued δ-contraction of Reich type, if there exist a, b, c ∈ R+ with a + b + c < 1 such that δ(T (x), T (y)) ≤ ad(x, y) + bδ(x, T (x)) + cδ(y, T (y)), for all x, y ∈ X. Reich proved that, if the metric space is complete, then FT = (SF )T = {x∗ }. The following multivalued generalized contraction condition was intro´ c, see Ref. 13. duced by Ciri´ Let (X, d) be a metric space and T : X → Pb (X). Suppose that there exists q ∈]0, 1[ such that, for each x, y ∈ X we have δ(T (x), T (y)) ≤ 1 q · max{d(x, y), δ(x, T (x)), δ(y, T (y)), (D(x, T (x)) + D(y, T (y)))}. 2 ´ c proved that if the metric space is complete then, FT = (SF )T = {x∗ }. Ciri´ An operator satisfying the above condition will be called a multivalued δ´ c type. contraction of Ciri´ Another well-posedness result with respect to H is: Theorem 3.5. Let (X, d) be a complete metric space and T : X → Pb (X) ´ c type. Then the fixed point problem be a multivalued δ-contraction of Ciri´ is well-posed with respect to Hd .

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´ c in Ref. 13 we have that FT = (SF )T = Proof. From a theorem by Ciri´ ∗ {x }. Let (xn )n∈N be a sequence in X such that H(xn , T (xn )) → 0, as n → +∞. Then d(xn , x∗ ) ≤ D(xn , T (xn )) + H(T (xn ), T (x∗ )) = D(xn , T (xn )) + δ(T (xn ), T (x∗ )) ≤ D(xn , T (xn )) 1 + q max{d(xn , x∗ ), δ(xn , T (xn )), δ(x∗, T (x∗ )), (D(xn , T (xn )) 2 + D(x∗, T (x∗ )))} = D(xn , T (xn )) 1 + q max{d(xn , x∗ ), δ(xn , T (xn )), D(xn , T (xn ))}. 2 1 ∗ Then we immediately get d(xn , x ) ≤ 1−q · H(xn , T (xn )) → 0, as n → +∞. The proof is complete. ¤ Since any multivalued δ-contraction of Reich type is a multivalued δ´ c type (with q := a + b + c), we immediately have: contraction of Ciri´ Theorem 3.6. Let (X, d) be a complete metric space and T : X → Pb (X) be a multivalued δ-contraction of Reich type. Then the fixed point problem is well-posed with respect to Hd . We finish this section by presenting two important remarks. Remark 3.3. Let (X, d) be a complete metric space and T : X → Pcl (X) be a multivalued operator such that FT = {x∗ }. Suppose that there exists a function γ : R+ → R+ such that γ(0) = 0 and γ is continuous in 0. Suppose that for each x ∈ X we have d(x, x∗ ) ≤ γ(D(x, T (x))). Then the fixed point problem is well-posed for T with respect to Dd . Indeed, let xn ∈ X, n ∈ N such that D(xn , T (xn )) → 0, as n → +∞. Then d(xn , x∗ ) ≤ γ(D(xn , T (xn ))) → 0, as n → +∞. ¤ By a similar argument we also have: Remark 3.4. Let (X, d) be a complete metric space and T : X → Pcl (X) be a multivalued operator such that (SF )T = {x∗ }. Suppose that there exists a function γ : R+ → R+ such that γ(0) = 0 and γ is continuous in 0. Suppose that for each x ∈ X we have d(x, x∗ ) ≤ γ(H(x, T (x))). Then the fixed point problem is well-posed for T with respect to Hd . 4. Asymptotic Regular Multivalued Operators Let (X, d) be a metric space and T : X → Pcl (X) be a multivalued operator. By definition, T is called asymptotic regular iff for each orbit (xn )n∈N of

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T , we have that d(xn , xn+1 ) → 0, as n → +∞. The purpose of this section is to study the connection between asymptotic regularity and well-posedness. In this respect, we have: Theorem 4.1. Let (X, d) be a metric space and T : X → Pcl (X) be a multivalued operator. Then the following assertions hold: (i) If T is a MP operator then each orbit of T converges to x∗ . (ii) If T is asymptotic regular and the fixed point problem is well posed for T with respect to D, then each orbit of T converges to x∗ . Proof. (i) Let (xn ))n∈N be an orbit of T starting from an arbitrary x ∈ X. Then, since T is a MP operator, we have: d(xn+1 , x∗ ) ≤ H(T n (x), x∗ ) → 0, as as n → +∞. (ii) Let (xn ))n∈N be an orbit of T starting from an arbitrary x ∈ X. Then D(xn , T (xn )) ≤ d(xn , xn+1 ) → 0, as n → +∞. Since the fixed point problem is well posed with respect to D we get the conclusion xn → x∗ , as n → +∞. ¤ The above results give rise to the following open question. Open question 4.1. Let (X, d) be a metric space and T : X → Pcl (X) be a multivalued operator such that FT = (SF )T = {x∗ }. In which conditions the following statements are equivalent: (a) T is a MP operator. (b) Each orbit of T converges to x∗ . (c) T is asymptotic regular and the fixed point problem is well posed for T with respect to D. 5. Non-self Multivalued Operators We will prove first the following abstract result. Lemma 5.1. Let (X, d) be a complete metric space and Y ∈ Pcl (X). Consider a multivalued operator T : Y → Pcl (X) such that FT = {x∗ }. Suppose that there exists a function γ : R+ → R+ such that γ(0) = 0 and γ is continuous in 0 and for each x ∈ Y we have d(x, x∗ ) ≤ γ(D(x, T (x))). Then the fixed point problem is well-posed for T with respect to Dd . Proof. Let xn ∈ Y , n ∈ N such that D(xn , T (xn )) → 0, as n → +∞. Then d(xn , x∗ ) ≤ γ(D(xn , T (xn ))) → 0, as n → +∞. ¤ In a similar way, we can prove:

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Lemma 5.2. Let (X, d) be a complete metric space and Y ∈ Pcl (X). Consider a multivalued operator T : Y → Pcl (X) such that (SF )T = {x∗ }. Suppose that there exists a function γ : R+ → R+ such that γ(0) = 0 and γ is continuous in 0. Suppose that for each x ∈ Y we have d(x, x∗ ) ≤ γ(H(x, T (x))). Then the fixed point problem is well-posed for T with respect to Hd . As an example we have: Theorem 5.1. Let (X, d) be a complete metric space and Y ∈ Pcl (X). Suppose that the multivalued operator T : Y → Pcl (X) is an a-contraction and (SF )T 6= ∅. Then the fixed point problem is well posed for T with respect to Dd and also with respect to Hd . Proof. Indeed, if x∗ ∈ (SF )T then, as before, we have FT = (SF )T = {x∗ }. Then for each x ∈ Y , we have d(x, x∗ ) ≤ D(x, T (x)) + H(T (x), T (x∗ )) ≤ 1 D(x, T (x)) + ad(x, x∗ ). As consequence d(x, x∗ ) ≤ 1−a D(x, T (x)) and the 1 conclusion follows from Lemma 5.1 and Lemma 5.2 (with γ(t) := 1−a t). ¤ Remark 5.1. It is not difficult to check that similar results hold: 1+b t. (a) if T is a Reich type operator with (SF )T 6= ∅, where γ(t) := 1−a ´ (b) if T is a Ciri´c type multivalued operator with (SF )T 6= ∅, where 1+q γ(t) := 1−q t. Theorem 5.2. Let (X, d) be a complete metric space, Y ∈ Pcl (X) and ´ c type. If (SF )T ={x∗ }, T : Y →Pb (X) be a multivalued δ-contraction of Ciri´ then the fixed point problem is well-posed with respect to Hd . Proof. Let us consider an arbitrary x ∈ Y . Then we have d(x, x∗ ) ≤ D(x, T (x)) + H(T (x), T (x∗ )) ≤ δ(x, T (x)) + δ(T (x), T (x∗ )) 1 ≤ q· max{d(x, x∗ ), δ(x, T (x)), δ(x∗, T (x∗ )), (D(x, T (x))+D(x∗, T (x∗ )))} 2 = δ(x, T (x)) + q · max{d(x, x∗ ), δ(x, T (x))} ≤ δ(x, T (x)) + q · (d(x, x∗ ), δ(x, T (x))). 1+q δ(x, T (x)), for each x ∈ Y . The conclusion follows 1−q 1+q from Lemma 5.2, taking γ(t) := t. ¤ 1−q

Hence d(x, x∗ ) ≤

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References 1. J.-P. Aubin and H. Frankowska, Set-Valued Analysis, (Birkhauser, Basel, 1990). 2. S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I-II, (Kluwer Academic Publishers, Dordrecht, 1997 and 1999). 3. W.A. Kirk, B. Sims (editors), Handbook of Metric Fixed Point Theory, (Kluwer Acad. Publ., Dordrecht, 2001). 4. A. Petru¸sel, Multivalued weakly Picard operators and applications, Scientiae Mathematicae Japonicae, 59 (2004), 167–202. 5. I.A. Rus, Generalized Contractions and Applications, (Cluj University Press, 2001). 6. I.A. Rus, Strict fixed point theory, Fixed Point Theory, 4 (2003), 177–183. 7. I.A. Rus, A. Petru¸sel, G. Petru¸sel, Fixed Point Theory 1950-2000: Romanian Contributions, (House of the Book of Science, Cluj-Napoca, 2002). 8. F.S. De Blasi, J. Myjak, Sur la porosit´e des contractions sans point fixe, C. R. Acad. Sci. Paris, 308 (1989), 51–54. 9. S. Reich and A.J. Zaslavski, Well-posedness of fixed point problems, Far East J. Math. Sci., Special Volume Part III, (2001), 393–401. 10. I.A. Rus, Picard operators and well-posedness of fixed point problems, to appear. 11. R.E. Smithson, Fixed points for contractive multifunctions, Proc. Amer. Math. Soc., 27 (1971), 192–194. 12. A. Petru¸sel, I.A. Rus, Multivalued Picard and weakly Picard operators, in Proc. Intern. Conf. on Fixed Point Theory and Appl., Valencia (Spain) 2003, (Yokohama Publishers, 2004), 207–226. ´ c, Fixed points for generalized multi-valued contractions, Mat. Vesnik, 13. L. Ciri´ 9 (1972), 265–272. 14. S. Reich, Fixed point of contractive functions, Boll. U.M.I., 5 (1972), 26–42. 15. R. Wegrzyk, Fixed point theorems for multifunctions and their applications to functional equations, Dissertationes Math., 201 (1982), 28 pp. 16. M. Frigon, A. Granas, R´esultats du type de Leray-Schauder pour les contractions multivoques, Topol. Math. Nonlinear Anal. 4 (1994), 197–208. 17. E. Matouˇskova, S. Reich, A.J. Zaslawski, Genericity in nonexpansive mapping theory, in Proc. of the 1st International School ”Advanced Courses of Mathematical Analysis”, A. Aizpuru-Tom´ as, ed., (C´ adiz, Spain, 2002), 81–98. 18. I.A. Rus, Picard operators and applications, Scientiae Mathematicae Japonicae, 58 (2003), 191–219. 19. A. Sˆınt˘ am˘ arian, Metrical strict fixed point theorems for multivalued mappings, Seminar on Fixed Point Theory, Preprint no. 3, (1997), 27–31.

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SOME REMARKS ON A NONLINEAR SEMIGROUP ACTING ON POSITIVE MEASURES EUGEN POPA Department of Mathematics, University Alexandru Ioan Cuza, Ia¸si, 700506, Romania E-mail: [email protected] Taking as starting point a recent paper of M. Baake1 , where a semigroup of positive, nonlinear operators acting on measures is studied, we present a few related results of potential theory. First, the associated resolvent is computed. It is found out that the resolvent coincides with the Laplace transform of the semigroup, as in the linear case. Let A be the infinitesimal generator of the semigroup. It is an everywhere defined (non-linear) operator. Moreover, it is proved that this resolvent has a natural extension to a larger set of (non-necessarily positive) measures, as the inverse of (αI − A) on positive measures. It is proved also that this resolvent satisfies the usual resolvent equation, as in the linear case. In order to achieve these computations, an extension of the formula established by Baake is needed. Next, a characterization for this type of resolvent is provided, in terms of the infinitesimal generator. A similar characterization is also given for the semigroup. Finally, the invariant and excessive elements with respect to the resolvent/semigroup, are described. Keywords: Resolvent; Semigroup; Measure.

1. Preliminaries Let n be a natural number and denote N := {0, 1, ..., n}; Xi , i = 0, 1, ..., n locally compact spaces; X := X0 × X1 × ...Xn . Let L := { 21 , 23 , ..., 2n−1 2 }. There exists a canonical correspondence between the strictly increasing subsets A := {α1 , ..., αp } ⊆ L, with αi < αi+1 and the partitions of N : NA := {A0 , A1 , ..., Ap }, where A0 := {0, ..., [α1 ]}; A1 := {[α1 ] + 1, ..., [α2 ]},..., Ap := {[αp ] + 1, . . . , n}. The ”recombination operator”, associated with such a partition is de-

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fined on a (signed, non-zero) measure x on X as: p 1 O RA (x) := πA (x) kωkp i=1 i Q where πAi is the projection on the space i∈A Xi . Let us recall the fundamental formula of Baake:

Theorem 1.1 (Ref. 2). For any ω positive measure and t ∈ [0, 1]: RA [tω + (1 − t)RA ω] = RA ω The evolution equation for which an explicit solution is provided is: ω˙ = ρA (RA − 1)ω (where ρA ≥ 0). 2. Results about the Associated Resolvent For sake of simplicity, we consider the case n = 1, but we do not ask to have locally compact spaces. Hence, let (X, X ) and (Y, Y) be measurable spaces. M resp. M+ denotes the set of all signed (resp. positive ) measures on (X × Y, X ⊗ Y) with finite total variation. We define a (non-linear) operator V : M → M by 1 V x := x1 ⊗ x2 kxk where x1 denotes the projection of the measure x ∈ M, x 6= 0 on X. With this operator, we associate a semigroup (of positive, non-linear operators) as Pt x = e−ct x + (1 − e−ct )V x From the expression of the semigroup (or directly from the equation) we get that the infinitesimal generator is: Pt x − x (e−ct − 1)(I − V ) = lim = c(V − I) t→0 t→0 t t Also, by taking the integral (in any sense), we get the Laplace transform of the semi-group (we cannot name it the resolvent yet!): Z ∞ Z ∞ Z ∞ e−αt Pt xdt = e−(α+c)t xdt + (e−αt − e−(α+c)t )V xdt = Ax := lim

0

0

0

=

c 1 x+ Vx α+c α(α + c)

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c 1 x+ V x. It is well deα+c α(α + c) fined on M. However, we want to consider only on positive measures Vα : M+ −→ M+ . Let us denote Mα := Vα (M+ ) ⊆ M+ . Hence, let us denote Vα (x) :=

Proposition 2.1. The resolvent associated with the semigroup (Pt )t>0 is (Vα )α>0 , i.e. Vα : M+ −→ Mα is the inverse of (αI − A) : Mα −→ M+ . Proof. Due to the fundamental formula: [(α + c)I − cV ] Vα ω = [(α + c)I − cV ]

=ω+

c c Vω− V α α

·

µ

1 c ω+ Vω α+c α(α + c)

¶ =

¸ c α ω+ Vω =ω α+c α+c

Hence Vα is injective and the relation: (αI − A) ◦ Vα = 1M+ holds. Of course, now: Vα ◦ (αI − A) = 1Mα follows.

¤

A simple calculation shows that also the resolvent equation is satisfied: Proposition 2.2. (i) For α, β > 0 we have: Vα = Vβ + (β − α)Vα ◦ Vβ (ii) The following forms of the resolvent equation also hold: [I − (α − β)Vα ] ◦ [I + (α − β)Vβ ] = I [I + (α − β)Vβ ] ◦ [I − (α − β)Vα ] = I (iii) The markovian property holds: αVα x(1) = x(1). (iv) V ◦ (αVα ) = V Proof.

·

1 c Vα Vβ x = I+ V α+c α(α + c) =

¸µ

1 c x+ Vx β+c β(β + c)

1 c x+ V x+ (α + c)(β + c) β(α + c)(β + c)

¶ =

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c + V αβ(α + c) +

µ

β c x+ Vx β+c β+c

¶ =

1 x+ (α + c)(β + c)

c c 1 Vx+ Vx= x+ β(α + c)(β + c) αβ(α + c) (α + c)(β + c) +

c(α + β + c) Vx αβ(α + c)(β + c)

Particularly Vα Vβ = Vβ Vα . Now Vβ x + (β − α)Vα Vβ x =

+

c β−α 1 x+ Vx+ x+ β+c β(β + c) (α + c)(β + c)

c(α + β + c)(β − α) V x = Vα x αβ(α + c)(β + c)

As αVα x =

ω + (β − α)Vα ω = ω +

α c x+ Vx α+c α+c

c(β − α) β+c c(β − α) c(β − α) ω+ Vω = ω+ Vω = α+c α(α + c) α+c α(α + c)

· ¸ β α(β + c) c(β − α) = ω+ Vω α β(α + c) β(α + c) we have: [I − (β − α)Vβ ] [ω + (β − ω)Vα ω] = ·

α+c c(β − α) = I− V β+c β(β + c) =ω+

¸·

β α

µ

α(β + c) c(β − α) ω+ Vω β(α + c) β(α + c)

¶¸ =

c(β − α) c(β − α) Vω− V ω = ω. α(β + c) α(β + c) ¤

Remark 2.1. If 0 < α < β then Mα ⊂ Mβ ⊂ M+ . Indeed, for each β+c c(β − α) ω ∈ M+ , let us choose ν := ω+ V ω. Then Vβ ν = Vα ω. α+c α(α + c)

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3. An Extension of the Resolvent We can do more. We will establish the injectivity of the operator αI − A on M+ . Let us denote Mα := (αI − A)(M+ ). Obviously M+ ⊆ Mα . Proposition 3.1. αI − A : M+ −→ Mα is a bijection. Proof. Denoting, for simplicity β := 1 + (α + c)ω 0 − cV ω 0 may be written as

α , the equality (α + c)ω − cV ω = c

β(ω − ω 0 ) = V ω − V ω 0 β(ω − ω 0 ) =

1 1 ω1 ⊗ ω2 − 0 ω 0 ⊗ ω20 ω(X × Y ) ω (X × Y ) 1

It suffices to consider this equality for rectangles A × B: β (ω(A × B) − ω 0 (A × B)) =

−

1 ω(A × Y )ω(X × B)− ω(X × Y )

1 ω 0 (A × Y )ω 0 (X × B) ω 0 (X × Y )

β (ω(A × B)ω(X × Y )ω 0 (X × Y ) − ω 0 (X × Y )ω 0 (A × B)ω(X × Y )) = = ω(A × Y )ω(X × B)ω 0 (X × Y ) − ω 0 (A × Y )ω 0 (X × B)ω(X × Y ) We get next: ω 0 (X × Y ) [βω(X × Y )ω(A × B) − ω(A × Y )ω(X × B)] = = ω(X × Y ) [βω 0 (X × Y )ω 0 (A × B) − ω 0 (A × Y )ω 0 (X × B)] hence: βω(X × Y )ω(A × B) − ω(A × Y )ω(X × B) = ω(X × Y ) =

βω 0 (X × Y )ω 0 (A × B) − ω 0 (A × Y )ω 0 (X × B) ω 0 (X × Y )

Taking A = X one obtains ω(X × B) = ω 0 (X × B); taking B = Y , one obtains ω(A×Y ) = ω 0 (A×Y ). Especially ω(1) = ω 0 (1). Finally ω(A×B) = ω 0 (A × B), hence ω = ω 0 . ¤ Corollary 3.1. Φα := (αI − A)−1 is the extension to Mα of Vα .

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Remark 3.1. If 0 < α < β then M+ ⊂ Mα ⊂ Mβ . Indeed, for each β+c c(β − α) ω ∈ M+ , let us choose ν := ω+ V ω. Then (βI − A)ω = α+c α(α + c) (αI − A)ν. In order to obtain an explicit formula for Φα , we need an extension of Baake’s formula. Lemma 3.1. Let us denote y := mx + nV x, with m, n ∈ IR and x ∈ M. We have: 2

V (mx + nV x) =

(mkxk + nx(1)) Vx kxkkyk

If x ≥ 0, then: V (mω + nV ω) =

(m + n)2 ω(1) Vω kyk

If moreover mω + nV ω ≥ 0, then: V (mω + nV ω) = (m + n)V ω As a particular case V (ω − V ω) = 0; more generally: ´ ³X ck (ωk − V ωk ) = 0. V Proof. Indeed, by definition: Vy =

1 kyk

µ ¶2 x(1) m+n kxkV x. kxk ¤

As an application of this formula, we obtain the explicit formula for Φα : Proposition 3.2. Φα x =

1 c kxk x+ Vx α+c α(α + c) x(1)

Proof. Since x ∈ Mα , we have a unique expression x = (α + c)ω − cV ω, with ω ∈ M+ . Hence x(1) = (α + c)ω(1) − cV ω(1) = αω(1) and it follows that ω 6= 0 ⇒ x(1) 6= 0. Let us confirm: 1 [(α + c)ω − cV ω] + Φα x = Φα (αI − A)ω = α+c

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+

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kxk c c V [(α + c)ω − cV ω] = ω − V ω+ α(α + c) x(1) α+c +

c kxk α2 Vω =ω α(α + c) x(1) ω(1)kxk

The converse composition (αI − A) ◦ Φα x = (αI − A)ω = x follows simply from the above computation and the notations.

¤

Proposition 3.3. Let 0 < α < β and x ∈ Mβ (⊂ Mα ). Then Φα (Φβ x) = Φβ (Φα x) and Φα x = Φβ x + (β − α)Φα (Φα x) . Proof. x ∈ Mβ is written uniquely as x = (β + c)ω − cV ω, with ω ∈ M+ . Since Φβ x = ω, we obtain: Φα (Φβ x) =

1 c ω+ Vω α+c α(α + c)

On the other hand: Φα x =

+

1 c kxk 1 x+ Vx= [(β + c)ω − cV ω] + α+c α(α + c) x(1) α+c

c kxk β+c c V [(β + c)ω − cV ω] = ω− V ω+ α(α + c) x(1) α+c α+c 2

+

c kxk [(β + c)ω(1) − cω(1)] β+c c(β − α) Vω = ω+ V ω. α(α + c) x(1) ω(1)kxk α+c α(α + c)

Now:

· Φβ (Φα x) = Φβ

·

1 c = I+ V β+c β(β + c) +

¸·

¸ β+c c(β − α) ω+ Vω = α+c α(α + c)

¸ β+c c(β − α) β+c ω+ Vω = ω+ α+c α(α + c) (α + c)(β + c)

c β 1 c c(β − α) V ω+ Vω = ω+ V ω = Φα (Φβ x) α(α + c)(β + c) β(β + c) α α+c α(α + c)

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hence we get finally: Φβ x + (β − α)Φα (Φβ x) = ω +

+

β−α c(β − α) β+c ω+ Vω = ω+ α+c α(α + c) α+c

c(β − α) V ω = Φα x. α(α + c) ¤

4. A Characterization of the Resolvent The resolvent has the same ”infinitesimal generator” as the semigroup: µ ¶ α c αc α(ω−αVα ω)=α ω− ω− Vω = (ω−V ω) −→ c(I−V )ω α+c α+c α+c Moreover, this operator satisfies the following equality: A [I − αA] = (1 − cα)A where α ∈ [0, c−1 ). Conversely, these properties characterize the resolvent (Vα )α>0 : Theorem 4.1. Let (Vα )α>0 be a family of (non-linear) operators, acting on positive finite measures on (X × Y, X ⊗ Y), such that the resolvent equation holds: Vα ω = Vβ ω + (β − α)Vα Vβ ω and there exists A(ω) := lim α(ω − αVα ω) α→+∞

(for each) ω which satisfies A [I − αA] = (1 − cα)A for α ∈ [0, c−1 ). Then the resolvent (Vα )α>0 is associated with the operator V := I+c−1 A i.e. 1 c Vα = I+ V α+c α(α + c) Proof. We show that (βI − A) ◦ Vβ = 1M+ , for any β > 0. Indeed: (βI − A)(Vβ ω) = βVβ ω − A(Vβ ω) = βVβ ω − lim α(Vβ ω − αVα Vβ ω) = α→+∞ ¸ · αβ α (αVα ω) − Vβ ω = = βVβ ω − lim α→+∞ α − β α−β = βVβ ω + ω − βVβ ω = ω

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Let us denote Wβ =

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c 1 I+ V . Then: β+c β(β + c) Wβ ◦ (βI − A) ◦ Vβ = Vβ = Wβ . ¤

A similar characterization holds for the semigroup. Let F0 be any set of numerical, positive, measurable functions, which separates M+ . Proposition 4.1. Let (Pt )t>0 be a semigroup of (non-linear) operators, acting on positive finite measures on (X × Y, X ⊗ Y), such that for each ω there exists Pt (ω) − ω A(ω) := lim t→0 t which satisfies A [I − αA] = (1 − cα)A for α ∈ [0, c−1 ). Suppose that: Pt ◦ Ps + e−ct I = e−ct Ps + Pt Suppose moreover that each map t 7→ Pt ω(f ) is right continuous on [0, +∞), for each ω and f ∈ F0 . Then the semigroup (Pt )t>0 is associated with the operator V := I + c−1 A i.e. Pt ω = e−ct ω + (1 − e−ct )V ω. Proof. It follows that Qt := e−ct I + (1 − e−ct )V is a semigroup. Moreover, if we define, for ω and f ∈ F0 fixed q(t) := Qt ω(f ˙ ), then q is derivable on [0, +∞) and q 0 (t) = Qt Aω(f ) = AQt ω(f ) = ce−ct (V − I)ω(f ). Let now define p(t) := Pt ω(f ). From the hypothesis, we obtain that p has a right derivative at each point from [0, +∞) and p0d (t) = ce−ct (V − I)ω(f ). Since both p(t) and q(t) tend to ω(f ) for t → 0, it follows that the two functions coincide. As F0 separates M+ , it follows that Pt = Qt . ¤ Remark 4.1. Both Pt and αVα are convex combinations of I and V . Each such map N satisfies a property of the type: N (λI + (1 − λ)N ) = µI + νN with µ and ν depending on N . More precisely, let N = mI +nV ; for N = Pt , α we have m = e−ct and n = 1 − e−ct ; while for N = αVα we have m = α+c c and n = . Now, the relation is: α+c N [(λ − m)I + (1 − λ)N ] = (1 − λm)N + m(λ − 1)I

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We remark that the sum of the coefficients in each member is 1−m; however, one of the coefficients may be negative. 5. About Excessive Elements The fact that a positive measure ω is invariant with respect to V , i.e. V ω = ω is obviously equivalent to the fact that it is invariant with respect to the resolvent (αVα ω = ω, ∀α > 0) or to the semigroup (Pt ω = ω, ∀t > 0). Also, the fact that a positive measure ω is excessive with respect to V , i.e. V ω ≤ ω is equivalent to the fact that it is excessive with respect to the resolvent (αVα ω ≤ ω, ∀α > 0) or to the semigroup (Pt ω ≤ ω, ∀t > 0). Indeed: e−ct ω + (1 − e−ct )V ω ≤ ω ⇐⇒ V ω ≤ ω resp. α c ω+ V ω ≤ ω ⇐⇒ V ω ≤ ω α+c α+c Proposition 5.1. A positive measure ω is invariant iff ω = µ ⊗ ν. 1 ω1 ⊗ ω2 . ω(1) Conversely, since π1 (µ ⊗ ν)=ν(1)µ, it follows that µ ⊗ ν is invariant. ¤

Proof. If ω is invariant, then ω =

Remark 5.1. Hence V (ε(x,y) ) = ε(x,y) . That means that if we try to associate with¢ V a kernel on functions, it would be the identity: V˜ f (x, y) = ¡ V (ε( x, y)) (f ) = f (x, y). In order to describe analogously the excessive measures, it seems necessary to select them from a smaller class, related to the product structure. Let us denote: M0 := {ω|ω is absolutely continuous with respect to ω1 ⊗ ω2 } Let us note some properties of this set. (i) M0 is a convex cone and µ⊗ν ∈ M0 (for any positive, finite measures µ and ν). (ii) Let ωn be an increasing sequence from M0 . If ω := supn ωn is a finite measure, then ω ∈ M0 . (iii) M0 is solid. More generally, let ω ∈ M0 and F be a positive, ω–integrable function. Then ω 0 := F.ω ∈ M0 . Indeed, we have ω10 = F1 .ω1 . If ω10 ⊗ω20 (A) = 0, then we split A into three parts, as follows: Let A1 := {x ∈ X|F1 (x)>0} and A2 := {y ∈ Y |F2 (y)>0}.

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We consider separately (A1 × Y ) ∩ A; (X × A2 ) ∩ A; and the difference to A. Now it is immediate that each of these subsets are ω 0 –negligible. As a conclusion, M0 contains any X positive measure, which is dominated Fn .(µn ⊗ νn ). by a (finite) measure of the form n

Within this class, the only excessive measures are the invariant ones: Theorem 5.1. Let ω ∈ M0 . Then ω is excessive iff ω = µ ⊗ ν. Proof. Since ω ∈ M0 we can write ω = F.(ω1 ⊗ ω2 ). As ω is V –excessive, we have Vω =

1 (F1 ⊗ F2 )(ω1 ⊗ ω2 ) ω(1)

Z (where F1 (x) :=

F (x, y)ν(dy)). Y

We get:

1 (F1 ⊗ F2 ) ≤ G, ω–a.e. Let us denote ω(1) G(x, y) := F (x, y) −

1 F1 (x)F2 (y) ω(1)

Z We have G ≥ 0, ω–a.e., while

G(x, y)ω1 (dx)ω2 (dy) = 0. Hence G = 0 X×Y

ω1 ⊗ ω2 –a.e. Finally ω = (f.ω1 ) ⊗ (g.ω2 ).

¤

A similar result holds in the topological case, for another class of measures. Proposition 5.2. Let us suppose that X and Y are locally compact spaces. A positive measure ω which does not charges any set with void interior is excessive iff ω = µ ⊗ ν. 1 ω1 ⊗ ω2 ≤ ω. From Radon–Nicodym’s ω(1) theorem, it follows that there exists a measurable function 0 ≤ F ≤ ω(1), such that ω1 ⊗ ω2 = F.ω. Further, the set [F 6= 0] differs by a ω–negligible set from supp ω1 × supp ω2 , since it has void interior. Hence, F = 0 on each of the sets (X \ supp ω1 ) × Y and X × (Y \ supp ω2 ). Also, we have: ω ((X \ supp ω1 ) × Y ) = 0, ω (X × (Y \ supp ω2 )) = 0. 1 if Hence, we can write ω = G.(ω1 ⊗ ω2 ), where G(x, y) = F (x, y) F (x, y) 6= 0 and G(x, y) = 0 elsewhere. Proof.

Let ω be excessive:

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Let us write once again that ω = G.(µ Z ⊗ ν) is V –excessive. We have π1 (G.(µ ⊗ ν)) = G1 .µ, where G1 (x) := G(x, y)ν(dy). Hence Y

1 (G1 ⊗ G2 )(µ ⊗ ν) Vω = ω(1) 1 (G1 ⊗ G2 ) ≤ G, ω–a.e. ω(1) The rest of the proof is as above.

We get:

¤

Remark 5.2. The condition (α + c)ω − cV ω ≥ 0 shows the interest to study the ”pseudo–excessive” elements: qV ω ≤ ω, where 0 < q < 1. For example, the measures of the form F.(µ⊗ν) are excessive if F is any positive, measurable, µ ⊗ ν–integrable function satisfying F ≥ C.F1 ⊗ F2 . Acknowledgment The author acknowledges support from grant CERES-2-CEx06-11-10; and partial support from grants: CNCSIS-GR214, CERES-CEx05-D11-23, CERES-2-Cex06-11-56. References 1. M. Baake: Recombination semigroups on measure spaces, Monatsh. Math., 146 (2005), 267–278. 2. M. Baake, E. Baake: An exactly solved model for mutation, recombination and selection, Can. J. Math., 55 (2003), 3–41. 3. E. Popa: Semidynamical systems on cones of positive measures, An. S ¸ tiint¸. Univ. Al.I. Cuza, Ia¸si, Sect¸. I.a, Mat., 49, 2 (2003).

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INTEGRAL INCLUSIONS IN BANACH SPACES USING HENSTOCK-TYPE INTEGRALS∗ B. SATCO Faculty of Electrical Engineering and Computer Science ”Stefan cel Mare” University Universitatii 13, 720229 Suceava E-mail: [email protected] In the present paper, we consider an Hammerstein integral inclusion, where the set-valued integral involved is of Henstock-type. An existence result is obtained via M¨ onch’s fixed point theorem, imposing a condition using a measure of noncompactness, as well as some uniform integrability conditions appropriate to Henstock integral. The Henstock integral is more general than classical integrals, therefore our result extends a large number of existence results in literature, given in single- or set-valued setting. Keywords: Hammerstein integral inclusion; Henstock integral; Set-valued integral.

1. Introduction In the last two decades, one can remark an increase of interest in studying differential and integral problems under hypothesis of integrability in a weaker sense than the classical ones (by classical meaning Lebesgue integral on R, respectively Bochner and Pettis integral in the vector case). More precisely, integrals of quite oscillating functions were taken into account. Thus, on the real line, significant results were obtained using the HenstockKurzweil integral (e.g. Chew and Flordeliza1 , Federson and Bianconi2 , Federson and T´aboas3 and Schwabik4 ) and then, in the general case of Banach spaces, similar problems were investigated under Henstock integrability assumptions (see Sikorska5 ) or imposing some Henstock-KurzweilPettis integrability conditions (e.g. Cich´on, Kubiaczyk and Sikorska6 and Satco7 ). ∗ The

research was supported by MEdC-ANCS, Research Contract CEEX No. 5954/ 18.09.2006.

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In the present work, we will consider in Banach spaces an integral that generalizes to set-valued case the Henstock vector integral. Using a characterization of Henstock integrable compact convex-valued multifunctions recently given by Di Piazza and Musial8 , we obtain an existence result for the Hammerstein integral inclusion Z 1 x(t) ∈ (H) k(t, s)F (s, x(s))ds, 0

where X is a separable Banach space, F : [0, 1] × X → P0 (X) and k : [0, 1] × [0, 1] → R. The result is obtained by applying a set-valued variant of M¨onch’s fixed point theorem given by O’Regan and Precup9 . It extends the results obtained in the study of similar problems in the set-valued case by O’Regan and Precup9 under Bochner integrability assumptions and by Castaing and Valadier10 in the Pettis setting. It also generalizes the single-valued case result obtained by Federson and Bianconi2 by considering Henstock-integrable functions. 2. Notations and Preliminary Facts We begin by recalling the Henstock-Kurzweil integral, a concept that extends the classical Lebesgue integral on the real line. A gauge δ on the unit interval [0, 1] provided with the σ-algebra Σ of Lebesgue measurable sets and with the Lebesgue measure µ is a positive function; a partition of n [0, 1] (that is a finite family (Ii , ti )i=1 of nonoverlapping intervals covering [0, 1] with the tags ti ∈ Ii ) is said to be δ-fine if, for each i ∈ {1, ..., n}, Ii ⊂ ]ti − δ(ti ), ti + δ(ti )[. Definition 2.1. A function f : [0, 1] → R is called Henstock-Kurzweil R1 (shortly, HK-) integrable if there exists a real, denoted by (HK) 0 f (t)dt, satisfying that, for every ε > 0, one can for any ¯ find a gauge δε such that, ¯ δε Z 1 n ¯X ¯ ¯ ¯ fine partition P = (Ii , ti )ni=1 of [0, 1], ¯ f (ti )µ(Ii ) − (HK) f (t)dt¯ < ε. ¯ ¯ 0 i=1

Let us remind the reader of the fact that Theorem 2.1. (Theorem 9.12 in Ref. 11) Any HK-integrable function is measurable and its primitive is continuous. The following notion was used in Ref. 11 in order to obtain convergence results:

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Definition 2.2. A collection K of real HK-integrable functions is said to be uniformly HK-integrable if, for any ε > 0, there is a gauge δε such that, for any δε -fine partition (Ii , ti )ni=1 and for any f ∈ K, one has ¯ ¯ n Z 1 ¯ ¯X ¯ ¯ f (ti )µ(Ii ) − (HK) f (t)dt¯ < ε. ¯ ¯ ¯ 0 i=1

We will make intervene a result that can be proved by a straightforward adaptation of the proof of Theorem 12.21 in Ref. 11: Lemma 2.1. Let (fn )n be an uniformly HK-integrable, pointwisely bounded sequence of real functions defined on [0, 1] and g : [0, 1] → R be a function of bounded variation. Then the sequence (gfn )n is uniformly HK-integrable. For more on this integral, we refer the reader to Ref. 11. Through this paper, X is a real separable Banach space with topological dual space X ∗ , B ∗ the closed unit ball of X ∗ and P0 (X) (Pc (X), resp. Pkc (X)) stands for the family of nonempty (nonempty and convex, resp. nonempty, compact and convex) subsets of X. D denotes the Hausdorff distance on Pkc (X). Let us now recall the set-valued Henstock integral (introduced in Ref. 8): Definition 2.3. A set-valued function F : [0, 1] → Pkc (X) is said to be Henstock integrable in Pkc (X) if one can find a set I ∈ Pkc (X) such that, for every ε > 0, there is a gauge δε satisfying, for any δε -fine partition n ((ai , bi ), ti )i=1 of [0, 1], the inequality Ã n ! X D I, F (ti )(bi − ai ) < ε. i=1

We denote I by (H) [0, 1].

R1 0

F (t)dt and call it the Henstock integral of F on

In the particular case when F is vector-valued, we obtain the vector Henstock integral, which is the straightforward generalization of HenstockKurzweil integral (see Ref. 12). For this vector integral, the Definition 2.2 (where the norm takes the place of modulus) gives the notion of uniformly Henstock-integrable family of functions. On the space of all Henstock-integrable X-valued functions, ° ° we can conZ b ° ° ° ° sider the Alexiewicz norm, kf kA = sup °(H) f (s)ds°. ° ° [a,b]⊂[0,1] a It was proved that

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Proposition 2.1. (Proposition 1 in Ref. 8) A multifunction F : [0, 1] → Pkc (X) is Henstock-integrable if and only if each countable subset of the collection {σ (x∗ , F (·)) , x∗ ∈ B ∗ } is uniformly HK-integrable. By SΓH we denote the family of Henstock-integrable selections of Γ. The following characterization was also given (let us recall, from Ref. 8, that a set-valued function F is said to be scalarly HK-integrable if, for each x∗ ∈ X ∗ , its functional support σ(x∗ , F (·))) is HK-integrable): Theorem 2.2. (Theorem 2 in Ref. 8) Let F : [0, 1] → Pkc (X) be a scalarly HK-integrable multifunction. Then the following conditions are equivalent: i) F is Henstock-integrable in Pkc (X); ii) SFH is nonempty and for every f ∈ SFH one can find a compact convex-valued Pettis-integrable function G such that, for each t ∈ [0, 1], F (t) = f (t) + G(t); iii) every measurable selection of F is Henstock-integrable. We will make use of the Hausdorff measure of noncompactness β which is defined, for any A⊂X, by the infimum of all r > 0 such that there is a finite number of ball covering A, of radius smaller than r. For its properties we refer the reader to Ref. 14. Let us end this introductory section by recalling a set-valued version of M¨onch’s fixed point theorem that will be used in the sequel. This result is obtained from Theorem 3.1 in Ref. 9, with the remark that the proof is quite similar: Theorem 2.3. Let D be a closed convex subset of a Banach space X and let G : D → Pc (D) satisfy the following conditions: i) Graph(G) is closed; ii) for some x0 ∈D, every M ⊂D such that M =conv ({x0 } ∪G(M )) is relatively compact. Then G has a fixed point in D. 3. Hammerstein Integral Inclusions Via Henstock Integral In what follows, consider the Hammerstein integral inclusion Z 1 x(t) ∈ (H) k(t, s)F (s, x(s))ds, 0

where F : [0, 1] × X → Pkc (X) and k : [0, 1] × [0, 1] → R.

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In order to establish the existence of continuous solutions, we will make use of some auxiliary lemmas. It is known that, on the real line, the HK-integrability is preserved under multiplication by real functions of bounded variation (whose space is denoted by BV ). We show in the sequel that the same is true for the Henstock integral in Banach spaces: Lemma 3.1. Let F : [0, 1] → Pkc (X) be Henstock-integrable and g be a real function of bounded variation on [0, 1]. Then gF is Henstock-integrable. Proof. By Proposition 2.1, each countable subset of {σ (x∗ , F (·)) , x∗ ∈ B ∗ } is uniformly HK-integrable. Moreover, any such countable family is a pointwisely bounded sequence (since |hx∗ , F (t)i| ≤ kF (t)k, for every x∗ ∈ B ∗ ). Then we are able to apply Lemma 2.1 in order to deduce that each countable subset of the collection {σ (x∗ , g(·)F (·)) , x∗ ∈ B ∗ } is uniformly HK-integrable. Again by Proposition 2.1, it follows that gF is Henstockintegrable. ¤ In particular, this result is available in the vector case. We also obtain the following Lemma 3.2. Let F be a k · kA -bounded family of Henstock-integrable functions on [0, 1] and let k : [0, 1] × [0, 1] → R be such that: i) for every t ∈ [0, 1], k(t, ·) ∈ BV ([0, 1]); ii) the function t 7→ k(t, ·) is continuous with respect to the norm kf kBV = |f (0)| + V (f ) on BV ([0, 1]), where V (f ) denotes the total variation of f. Then

½

Z (H)

¾

1

k(·, s)y(s)ds, y ∈ F 0

is equi-continuous. Proof. Let us remark that, as k is continuous on a compact set, it is bounded, therefore sup kk(t, ·)kBV < ∞. Fix c ∈ [0, 1] and ε > 0. There is t∈[0,1]

δε,c >0 such that, for any t with |t − c|<δε,c , we have kk(t, ·) − k(c, ·)kBV <

ε . 2sup kykA y∈F

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Then

° Z ° °(H) °

° ° k(t, s)y(s)ds − (H) k(c, s)y(s)ds° ° 0 0 ° ° Z 1 ° ° =° (k(t, s)−k(c, s)) y(s)ds° °(H) ° 0 ¯ ¯ Z 1 ¯ ¯ ∗ ¯ (k(t, s) − k(c, s)) hx , y(s)i ds¯¯ = sup ¯(HK) Z

1

x∗ ∈B ∗

1

0

whence, applying an integration by parts result, we obtain that ° ° Z 1 Z 1 ° ° °(H) ° k(t, s)y(s)ds − (H) k(c, s)y(s)ds ° ° 0 0 ¯ ¯ Z 1 ¯ ¯ hx∗ , y(s)i ds¯¯ ≤ sup ¯¯[k(t, 1) − k(c, 1)] (HK) x∗ ∈B ∗

¯Z ¯ + sup ¯¯ ∗ ∗ x ∈B

0

1

µ

Z (HK)

0

s

∗

hx , y(τ )i dτ 0

¶

¯ ¯ d (k(t, s) − k(c, s))¯¯

≤ |k(t, 1) − k(c, 1)| kykA + kykA kk(t, ·) − k(c, ·)kBV ≤ 2kykA kk(t, ·) − k(c, ·)kBV < ε.

¤

Lemma 3.3. Let F be an uniformly Henstock-integrable and pointwisely © ª of X-valued functions defined on [0, 1]. Then R · bounded family (H) 0 f (t)dt, f ∈ F is an equi-continuous family and F is k·kA -bounded. Proof. Fix c ∈ [0, 1] and ε > 0. By the pointwise boundedness assumption, there exists Mc < +∞ such that, for all f ∈ F, kf (c)k ≤ Mc and, thanks to the uniform Henstock integrability hypothesis, there also exists a gauge δε satisfying ° µ ¶¸° Z ci+1 Z ci k · °X ° ° ° f (ti )(ci+1 − ci ) − (H) f (t)dt − (H) f (t)dt ° < ε, ° ° ° 0 0 i=1

for each f ∈ F and each δε -fine partition.

¶ ε , Then every t ∈ [0, 1] with |t − c| ≤ ηε, c , where ηε, c = min δε (c), Mc satisfies that the interval (t, c) with the corresponding tag c is an element of a δε -fine partition of [0, 1]. Then, by Lemma 11 in Ref. 12, µ

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° ° Z t Z c ° ° °(H) ° f (s)ds − (H) f (s)ds ° ° 0 0 ° ° Z t Z c ° ° ° f (s)ds − (H) f (s)ds − f (c)(t − c)° ≤ °(H) ° + kf (c)(t − c)k 0

≤ 2ε,

0

∀f ∈ F ,

and so, the equi-continuity is verified. In order to obtain the k·kA -boundedness it suffices to cover the compact [0, 1] with open intervals like before and extract a finite sub-cover. ¤ The following classical result will be used (in the case when the weak topology is considered, a proof can be found in Ref. 7): Lemma 3.4. For any sequence (y n )n of measurable selections of a Pkc (X)-valued measurable multifunction Γ, there exists a sequence zn ∈conv {y m , m≥n} a.e. convergent to a measurable y. We give now the main result of the paper. Theorem 3.1. Let F : [0, 1] × X → Pkc (X) and k : [0, 1] × [0, 1] → R satisfy the following conditions: 1) there exists a positive constant c such that, for any bounded A ⊂ X, β(F ([0, 1] × A)) ≤ cβ(A); 2) for every x ∈ C([0, 1], X), the multifunction F (·, x(·)) is Henstockintegrable in Pkc (X); n o 3) the family F = y ∈ SFH(·,x(·)) , x ∈ C ([0, 1], X) is uniformly Henstockintegrable and pointwisely bounded; 4) for any t ∈ [0, 1], F (t, ·) is upper semicontinuous; 5) for every t∈[0, 1], k(t, ·)∈BV ([0, 1]), the function t 7→ k(t,·) is conti1 nuous with respect to the norm k·kBV and M = sup kk(t, ·)kBV < . c t∈[0,1] Z 1 Then the inclusion x(t) ∈ (H) k(t, s)F (s, x(s))ds has a continuous solution.

0

Proof. By hypothesis 3) and Lemma 3.3 it follows that we are able to find a positive° constant M such that, for any t ∈ [0, 1] and y ∈ F , ° Rt ° ° °(H) 0 y(s)ds° ≤ M . Consider now the closed and convex set

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)

( K=

x ∈ C([0, 1], X) : sup kx(t)k ≤ 2M M

.

t∈[0,1]

The multifunction Ξ : K → P0 (K) defined by ½ ¾ Z 1 H Ξ(x) = y ∈ K : y(t)=(H) k(t, s)y(s)ds, ∀t∈[0, 1], where y ∈ SF (·,x(·)) 0

has nonempty and convex values. Indeed, as F (·, x(·)) is measurable, it has at least one measurable selection, which is, by Theorem 2.2, Henstockintegrable. We prove, in the first place, that Graph(Ξ) is C ([0, 1], X)-closed. Consider (xn , yn )n ⊂ Graph(Ξ) uniformly convergent to (x, y). For every n ∈ N, we can find y n ∈ SFH(·,xn (·)) such that, for each t ∈ [0, 1], R1 yn (t) = (H) 0 k(t, s)y n (s)ds. Since xn → x uniformly, for any s ∈ [0, 1], the set {xn (s), n ∈ N} is relatively compact. Hypothesis 1) gets that β(F ({s} × {xn (s), n[∈ N})) ≤ cβ({xn (s), n ∈ N}) = 0, which means that the set Γ(s) = F (s, xn (s)) is relatively compact. Using Lemma 3.4, we are able n∈N

to find a measurable function y and a sequence zn ∈ conv {y m , m ≥ n} such that (zn )n be a.e. convergent to y. On the other hand, from 5) it follows that, for any neighborhood V of the origin, there exists ns,V ∈ N such that, for every n ≥ ns,V , F (s, xn (s)) ⊂ F (s, x(s)) + V . Obviously, the preceding y is a measurable selection of the Pkc (X)-valued Henstock-integrable multifunction F (·, x(·)), so it is Henstock-integrable. By 3), the sequence (zn (·))n is uniformly Henstock-integrable and pointwisely bounded. We are then able to apply a passage to the limit result R(Theorem 4 in Ref. R t 15) and deduce that kzn − ykA → 0, therefore t (H) 0 zn (s)ds → (H) 0 y(s)ds uniformly. That is to say that, for every ε>° 0, there exists nε ∈ N such that, ° for any n ≥ nε and any t ∈ [0, 1], one Rt Rt ° ° has °(H) 0 zn (s)ds − (H) 0 y(s)ds° < ε. From the integration by parts it follows, after some computations similar to those in the proof of Lemma 3.2, that ° ° Z 1 Z 1 ° ° °(H) k(t, s)zn (s)ds − (H) k(t, s)y(s)ds° ° ° 0 0 ° ° Z 1 ° ° ≤° zn (s) − y(s)ds° °k(t, 1)(H) ° 0 °Z 1 µ ° ¶ Z s ° ° ° ≤ 2M ε, +° (H) z (τ ) − y(τ )dτ dk(t, s) n ° ° 0

0

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for every n ≥ nε and every ³ t R∈ [0, 1]. ´ 1 In other words, (H) 0 k(·, s)zf (s)ds uniformly converges to n R1 (H) 0 k(·, s)y(s)ds. At the same time, we have that Z

1

k(t, s)zn (s)ds → y(t), 0

R1 whence y(t) = (H) 0 k(t, s)y(s)ds, for all t. Thus, Graph(Ξ) is closed. Finally, consider an arbitrary set M ⊆ K, with M = conv({x0 } ∪ Ξ(M) and prove that M is relatively compact. The equi-continuity is satisfied by Lemma 3.2. It suffices to show that, for every t ∈ [0, 1], M(t) is relatively compact, that is, β(M(t)) = 0 and then, applying Ascoli’s theorem, will follow that M is relatively compact. Using a mean result and hypothesis 1), one obtains that β(M(t)) = β (Ξ(M)(t)) ≤ M β (F ([0, t] × M([0, t]))) ≤ M cβ(M([0, t])) ≤ M cβ(M([0, 1])), whence β(M([0, 1])) ≤ M cβ(M([0, 1])). Since, by 5), M c < 1, one deduces that β(M([0, 1])) = 0, and so, for each t, β(M(t)) = 0. The assumptions of Theorem 2.3 are satisfied, therefore Ξ has a fixed point, which is a continuous solution to our integral inclusion. ¤ In the particular case k(t, s) = 0, ∀s ≥ t in [0, 1], we obtain the following Corollary 3.1. Under the assumptions of Theorem 3.1, the Volterra integral inclusion Z

t

x(t) ∈ (H)

k(t, s)F (s, x(s))ds 0

has a continuous solution on [0, 1]. Remark 3.1. Our main theorem extends the existence result for Hammerstein inclusions obtained in Ref. 9 under Bochner integrability assumptions, as well as the result proved for the Volterra equations in Ref. 2 in Henstock integrability setting, under contraction assumptions, using the successive approximations method. It also generalizes R t the existence result given in Ref. 10 for the integral inclusion x(t) ∈ 0 F (s, x(s))ds in the Pettis integrability setting.

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References 1. T.S. Chew, F. Flordeliza, On x0 = f (t, x) and Henstock-Kurzweil integrals, Differential Integral Equations, 4, 4 (1991), 861–868. 2. M. Federson and R. Bianconi, Linear integral equations of Volterra concerning Henstock integrals, Real Anal. Exch., 25, (1) (1999/2000), 389–418. 3. M. Federson, P. T´ aboas, Impulsive retarded differential equations in Banach spaces via Bochner-Lebesgue and Henstock integrals, Nonlinear Anal., 50, 3 (2002), Ser. A: Theory Methods, 389–407. 4. S. Schwabik, The Perron integral in ordinary differential equations, Differential Integral Equations, 6, 4 (1993), 863–882. 5. A. Sikorska-Nowak, Retarded functional differential equations in Banach spaces and Henstock-Kurzweil integrals, Demonstratio Math., 35, 1 (2002), 49–60. 6. M. Cich´ on, I. Kubiaczyk, A. Sikorska, The Henstock-Kurzweil-Pettis integrals and existence theorems for the Cauchy problem, Czechoslovak Math. J., 54(129) (2004), 279–289. 7. B. Satco, Volterra integral inclusions via Henstock-Kurzweil-Pettis integral, to appear in Discuss. Math. Differ. Incl. Control Optim. 8. L. Di Piazza and K. Musial, A decomposition theorem for compact-valued Henstock integral, Monatshefte f¨ ur Mathematik, 148 (2006), no. 2, 119–126. 9. D. O’Regan and R. Precup, Fixed Point Theorems for Set-Valued Maps and Existence Principles for Integral Inclusions, J. Math. Anal. Appl., 245 (2000), 594–612. 10. C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math., 580, (Springer-Verlag, Berlin, 1977). 11. R.A. Gordon, The Integrals of Lebesgue, Denjoy, Perron and Henstock, Grad. Stud. in Math., 4 (1994). 12. S.S. Cao, The Henstock integral for Banach-valued functions, SEA Bull. Math., 16 (1992), 35–40. 13. L. Di Piazza and K. Musial, Set-valued Kurzweil-Henstock-Pettis integral, Set-Valued Anal., 13, 2 (2005), 167–179. 14. A.A. Mitchell and C. Smith, An existence theorem for weak solutions of differential equations in Banach spaces, Nonlinear equations in abstract spaces, in Proc. Internat. Sympos., Univ. Texas, Arlington, Tex. (1977), (Academic Press, New York, 1978), 387–403. 15. L. Di Piazza, Kurzweil-Henstock type integration on Banach spaces, Real Anal. Exch., 29, (2) (2003/2004), 543–556. 16. H. M¨ onch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal., 4 (1980) 985–999.

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GOOD INTERMEDIATE-RANK LATTICE RULES BASED ON THE WEIGHTED STAR DISCREPANCY VASILE SINESCU and STEPHEN JOE Department of Mathematics, University of Waikato, Hamilton, New Zealand E-mail: [email protected], E-mail: [email protected] We study the problem of constructing good intermediate-rank lattice rules in the sense of having a low weighted star discrepancy. The intermediate-rank rules considered here are obtained by “copying” rank-1 lattice rules. We show that such rules can be constructed using a component-by-component technique and prove that the bound for the weighted star discrepancy achieves the optimal convergence rate. Keywords: Intermediate-rank lattice rules; Weighted star discrepancy; Component-by-component construction.

1. Introduction Integrals over the d-dimensional unit cube given by Z Id (f ) = f (x) dx [0,1]d

may be approximated by rank-1 lattice rules. These are quadrature rules defined by ¾¶ n−1 µ½ 1X kz f . (1) n n k=0

d

Here, z ∈ Z is the generating vector having all the components conveniently assumed to be relatively prime with n, while the braces around the vector indicate that we take the fractional part of each component of the vector. In general terms, the “rank” of a lattice rule represents the minimum number of generating vectors required to produce the quadrature points. For d-dimensional integrals, lattice rules may have rank up to d. Further

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details on the definition and the representation of lattice rules can be found in Ref. 1 and 2. In some practical applications, the first variables are the most important. Hence, it seems natural to consider lattice rules obtained by “copying” rank-1 lattice rules. If ` ≥ 1 is an integer satisfying gcd(`, n) = 1 and r is a fixed integer taken from the set {0, 1, ..., d}, then we can define the following lattice rule: ¾¶ `−1 `−1 n−1 X X µ½ kz 1 X (m1 , ..., mr , 0, ..., 0) QN,d (f )= r ... f + . (2) ` n m =0 m =0 n ` r

1

k=0

For r ≥ 1, this lattice rule is a rank-r lattice rule or “intermediate-rank lattice rule”. Let’s remark that the lattice rule (2) has N = `r n distinct points and is obtained by copying the rank-1 lattice rule (1) ` times in each of the first r dimensions. It is easy to observe that when r = 0 or ` = 1, the lattice rule (2) is reduced to the rank-1 lattice rule (1). Such intermediate-rank lattice rules have been previously studied in Refs. 3,4, and 1. Here, in order to construct these intermediate-rank lattice rules, we employ the “weighted star discrepancy” as a measure of “goodness”. An unweighted star discrepancy (corresponding to an L∞ maximum error) has been previously used in Ref. 5 and in more general works such as Ref. 6 or 1, while the weighted star discrepancy has been used in Ref. 7,8, and 9. 2. Bounds for the Weighted Star Discrepancy Let’s observe first that the quadrature points of the lattice rule (2) can be rewritten as: ½ ¾ kz (m1 , ..., mr , 0, ..., 0) y + = t, n ` N where y t /N , 0 ≤ t ≤ N − 1, are in [0, 1)d . Of course, these points are a reordering of the N -points of the rank-r lattice rule defined by (2). Hence the lattice rule (2) may be rewritten as QN,d (f ) =

N −1 1 X ³ yt ´ . f N t=0 N

In order to introduce the weighted star discrepancy, let the set of quadrature points {y t /N, 0 ≤ t ≤ N −1} be denoted by PN . Then the star discrepancy of PN is defined by ∗ DN (PN ) :=

sup |discr(x, PN )| ,

x∈[0,1)d

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where discr(x, PN ) is the local discrepancy given by d

discr(x, PN ) :=

A([0, x), PN ) Y − xj . N j=1

Here A([0, x), PN ) represents the counting function, namely the number of points in PN which lie in [0, x) with x = (x1 , x2 , ..., xd ). The star discrepancy gives a measure of the uniformity of the distribution of the quadrature points. Let now u be an arbitrary subset of D := {1, 2, ..., d − 1, d} and denote its cardinality by |u|. For the vector x ∈ [0, 1]d , let xu denote the vector from [0, 1]|u| containing the components of x whose indices belong to u. By (xu , 1) we mean the vector from [0, 1]d whose j-th component is xj if j ∈ u and 1 if j 6∈ u. Now let us introduce a set of non-increasing positive weights {γj }∞ j=1 which describes the decreasing importance of the successive coordinates xj and set Y γu = γj . j∈u

From Zaremba’s identity (see for instance Ref. 10 or 11) and by applying H¨older’s inequality for integrals and sums, we obtain |QN,d (f ) − Id (f )|   Z X ≤ sup γ u |discr((xu ,1), PN )| sup γ −1 u u⊆D

|u|

u⊆D xu ∈[0,1]

¯ |u| ¯ ¯∂ ¯ ¯ f ((xu ,1))¯¯ dxu . ¯ [0,1]|u| ∂xu

∗ Thus we can define a weighted star discrepancy DN,γ (PN ) by X ∗ (PN ) := γ u sup |discr((xu , 1), PN )| . DN,γ u⊆D

(3)

xu ∈[0,1]|u|

From Ref. 6, we make use of Theorem 3.10 and Lemma 5.21, together with the arguments leading to Theorem 5.6, to obtain the following inequality: sup

|discr((xu , 1), PN )| ≤ 1 − (1 − 1/N )|u| +

xu ∈[0,1]|u|

where

 N −1 1 X Y RN (PN , u) = 1+ N t=0 j∈u

X0 N −N 2
RN (PN , u) , 2

(4)

 e2πihyt,j /N  − 1. |h|

In the above yt,j is the j-th coordinate of y t , while the 0 in the sum indicates we omit the h = 0 term.

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Let us mention here that from the general theory on lattice rules (for example, see Ref. 6 or 1), it will follow that RN (PN , u) ≥ 0 for any u ⊆ D. From (3) and (4), we see that the general weighted star discrepancy satisfies the inequality µ ¶ X RN (PN , u) ∗ (PN ) ≤ γ u 1 − (1 − 1/N )|u| + DN,γ . (5) 2 u⊆D

Further bounds on the weighted star discrepancy may be obtained by making use of (5). If the weights γj are summable, that is, ∞ X

γj < ∞,

j=1

then from Ref. 8, Lemma 1, we obtain: ∞ ³ ´ max(1, Γ) Y X max(1, Γ) P∞ γ u 1 − (1 − 1/N )|u| ≤ (1 + γj ) ≤ e j=1 γj , rn N ` j=1 u⊆D

where Γ :=

∞ X j=1

γj < ∞. 1 + γj

The complete proof of this result may be found in Ref. 8. Thus we obtain ³ ´ X γ u 1 − (1 − 1/N )|u| = O(n−1 ), (6) u⊆D

where the implied constant depends on `, r and the weights. We have from Ref. 8 that X u⊆D

 N −1 d 1 X Y γ u RN (PN , u) = βj + γj N t=0 j=1

where βj = 1 + γj . If we set e2N,d (z) =

X

X0 N −N 2
 d e2πihyt,j /N  Y − βj , |h| j=1

γ u RN (PN , u),

u⊆D

then we see that we have

 N −1 Y d X 1 βj + γj e2N,d (z) = N t=0 j=1

X0 N −N 2
 d e2πihyt,j /N  Y − βj . |h| j=1

(7)

Let’s remark that the dependency on z in e2N,d (z) makes sense as the vectors y t actually depend on z.

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In research papers such as Refs. 12 or 3, it was proved that when n is prime, the quantity (7) is identical to a quadrature error obtained from applying a rank-1 lattice rule to a certain integrand. Working with such a quadrature error simplifies in general the analysis of the problem and also has some computational advantages. Indeed, from (2) and (7) we have   r n−1 `−1 `−1 Y 2πih(kzj /n+mj /`) X X X X 0 1 1 e βj + γj  e2N,d (z) = ... n `r m =0 m =0 j=1 |h| N N k=0

×

r

d Y

−

1



2



βj + γj

j=r+1

X0 N −N 2
 n−1 d 1 X (`,r) Y  βj + γj Ak = n j=r+1 k=0

2

d e2πihkzj /n  Y − βj |h| j=1

X0 N −N 2
 d e2πihkzj /n  Y − βj , |h| j=1

where (`,r) Ak

 `−1 `−1 Y r X 1 X βj + γj = r ... ` m =0 m =0 j=1 r

1

X0 N −N 2
 e2πih(kzj /n+mj /`)  . |h|

(`,r)

Before expanding Ak

, let’s also observe that  `−1 ³  `, h ≡ 0 (mod `), ´m  X 2πih/` e =   0, otherwise. m=0

Then we have (`,r) Ak

  `−1 r X0 e2πih(kzj /n+m/`) Y 1 X  = βj + γj ` |h| N N m=0 j=1 − 2
r Y 1  `βj + `γj `  j=1

X0 N −N 2
e2πihkzj /n    |h| 

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=

r Y j=1 r Y



 X0

βj + γj

−N 2



βj + γj = ` j=1

e

X0 N N
2πiq`kzj /n

|q|`



 e2πih`kzj /n  . |h|

Going back to the expression of e2N,d (z), we obtain   n−1 d d 2πihkˆ zj /n X Y X 0 1 e   Y e2N,d (z) = β + γ ˜ − βj .   j j n |h| ˜ ˜ j=1 j=1 N N k=0

−

j 2

(8)

j 2

In the above, the following notations have been introduced: ½ γj /`, 1 ≤ j ≤ r, γ˜j = γj , r + 1 ≤ j ≤ d. Next, ½ ˜j = N

N/` = `r−1 n, 1 ≤ j ≤ r, N, r + 1 ≤ j ≤ d.

ˆ = (ˆ Finally, z z1 , zˆ2 , ..., zˆd ), with ½ `zj , 1 ≤ j ≤ r, zˆj = zj , r + 1 ≤ j ≤ d. Then by denoting



fN (x) =

d Y

 X0

 βj + γ˜j

j=1

−

˜ N j 2

˜ N j 2

e2πihxj  , |h|

it is easy to observe that e2N,d (z)

µ ¶ Y n−1 d 1X k ˆ − = fN z βj . n n j=1 k=0

e2N,d (z)

(which is based on a rank-r lattice rule with Now it is clear that N = `r n points) can be obtained from applying a modified n-point rank-1 lattice rule to fN . Next, we are looking to obtain a result for the mean of the quantities Such a result, together with (5) and (6), will allow us to deduce a certain bound for the weighted star discrepancy. This mean will be taken

e2N,d .

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ˆ . Because z ˆ is known when z is known, the mean over all possible values of z will be actually considered for all possible values for z. Each component zj , 1 ≤ j ≤ d, of the vector z can be taken from the set Zn := {1, 2, ..., n−1} because we only take the fractional part of each component of the vector. Thus, for prime n, the mean MN,d,γ is defined by MN,d,γ :=

X 1 e2N,d (z). d (n − 1) d z∈Zn

An expression for MN,d,γ is given in the next theorem. Theorem 2.1. If n is prime, ` is a positive integer such that gcd(`, n) = 1 and r is an integer chosen such that 1 ≤ r ≤ d, then MN,d,γ =

+

d ´ 1 Y³ βj + γ˜j SN˜j n j=1 d µ d ´¶ Y n−1 Y γ˜j ³ βj − SN˜j − SN˜j /n − βj , n j=1 n−1 j=1

(9)

where Sn =

X0 n −n 2 ≤h< 2

1 . |h|

Proof. Using the definition of the mean and separating out the k = 0 term in (8), we obtain: MN,d,γ =

d d ´ Y 1 Y³ βj + γ˜j SN˜j + ΘN,γ − βj , n j=1 j=1

(10)

where µ ¶ X n−1 X 1 k ˆ z f N d n(n − 1) n d k=1 z∈Zn   n−1 d n−1 1 XY 1 X  =  βj + γ˜j n n − 1 z =1 j=1

ΘN,γ =

k=1

j

 =

1 n

n−1 d XY k=1 j=1

 βj +

n−1 X

γ˜j n−1z

j =1

 X0 −

˜ N j 2

X0 −

˜ N j 2

e

e ˜ N j 2

2πihkˆ zj /n

|h|

˜ N j 2



2πihkˆ zj /n

|h|

 .

 

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For 1 ≤ k ≤ n − 1 and for any j ≥ 1, consider now n−1 X

Tn (k, j) =

zj =1

X0 ˜ N − 2j

e2πihkˆzj /n . |h|

˜ N
(11)

By separating out the terms for which h ≡ 0 (mod n) and replacing h by nq, we obtain Tn (k, j) =

n−1 X zj =1

=

X0 ˜ N

˜ N

− 2j
n−1 X zj =1

n−1 X 1 + |h| z =1 j

X0 −

˜ N j 2

˜ N

e2πihkˆzj /n |h|

˜ N

− 2j
X0

1 + n|q|

˜ N j 2

X0

˜ N

˜ N

− 2j
n−1 1 X ³ 2πihk/n ´zˆj e . |h| z =1 j

If zˆj = `zj , then n−1 X

³

e2πihk/n

´zˆj

n−1 X

=

zj =1

³

e2πihk`/n

´zj

.

zj =1

Since n is prime and gcd(`, n) = 1, then when h 6≡ 0 (mod n), it follows that hk` 6≡ 0 (mod n). It is then easy to check that n−1 X

³

e2πihk`/n

´z j

= −1.

zj =1

When zˆj = zj , the sum is the above with ` = 1 and has the same value of −1. Replacing in the expression of Tn (k, j) we obtain: X0 n−1 1 SN˜j /n − . Tn (k, j) = n |h| ˜ ˜ N

N

− 2j
The last term of the sum may be written as: X0 X0 X0 1 1 = − |h| |h| ˜ ˜ ˜ ˜ ˜ N N N N

N

− 2j
−

j 2

= SN˜j −

1 n

j 2

−

X0 ˜ N

˜ N

− 2nj
j 2

˜ N j 2

1 n|q|

1 1 = SN˜j − SN˜j /n . |q| n

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Thus we obtain: n−1 1 SN˜j /n − SN˜j + SN˜j /n = SN˜j /n − SN˜j . n n Using now (12), we see that n−1 d µ ´¶ γ˜j ³ 1XY βj + SN˜j /n − SN˜j , ΘN,γ = n n−1 j=1 Tn (k, j) =

(12)

k=1

and by replacing in (10), we obtain the desired result. From this theorem, we can deduce the following:

¤

Corollary 2.1. If n is a prime number, ` is a positive integer such that gcd(`, n) = 1 and r satisfies 1 ≤ r ≤ d, then there exists a z ∈ Znd such that e2N,d (z) ≤

d d ´ ´ 1 Y³ 1 Y³ ˜j . βj + γ˜j SN˜j ≤ βj + 2˜ γj ln N n j=1 n j=1

Proof. Since βj = 1 + γj for any 1 ≤ j ≤ d, it will follow from Ref. 13, Lemmas 1 and 2, and the arguments used in Ref. 8 that d µ d ´¶ Y n−1 Y γ˜j ³ βj − SN˜j − SN˜j /n − βj ≤ 0. n j=1 n−1 j=1 ˜j for any N ˜j ≥ 2 Using this in (9) together with the fact that SN˜j ≤ 2 ln N (see also Refs. 8 and 13), we obtain MN,d,γ ≤

d d ´ ´ 1 Y³ 1 Y³ ˜j . βj + γ˜j SN˜j ≤ βj + 2˜ γj ln N n j=1 n j=1

Clearly there must be a vector z ∈ Znd such that e2N,d (z) ≤ MN,d,γ . This, together with the previous inequalities completes the proof.

¤

From (5), (6) and Corollary 2.1, it follows that there exists a generating vector z such that d ´ 1 Y³ ∗ ˜j , DN,γ (z) ≤ O(n−1 ) + βj + 2˜ γj ln N 2n j=1 with the implied constant depending on `, r and the weights, but independent of the dimension. As the above bound has a ln n dependency, it would appear that the weighted star discrepancy has the order of magnitude of

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O(n−1 (ln n)d ), a result which is widely believed to be the best possible in an unweighted setting (see Refs. 14 or 6 for details). However, in our case, under the assumption that the weights are summable, it follows from Ref. 7, Lemma 3, or Ref. 8, Lemma 2, that there exists a generating vector z such that the weighted star discrepancy achieves the strong tractability error bound ∗ DN,γ (z) = O(n−1+δ ),

for any δ > 0, where the implied constant depends on δ, `, r and the weights but is independent of n and d.

3. Component-by-component Construction of the Generating Vector In this section we show that intermediate-rank lattice rules of the form (2) that have good bounds for the weighted star discrepancy, can be obtained by making use of the so-named “component-by-component”(CBC) construction of the vector z. This idea has been successfully used in several research papers such as Refs. 4,5,8, and 9 and is based on finding each component one at a time. The result is based on the following: Theorem 3.1. Consider n a prime number, ` a positive integer such that gcd(`, n) = 1 and r chosen such that 1 ≤ r ≤ d. Assume there exists a vector z in Znd such that e2N,d (z) ≤

d ´ 1 Y³ βj + γ˜j SN˜j . n − 1 j=1

Then there exists a zd+1 ∈ Zn such that: e2N,d+1 (z, zd+1 ) ≤

d+1 ´ 1 Y³ βj + γ˜j SN˜j . n − 1 j=1

Such a zd+1 can be found by minimizing e2N,d+1 (z, zd+1 ) over Zn . Proof. When we add a new component, we obtain from (8) that

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 e2N,d+1 (z, zd+1 ) =

1 n

n−1 Y X d+1



−

 =

1 n

X0

 βj + γ˜j

k=0 j=1 n−1 d XY

339

˜ N j 2

k=0 j=1

˜ N j 2

X0

 βj + γ˜j −



˜ N j 2

d+1 e2πihkˆzj /n  Y βj − |h| j=1

 e

−

˜ N d+1 2

|h|

˜ N j 2

X0

 × βd+1 + γ˜d+1

2πihkˆ zj /n

  

e

2πihkˆ zd+1 /n

|h|

˜ N d+1 2

d+1

 Y βj . − j=1

From (8) and by separating out the k = 0 term in the above, we see that we can write d ³ ´ γ˜d+1 SN˜d+1 Y e2N,d+1 (z, zd+1 ) = βd+1 e2N,d (z) + βj +˜ γj SN˜j n j=1    n−1 d 2πihkˆ zj /n 2πihkˆ zd+1 /n X Y X X 0 0 e e γ˜d+1    + βj + γ˜j  . n |h| |h| ˜ ˜ ˜ ˜ j=1 N N N N k=1

−

j 2

j 2

−

d+1 2

d+1 2

We next average e2N,d+1 (z, zd+1 ) over all possible values of zd+1 ∈ Zn and consider: n−1 X 1 e2 (z, zd+1 ). Avg(e2N,d+1 (z, zd+1 )) = n − 1 z =1 N,d+1 d+1

As the other terms that occur in the expression of the average are independent of zd+1 , we next focus on the quantity n−1 ´ X X0 e2πihkˆzd+1 /n 1 ³ 1 = SN˜d+1 /n − SN˜d+1 , n − 1 z =1 N˜ |h| n−1 ˜ N d+1

−

d+1 2

d+1 2

where we made use of (11) and (12). By replacing this equality in the expression of the average, we see that Avg(e2N,d+1 (z, zd+1 )) is given by: βd+1 e2N,d (z) +

d ³ γ˜d+1 SN˜d+1 Y

n

´ βj + γ˜j SN˜j

j=1



+

γ˜d+1 (SN˜d+1 − SN˜d+1 /n ) n(n − 1)

 × −

 n−1 d XY k=1 j=1

 X0

 βj + γ˜j −

˜ N j 2

e ˜ N j 2

2πihkˆ zj /n

|h|

  .

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Next,  −

1 n

n−1 d XY



 βj + γ˜j

k=1 j=1

−

 =−

1 n

X

n−1 d XY

˜ N j 2

− d Y

βj +

j=1

e

|h|

˜ N j 2

˜ N j 2

  

X

 βj + γ˜j

k=0 j=1

= −e2N,d (z) −

2πihkˆ zj /n

˜ N j 2

d ´ e2πihkˆzj /n  1 Y ³ βj + γ˜j SN˜j + |h| n j=1

d d ´ ´ 1 Y³ 1 Y³ βj + γ˜j SN˜j ≤ βj + γ˜j SN˜j . n j=1 n j=1

In the last step we used e2N,d (z) ≥ 0, as RN (PN , u) ≥ 0 for any u ⊆ D (see the previous section). Using also that SN˜d+1 − SN˜d+1 /n ≤ SN˜d+1 and the hypothesis, we now obtain: Avg(e2N,d+1 (z, zd+1 )) ≤ βd+1 e2N,d (z) + +

d ³ ´ γ˜d+1 SN˜d+1 Y βj + γ˜j SN˜j n j=1

d ³ γ˜d+1 SN˜d+1 Y

n(n − 1)

´ βj + γ˜j SN˜j

j=1

¶ d ³ ´µ γ˜d+1 SN˜d+1 Y 1 + βj + γ˜j SN˜j 1+ n n−1 j=1

=

βd+1 e2N,d (z)

≤

d d ³ ´ γ˜d+1 S ˜ ´ Y βd+1 Y ³ Nd+1 βj + γ˜j SN˜j + βj + γ˜j SN˜j n − 1 j=1 n − 1 j=1

=

d ´³ ´ 1 Y³ βj + γ˜j SN˜j βd+1 + γ˜d+1 SN˜d+1 . n − 1 j=1

Clearly, the zd+1 ∈ Zn chosen to minimize e2N,d+1 (z, zd+1 ) will satisfy e2N,d+1 (z, zd+1 ) ≤ Avg(e2N,d+1 (z, zd+1 )). This, together with the previous inequality completes the proof.

¤

From this theorem we can deduce the following: Corollary 3.1. Consider n a prime number, ` a positive integer such that gcd(`, n)=1 and r chosen such that 1 ≤ r ≤ d. Then for any m=1, 2, ..., d,

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there exists a z∈Znm such that e2N,m (z1 , z2 , ..., zm )≤ where e2N,m (z1 , z2 , ..., zm ) =

1 n

k=0 j=1

m ´ 1 Y³ βj +˜ γj SN˜j , n−1 j=1



 n−1 m XY

341

X

 βj + γ˜j −

˜ N j 2

˜ N j 2

m e2πihkˆzj /n  Y βj . − |h| j=1

We can set z1 = 1 and for every 2 ≤ m ≤ d, zm can be chosen by minimizing e2N,m (z1 , z2 , ..., zm ) over the set Zn . Proof. If m = 1, then by expanding the expression of e2N,1 (z1 ) and using well-known results for geometrical series, we obtain that e2N,1 (z1 ) = 0 for any z1 ∈ Zn . The result then follows straight from Theorem 3.1. ¤ Component-by-component (CBC) algorithm The generating vector z = (z1 , z2 , ..., zd ) of a lattice rule (2) that satisfies the bound from Corollary 3.1 can be constructed as follows: 1. Set the value for the first component of the vector, say z1 := 1. 2. For m = 2, 3, ..., d, find zm ∈ Zn such that e2N,m (z1 , z2 , ..., zm ) is minimized. Clearly each e2N,m (z1 , z2 , ..., zm ) can be evaluated in O(n2 m) operations with a constant depending also on ` and r. This cost can be reduced to O(nm) by using asymptotic techniques as presented in Ref. 15 (see also Ref. 8, Appendix A). Thus the total complexity of the algorithm will be O(n2 d2 ). This can be reduced to O(n2 d) if we store the products during the construction at an extra expense of O(n) storage. In fact, this order of complexity can be further reduced to O(nd log n) by making use of the fast CBC algorithm proposed by Nuyens and Cools in Ref. 16. Their approach was based on minimizing a function of the form µ½ ¾¶¶ n−1 d µ 1 XY kzj 1 + γj ω − 1. n n j=1 k=0

From (8), we know that e2N,d (z) is obtained by applying a rank-1 lattice rule to a modified function, so the techniques used in Ref. 16 will also work here with some modifications. References 1. I.H. Sloan and S. Joe, Lattice Methods for Multiple Integration, (Clarendon Press, Oxford, 1994).

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2. I.H. Sloan and J.N. Lyness, The representation of lattice quadrature rules as multiple sums, Math. Comp., 52 (1989), 81–94. 3. S. Joe and S.A.R. Disney, Intermediate rank lattice rules for multidimensional integration, SIAM J. Numer. Anal., 30 (1993), 569–582. 4. F.Y. Kuo and S. Joe, Component-by-component construction of good intermediate-rank lattice rules, SIAM J. Numer. Anal., 41 (2003), 1465– 1486. 5. S. Joe, Component by component construction of rank-1 lattice rules having O(n−1 (ln n)d ) star discrepancy, in Monte Carlo and Quasi-Monte Carlo Methods, 2004, H. Niederreiter (Ed.), (Springer, 2006), 293–298. 6. H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, (SIAM, Philadelphia, 1992). 7. F.J. Hickernell and H. Niederreiter, The existence of good extensi ble rank-1 lattices, J. Complexity, 19 (2003), 286–300. 8. S. Joe, Construction of good rank-1 lattice rules based on the weighted star discrepancy, in Monte Carlo and Quasi-Monte Carlo Methods 2004, H. Niederreiter and D. Talay (Eds), (Springer, 2006), 181–196. 9. V. Sinescu and S. Joe, Good lattice rules based on the general weighted star discrepancy, Math. Comp., posted on December 12, 2006, PII S00255718(06)01943-0 (to appear in print). 10. I.H. Sloan and H. Wo´zniakowski, When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?, J. Complexity, 14 (1998), 1–33. 11. S.K. Zaremba, Some applications of multidimensional integration by parts, Ann. Polon. Math., 21 (1968), 85–96. 12. S. Joe, Bounds on the lattice rule criterion R, Math. Comp., 61 (1993), 821– 831. 13. H. Niederreiter, Existence of good lattice points in the sense of Hlawka, Monatsh. Math., 86 (1978), 203–219. 14. G. Larcher, A best lower bound for good lattice points, Monatsh. Math., 104 (1987), 45–51. 15. S. Joe and I.H. Sloan, On computing the lattice rule criterion R, Math. Comp., 59 (1992), 557–568. 16. D. Nuyens and R. Cools, Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces, Math. Comp., 75 (2006), 903–920.

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ON THE MINIMUM ENERGY PROBLEM FOR LINEAR SYSTEMS IN HILBERT SPACES ALINA ILINCA VIERU∗ Department of Mathematics, ”Gh. Asachi” Technical University, Ia¸si Bd. Carol I, No.11, Ia¸si 700506, Romania E-mail: vieru [email protected] In this paper we consider the abstract minimization problem: min {kuk ; Du + f ∈ B (0, r)} , where D is a linear bounded operator in Hilbert spaces, B (0, r) is a closed ball of center 0 and radius r and f is given such that f ∈ / B (0, r). Under suitable hypothesis we find that the solution is given by u = D∗ (−z), where z ∗ is the solution of an appropriate equation. Then we give an application to the controllability with minimum energy of a linear control system in Hilbert spaces. Keywords: Minimum energy; Linear systems; Hilbert spaces

1. Introduction Let X and U be two Hilbert spaces. Consider the linear control system y 0 (t) = Ay (t) + Bu (t) , t ≥ 0,

(1)

where A generates a C0 -semigroup S (t), t ≥ 0, on X, B ∈ L (U, X), u ∈ L2loc ([0, ∞) ; U ). The solution of (1) with y (0) = x, denoted by y (·, x, u), is understood in the mild sense, i.e., Z t y (t, x, u) = S (t) x + S (t − s) Bu (s) ds, t ≥ 0. 0

Let T > 0. Define the operator V : L2 (0, T ; U ) → X, by Z T Vu= S (s) Bu (s) ds. 0 ∗ This

work was supported by the grant CEEX 47/2005

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Obviously, V is a bounded linear operator. In Ref. 1 we considered the problem of finding the minimum norm solution of the equation S (T ) x + V u = c, where c is a given element in X such that c 6= S (T ) x. Under suitable hypothesis regarding c, we transferred this problem to the problem of solving an appropriate equation. Here, we extend the approach in Ref. 1 and, instead of c, we take a closed ball BX (0, r) , r > 0 as a target. We are interested in finding the form of the minimum norm control by which the ball BX (0, r) is attained from a given state of the system x, at a given time T , i.e. S (T ) x + V u ∈ BX (0, r) . As in Ref. 1, in order to deal with these problems, it is useful to provide an abstract framework for which the above mentioned problems will be a special case. Then, we shall apply the abstract results to the control system (1). 2. Main Results Let U be a Hilbert space, Z a Banach space, D ∈ L (U, Z), f ∈ Z a given element and B (0, r) a closed ball of center 0 and radius r, in Z, such that f∈ / B (0, r). Here, we denote by L (U, Z) the space of all linear and bounded operators from U to Z. Further, R (D) denotes the range of the operator D. We shall consider the abstract problem: find u ∈ U such that Du + f ∈ B (0, r) .

(2)

Let us denote B the closed ball of center −f and radius r in Z and suppose that 0 ∈ / B. Then, the problem (2) may be written equivalently Du ∈ B.

(3)

Of course, problem (3) has solutions if and only if R (D) ∩ B 6= ∅. Our aim here is to find ways of obtaining the minimum norm solution. Suppose that (3) is solvable and consider the minimization problem min kuk , Du ∈ B.

(4)

Let C = {u ∈ U ; Du ∈ B}. Since we supposed that the problem (3) is solvable, C is nonempty. Moreover, it is easy to see that C is closed and convex. The minimization problem (4) becomes min kuk , u ∈ C. It is clear that the minimization problem (5) has a unique solution.

(5)

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The constrained minimization problem (5), equivalent to min

1 2 kuk , u ∈ C, 2

can be replaced by an equivalent unconstrained minimization problem µ ¶ 1 2 min kuk + iB (Du) , u ∈ U , (6) 2 where iB is the indicator function of the set B. In order to handle our problem, we apply a general result from Ref. 2, Proposition 2.1 below. As usual, dom f is the domain of a function f : U → R, f ∗ is the conjugate of f and ∂h (x∗ ) is the subdifferential of a convex, proper function h : U ∗ → R at x∗ ∈ dom h (see Ref. 2). Proposition 2.1 (Aubin, 1982). Let f and g be two lower semicontinuous convex proper functions defined on the Banach spaces U and V respectively and let L : U → V a linear continuous operator. Consider also the strategy set X = {x ∈ U ; Lx ∈ Y }, where Y is a closed convex subset of V . Then x minimizes x 7→ f (x) + g (Lx) on X and p is a Lagrange multiplier if and only if ½ x ∈ ∂f ∗ (−L∗ p) ∩ L−1 ∂g ∗ (p) (i) (7) (ii) 0 ∈ ∂g ∗ (p) − L∂f ∗ (−L∗ p). Applying Proposition 2.1 to our problem we get Theorem 2.1. Let U be a Hilbert space. If there exists z ∗ ∈ dom i∗B such that −DD∗ z ∗ ∈ ∂i∗B (z ∗ ) ,

(8)

then the solution of (6) is u = D∗ (−z ∗ ). 2

Proof. Taking f = 21 k·k , g = iB and L = D in Proposition 2.1 we obtain that u is the solution of problem (6) and z ∗ is a Lagrange multiplier if and only if ½ (i) u ∈ ∂f ∗ (−D∗ z ∗ ) ∩ D−1 ∂i∗B (z ∗ ) (9) (ii) 0 ∈ ∂i∗B (z ∗ ) − D∂f ∗ (−D∗ z ∗ ). 2

It is known that the conjugate function of f is f ∗ : U ∗ → R, f ∗ (u) = 21 kuk , that f ∗ is differentiable on the Hilbert space U and ∂f ∗ (u) = u, for any u ∈ U.

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Then, relations (9) can be written (see also Propositions 5.2.3, 5.2.5 in Ref. 2)  u = D∗ (−z ∗ )  (i) (ii) z ∗ ∈ dom i∗B  (iii) −DD∗ z ∗ ∈ ∂i∗B (z ∗ ). So, using the previous equivalence, we obtain that, if z ∗ ∈ dom i∗B is a solution of −DD∗ z ∗ ∈ ∂i∗B (z ∗ ), then u = D∗ (−z ∗ ) is the solution of (5) and z ∗ is a Lagrange multiplier. ¤ The next result gives the form of the subdifferential of the conjugate of the indicator of B. Lemma 2.1. For any z ∗ ∈ dom i∗B , we have ½ ¾ ∗ ∗ ∗ ∗ ∂iB (z ) = z ∈ B; hz , zi = sup hz , yi . y∈B

Proof. It is easy to see that the conjugate of the indicator of B, i∗B : Z ∗ → R, defined by i∗B (z ∗ ) = supz∈B hz ∗ , zi, is a sublinear function. Using the expression of the subdifferential for a sublinear function (see Ref. 3), we have that ∂i∗B (z ∗ ) = {z ∈ ∂i∗B (0) ; hz ∗ , zi = i∗B (z ∗ )} , for any z ∗ ∈ domi∗B . Moreover, ½ ¾ ∂i∗B (0) = z ∈ Z; hz ∗ , zi ≤ sup hz ∗ , yi , ∀z ∗ ∈ Z ∗ = B, y∈B

which ends the proof.

¤

Now, we can prove the following result, where, by πB we denoted the projector of best approximation for the closed ball B. Theorem 2.2. Let U be a Hilbert space. If z ∗ ∈ Z ∗ is a solution of the equation −DD∗ z ∗ = πB (λz ∗ − DD∗ z ∗ ) , for some λ > 0, then u = D∗ (−z ∗ ) is the solution of (4).

(10)

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Proof. Let z ∗ ∈ Z ∗ be a solution of (10). Then, −DD∗ z ∗ ∈ B and hλz ∗ , y + DD∗ z ∗ i ≤ 0,

(11)

for any y ∈ B, for some λ > 0 (in fact for any λ > 0) and further, the inequality (with λ = 1) becomes hz ∗ , yi ≤ hz ∗ , −DD∗ z ∗ i , for all y ∈ B. Then, (11) is equivalent with −DD∗ z ∗ ∈ B and hz ∗ , −DD∗ z ∗ i = sup hz ∗ , yi y∈B

and, by Lemma 2.1, this is equivalent with −DD∗ z ∗ ∈ ∂i∗B (z ∗ ) , that is condition (8). Now, using Theorem 2.1 we obtain the conclusion. ¤ Remark 2.1. In fact, Theorem 2.2 transfers the problem of finding the solution of (4) to the problem of solving the equation (10). Remark 2.2. If we return to the ball B (0, r), equation (10) becomes −DD∗ z ∗ + f = πB(0,r) (λz ∗ − DD∗ z ∗ + f ) ,

(12)

for some λ > 0, and in this case, the solution of the minimization problem min kuk , Du + f ∈ B (0, r)

(13)

is also u = D∗ (−z ∗ ). Remark 2.3. In case D is surjective, it is shown in Ref. 4 that the equation (12) has a unique solution −1

h³

z ∗ = − (DD∗ )

´

(y0 − f ) ,

i −1 −1 where y0 = πB(0,r) 1 − λ (DD∗ ) y0 + λ (DD∗ ) f , for arbitrary small λ > 0, and then, the solution of problem (13) is given by −1

u = D∗ (DD∗ )

(y0 − f ) .

Remark 2.4. Obviously, a necessary condition for the solvability of (12) is R (DD∗ ) ∩ (B (0, r) − f ) 6= ∅, which is clearly satisfied when D is surjective. Now, we are interested to provide equivalent conditions for the solvability of (12) .

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Proposition 2.2. Let U be a Hilbert space. Equation −DD∗ z ∗ + f = πB(0,r) (λz ∗ − DD∗ z ∗ + f )

(14)

has solutions for some λ > 0 if and only if there is z ∗ ∈ Z ∗ such that −DD∗ z ∗ + f ∈ B (0, r) and there is µ > 0 such that z ∗ = µ (−DD∗ z ∗ + f ), kz ∗ k or, equivalently, there is z ∗ ∈ Z ∗ such that z ∗ = (−DD∗ z ∗ + f ). r Proof. Let z ∗ ∈ Z ∗ be a solution of equation (14). This is equivalent to −DD∗ z ∗ + f ∈ B (0, r) and λz ∗ − DD∗ z ∗ + f = η (−DD∗ z ∗ + f ) for some η > 1. The second statement is equivalent to z ∗ = µ (−DD∗ z ∗ + f ), for some µ > 0, that ends the proof. ¤ From Proposition 2.2 and Remark 2.2 we obtain the following result. Corollary 2.1. Let U be a Hilbert space. Let z ∗ ∈ Z ∗ be a solution for −DD∗ z ∗ + f ∈ B (0, r), such that z ∗ = µ (−DD∗ z ∗ + f ), for some µ > 0. Then, the minimum norm solution of the problem Du + f ∈ B (0, r) is u = D∗ (−z ∗ ). Return now to the initial system (1). Assume that the ball B (0, r) is attained, that is R (V ) ∩ B (0, r) 6= ∅. By Theorem 2.2 we have the following result. Proposition 2.3. Let U and X be two Hilbert spaces. If x ∈ X is a solution of the equation Z T − S (s) BB ∗ S ∗ (s) xds + S (T ) x 0 Ã ! Z (15) T

= πB(0,r)

λx −

S (s) BB ∗ S ∗ (s) xds + S (T ) x ,

0

for some λ > 0, then, the minimum norm solution by which the ball B(0, r) is attained from x, in time T , with controls from L2 (0, T ; U ) is u = B ∗ S ∗ (·) (−x) . Moreover, applying Proposition 2.2, we obtain the following equivalence. Proposition 2.4. Let U and X be two Hilbert spaces. Equation (15) has solutions if and only if there is x ∈ X such that Z T − S (s) BB ∗ S ∗ (s) xds + S (T ) x ∈ B (0, r) 0

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and there is µ > 0 such that Ã Z

!

T

x=µ −

349

∗

∗

S (s) BB S (s) xds + S (T ) x . 0

And from Corollary 2.1 we obtain Corollary 2.2. Let U and X be two Hilbert spaces. Let x ∈ X Z T be such that − S (s) BB ∗ S ∗ (s) xds + S (T ) x ∈ B (0, r) and 0 Ã Z ! T ∗ ∗ x=µ − S (s) BB S (s) xds + S (T ) x , for some λ > 0. Then, the 0

minimum norm solution by which the ball B (0, r) is attained from x, in time T , with controls from L2 (0, T ; U ) is u = B ∗ S ∗ (·) (−x) . 3. Application We give an example that refers to a system in a finite dimensional setting. Let X = R2 , U = R and take the system (1) with µ ¶ µ ¶ µ ¶ 10 0 x1 A= , B= and x = . 12 1 x2 Proposition 3.1. Assume that r2 − e2T x21 ≥ 0.

(16)

Then, the minimum norm control by which the ball B (0, r) is attained is given by u = −z2 e2T , where

µ q ¶ ¡ 2T ¢ ¢ 1¡ T 2T 2 2 2T z2 = ± r − e x1 − e − e x1 − e x2 / e4T − 1 . 4

Proof. We intend to apply Corollary 2.2. We have Ã ! et 0 tA S (t) = e = , ∀t ≥ 0. −et + e2t e2t

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RT The operator V V ∗ : R2 → R2 , V V ∗ = 0 S (t) BB ∗ S ∗ (t) dt is of the form   0 0 VV∗ = ¢, 1 ¡ 4T 0 e −1 4 therefore, for z = (z1 , z2 ) ∈ X,   eT x1 . −V V ∗ z + S (T ) x =  1 ¡ ¢ ¡ ¢ 1 − e4T z2 + e2T − eT x1 + e2T x2 4 Because condition (16) is assumed, there exists z = (z1 , z2 ) such that −V V ∗ z + S (T ) x ∈ B (0, r). Suppose also that z satisfies z = µ (−V V ∗ z + p S (T ) x) for some µ > 0. This is equivalent to z1 = µeT x1 and z2 = ±µ r2 − e2T x12 . Using Corollary 2.2 we obtain that the minimum norm control is µ ¶µ ¶ ¡ ¢ eT −eT + e2T −z1 u = V ∗ (−z) = 0 1 = −z2 e2T . 0 e2T −z2 ¤ References 1. A. Vieru, On the minimum energy problem for linear systems, to appear in J. Analysis Appl. 2. J.P. Aubin, Mathematical Methods of Game and Economic Theory, (NorthHolland Publishing Co., Amsterdam-New York, 1982). 3. C. Z˘ alinescu, Convex Analysis in General Vector Spaces, (World Scientific Publishing Co., New York, 2002). 4. K. Kassara, Feedback spreading control laws for semilinear distributed parameter systems, Systems Control Lett., 40 (2000), 289-295.

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AUTHOR INDEX Agamennoni, O.E., 113 Alzabut, J.O., 1

Mordukhovich, B.S., 237 Munteanu, M., 255

Bakr, A.A., 79 Biagiola, S.I., 113 Bˆırsan, M., 11 Bostan, M., 21 Buliga, L., 227 Burlic˘ a, M., 31

Namah, G., 21 Niculescu, C.P., 267

Castro, L.S., 113 Cernea, A., 45 Chalishajar, D.N., 55 Donchev, T., 69 Elaiw, A.M., 79 Figueroa, J.L., 113 Fursikov, A.V., 93 Garcia, A.G., 113 Ghergu, M., 127 Ghiba, I.-D., 139 Goreac, D., 153 Groza, G., 165 Ibrahim, F.S., 79 Jitara¸su, N., 177 Joe, S., 329 Luca-Tudorache, R., 185 Marinoschi, G., 199 Maticiuc, L., 217 Megan, M., 227

Pavarino, L.F., 255 Pavel, N.H., 275 Perjan, A., 281 Petru¸sel, A., 295 Pop, N., 165 Popa, E., 307 R˘ adulescu, V., 127 R˘ a¸scanu, A., 217 Ro¸su, D., 31 Rovent¸a, I., 267 Rus, I.A., 295 Satco, B., 319 Seidman, T.I., 237 Sinescu, V., 329 Stoica, C., 227 Vieru, A.I., 343