Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari
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Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari
277
Springer Berlin Heidelberg NewYork Barcelona Hong Kong London Milan Paris Tokyo
A. Sasane
Hankel Norm Approximation for Inˇnite-Dimensional Systems With 19 Figures
13
Series Advisory Board
A. Bensoussan · P. Fleming · M.J. Grimble · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · J.N. Tsitsiklis
Author Amol Sasane University of Twente Signals, Systems and Control Dept. Faculty of Mathematical Sciences Postbus 217 7500 AE Enschede The Netherlands
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek – CIP-Einheitsaufnahme Sasane, Amol: Hankel norm approximation for infinite-dimensional systems / Amol Sasane. Berlin ; Heidelberg ; NewYork ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2002 (Lecture notes in control and information sciences ; 277) (Engineering online library) ISBN 3-540-43327-9
ISBN 3-540-43327-9
Springer-Verlag Berlin Heidelberg New York
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution act under German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2002 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Digital data supplied by author. Data-conversion by PTP-Berlin, Stefan Sossna e.K. Cover-Design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN 10869676 62/3020Rw - 5 4 3 2 1 0
Preface This book is aimed primarily at system theorists working on the approximation of systems described in terms of linear partial diÿerential equations by systems described in terms of linear ordinary diÿerential equations. However, it will be of interest also to functional analysts with an interest in the classical problems of interpolation theory of analytic functions. Model reduction is an important engineering problem in which one aims to replace an elaborate model by a simpler model without undue loss of accuracy. The accuracy can be mathematically measured in several possible norms and the Hankel norm is one such. The Hankel norm gives a meaningful notion of distance between two linear systems: roughly speaking, it is the induced norm of the operator that maps past inputs to future outputs. It turns out that the engineering problem of model reduction in the Hankel norm is closely related to the mathematical problem of þnding solutions to the sub-optimal Nehari-Takagi problem, which we call \the sub-optimal Hankel norm approximation problem" in this book. Although the existence of a solution to the sub-optimal Hankel norm approximation problem has been known since the 1970's, in this book, we give explicit solutions and, in particular, we give new formulae for several large classes of inþnite-dimensional systems. The approach taken in this book is as follows. First we give a \frequency domain" solution to the sub-optimal Hankel norm approximation problem. We start with a complex matrix-valued function G deþned on the imaginary axis satisfying certain assumptions. In particular, we demand the existence of a solution to a certain Jÿspectral factorization problem. We then give a solution to the sub-optimal Hankel norm approximation problem in terms of the Jÿspectral factor. Furthermore, we give a parameterization of all solutions in terms of the Jÿspectral factor, the parameterizing set being the unit ball in a certain Hardy space. In this manner we give purely \frequency domain" solutions to the sub-optimal Hankel norm approximation problem in Chapter 4 of this book.
VI
Preface
Subsequently, we give \state-space" solutions to the sub-optimal Hankel norm approximation problem for important classes of inÿnite-dimensional linear systems. In Chapter 5 we consider the case where G is the transfer function of a well-posed linear system given by a triple of operators (A; B; C ). Under certain assumptions we solve the sub-optimal Hankel norm approximation problem for G = C (sI ÿ A)ÿ1 B by constructing a J ÿspectral factor in terms of the system parameters (A; B; C ) and verifying that this constructed J ÿspectral factor satisÿes the assumptions demanded in Chapter 4. In this manner, we obtain \state-space" solutions to the sub-optimal Hankel norm approximation problem for two important classes of well-posed linear systems: the smooth PritchardSalamon class of exponentially stable inÿnite-dimensional systems and the class of exponentially stable analytic systems. Finally, we also solve the sub-optimal Hankel norm approximation problem for certain classes of inÿnite-dimensional systems with a non-exponentially stable semigroup and for regular linear systems. This book is a slightly adapted and extended version of my Ph.D. thesis. It is the outcome of a period of four years at the Department of Mathematics of the University of Groningen, The Netherlands. My advisor during this period was Professor Ruth Curtain, and the results in this thesis are the product of cooperation with her. I would like to express my deepest gratitude for this. Finally, I wish to thank my family for everything.
Contents 1
Introduction 1.1
The sub-optimal Nehari problem
1.2
The sub-optimal Hankel norm approximation problem
. . . . . .
5
1.3
The model reduction problem . . . . . . . . . . . . . . . . . . . .
6
1.3.1
The Hankel norm of systems
. . . . . . . . . . . . . . . .
6
1.3.2
The model reduction problem . . . . . . . . . . . . . . . .
8
1.3.3
Hankel norm approximation and model reduction . . . . .
8
1.4
2
3
1
Organization of the book
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
4
10
Classes of well-posed linear systems
13
2.1
Well-posed linear systems
. . . . . . . . . . . . . . . . . . . . . .
14
2.2
Duality theory
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.3
The Pritchard-Salamon class
2.4
The analytic class
2.5
. . . . . . . . . . . . . . . . . . . .
20
. . . . . . . . . . . . . . . . . . . . . . . . . .
25
Transfer function algebras . . . . . . . . . . . . . . . . . . . . . .
52
Compactness and nuclearity of Hankel operators
63
3.1
Compactness of Hankel operators . . . . . . . . . . . . . . . . . .
66
3.2
Nuclearity of Hankel operators
74
3.3
L1 ÿerror of sub-optimal Hankel norm approximants .
. . . . . . . . . . . . . . . . . . . . . . . . .
80
VIII
4
Characterization of all solutions
85
4.1
The sub-optimal Hankel norm approximation problem . . . . . .
85
4.2
Main assumptions and a few useful consequences
89
4.3
All solutions to the sub-optimal Hankel norm approximation
4.4
5
6
7
8
Contents
. . . . . . . . .
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
A comparison with existing results
99
State space solutions
. . . . . . . . . . . . . . . . .
101
5.1
J ÿspectral factorization for the
Pritchard-Salamon class . . . . . 101
5.2
J ÿspectral factorization for the
analytic class . . . . . . . . . . . 104
The non-exponentially stable case
109
6.1
A few useful results . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2
The class of systems and the main result . . . . . . . . . . . . . . 111
6.3
State-space solutions . . . . . . . . . . . . . . . . . . . . . . . . . 114
The case of regular linear systems
119
7.1
Reciprocal systems . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.2
The sub-optimal Hankel norm approximation problem . . . . . . 121
7.3
The sub-optimal Nehari problem
Coda
. . . . . . . . . . . . . . . . . . 123
127
Bibliography
131
Index
139
Standard notation
141
Chapter 1 Introduction Systems and control theory is a subject in which beautiful and deep mathematical theorems hav e exactly matched the needs of a vital branch of technology. This conjunction of mathematics and engineering has been enric hing for both ÿelds.
In this book w e present a solution to the sub-optimal Hankel norm
approximation problem which has been studied b y both mathematicians and engineers. In this c hapter, among other things, w e explain the problem and the relev ance of its solution to the important engineering problem of model reduction, in which the aim is to replace a high order (possibly inÿnite-dimensional) system by a low order model, without undue loss of accuracy. Finally we will giv e a brief outline of the chapters in this book. But ÿrst we give a brief introduction to the history of the problem, based on Young [93]. In 1957, Zeev Nehari [56] studied the following natural problem. Suppose we are given a sequence does the inÿnite
fangnÿ0 of complex numbers. 2 3 a0 a1 a2 : : : 6 a1 a2 a3 : : : 7 H =6 64 a2 a3 a4 : : : 7 7 5 . . .
Hankel matrix 1
. .
. .
. .
..
Under what conditions
(1.1)
.
`
deÿne a bounded linear operator on 2 (N )? He found a striking answer: a necessary and suÆcient condition is that there
'þ on the unit circle T = fz 2 ÿ j jz j = 1g (parameterized as eiÿ j 0 ÿ ÿ ÿ 2þ ) suc h that th F ourier coeÆcient ' the n ^(n) = an ; for all n þ 0: (1.2) 1 That is, a matrix fckl gk lÿ0 , where ckl depends only on k + l and so ckl = ak+l for some sequence fan gnÿ0 .
should exist an essen tially bounded function
C
;
A. Sasane: Hankel Norm Approximation for Infinite-Dimensional Systems, LNCIS 277, pp. 1−12, 2002. Springer-Verlag Berlin Heidelberg 2002
2
Chapter 1.
Introduction
Hermann Hankel (1839-1873).
(Courtesy of
http://www-groups.dcs.st-andrews.ac.uk/ÿhistory)
Furthermore, when (1.2) is satisÿed,
kH kL(`2 (N))
= inÿmum of the
2 6 H =6 6 4
Example 1.0.1 If
1 1 2 1 3
. . .
1 2 1 3 1 4
. . .
L1 þ norms of all such ': 1 3 1 4 1 5
. . .
::: ::: ::: ..
(1.3)
3 7 7 ; 7 5
.
ÿ þ ÿ. By taking ' eiÿ i(ÿ þ þ)eiÿ , we have a function with L1þnorm at most ÿ, and Z 2þ 1 '^(n) = i(ÿ þ þ)eiÿ eÿinÿ dþ 2ÿ 0 ý ü2þ Z 2þ þi(ÿ þ þ)eÿi(n+1)ÿ i = þ eÿi(n+1)ÿ dþ 2ÿ (n + 1) 2 ÿ ( n + 1) 0 0 the
Hilbert matrix, Schur proved that kH kL(`2 (N))
= for all
1
n+1
=
=
;
ÿ
n ý 0.
We can reformulate the above result as follows. Let
G(z ) =
1 X
ÿ1
n=
gn z n
and
gn = an
for all
n ý 0:
Introduction
3
In (1.3), the right hand side is the inÿmum of on the circle such that
k'k1
over all
L1 ÿfunctions '
'^(n) = an ; n þ 0: This class of
'
is exactly the coset
G + H1 (D )?
in
L1 (T),
where
H1 (D )?
L1 (T)) of functions whose nonnegative Fourier coeÆcients fz 2
is the subspace (of
vanish, which is the space of functions which are analytic and bounded in
C
j jz j > 1g.
Hence
kH kL(`2 (N))
inf kG + K k1 K 2H1 (D)? ÿ ?þ : distL1 (T) G; H1 (D )
=
=
H1 (D )?
So Nehari's theorem gives us an expression for the distance of a bounded function on the circle from
given function in
H1 (D )?
as the norm of a Hankel matrix, and so it gives a
lower bound on the error if we are ÿnding an approximation in
L1 (þ).
of a
1 (ÿ)
L
1(D )?
H
ÿÿ2 ÿÿ6 ÿÿ4 ÿ
G(z )
=
P
1
n=
1
a0
a1
:::
a1
a2
:::
gn z
. . .
n
+
. . .
..
P1
.
3ÿÿ 75ÿÿÿ ÿÿ
n=0 an z
n
Figure 1.1: Nehari's theorem.
L1 ÿapproximation on
More detailed information relating to this type of
the
basis of Hankel operator theory was subsequently obtained by V.M. Adamjan, D.Z. Arov and M.G. Krein [2], D. Sarason [74] and S.R. Treil [82]. An excellent introduction to the sub ject is Young [93], upon which this chapter is based, and a survey of results can be found in Nikol'skii [57]. The fact that results on such
L1ÿapproximation problems were relevant to the preoccupations of some
engineers was realized by a number of mathematicians in the seventies.
This
stimulated further research and the birth of an intensive period of research and
ÿ
interaction by both engineers and mathematicians (see Francis [32]).
the
H1 ÿera
of the 80's
4
Chapter 1.
1.1
The sub-optimal Nehari problem
ÿ
A reformulation of the question answered
of
2
by Nehari is the following: Given a
p m matrix-valued function G on the imaginary axis, ÿnd the distance
G from H1 (C pÿm )? in the L1 þnorm, where H1 (C pÿm )? denotes the space
bounded of
Introduction
p
ÿm
matrix-valued functions which are bounded and analytic in the open
left half-plane. That is, ÿnd
ÿ
L1 G; H1
dist
ÿm ý? ü
þ
Cp
=
=
where we use the notation
K (þ )2H1 (C pÿm ) inf
K (þ )2H1 (C pÿm ) inf
H1 (C pÿm )
kG iý (
K (iý)k1
) +
esssup
kG i! pÿm
!2R
for the space of
(
)+
K (i!)k;
matrix-valued
functions which are bounded and analytic in the open right half-plane. Another
G in the L1 and we seek the closest point or points in the closed subspace
way of phrasing the question is to say that we are given a point
H1 (C pÿm )? .
Banach space
It is perhaps not surprising that in any Banach space other than
a Hilbert space, best approximation problems are usually diÆcult and one does not expect to solve them exactly.
Fortunately for control theory, the Nehari
problem is one of the rare cases which does admit a precise solution. Let
H2 (C k ) denote the set of all analytic functions f û
kf k
2 := sup 0
ÿ>
1
ÿ
2
1 kf þ1
Z
:
þ + i!)k d! 2
(
ú 21
G 2 L1 (iR; C pÿm ) we deÿne the Hankel operator m p by HG , acting from H2 (C ) to H2 (C ), as follows: For
HG f = þ(MG fþ )
!C
k
such that
< 1:
with symbol
G,
denoted
f 2 H2 (C m ) ;
L2 (iR; C m ) induced by G, þ is the orp p thogonal pro jection operator from L2 (iR; C ) onto H2 (C ) and fþ (s) := f (þs). The operator HG is related to the inÿnite Hankel matrix H in (1.1): HG has
where
MG
for
C+ 0
is the multiplication map on
G 2 L1 (iR; C pÿm ),
a matrix of that form with respect to a suitable orthonormal basis. theorem says that for any
L1
dist
G; H1 (C pÿm )?
þ
ý
=
Nehari's
kHGk:
One is of course interested not only in the value of the distance, but also in characterizing the set of
K 's for which the distance is attained. In the case of K (see Glover [34]).
rational transfer functions there are explicit solutions for
However, for control applications, the focus is on the sub-optimal Nehari problem. The
2 The
sub-optimal Nehari problem
is the following:
matrix case was settled by Adamjan, Arov and Krein in [1].
1.2.
The sub-optimal Hankel norm approximation problem
If
ÿ >
kH k; G
5
Let G(iÿ) 2 L1 (R; C pÿm ): then ÿnd K (þÿ) 2 H1 (C pÿm ) such that kG(iÿ) + K (iÿ)k1 ý ÿ:
The solutions K to the sub-optimal Nehari problem are the key to the design of controllers which maximize robustness with respect to uncertainty or minimize sensitivity to disturbances of sensors (see McFarlane and Glover [54] and Georgiou and Smith [33]). For the purpose of \model reduction", a notion to be explained later in this chapter, one considers a natural generalization of the Nehari problem, and this problem turns out to be much harder.
1.2
The sub-optimal Hankel norm approximation problem
Let H1;l (C pÿm ) denote the space of all p ü m matrix-valued functions K of a complex variable deÿned in the right half-plane such that K = Gþ + F , where F is an element in H1 (C pÿm ) and Gþ is the transfer function of a ÿnite-dimensional system with MacMillan degree 3 at most l, with all its poles in the open right half-plane. H1;l (C pÿm ) is a subset of L1 (iR; C pÿm ). Now we recall the notion of singular values of a bounded linear operator from a Hilbert space H1 to a Hilbert space H2 . For k 2 f1; 2; : : :g the kth singular value of an operator H 2 L(H1 ; H2 ) (denoted by ÿk (H)) is deÿned to be the distance with respect to the norm in L(H1 ; H2 ) of H from the set of operators in L(H1 ; H2 ) of rank at most k þ 1. Thus ÿ1 (H) = kHk, and ÿ1 (H) û þ ÿ2 (H) û ÿ3 (H) û : : : û 0. If H is compact, then H H is compact and nonnegative, and so the spectrum of Hþ H consists of a pure point spectrum with countably many nonnegative eigenvalues. The square roots of these eigenvalues are then the singular values of H. If the Hankel operator with symbol G 2 L1 (iR; C pÿm ) is compact, we sometimes denote the singular values of HG , ÿk (HG ), simply by ÿk (G). The ÿk (G)'s are then referred to as the Hankel singular values of G. 3 The
MacMillan degree
of a proper rational function G is deÿned to be the minimal state
dimension of all possible (A; B; C; D)'s that realize G, that is
MacMillan degree of G =
min
2 Cn; mÿn ; B 2 C nÿp ; C 2 C pÿm D 2 C
A
fn j G(s) = C (sI ÿ A)þ1 B + Dg:
6
Chapter 1.
The following theorem
4
Introduction
of Adamjan, Arov and Krein [1] is a natural extension
of Nehari's theorem:
Theorem 1.2.1 For any G
2 L1 (iR; C pÿm ), ÿ k1 = ÿl+1 (G):
þ 2 1 Cÿ k ÿ
K(
)
inf
H
;l (
p
G(i ) + K (i )
m)
The Nehari theorem corresponds to the case
ÿ
üÿ
ü þÿ
ÿ
K ( ) = G ( ) + F ( ), where G (
l
=
0.
In Theorem 1.2.1, if
) is the rational transfer function of a ÿnite-
ÿm
dimensional system with MacMillan degree at most
þÿ) 2
in the open right half-plane, and F (
1
H
(
be an optimal Hankel norm approximant of G.
Cp
l
and with all its poles
), then
þGü (ÿ)
is said to
The sub-optimal Hankel norm
approximation problem is the following:
ÿ 2 L1 (R; C pÿm ): pÿm ) ÿl+1 < ÿ < ÿl ; then ÿnd K (þÿ) 2 H1 l (C such that kG(iÿ) + K (iÿ)k1 ý ÿ: Let
If
G(i )
;
K is then called a solution of the sub-optimal Hankel norm approximation problem.
1.3
The model reduction problem
An elaborate model of a physical system can sometimes be replaced by a simpler one, which is easier to analyze and use, without a signiÿcant loss of accuracy. In this section we shall see how a solution to the sub-optimal Hankel norm approximation problem gives rise to a ÿnite-dimensional reduced model to the original inÿnite-dimensional system.
1.3.1
The Hankel norm of systems
In practice, many physical systems and engineering devices are modelled by mathematical systems of the type
4 the
x(t)
=
Ax(t) + Bu(t);
y(t)
=
C x(t);
matrix case was proved in Kung and Lin [50].
ÿ t
ü 0;
1.3.
7
The model reduction problem
where T (t)
is the inÿnitesimal generator of a strongly continuous semigroup (C m ; X ) and C (X; C p ). 5 0 on the Hilbert space X , B
f gtÿ
2L
A
2L
If A is exponentially stable (that is, there exists an ÿ < 0 and a M > 0 ÿÿt for all t 0), then the impulse response h( ) = such that T (t) Me C T ( )B is an element belonging to L1 ([0; ); C pþm ), and h induces a compact time-domain Hankel operator þh (L2 ([0; ); C m ); L2 ([0; ); C p )), deÿned m as follows: given u L2 ([0; ); C ),
k kÿ
ý
2
1
(þh u)(t) =
Z
þ 1 2L 1
1
0
ý
1
( + þ )u(þ )dþ for all
h t
t
þ0
:
ü
The impulse response h has the Laplace transform G(s) = C (sI A)ÿ1 B with pþm ). This induces a compact Hankel operator G( ) : iR C pþm L1 (iR; C HG with symbol G, which is related to þh via Laplace transformation:
ý
!
2
ÿ
H
þ
G f^(s) = þd h f (s) for all
s
in the open right half-plane;
k k k kÿk k ÿk k üý 2 1
and þh = HG G 1 h L1 . If uý ( ) L2 ([0; ); C m ), and we have for t (þh u) (t) =
1
Z
0
ý2
þ0 u
( + þ )u(þ )dþ =
CT t
Z
L2
ü1 0] C
((
0
ÿ1
;
(
CT t
ü
;
)
ý
m ),
ý(
þ u
then u( ) :=
)
þ dþ :
Roughly speaking, if we run the system ( ) + Buý (t); ()
() = () =
x t
Ax t
y t
Cx t ;
ü1
ü1
from time to 0 with input function uý and \initial state x( ) = 0", and then cease to input anything and observe the output y ( ) from time 0 onwards, then (þh u) (t) = y (t) for all t 0 (see Figure 1.2).
ý
þ
u( )
y( )
ÿh
0
0
t
t
Figure 1.2: The time-domain Hankel operator. 5 More and
Y
generally, the input space
, respectively, and
B
and
C
Cm
and the output space
could be
unbounded.
Cp
could be Hilbert spaces
U
8
Chapter 1.
Introduction
In other words, ÿh is the mapping from past inputs to future outputs. The
Hankel norm of a system with transfer function kH k.
is deþned to be
G
G
(sometimes denoted by
kGk
H)
It is now evident that this is a physically meaningful
notion of closeness of systems. To say that
ÿ is close to
G
G
in the Hankel norm
means that the corresponding systems acting on a past input signal give future outputs which are close to each other.
1.3.2
The model reduction problem
In some areas of engineering, linear inþnite-dimensional state-space models of dynamic systems are derived and it is desirable to replace them by þnitedimensional systems without incurring too much error. This would make the subsequent control system design computationally less demanding and possibly numerically more reliable. A wide variety of methods for model reduction have been proposed over the years; here we list a few: 1. Balanced model reduction: See for example Moore [55] and Pernebo and Silverman [64] for the þnite-dimensional case and Curtain and Glover [36], Ober [58], Young [92] for the inþnite-dimensional case.
2
\
h L1 L2 with a nuclear Hankel operator: See Curtain, Glover and Partington [37].
2. Approximation of systems with
3. Other miscellaneous techniques: See Curtain [12], Glover, Lam and Partington [38], [39], Gu, Khargonekar and Lee [43]. However, in this book, we consider the
the Hankel norm:
Given a stable
G
þnd a stable rational such that
model reduction problem with respect to
and a nonnegative integer
ÿ
G
l;
of MacMillan degree at most
kG ÿ Gÿ k
H
l
is small.
We explain the relation of this question to the problem of sub-optimal Hankel norm approximation in the next section.
1.3.3
Hankel norm approximation and model reduction
Suppose that
+1
ÿl
G
2
< ÿ < ÿl , K
a compact Hankel operator and that if 1 ( R C pþm ) phas (ÿþ) 2 1 l (C þm ) is a solution to the sub-optimal Hankel L
i
;
H
;
norm approximation problem, that is,
kG(iþ) + K (iþ)k1 ý ÿ:
1.3.
ÿ
Let
9
The model reduction problem
K(
) =
ÿ (ÿ) +
G
ÿ
F(
), where
G
ÿ (þÿ) is the rational transfer function of a þÿ 2 1 þ þ ÿÿ
ÿnite-dimensional system with MacMillan degree at most l and with all its poles in the open right half-plane, and F ( ) H (C p m ). Then G ( ) (called a sub-optimal Hankel norm approximant) is a reduced order model of the original
system
k
G
G
in the Hankel norm, with error at most equal to
ÿ kH = k
+G
G
ÿ
+G +F
kH k =
G
+K
kH ý k ÿ G(i
ÿ.
Indeed, we have
ÿ k1 ý
) + K (i )
ÿ:
(1.4)
It is not clear that closeness in the Hankel norm is an appropriate criterion for a reduced order model. Most robust control designs are robust with respect
1 þnorm:
to the stronger
L
k kH ý k k1 G
G
. However, in practice the Hankel
norm approximants performed well. The explanation of this fortunate fact was provided by Glover in his seminal paper [34], which was generalized in [37].
P1
2
In Glover et al. [37] it was shown that if h L1 ([0; L2 ([0; ); C p m ), and if ÿk < , then for a given integer k=1
1
þ
1
ists an optimal Hankel-norm approximation
0
G
k þ 0 kH ý k þ 0 k1 ý G
G
G
G
of order
1 X
= +1
l,
1
þ
); C p m )
l,
\
there ex-
such that
ÿk (G):
k l
It is very commonly the case that the Hankel operator has a few sizeable singular values and the remaining tail away very quickly to zero. In such a case, the right hand side can be made very small, and one is assured that an optimal Hankel norm approximant is also good with respect to the
L
1 þnorm.
This norm is
extremely desirable because there exist many robustness theories with respect to this norm. In this book, we only consider the sub-optimal Hankel-norm approximation problem. In (1.4) above we showed how this yields a sub-optimal Hankel norm approximant
þ ÿÿ G
( ) which is close in the Hankel norm. However, for control
applications, we prefer closeness in the
1 þnorm.
L
Since we were unable to
ÿnd any results on this, even in the ÿnite-dimensional literature, we prove (in Section 3.3) that the error bound in Corollary 7.5 (see page 69, Partington [62])
P1
for an optimal Hankel norm approximant also holds for a sub-optimal Hankel norm approximant: If h L1 ([0; ); C p m ) and , then any subk=1 ÿk < optimal Hankel norm approximant G as in (1.4) of MacMillan degree l satisÿes
2
k ÿ G(i
1
ÿ
ÿ
ÿ k1 ý
) + G (i )
þ
4l
ÿl (G)
tends to zero as
l
!1
:= 4l
ÿl (G)
1 X
= +1
ÿ k ( G ):
k l
Moreover we will show that the error bound El
+2
1
+2
1 X = +1
ÿk (G)
k l
. So the sub-optimal Hankel norm approximant is also
a good approximation of
G
in the
L
1þnorm.
10
Chapter 1.
Introduction
While the natural formulation of the sub-optimal Hankel norm approximation problem is in terms of the transfer function, most systems in control applications are modelled in state-space form in terms of parameters
A, B , C .
The Hankel norm approximation problem has been solved for the rational case in Glover [34], Ball and Ran [6] and for the special class of Pritchard-Salamon inÿnite-dimensional systems in Curtain and Ran [17]. Our aim in this book has been two-fold:
1. To obtain suÆcient conditions for the existence of a solution to the suboptimal Hankel norm approximation problem in terms of a solution to a key
J ÿspectral factorization problem.
This allows a complete param-
eterization of all solutions in terms of the spectral factor (see Chapter 4). 2. For certain classes of well-posed linear inÿnite-dimensional systems, we use the frequency domain results established in Chapter 4 to obtain explicit solutions to the sub-optimal Hankel norm approximation problem in terms of the state-space parameters
A, B , C .
The classes considered
are exponentially stable, smooth Pritchard-Salamon systems (Section 5.1 of Chapter 5), exponentially stable analytic systems (Section 5.2 of Chapter 5), non-exponentially stable systems (Chapter 6) and regular linear systems (Chapter 7).
The above strategy is reminiscent of the strategy in Ball and Ran [6], Ran [67] and Curtain and Ran [17] with one important diþerence. In these three references, the starting point was to quote a result (Theorem 4.4.1 of Section 4.4 in Chapter 4) which was claimed to be a corollary of the results in Ball and Helton [5]. Since it is not directly clear to non experts and certainly not to the engineering community how this result follows from [5], we found it worthwhile to derive a simpler frequency domain result from ÿrst principles using elementary mathematics in Chapter 4.
We remark that an analogous
strategy has been followed earlier to solve the special case of the sub-optimal Nehari problem in Curtain and Zwart [27], [26], Curtain and Oostveen [16] and Curtain and Ichikawa [14].
1.4
Organization of the book
In this section, we give an overview of the results in this book.
Chapter 2:
We ÿrst deÿne well-posed linear systems, which forms the general
framework for studying inÿnite-dimensional systems with a state space representation. We recall a number of deÿnitions concerning this class of systems
1.4.
11
Organization of the book
which are relevant to the remainder of this book. Next we discuss two important classes of well-posed linear systems: 1. the Pritchard-Salamon class, and 2. the analytic class, which are central in this book. Some of their properties, which will be used in Chapter 5 are listed. Finally, in the last section of this chapter we will introduce several frequency domain spaces and prove a few elementary lemmas which will be used later.
Chapter 3:
In the theory of approximation of inÿnite-dimensional systems, the
nuclearity and compactness properties of the Hankel operator play an essential role. In this chapter we investigate the relationships between the exponential (or strong) stability of certain classes of regular linear systems and the compactness and nuclearity properties of the Hankel operator by means of considering illustrative examples. New suÆcient conditions for nuclearity are given for exponentially stable regular linear systems with an analytic semigroup. In the last section of this chapter we will derive a bound on the
L1 ÿerror of a sub-optimal
Hankel norm approximant of a system with a nuclear Hankel operator.
Chapter 4:
In this chapter, ÿrst we prove a few useful consequences of the main
(frequency domain) assumptions S1-5 in this chapter, and subsequently, under these assumptions, we give a characterization of all solutions to the sub-optimal Hankel norm approximation problem. These derivations are new. Later in Chapter 5, we will show that for the Pritchard-Salamon class and the analytic class, there do exist
J ÿspectral factors for each class satisfying
the assumptions in this chapter, hence solving the sub-optimal Hankel norm approximation problem for these classes of inÿnite-dimensional systems. We
J ÿspectral factor þ in terms of the stateA B , C ) of the original system.
will give an explicit formula for a space parameters ( ,
Chapter 5:
In Section 5.1, the sub-optimal Hankel norm approximation prob-
lem is solved for the smooth Pritchard-Salamon class of exponentially stable inÿnite-dimensional systems with ÿnite-dimensional input and output spaces by using the results from Section 2.3 of Chapter 2 and Chapter 4: we give a formula for a
J ÿspectral factor þ in terms of the state-space parameters, and
check that it has all the properties required in Chapter 4. These results were obtained earlier in Curtain and Ran [17] by appealing to the Ball and Helton result [5]. Next, in Section 5.1, the sub-optimal Hankel norm approximation problem is solved for the analytic class of exponentially stable inÿnite-dimensional systems with ÿnite-dimensional input and output spaces by using the results
12
Chapter 1.
Introduction
from Section 2.4 of Chapter 2 and Chapter 4. Again, we give a formula for a
J ÿspectral
factor ÿ in terms of the state-space parameters, and check that it
has all the properties required in Chapter 4. The results of this chapter are new and include the results on the Nehari problem in Curtain and Ichikawa [14] as
a special case.
Chapter 6:
In the previous chapters the sub-optimal Hankel norm approxi-
mation problem was solved for various classes of inþnite-dimensional systems under the assumption that
A
generates an
exponentially
stable strongly contin-
uous semigroup. In this chapter, we will solve the sub-optimal Hankel norm approximation problem for non-exponentially stable inþnite-dimensional systems in terms of a solution to the sub-optimal Hankel norm approximation problem for an exponentially stable system. The exponentially stable system is obtained by shifting the generator of the semigroup of the original system. The results in this chapter are new.
Chapter 7:
In this chapter, we will solve the sub-optimal Hankel norm approx-
imation problem for regular linear systems with generating operators (A; B; C ) satisfying some mild assumptions. Furthermore, in the special case of the suboptimal Nehari problem, we will also give an explicit parameterization of all solutions. The class of regular linear systems is a large class of inþnite-dimensional state-linear systems allowing for unbounded inputs and outputs; it includes the Pritchard-Salamon class and the analytic class. But the solution obtained in this chapter is diýerent from the form in the preceding chapters.
Chapter 2 Classes of well-posed linear systems
In this chapter we recall the notion of w ell-posed linear systems, which forms the general framework for studying inÿnite-dimensional systems in a state space con text. In particular, we introduce tw o importan t classes of well-posed linear systems which are central in this book and list some of their properties, which will be used in the sequel. Figure 2.1 shows the hierarchy of the various classes of systems that we will encounter in this chapter. Finally, in the last section of this chapter we will introduce several frequency domain spaces and prov e a few elementary lemmas which will be used in what follows.
well posed
regular well posed Pritc hard Salamon smooth Pritchard Salamon A A; B; C
bounded A
generates an
analytic semi group on
generates a
strongly continuous semigroup B
and
C
bounded
Z,
2 L(U; ZÿB ), C 2 L (Z ÿ C ; Y )
B
Figure 2.1: Various classes of systems.
A. Sasane: Hankel Norm Approximation for Infinite-Dimensional Systems, LNCIS 277, pp. 13−62, 2002. Springer-Verlag Berlin Heidelberg 2002
14
Chapter 2.
2.1
Classes of well-posed linear systems
Well-posed linear systems
We assume that
is a Hilbert space and
:
A
0 on space X1 as D (A) with the norm x 1 = (ÿI ÿxed1 . The ÿ Hilbertÿspace ÿ Xÿ1 is the completion of 1 xÿ. We have x ÿ1 = ÿ(ÿI A)
kk
ÿ
T (t) t
þ þ
X1
! k
D (A)
f gÿ kk k ÿ
X
of a strongly continuous semigroup
X.
A)x
ÿ1 , see Weiss [86].
A,
We assume that
whose domain is
U
X,
so that
are bounded from
L2 ([0;
Z þt u :=
t
0
1 ÿ
); U ) to
T (t
x(t)
and
u
=
2
X -valued
1
is called
B
We assume that called an
ý
ÿ
2
Y
x(t)
=
T (t)x0
bounded
if
B
2L
t
0
We regard
ý
Kt
Z
The operator
þ
(
ÿ1 ) !
X
u
ý 2
f g
1
L2 ([0;
); U ):
(2.2)
k
ý ý
t
2
ÿ
0,
+ þt u:
(2.3)
(U; X ) (and
for
0 such that
C T (ý )x0
k ü k k 2
dý
2
x0
Kt
unbounded
2
f gþ T (t) t
for all
C
otherwise).
2L
(X1 ; Y ) is
0 , if for every
x0
2
t >
D (A):
0 there
(2.4)
2L
C is called bounded if it can be extended such that C (X; Y ). loc ([0; ); Y ) as a Frý echet space with the seminorms being the
L2
1
2N
L2 norms on the intervals [0; n], n that there is a continuous operator ü :
(üx0 )(t) =
.
X
C T (t)x0
Then the admissibility of C means loc L2 ([0; ); Y ) such that
!
for all
x0
2
1
D (A):
The operator ü is completely determined by (2.5), because 1 This
X
2L
is another Hilbert space. The operator
admissible observation operator
exists a number
ÿ
for all
function of t. We have that for all
The operator
:
X 1 , but the result is in X . If x is the solution 0, which is an equation in X 1 , with x(0) = x0 X ); U ), then x(t) X for all t 0. In this case, x is a continuous
Ax(t) + Bu(t), t
L2 ([0;
T (t) t
ý )Bu(ý )dý ;
The above integration is done in of
f gþ
for
X
A
B (U; Xÿ1 ) is said 0 , if the input maps þt tþ0 for all ÿnite t 0, where
is a Hilbert space. The operator
admissible control operator
is
(2.1)
f gþ
X
to be an
þ(A)
T (t) t 0 extends to a The generator of this extended
ÿ1 , denoted by the same symbol.
X
semigroup is an extension of
2
ÿ
X
densely and with continuous embeddings. The semigroup semigroup on
, where
with respect to the norm
X
ÿ1 ;
X
is the generator
X
We deÿne the Hilbert
norm is equivalent to the graph norm.
D (A)
(2.5) is dense in
X.
2.1.
B B
for
L L
is bounded.
f g ÿ0 T (t) t
If
f g ÿ0 1
is said to be an inÿnite-time admissible control operator for
is admissible and for any
X)
15
Well-posed linear systems
B
2L
2
u
1
); U ), the map
7!
t
T (t) t
ÿt u (from [0;
1
ÿ 0 k2 þ k 0k2
0
k
0 such that
K
C T (ÿ )x
dÿ
K
for all
x
x
2
0
D (A):
ÿ1 ) is an admissible control operator for f ( )g þ0 and 2 f ( )g þ0, then the transfer
(U; X
T t
(X1 ; Y ) is an admissible observation operator for
T t
C
t
t
functions of the system þ given by the triple (A; B; C ) are solutions
(U; Y ) of
ý
G(s)
for
and
s
ý
T ( t) t
G(ý )
=
ý ý (s
, (
I
ÿ1
A)
B
(U; X )
B
L
a
ý
(U; Y )
A)
I
,
and
Re(s)
s
ý
ýI
A)
ÿ1
:
þ(A)
!
(2.6)
B
is an admissible control operator for
B
analytic function. Both ( =
ÿ1 (
G
valued analytic function and since
T (t) t
right half-plane
A)
L ý f g þ0 ü ý ü ý ÿ1 ü ý f 2C j g is an
an admissible observation operator
C+ ÿ
ý
ý )C (sI
in þ(A). We remark that since
f g þ0 ü ý
if
) to
is said to be an inÿnite-time admissible observation operator
C
if there exists a
Z
2([0;
L
C( I
C( I
> ü
.
A)
A)
ÿ1 is a L(
ÿ1
X; Y
ý
)
C
is
valued
are analytic on some
Thus any transfer function is
valued function which is analytic in some
transfer functions diýer only by an additive constant, is that they need not necessarily be bounded on any
C+ ÿ . Moreover any two
D
2L
(U; Y ). The point
C+ ÿ . We impose this as
an extra assumption on the triple (A; B; C ) and call this well-posedness. The system þ given by the triple (A; B; C ) is said to be well-posed linear system if is an admissible control operator for operator for
f g þ0 T (t) t
f g þ0 T (t) t
,
C
B
is an admissible observation
and its transfer functions are bounded on some half-plane
C+ ÿ . For a well-posed linear system, the operator from the initial state and the
input function to the ünal state and the output function is bounded. The input and output functions
u
and
are locally
y
respectively. The state trajectory
x
2 functions with values in U
L
is an
X -valued
property mentioned earlier means that for every that
k k2 x(t)
(with
ct
Z +
independent of
t >
ÿ
k k 2 þ 2 k k2 0 t
y (ÿ )
x(0)
dÿ
and of
ct
u).
x(0)
and in
Y
,
function. The boundedness
Z +
0 there is a
k k2 0
þ
t
u(ÿ )
ct
dÿ
ÿ
0 such (2.7)
For the detailed deünition, background
and examples we refer to Salamon [72], [73], Staýans [75], [76], [79], Weiss [89], [88], Weiss and Rebarber [68] and Weiss2 [90]. ÿ (A; B; C ) u
y
Figure 2.2: Well-posed linear systems.
f g þ0 T (t) t
U
is called the semigroup of þ and
is called the input space ,
Y
A
is called its inÿnitesimal generator.
is called the output space and
X
is called the
16
Chapter 2.
state-space
called
Classes of well-posed linear systems
of the well-posed linear system. The well-posed linear system ÿ is if the limit lim G(ÿ)v = Dv (2.8)
regular
!+1
ÿ
2
exists for every v U , where ÿ is real (see [87], [89]). In this case, the operator (U; Y ) is called the feedthrough operator of ÿ, and there holds
D
2L
( ) = Cÿ (sI
G s
where
C
)ÿ1 B + D;
A
ÿ is an extension of C given by C
(ÿ
ÿ
2 R), for all
x
C ÿ(ÿI ÿ x0 = !lim +1 ÿ
ÿ
)ÿ1 x0
(2.9)
A
0 in the domain D
(Cÿ ) =
f 0 2 j the limit in (2.9) existsg x
X
(see Staþans [79]). The operators A, B , C , D are called the generating operators of ÿ, and we write ÿ = (A; B; C; D).
2.2
Duality theory
We will use the notion of \duality" in this book following the notation and results from Keulen [84]. For a short introduction about representations of dual spaces, duality pairings and pivot spaces we refer to Aubin [4]. The dual space of a Hilbert space Z is just the linear space of bounded linear functionals, that is, (Z; C ). We shall denote (Z; C ) by Z d . The vector space d 0 Z d, Z is a Banach space with the dual norm d deýned as follows: If z then 0 z (z ) 0 z d = sup :
L
kk
z
2
Z;z
L jj þ jj j j =6 0 k k
2
z
If Z is a Hilbert space, then there exists a surjective isometry J from Z to Z d , deýned by (J (z1 )) (z2 ) = z2 ; z1 Z for all z2 Z . The dual space Z d is a Hilbert space with the inner product deýned as follows: if z10 and z20 Z d , then
h
h 10 20 i z ;z
Zd
=
i
2
ÿ1 20 ÿ1 10
þ
ÿ
J
! !
z ;J
z
Z
= z10
ý
J
ÿ1 ( 20 ) z
2
ü
:
d We call the isometry J : Z Z the duality map 2 from Z to its dual Z d . ÿ 1 d d The inverse map J : Z Z is denoted by þZ and the inner product on Z satisýes 0 0 0 0 Z for all z10 and z20 in Z d: z1 ; z2 Z d = þZ z2 ; þZ z1
h
i
h
i
2 The duality map is conjugate linear, that is, it satisÿes ÿ2 J (z2 ) for all ÿ1 and ÿ2 in C , and all z1 and z2 in Z .
J (ÿ1 z1
+
ÿ2 z2 )
=
ÿ1 J (z1 ) +
2.2.
17
Duality theory
hÿ ÿi
The duality pairing
;
h i 0
=
z ; z hZ d ;Z i
Let
Z
0
z
0
2L
ÿZ j
tation of
h
(z ) =
Z
d
Z
0
onto
Z
d
z
0
2
i
0
z; ÿZ z
for all
Z;
0
þ! þ!Z j
Z
ÿ
d
Z.
=
Z
0
and
2
z
Z.
h
i
0
d
Z
Z
d
and
f g 0
The pair
hÿ ÿi
. We deÿne the duality pairing z ; z hZ 0 ;Z i
2
0
z
2
Z:
be a conjugate linear map which is an
j
(Z 0 ; Z ) is an isometry onto
h i
z
, that is, Z
0
for all
d ;Z i is deÿned by
be a Hilbert space and let
isometry from
and
hZ
;
Z ;j
h
0
= (j z )(z ) =
j z ; z hZ ;Z i
In this case, we also call
Z
is called a represen-
by
hZ 0 ;Z i
0
z; ÿZ j z
0
i
Z;
the dual space of
Z
(this
is standard abuse of terminology). We say that Z is identiÿed with its dual if the representation for Z d is chosen ÿ1 to be Z; ÿZ . It follows that in this case, the duality pairing corresponds to
f
g
the inner product on
Z:
h
i
=
z1 ; z2 hZ;Z i
for all
z1
and
z2
in
Z.
If
Z
h
z2 ; ÿZ ÿ
ÿ1
Z
i h =
z1 Z
i
z2 ; z1 Z
is identiÿed with its dual,
Z
is called a pivot space.
Usually, it is assumed that whatever choice is made for the representation
f g L L 0
Z ;j
of
(Z 0 )d =
given by
(Z; C ), the representation of the dual of this dual space (that is,
f
(Z 0 ; C )) is given by
cZ
=
ÿ1
ÿ
Z0
j
ÿ 1 ÿ1
ÿ
Z
Z; cZ
:
0 d
(Z ) (we could also say that
h
z
00
Z
00
;z
0
i
=
g
, where
Z þ! þ!Z þ! ÿ
Z ).
0
Z
0
z
00
z ;z
0
i
ÿ1
ÿ
Z0
ý
00
j
0
z ;j
ý
i
to
Z
d
Z
hZ;Z 0 i
;z
0
i
d
h(Z 0 ) ;Z 0 i
0
ÿ
ÿ1
ÿ1 ÿ1
ÿ
Z
ÿ1 ÿ1
ÿ
Z
z
00
; jz
=
h
=
(j z )(z )
Z
0
ÿZ j z ; z 0
h
= Z
0
;z
00
ÿ
=
h
ÿ
d
(cZ z )(z )
=
with
00
cZ z
=
Z
Z
h h
=
hZ 00 ;Z 0 i
=
given by
j
Z
This implies that
=
We identify the bidual of
is the isometry from
cZ
0
z ;z
00
0
i
z
z
00
00
ü
Z
þ
ü
0
(z )
Z0
d
Z
00
00
i
hZ 0 ;Z i :
itself and the duality pairing
hZ;Z 0 i
=
h
0
z ;z
00
i
hZ 0 ;Z i :
hÿ ÿi ;
hZ;Z 0 i
is
18
Chapter 2.
If we have two Hilbert spaces Here, we identify an element Similarly, if
w
2
0
W
d
2
j
z
satisÿes
and
W 0
w
0
Z
z
2
0
d
Z
and
2
w
h i 0
Z
=
Z
0
for
w1
h
0
and
in
w2
Z
=
Z
0
0
0
w1 ; w2
i
0
2
Z
d
,
f
ÿZ
0
W ;
k k ÿZ z
is given by
g
d
Z
#
Z
=
to
Z.
=
W0
Z
,
Z
1
ÿZ z
d
0
W
; w)
Z
Z
=
Z
0
Z
W
image of
V
d
i
0
we have
W
,
ü
= W
ÿZ
0
1
ÿ
z
0
û
7!
n
h 0
0
0
v1 ; v2
V ;
(ÿZ
i
Z
!
,
Let
W
W
W
0
j h i
1
ÿ
V
d)
D
o
(ÿZ
!
Z is continuous.
j
V
0
ýý
Zd
ü
ÿZ
(w ) =
d)
0
0
which is a repre-
, as follows. Recall
þ
0
w1
Zd
kk
=
0
z
1
ÿ
ü
w
1
ÿ
ÿ
Z
0
û
Wd:
to
ÿZ
V
1
ÿ
to mean W
ú
0
of
0
v2 ;
(ÿZ
ÿ
W
d
z
0
û
h i
(w ) =
V
with
V d
w; z
!
Z ,
such that to
ÿZ
hþ þi ;
j
(ÿZ
j
1
ÿ
V d)
1
ÿ
V
d)
v
0
hW 0 ;W i
(w )
is a representation of =
then
be the completion of
.
=
V0
we use the notation ,
x : W
W,
W :
d denote the restriction of
0
x
!
0
so that the duality pairing
=
w ; w hW 0 ;W i
v ; v hV 0 ;V i
3 Here
2
. Furthermore,
V
d
V
V 0
0
Z:
, we have
!
,
Z
=
and deÿne
Z
V
under this map. Furthermore, deÿne the inner product on
Then the pair is given by
j
ÿZ V
z
.
(W; C ).
. Furthermore, we can choose the pair
d
and we can deÿne a representation
as follows. Let
0
d
d
If in addition we have a third Hilbert space Z
w
W
0
,
ÿ
1
ÿ
W
!
0
ýý ÿ
0
w2 ; ÿ
!
d
W
0
W ,
Z
to
h
0
=
1
ÿ
for all
! L ,
0
!
ÿ
as a representation of
z ; w hW 0 ;W i
d
W
Z
d
=
has a unique conjugate linear extension
ÿZ
2 ý h i ! !
and for all (z
Finally,
0
is an isometry from
(ÿZ )ÿ1
ÿ
. We have
Z
Thus it follows that and
z
W
Z
d
z ; w hW d ;W i :
W ,
=
W0
ÿZ
z
z
then
h i
(w ) =
such that
W
Z
For all
0
Z,
0
is also denoted by
Z
with respect to the inner product
Z
0
(a constant)
to
denotes the canonical map from
ÿZ
=
z
! j 2 jj jj ,
is identiÿed with its dual, we can deÿne a space
sentation of the dual of that
jÿ
0
W
(Z; C ) with
=
we have
W
z ; w hZ d ;Z i
Now if
w
with 3
L
Z
(z )
the unique continuous extension of for all
d
Classes of well-posed linear systems
ù
V
0
v1
d
E
Zd
0
d
!
V 0
,
!
,
V
as the V
0
by
:
and the duality pairing
(v ):
Z , W is dense in Z , and the canonical injection
2.2.
19
Duality theory
2 ÿ
Hence for all (v 0 ; z ) 0
=
v ; z hV 0 ;V i
(v
0
V
; w)
0
! 2 ÿ ,
Z
V
0
=
0
we have
Z
ÿ
h i
and
0
V
!
Z ,
V
j
(ÿZ
V d)
1
ÿ
0
þ (z ) =
ý
ÿ
1
ÿ
Z
v
0
ü
h i
(z ) =
z; v
h i 0
h
=
v ; w hW 0 ;W i
w; v
Vd
V
0
0
=
Z
v ; w hV 0 ;V i :
Wd ÿZ
ÿZ Z
0
0
=Z
W
0
Figure 2.3: Interlacing of the dual spaces. and
X
representations of
hþ þi ;
hY 0 ;Y i
that A
ÿY k
from
2.
D (A
h
0
0
Y
ý
û )=
to
X
Y
2 j 9 2h h i 0
y
0
i
Y
A y ; x hX 0 ;X i
The adjoint of
A
x
0
=
þ
h
2
)
=
i
þ
x; A y X
þ
A
0
) from 0
y
=
h
=
0
D (A)
1
ÿ
(ÿY
ÿ
k)
x
0
2
0
D (A
1
Y
0
0
Y ;k
;
to
D (A)
)
ý
) and all
Y
to
);
1
ý 2
)
x ; x hX 0 ;X i
þ
ÿ
(ÿX j )
ù
=
2
0
D (A
h i
=
D (A
(ÿX j )A (ÿY k ) x
from
A
for all
k ) D (A
ù
f hþ þi
and
is an isometry from
ÿX j
i
0
) and
g
X
g
hX 0 ;X i 0
to
X
are and and
. The dual of a densely deÿned operator
Y
such that for all
0
(ÿY
=
D (A þ
0
0
A y
y
X
y ; Ax hY 0 ;Y i
þ
so that for all
to
y ; Ax hY 0 ;Y i
(denoted by
D (A
0
is an operator
Y
0
X ;j
, with the pairings denoted by
, respectively. Note that now
follows: 1.
and
d
is an isometry from
D (A)
0
X
f
are Hilbert spaces and that
Y
d
Z
i h i
Zd ÿZ jV d
Suppose that
0
. We depict this schematically in Figure 2.3. For all
we have
W
v
for all
y
y
X
2
0
, deÿned as
ú , and
2
x
X
D (A).
is related to
þ
D (A
A
0
by
);
we have 0
(ÿX j )A (ÿY k )
ø
y; Ax
i 2L
1
ÿ
ø
y; x
hX 0 ;X i
hY 0 ;Y i
Ax; y Y :
2L
For a bounded linear operator T (X; Y ) we have T 0 (Y 0 ; X 0 ), T þ = 0 ÿ1 (ÿX j )T (ÿY k ) (Y ; X ). Owing to the identiÿcation of the bidual of a Hilbert
2L
space with itself, we have that an operator
T
2L
(X; Y ) satisÿes (T 0 )0 =
The dual of a linear densely deÿned closed operator denoted by
0
A
:
0
D (A
) (
ý
Y
0
)
!
X
0
A
:
D (A)
(
T.
ý ! X)
Y
,
is closed and densely deÿned. Hence the
20
Chapter 2.
dual of
A
0
, denoted by (A0 )0 :
and moreover,
D ((A
0 0
))=
0 0
D ((A
)) (
ÿ
0 0
D (A)
and (A )
X
Classes of well-posed linear systems
00
=
x
=
X)
!
Y
for all
Ax
00
x
=
2
Y
is well-deÿned
D (A).
Since we identify the bidual of a Hilbert space with itself, it follows that the dual of an operator T
2L
T
0
2L
(X; X 0 ) satisÿes
(X; X ) self-dual if
T
is self-adjoint and satisÿes
2.3
=
h
T
0
T
2L
0
(X; X 0 ). We call an operator
. In this case the operator
i
h
=
y; S x X
i
:= (ÿX j )T
S
T x; y hX 0 ;X i .
2L
(X )
The Pritchard-Salamon class
In this section we introduce a special class of well-posed linear systems for which there exists a rich literature (see Salamon [72], [73], Pritchard and Salamon [66], Curtain et al. [15]). Most systems containing delays and some systems described by partial diþerential equations ÿt into this framework. This class generalizes the class of state linear systems with bounded input and output operators studied in [27]. Let
V
and
be separable Hilbert spaces with continuous, dense injections and
W
which satisfy
! !
W ,
Suppose ÿ þthat T
W
(t)
that
tÿ0
,
ÿ
A
Z ,
V:
is theþ inÿnitesimal generator of strongly continuous semigroups
T
Z
(t) T
tÿ0
V
(t)
j
and
Z
=
ÿ
T
T
Z
V
þ
(t )
on
tÿ0
(t) and
T
Z
W, Z
j
(t)
W
=
and
T
W
V
, respectively, such
(t):
Since these semigroups are consistent, we shall simply use the notation
f g
T (t) tÿ0 .
Assume further that
U
and
Y
are separable Hilbert spaces (the
input and output spaces), respectively.
1.
2L f g
(U; V ) is a Pritchard-Salamon admissible control operator for
B
T (t) tÿ0
for all 2.
u
2L f g C
if there exist a
ýZ ý ý ý
2
t
T (t
0
þ
t >
ý )Bu(ý )dý
ý ý ý ý
þ >
0 such that
ý kk þ
W
u L2 ([0;t);U )
(2.10)
L2 ([0; t); U ).
(W; Y ) is a Pritchard-Salamon admissible observation operator for
T (t) tÿ0
if there exist a
t >
z
2
0 and a constant
k ü k CT (
for all
0 and a constant
W.
)z
L2 ([0;t);Y
)
ü >
ý kk ü
z
V
0 such that (2.11)
2.3.
21
The Pritchard-Salamon class
Remarks: In the above deÿnition,
it is usual to take either
t
Z
=
V
or
Z Z
is not essential and in the applications
W.
=
Furthermore, we remark that if
(2.10) holds for some , then it can be shown that it holds for all
ÿ
t
t
t > 0, where
will depend on . Similarly, if (2.11) holds for some , then it can be shown
t > 0, where þ will depend on t in general.
that it holds for all
See for example,
Remark 2.10, page 11, Curtain et al. [15].
Under the above assumptions, the state linear system þ( Pritchard-Salamon system for any
D 2 L(U; Y ).
A; B; C; Dÿ) is þcalled a D AV ,! W ,
If, in addition,
A; B; C; D) is called a smooth Pritchard-Salamon system.
then þ(
When confusion may arise we use superscripts, for example denote the operators on on
X
by
!0X
to
V , Z or W . We denote the growth bound4 bounds !0W and !0V are diýerent in general). We
X
(the growth
AX , T X (t),
=
now enumerate a few properties of Pritchard-Salamon systems which have been proved in Curtain et al. [15] and in Curtain and Zwart [26] and will be used in the proofs.
1. For a smooth Pritchard-Salamon system,
ÿ þ ÿ þ ý AW = ý AV .
L 2 L Y; V fT t gtÿ0
2. Let
(
ÿ
ýp AW
þ
ÿ
ýp AV
=
þ
and
) be a Pritchard-Salamon admissible input operator for
A; B; C; D) is a Pritchard-Salamon sysA + LC; B; C; D) is also a Pritchard-Salamon system, where A + LC is the inÿnitesimal generator of fTLC (t)gtÿ0 which is the unique ( )
and assume that þ(
tem. Then þ( solution of
TLC (t)z = T (t)z + Furthermore,
fTLC t gtÿ0 ( )
We note that
TLC (t ÿ ü )LCT (ü )zdü:
0
is also the unique solution of
TLC (t)z = T (t)z + trol operator.
Z t
Z t
T (t ÿ ü )LCTLC (ü )zdü:
0
L 2 L(Y; W ) is always a Pritchard-Salamon admissible con-
A; B; C; D) has a well-deÿned G(s) given byý y^(s) = G ( u(s), for all s 2 C +ÿ := fs 2 ü s)^ V W j Re(s)> ûg, where û > max !0 ; !0 , and
3. A smooth Pritchard-Salamon system þ( transfer operator
C
G(s) = D + C sI ÿ AV ÿ
4 The growth bound of a semigroup fT (t)g
inf
t>0
log
kT (t)k
t
tÿ0
,
þÿ1
B:
denoted by
!0 ,
is deÿned to be
22
Chapter 2.
Furthermore,
ÿ
C
ÿþ
ÿ
L
+
I
A
2 !L
B
ý
þ
C ÿ ; (U; Y ) =
H1
þÿ1
V
G
+
Cÿ
:
(C + ; ÿ
H1
W
0
B
0
.
2L
(V 0 ; U ),
C
0
j
(U; Y )
4. Let us assume that the Hilbert spaces duals. Then
Classes of well-posed linear systems
2L
L
(U; Y )), where is analytic, and
G
sups2C + ÿ
Z, U , Y
(Y ; W 0 ),
D
0
k
k
G(s) L(U;Y ) <
D
If
(Y ; U ) and
V
! !
0
,
Z
) is also a smooth Pritchard-Salamon system, and
Its
and its
observability map
2 2L 2L 1 L ý ý W.
z
The
(L2 ([0;
B C2L
Z
(A )
ú
C
(L2 ([0;
); U ); L2 ([0;
where
H1 (
(V ; L2 ([0;
C
z
=
1 ÿ
1
0, taking values in
right half-plane Re(s) half-plane Re(s)
>
0, and
CT (
þt
e
G
L
)B
.
and
exponen-
2L BB CC CB þ 2
(W 0 ; W ) and
LB
=
0
,
LC
=
G
C (sI
A)
0
ÿ1
. Its
observability
Hankel opera-
. For an exponen-
B
c
L
H1 (
(U; Y )),
deÿned in the closed right half-
(U; Y ), such that
G
is analytic in the open
is bounded and continuous in the closed right
0. Moreover,
ÿ 2
CT (
tain et al.
,
)z
); Y )) is deÿned by ý =
jslim j!1
(That
V
0
0
); Y )) is deÿned by
(U; Y )) denotes the space of maps
plane Re(s)
V
B
); U ); W ) is deÿned by
controllability Gramian LB 0
!
,
,
,
T (s)Bu(s)ds
0
tially stable, smooth Pritchard-Salamon system, c
1
0
0
1
=
(V ; V ) are deÿned by
Gramian LC
ý
B2L
controllability map
u
tor
D
0 W
0
then we say that the Pritchard-Salamon system þ(A; B; C; D ) is
tially stable.
for
û
generates an exponentially stable strongly continuous semigroup on
A
W,
:
are identiÿed with their
2L
If þ(A; B; C; D ) is a smooth Pritchard-Salamon system, then þ(A0 , 0
1
ü
k
G(s)
k
= 0:
(2.12)
Re(s)þ0
L2 ([0;
[15], where
1 L ÿ
CT (
);
(U; Y )) for some
ÿ
>
0 is shown in Cur-
)B has to be interpreted in the sense of Re-
mark 2.10.(iv) [15].
Using the Cauchy-Schwarz inequality, it follows that
CT (
(U; Y )). Finally, using Property A.6.2.g, page 636, Cur-
ÿ 2 )B
L1 ([0;
1 L );
tain and Zwart [27], we obtain the desired result.) We now quote the following lemma from Curtain and Zwart [26] on Lyapunov equations:
Lemma 2.3.1 Suppose that þ(A; B; C; D ) is an exponentially stable, smooth Pritchard-Salamon system. Then
2.3.
1.
23
The Pritchard-Salamon class
L
B
and
L
L
are the unique solutions in
C
of the following Lyapunov equations
h
z1 ; (ALB
1 2
for all z ; z
2
ÿ
(A0 )W
D
h1
z ;
1 2
for all z ; z
2
D
ý
A
0
(A
+ LB A + BB )z2 0
þ 0
+ LC A + C
LC
and ÿ
2L
(V ; W )
2L
ÿ
2L \L \L (V )
(LB LC ) =
ÿ
(V ; V 0 )
respectively
=0
hW 0 ;W i
(2.13)
0
C )z
2
i
=0
hV ;V 0 i
(2.14)
. Moreover,
ÿ
W
ý
D
ü
V
A
(W ),
(LB LC )
0 W
(A )
D
ÿ
V
i
L
and
;
LC
2. LB LC
0
ü V
LB
and
(W 0 ; W )
0
þ ;D
ÿ
(I + þLC LB )
ÿ
0 W
0
þþ :
2L \L ÿ 2 2 L \L 2 L \L
LC LB
1 1
ÿ
üþ
V
A
(A )
;D
and for
(I + þLB LC )
ý
(W 0 ; V 0 )
1
( LB LC ) ,
ÿ
ÿ
(V )
(V 0 )
\L
(W 0 ),
there holds
(W );
0
0
(W )
(V )
and
1
(I + þLB LC )
ÿ
(I + þLC LB )
ÿ
1
2 L ÿý 2 L
ý
D
üü
V
A
ÿ
D
;
0 W
(A )
0
þþ :
From Curtain and Zwart [26], we have
Lemma 2.3.2 Suppose that ÿ(A; B; C ) is an exponentially Pritchard-Salamon system. Then ý (þ r
k k2
(þþ þ) = þ
If
U
=
þ
, where r (
C m and
Y
=
)
þ
nf g
þ)
0 =
ý
(LB LC )
denotes the spectral radius.
nf g
stable,
0
smooth
and r (LC )
=
C p , then the Hankel operator of an exponentially stable
Pritchard-Salamon system is compact (Lemma 3.5 in Curtain et al. [15] and Glover et al. [37]).
Lemma 2.3.3 Suppose that ÿ(A; B; C ) is an exponentially
ÿ1 ÿ1 ÿ [ 1
Pritchard-Salamon system with input space
+1
ýl
< ý < ýl .
The operator J
trum ý (J ) contained in consists of exactly
l
(
;
:=
Æ)
I
(Æ;
þ2
)
negative eigenvalues.
Cm
þþ þ
2L
1
stable,
and output space
(L2 ([0;
for some Æ >
0,
); C m )
and ý (J )
Cp
smooth and let
\ ÿ1 ÿ
has a spec-
(
;
Æ)
24
Chapter 2.
Classes of well-posed linear systems
ÿþ ÿ
ýÿ ü ú with 0 as the accu-
ÿÿ ÿ is compact and has a pure point spectrum with 0 as the accumuþ1 1 ÿÿ ÿ ÿI , it is easy to see lation point. Considering the resolvent I ÿ2 1 that J has a spectrum which is a shifted version of the spectrum of ÿ 2 ÿÿ ÿ. Proof
û
Finally, since ÿÿ ÿ has a pure point spectrum mulation point, and since
þl+1 < þ < þl , J
2
2
þ1 ; þ2 ; : : :
has exactly
l
negative eigenvalues.
Finally we give a simple example5 of a delay system which can be modelled as a smooth Pritchard-Salamon system.
ÿsÿ
e
Example 2.3.4 Let G(s) =
of
G.
Let
Z
=
1;2
þý ;
(
0. We will þrst give a realization
ý >
ÿ L2 (þý ; 0) (with the obvious inner product).
C
Sobolev space
W
, where
s+1
8 < 0) = :
2 L2 (þý ; 0) j
f
9 = distributions) is a regular distribution ; with 2 (þ 0) the derivative of
Tg ;
equipped with the inner product
g
Z h
f
L2
(in the sense of
W 1;2 (þþ;0) =
f (x)g (x)
;
ý;
0
hf; gi
Consider the
0
+ f (x)g 0 (x)
i
dx:
þþ
(With this inner product,
W
1;2
þý ; 0) is a Hilbert space: see for example, Propo-
(
sition 5, page 55, Yosida [91]. Furthermore, from Corollary 7.3 [80], it follows that
W
1;2
þý ; 0) ý C (þý ; 0).)
(
ù
Deþne
A
÷ù
with D (A)
=
ø
z1 z2
Then it can be shown that
A
z1
ø ùþ =
z2
z2
W.
5
ÿ
Let
W
z ~1
y (t)
= =
tý0
øô
z ~2
B
D (A)
z (t)
z (t
ÿ
+ u(t); ÿ ):
:
=
on
Z
(see for instance, Example
hz2 ; z~2 i
W 1;2 (þþ;0)
W
ù ø
ÿ
= z2 (0)
with the inner product
tý
Bu z (t)
fT (t)g
:=
fT (t)g 0 restricts to a 2 L(C ; Z ) be deþned as follows:
is a Hilbert space and on
ö z1
is the inþnitesimal generator of an exponentially
øù ;
;
0
z2
;
2.8, van Keulen [84]). The space z1
ø
2 Z j z2 2 W 1 2 (þý ; 0);
stable strongly continuous semigroup
õù
z2 (0)
=
u
0
;
strongly continuous semigroup
2.4.
25
The analytic class
Finally, if
:
C
D (A)
!C
is given by
ÿ
then
2L
and
B
C
ÿ
= z2 (
z2
(W; C ). We now choose
C
is explained that
þ
z1
C
=
V
Z.
ÿ );
In Pritchard and Salamon [66] it
are Pritchard-Salamon admissible control and obser-
vation operators, respectively. Finally we note that satisÿed since
=
W
D (A),
ý Vü A
!
,
W
is trivially
and so in fact we have a smooth Pritchard-Salamon
system, with a transfer function
2.4
D
eÿÿs . s+1
ÿ
The analytic class
In this section we introduce the class of exponentially stable analytic inÿnitedimensional systems and establish some basic system theoretic properties. First we recall the basic properties of exponentially stable analytic semigroups using the notation from Staþans [77] (see also Pazy [63]). 1. Let
ÿ
A
be the inÿnitesimal generator of an exponentially stable analytic
semigroup
f gtÿ 2R T (t)
in
0
Z.
k kZ
ý ü , we let6 Zÿ = Aÿÿ Z be D Aÿ with the norm û û þ ú ÿ ÿ ù ûAÿ z û and inner product z1 ; z2 Zþ = A z1 ; A z2 Z . Z
2. For each
3. The restrictions of for
þ <
Z
A
A
to
i
ÿ for
Z
þ
2R
ÿ
þ >
0 (which we still denote by
ÿ for all
ÿ 2L
ÿ
h
0 and the extensions of
A)
þ =
z
ÿ
A
to
ÿ
Z
generate analytic semigroups in
. The generator of the semigroup
(Zÿ +1 ; Zÿ ).
f gtþ T (t)
0
on
ÿ is then
Z
These semigroups are all similar to each other, and they commute with
þ for all ý T (t) tþ0 .
f g
A
2R
. We therefore denote all of them by the same symbol
2R
4. For each
t >
0 and
5. For each
þ
0, there exist
þ
þ
,
T (t)
ÿ into
maps
K1 >
û ÿ û ûA T (t)û
Z
0 and
ý
K1
ü >
ÿýt e ÿ
t
\þ2R
Z
þ.
0 such that
;
t >
0
where the norm represents the operator norm in any one of the spaces If
þ <
0, then there exist
K2 >
û ÿ û ûA T (t)û
0 and
L(Z )
ý
ü >
0 such that
ÿýt ;
K2 e
ÿ.
Z
t >
0:
(This follows from Lemma 6.3, page 71, Pazy [63].) 6 For
the deÿnition of
Aÿ
are its related properties, we refer the reader to
x2.6
of Pazy [63].
26
Chapter 2.
6. The map
t
7!
T (t)
1 !L 2R
: (0;
)
Classes of well-posed linear systems
(Z ) is continuous in the norm topology.
(See for example, Proposition 2.1.1.(iv), page 35, Lunardi, [53].) This implies that for any
, the map
ÿ
7!
t
continuous in the norm topology.
T (t)
: (0;
7. The same conclusion as above can be repeated with 0
give another chain of Hilbert spaces 0
We identify that
=
Z0
where
0
=
Z0
Z.
For
h
z1 ; z2
hÿ ÿi ;
with the dual of
Zÿ
2
z1
i
hZ
Zÿ
and
D
=
ÿÿ ;Zÿ i 0
= (A )
by using
Zÿÿ
0 ÿÿ
Z
0 ÿÿ
z2
2
Z
(Zÿ ) is
replaced by
A
A
0
to
with similar properties.
we have
0 ÿÿ
ÿ
(A )
A z2 ;
)
as the pivot space. Note
Z
Zÿ ,
1 !L
E
z1
; Z
denotes the duality pairing.
0 ;Zÿ i ÿÿ
hZ
Next we analyze the system theoretic properties of the class of systems described by the triple (
A1.
þ
A
þ
A; B; C )
B
2L
C
f g
T (t) tþ0
on the Hilbert space
(U; Zþ ), where
satisfying A3.
2L
þ
1
< þ
ý
(Z; Y ), where
ý
under the assumptions
satisfying
þ
1
Z.
is a Hilbert space and
U
þ
is a ÿxed number
0. is a Hilbert space.
Y
2L ! ! !
We note that the assumption A2 implies that for every
Z,
is the inÿnitesimal generator of an exponentially stable analytic
semigroup
A2.
on the Hilbert space
< ý
ý
þ: Z ,
A
þ
Zþ ,
B
Zý ,
(U; Z ) and
B
Zÿ1 .
2L
(U; Zý )
Remark: We remark that what follows is similar to what was done in Curtain
and Ichikawa [14], but there are diþerences in assumptions (see Table 2.1). First we derive several results about
B
Lemma 2.4.1.1 below we prove that if
1
then the controllability map
B B
:
u
L2 ([0;
Z
=
ý
and its controllability Gramian. In 1 ý < 2 + þ, ); U ) Zý given by is a real number satisfying
!
1
0
T (t)Bu(t)dt
is a well-deÿned bounded linear map, that is,
f g
B
(2.15)
is an admissible input operator
(see Weiss [86]) for the semigroup T (t) tþ0 on the Hilbert space Zý , where 1 ý < 2 + þ. Moreover, in part 2. we show that if ý and ü are real numbers satisfying ý ü < 1 + 2þ, ý < 1 + þ, ü > (1 + þ), then the controllability
Gramian
þ
LB
:
0
Zü
!
Zý
given by
Z
LB z
=
þ
1 0
0
0
T (t)BB T (t) zdt
(2.16)
2.4.
1.
27
The analytic class
Curtain and Ichikawa [14]
Present approach
They considered
Here we simply assume that
A
to be a
ÿ
uniformly strongly elliptic
A
operator of order two with
is the inÿnitesimal
generator of an exponentially
smooth coeÆcients
stable, analytic semigroup on
deÿned on a bounded open domain þ of d .
Z.
Thus
ÿ
R
is the inÿnitesimal
A
generator of an exponentially stable, analytic, compact semigroup on 2.
B
2L
(U; Zÿ ),
=
2 (@ þ).
where 3.
U
Z.
L
ÿ
1
ÿ 34
< ÿ <
Thus
2L
B
.
U
where is
=
U
(U; Zÿ ),
ÿ
1
þ
< ÿ
0.
Cm
inÿnite-dimensional, unless þ is, for example, (0; 1), in which case
U
can be identiÿed with
C 2.
Table 2.1: Diýerences in assumptions.
is a well-deÿned bounded linear map from
þ
Z
0
to
ý.
Z
B
Lemma 2.4.1 and LB given by (2.15) and (2.16) are well-deÿned and have the following properties:
1.
B2L
(L2 ([0;
1
B 2L
); U ); Zý ), and
satisfying þ < 12 + ÿ. ý ü LB Zþ ; Z ý for all
0
ÿ
ý ; L2 ([0;
0 ÿ
Z
þ
1
); U ) , for all
þ
2L and satisfying ÿ , , ÿ . ý ü 1 and ÿ 21 ÿ and it is self-dual 3. B BB 2 L for ý þ 2 þ ÿ 1 in L . ý ý for 2 þ þ ÿ ÿ 4. B 2 L ý ý +1 \L ý 1 ý is a solution of the Lyapunov equa2.
0
L
ý
þ
þ
ý <
1 + 2ÿ
þ <
1+
ÿ
(1 + ÿ)
ý >
0
=
0
Z ;Z
0
Zÿ ; Z
L
Z
0 ÿ
þ <
for any
z
2
ý >
;Z
Z
0 ÿ
ÿ
;Z
B z + LB A z = BB z
AL
ý,
0 Zÿ
where
ÿ
+ÿ
þ <
tion
+ÿ
0
ÿ
1
< þ < ÿ.
0
in
Z
ý
(2.17)
28
Chapter 2.
Proof 1.
Let
2 2
u
T (t)Bu(t)
if
1 2
ÿ <
1
L2 ([0; Zþ
); U ). Since for
for all
2R
ÿ þ ÿ ÿA T (t)Bu(t)ÿ
ý
ÿ þ ÿA
=
ÿ
ÿ
T (t)
ÿ ÿ
L
0
þZ
ý Case 1:
ÿ þ ÿA
ÿ
ÿ
ÿ2 T (t)ÿ
þ
Z
Z
ý
ý
þZ
ý
þZ
ý
ÿ þ ÿA
ü
ÿ
ÿ
ÿ
0 2
ý
t
0
þ <
Z
2(þ
ü
ÿ)
R
1 ÿ
0
Z 0
ý
e
ý
ý ý
þZ
dt <
L1 ([0;
1
); Z )
kL
k k
kk
1
kk
1
(U;Z ) u(t)
u L2 ([0;
U dt
k
ÿ
kL
k
ÿ
kL
);U ) A B
(U;Z ) :
dt
(Z )
ý 21
u L2 ([0;
kk
dt
u L2 ([0;
1
þZ
2
2(þ
ÿ
);U ) A B
e
ÿ ÿ
ý 21
2üt
dt
ÿ)
(U;Z )
kL
ÿ2ÿt ÿý) dt <
t2(þ
0
K1 e t
0
1
R
k
1
);U ) A B
(U;Z ) :
1
, and so
kk
1
u L ([0; 2
k
ÿ
);U ) A B
kL
(U;Z ) :
(2.18)
0. In this case,
þZ
2üt
B
ý 12
ÿ þ ÿ ÿA T (t)Bu(t)ÿ
0
But
it follows that
dt
1, we have
ý
dt
Z
L
2üt
þ) <
Z
ÿ2 T (t)ÿ
ÿ ÿ
K1 e
0
Case 2:
ÿ
dt
(Z )
ÿ þ ÿ ÿA T (t)Bu(t)ÿ
ÿ þ ÿ ÿA T (t)Bu(t)ÿ
ý
Zþ ,
þ
)Bu( )
0. In this case,
satisfying 2(ÿ
ÿ
A
ý 21
0
For
k
L
0
ü û
ÿ
ý
(Z )
ÿ \ 2R þ þ2
þ
A T(
dt
Z
Z0 ý
T (t)Zÿ
. Next we show that
ÿ
+ þ:
Z
0,
t >
Classes of well-posed linear systems
ý
ÿ þ ÿA
ÿ
ÿ
ÿ2 T (t)ÿ
0 ý
0
ÿ
2 K2 e
2üt
Z
dt
ý 12
L
ý 21
(Z )
kk
u L2 ([0;
dt
1
kk
k
1
k
1
u L2 ([0;
dt
ÿ
ÿ
);U ) A B
);U ) A B
kL
kL
(U;Z )
(U;Z ) :
, and so
ÿ þ ÿ ÿA T (t)Bu(t)ÿ
Z
dt
ý
1
þZ 0
ÿ
2 K2 e
2üt
ý 21 dt
kk
u L ([0; 2
1
k
ÿ
);U ) A B
kL
(U;Z ) :
(2.19)
2.4.
29
The analytic class
ÿ
ÿ
From the above two cases, it follows that
ÿ2
1
)Bu( ) L1 ([0; ); Z ). Consequently, (see for example Theorem A.5.23, page 628, Curtain and Zwart [27]: with
f , A, D (A), Z1 , Z2
Zÿ , Z
and
ÿ
ÿ
A T(
ÿ
and
Af
in the theorem replaced by
)Bu( ), respectively)
1
Z
ÿ
A
T (t)Bu(t)dt
=
L2 ([0;
ÿ
ÿ
Z
1
ý
); U )
=
0
and 0
T (t)BB T (t)
); U ); Zÿ ).
for
ý.
ý
ÿ <
1 2
t
>
0 we have
2
L1 ([0;
1
); Z ) for
z
2
Z
ÿ ÿ ÿ ÿA T (t)BB 0 T (t)0 (A0 )ÿþ z ÿ
0
=
Suppose that
ÿ
ü û ü ü û 0,
þ
k
þ
ý
0.
ÿ 1
ÿý ÿ1
ý
0
1
ý
(1 + 2 )
þ
ÿ(1 + 2ý) ÿ1
Z
z
2
ý
0
);U )
0
0
Z,
ÿ ÿ ÿ ÿA T (t)BB 0 T (t)0 (A0 )ÿþ z ÿ
Z
dt
0
T (t)BB T (t) Zþ Z.
and
ý
For
z
2
ÿ
Z,
dt
L(Z )
Then we have for
Zÿ ,
B 2 0
+ þ.
k k kk
=
,
1
u L2 ([0;
Consequently, we also obtain that
Z
Case 1:
þ kk M
Z
Z0 ý ÿ ÿ ÿAÿ ÿü T (t)Aü BB 0 (A0 )ü T (t)0 (A0 )ÿüÿþ z ÿ dt Z Z0 ý ÿ ÿ ü 0 0 ü ÿAÿ ÿü T (t)ÿ A B L(U;Z ) B (A ) L(Z;U ) L(Z ) ÿ0 0 0 ÿüÿþ ÿ ÿ ÿT (t) (A ) z Z dt:
=
ÿ
ÿ
We ÿrst investigate the values of
(A0 )ÿþ z
Z
A
A T (t)Bu(t)dt;
ÿ Z 1 ÿ ÿ ÿ ÿ ÿA T (t)Bu(t)dtÿ ÿ ÿ 0
From property (4), for every ÿ
ÿ
)Bu( ),
Moreover, from (2.18) and (2.19) above, there exists a
(L2 ([0;
for any real A
þ.
B2L 1
0 ; ÿÿ
Z
2.
+
ÿZ 1 ÿ ÿ ÿ ÿ T (t)Bu(t)dtÿ ÿ 0 ÿ
and so
L
1 2
ÿ
T(
0
0 such that
M >
þ
ÿ <
1
Z
0
for every
A T(
k
ý
Zÿ ,
for which we have
30
Chapter 2.
ÿZ
ÿ ÿ where L1 ([0;
1
M1
kk z
ü
û
dt
k
A
ü
B
k
L(U;Z )
2
z
Z
Suppose that
k
B
0
0 ü
(A )
k
L(Z;U )
kk z
Z
Z;
is a constant independent of
); Z ) for
Case 2:
ý
t
þ
þt
K1 e
0 M1
ÿ2 ÿ2 ÿ 2
ÿ
Classes of well-posed linear systems
þ þ þ ý þ þ
and þ
2ý
þ
0,
ý
ÿ.
ü <
ý
ý
Thus
0
0
A T (t)BB T (t)
(A0 )ÿû z
1.
ü <
2
0.
ý 1
ÿþ ÿ1
1+þ
0
þ
ÿ
1
ÿ1 Then we have for
Z
ÿ ÿ where L1 ([0;
1
M2
Case 3:
z
2
Z,
ý ý ý ýA T (t)BB 0 T (t)0 (A0 )ÿû z ý dt Z þ ÿZ0 ÿ ÿ2þt ÿ
K1 K2 e
kk
M2 z
dt
ý ÿü
t
0
k
A
ü
B
k
z
2
k
B
0
Z
Suppose that
and þ
þ
þ
þ
ý <
ý <
0,
ÿ.
Thus
ý
1.
þ þ ý ý
ü
0.
1
ÿþ ÿ1
þ
0
1
ÿ(1 + þ) ÿ1
Z
0
z ÿ
2
k
L(Z;U )
kk
z Z
Z,
ý ý ý ýA T (t)BB 0 T (t)0 (A0 )ÿû z ý
Z
dt
0
0
A T (t)BB T (t)
ý
Then we have for
0 ü
(A )
Z;
is a constant independent of
); Z ) for
L(U;Z )
ÿ
(A0 )ÿû z
2
2.4.
ÿZ
ÿ ÿ where L1 ([0;
31
The analytic class
1
M3
ÿ2 ÿ ÿ t ý
kk
þ
þt
K1 K2 e
0
dt
ü
k
A
ý
B
k
L(U;Z )
k
B
0
0 ý
(A )
k
L(Z;U )
kk z
Z
M3 z Z ;
is a constant independent of
); Z ) for
Case 4:
ÿ
2
z
and
Z
Suppose that
ü
þ
þ þ þ
0,
þ <
ÿ.
û
Thus
0
A T (t)BB T (t)
0
(A0 )ÿü z
2
(A0 )ÿü z
2
1.
ý <
þ þ ý þ
0.
ý
ÿ 1
ÿý ÿ1
0
ý
þ
1
ÿ1 Then we have for
z
Z
ÿ ÿ where L1 ([0;
If
z
Z
2
1
1
M4
0
Z,
ý û ý ýA T (t)BB 0 T (t)0 (A0 )ÿü z ý dt Z ÿZ0 ÿ þ ÿ
2
K2 e
ÿ2þt
k
dt
kk
0
ý
A
B
k
L(U;Z )
k
B
0
0 ý
(A )
k
L(Z;U )
kk
z Z
M4 z Z ;
is a constant independent of
); Z ) for
Zü ,
2
2
z
then (A0 )ü z
2
Z
ý û ý ýA T (t)BB 0 T (t)0 z ý
0
M >
k k
LB z Zþ
Thus
û
0
0
A T (t)BB T (t)
and applying the above results,
Z
for some constant
ÿ.
Z.
Z
ý ý ÿ ÿ k k 2 û
=
0
0
0. So if =
1
A T (t)BB T (t)
dt
z
0 ü
M
(A )
0
then
Zü ,
z
Z
=
M
kk z
0 ÿü
(A )
Z0 ;
ÿ
Z
(using Theorem A.5.23, page 628, [27])
Z
1
0 M
ý û ý ýA T (t)BB 0 T (t)0 z ý
kk
z Z0 ; ÿ
Z
dt
0 ü
(A )
ý Z 1 ý ý û ý 0 0 ýA T (t)BB T (t) zdtý ý ý ýZ 1 0 ýZ ý ý û 0 0 ý A T (t)BB T (t) zdtý ý ý 0
ÿ ÿ
0
z
ý ý
Z
dt
32
Chapter 2.
for
ÿ
ÿ2 ÿ þ
ý <
1,
ÿ <
1 + þ,
ý >
Figure 2.4.
ÿ(1 +
Classes of well-posed linear systems
þ).
We depict this region (shaded) in
ý 1
1+þ
0
ÿ1
1 + 2þ
þ
ÿ
1
ÿ(1 + 2þ) ÿ(1 + þ) ÿ1 Figure 2.4: The feasible region of parameter values for
L
B.
We note that the point (1 + 2þ; 0) lies to the right of the point (þ; 0) since 1+þ
0. However, whether the point (1 + 2þ; 0) lies to the right or to the left
>
or coincides with the origin depends on the value of
ÿ1
ÿ 12
< þ <
þ
ÿ 12
=
:
left;
ÿ 12
:
coincides;
0
:
right:
ÿ
< þ
(In Figure 2.4, we have shown the case when
3.
L
That
ý
BBü 2 L ÿ 0
þ ; Zþ
0 Z ÿ
ÿ 0
Z ;Z
for
ÿ <
þ
þ
for
1 + 2
ÿ <
L
It can be easily shown that if
0
þ þ
B þ þ (
B2L
z
2
0
for
1 BB þ
(L2 ([0;
0
But since
ÿ
0
z
ý >
Z
=
ý <
ÿ1
ÿ
T (t)B
þ ÿ
B
þ
and
þ
=
=
for
Zÿ
þ þ þ
satisÿes
1 2
1 2
ý >
BB 2 L ÿ 0
þ
0
0
B T (t) z;
B
B
Z
0
zdt
þ
=
0
þ
ÿ 0
Z ;Z
ÿ <
then 0 z = 1 < 2 + þ,
þ,
1
0
2
z
z ) (t)
1 2 ); U ); Zþ ) for
ÿ
Z
þ
< þ
ÿ
0.)
and is self-dual in
1 2 + þ is a consequence of part 1 above.
So it remains to prove the equality 1 þ: ý > 2
Thus if
þ:
þ
t
B T(
ý ü 2
1 2 +
þ
and
0:
)0 z
L2 ([0;
0
0
T (t)BB T (t) zdt
1 2 + þ + 2 + þ = 1 + 2þ,
ÿ <
1 + þ, then 2
1
0
for
ÿ <
1 2 +þ
<
1 2
); U ), and since
þ
Z :
1 + þ,
ý >
þþ 1 2
þ >
2.4.
ÿ1 ÿ
ÿ,
Thus
it follows from part 2 that
ÿA
1
Z
BB 2 L 0
=
LB
From A2,
4.
33
The analytic class
B
0
0
0
ÿ
0
T (t)BB T (t) zdt
0
Zþ ; Zÿ
þ
for
2Z
LB z
ÿ:
1 1 2 + ÿ and ý > ÿ 2 ÿ ÿ.
þ <
(A0 )ÿ belongs to
=
L(Z; U ) for þ
ÿ1 < þ þ ÿ.
satisfying
Since
is the inÿnitesimal generator of an exponentially stable semigroup on the
Hilbert space
Z,
it follows from the standard theory for Lyapunov equations
(see for example Theorem 4.1.23, page 140, Curtain and Zwart [27]), applied to
ÿ
þ
ÿA; B (A ) ; ÿ with state space Z , that there exists a unique self-adjoint solution P 2 L(Z ) of the operator Lyapunov equation ý ü hA z1 ; P z2 i + hP z1 ; A z2 i = B (A ) z1 ; B (A ) z2 ;
þ
0 ÿ
0
0
0
where
1
and
z
z
2
Z
=
If
w
1
and
2
0
A
0 ÿÿ
(A )
1
ÿ
1 0
2
)
w
ü Z
We now proceed to relate û part 2, that
then
w
2Z
between
0 ÿÿ
hw1 ; AL
P
+
0 ; ÿÿ
Z
to
0 ÿ
(A )
0
T (t) zdt
for
0 ÿÿ
P (A
ú
LB
)
1
w ;A
0
0 ÿÿ
(A )
ÿÿ
2 ih
0
Zÿÿ ; Zÿ
+1 ý Z
ÿ
hw1 ; AL
+1
2 D(A ).
2
ü
deÿned in part 2.û If
Z
0
hB w1 ; B w2 i 0
=
0
U
:
(2.20)
ÿ
and so
þ < ÿ,
ú we
0 ÿÿ
ALB
have, using
Z
0
ú
ù
ÿ
i
=
A
ÿÿ
1 0
1
A w ;
ÿ
1 0
0
Z
1
2
0
0
û
Z0
0
ÿ
belongs to
ù ÿ ;Zÿ
ÿ
i
=
0
ù =
1
A w ;
0
1
A w ;
ø
0
Z
1 0
Z ÿÿ
ýZ
0 ÿÿ
0 ; ÿÿ 0 ÿÿ
0
0
1 0
ÿ
h
ÿ
i
0
ÿ +1
Z0
ÿ
1
. Since
ÿ ÿ1
0 ÿÿ ÿ
ú
;Z
. From part 2,
2
2
i
T (t)BB T (t) w dt
ø
h
12
2 w2 2
LB
LB
0
ÿ
0
ÿ
1
Zÿ ÿ ;Z
h
Zÿ ;Z
ÿ
i
ÿ +1 :
;
w
ø
T (t)BB T (t) w dt
0 0
Z
0 Zÿ ;Z
, it follows that
1
Z
2
h
ø
T (t)BB T (t) w dt
0 ÿÿ
w
2
0
T (t)BB T (t) w dt
0
and since
2 ih
ÿÿ
;Z
ÿÿ
Z
w ;A
ù
and so Bw
0
ÿ
0
0 Zÿ ;Z
0
ÿÿ
2 Z:
z
w
2 L (Z +1 ; Z ) and A 2 L Z ÿ þ ÿ þ 1 = D (A ) , A w1 belongs to D (A ) = Z
0
Zÿ
0
0 ÿÿ
ÿ
where we have used
L
ý
Zÿ
=
1û
is given by
P
then (A0 )ÿÿ w1 and (A0 )ÿÿ w2 belong to
0
Bw
Z
U
2 L Z ; Z . If w 2 Z1 , +1 ý Z . If hü; üih ÿÿ ÿ i denotes the duality pairing , since Z1 and Z , then for w1 and w2 in Z1 , with ÿ1 < þ < ÿ, we have 2L
LB
0 ÿÿ
Z
) and
T (t)A BB
Z ÿÿ ,
0 ÿÿ
w ; P (A
D (A
0
Consequently, we have
ý
0
1
belong to
w
0 ÿ
0
Z
belong to Pz
0 ÿ
0
Z
i
34
Chapter 2.
Hence
h since
w1 ; ALB w
0
A w
2
1
ÿ
Z
0
1
2
=
Z0 1 ý
= since
0
2L
ÿÿ
A
ý
1
h
0 ÿÿ
0
(A )
A
1
A
ý
0
A
0
0 ÿÿ
(A )
ÿ
0
(A0 )ÿÿ w1
Z
w ;
ÿ
ÿ
1
0
1
2
0
ih
)
Z0
i
ÿ ;Zÿ
ÿ
(Here, using part 2, we consider 0
LB A
2L
ù
0 ; ÿÿ
Z
i
0 ÿÿ
2
(A )
Z.
2
1
0
for
z
1 0
LB A
2
and
z
and
in
BB
h1
0
ih
Z
2L 2 ih
1
z
and
ÿ
ù+1
(
2
+
0
0 ÿÿ
(A )
T (t)
0 ÿ
0
2
w
0 ÿÿ
(A )
0
(A )
2
T (t)
1
in
0 ÿÿ
)
ù
2L
0
0 ÿÿ
ÿ
ÿ
in
ü
h
w
0 Zÿ ;Z
2
0 ÿÿ
(A )
Z
0
ÿÿ
dt;
ÿ
ÿ
ü
Z
i
dt
dt;
ú
2
w dt Z
1
w ;A
0
0 ÿÿ
(A )
ø
1
0
, we have
Zÿÿ ÿ ; Zÿ
2
w
for
ü Z
(2.22)
:
< þ;
ÿ
we have
0
ÿ
Zÿ ;Z
ÿ
0 ÿÿ
Z
i h1 =
+1
satisÿes
LB 0
z ; BB z
is dense in
2
ih
Z
0
ÿ
ÿ
Zÿ ;Z
0 ÿÿ
i
, and
ALB ,
, it follows that
Zÿ
+
ih
). But since
i h1
0 ÿÿ
Z
2
0
0
2
z ; LB A z
ih
0
ÿ
Zÿ ;Z
ÿ
i h1 =
0
2
z ; BB z
ih
0
ÿ
Zÿ ;Z
ÿ
i
.
Next we derive some properties of the given by and the
i
(2.21)
P (A
z ; LB A z
Z
0 Zÿÿ ;
Zÿ ;Z
z
ÿ ;Zÿ
As a result we obtain that
w
ý
i h1 ÿ ø
ÿ
0 ÿÿ
z ; ALB z
holds for
0
Zÿ ;Z
Z
ÿ
.)
Zÿ
2
z ; ALB z
Z0
Consequently,
So by substituting (2.21) and (2.22) in (2.20), we see that
h1
ih
:
and
LB
ø
ü
w
w
=
2
0
Zÿ .
T (t)
0 ÿ
0
ÿ
w ; P (A
w ; LB A w
0 ÿ
(A )
T (t)A BB
Similarly, it can be shown that for
h
0
ÿ
0 ;Z Zÿ
1
2
2
i
T (t)A BB
1
0 ÿÿ
(A )
ÿ
1
ih
0
0
w ; T (t)A BB
A
2
w ; ALB w
=
ÿ
ÿÿ
(Z; Zÿ ) and
û =
0 Zÿ ;Z
1
0
1
0
A w ; T (t)BB T (t) w
T (t)BB T (t) w
ih
A w ;A
h
0
0
and
w ; ALB w
Z
=
ÿ ;Zÿ
þ
1
Z
ih ÿ i
(A0 )ÿÿ
D
h
2
Classes of well-posed linear systems
observability Gramian LC z
C
þ
z
=
CT (
:
Z
Z
LC
=
observability map C :
Z
0
!
)z;
Z
!
2 ([0;
L
1
); Y )
(2.23)
given by
1 0
0
T (t) C C T (t)zdt:
(2.24)
2.4.
35
The analytic class
f gÿ
Owing to the smoothing property of
T (t) t
smoothness too.
Lemma 2.4.2
erties: 1.
C2L
C and
isfying
ÿ
ÿ >
1 . 2
1
LC
0
Zÿ ; Zþ
0
þ
0
=
LC
0
Zÿ ; Zþ
0 Z ÿÿ
Zÿ ;
0
ý >
LC
Proof
1. If
z
2
z
Zÿ ,
Z
ü
0
2
and
LC
inherit some
ü
þ
and
þ
1
0 ); Y ); Zÿ , for all ÿ
ÿ
1,
ÿ <
1 2
þ <
\ Lý
0
0
1
þ <
ü
k
k
þ
Z
kk k k Z û û kk Z û û þ kk Z k k k k ûû
C T (t)z Y dt
ü
=
þ
1 < ÿ 2
Z
ü
0. If
k
C T (t)z
0
z
k
2
2
Y dt
2
C
C
C
Zÿ ,
L(Z;Y
Z
dt
A
)
ÿÿ
ÿ
T (t)A z
0
L(Z;Y
2
L(Z;Y
A
)
ÿÿ
T (t)
Z
ÿ
A z
û2 û
ü
2
z Z ÿ
ÿÿ
A
dt
k k
L(Z )
0
)
û2 û
û2 û
ü
2
T (t )
0
L(Z )
then
þ k k kk þ k k kk þ kk C
2
M1
M1 >
2
ü
2
C
for some constant
) T (t)z
L(Z;Y
0
2
L(Z;Y
)
2
z
z L(Z;Y ) z
2
2
Z ÿ
Z
2
ÿ
Z
ÿ;
Z
0 (independent of
ü ),
since
ü
Z
2
ÿ2ýt
K1 e
ÿ2ÿ
t
0 1
2
t
ÿ
2ÿ
<
dt
ÿ2ýt
K1 e
ÿ2ÿ
0
1.
(2.25)
Zÿý
then 2
sat-
is a solution of the
0
in
ÿ
and it is self-dual in
Zýÿ1 ; Zÿý
C Cz
ÿ,
ÿ >
Zý .
C
ÿ
0
1 2
+ LC Az =
A LC z
=
Case 1:
ÿ
ÿ >
0
L2 ([0;
satisfying
Zý ; Zÿý+1
Lyapunov equation
for any
ÿ
1 2
ÿ >
ÿ
C 2L
); Y )) and
ü 2L for all and ý ü 3. for ÿ C C þ2 L L for ÿ. 4. If , then 2 Lý 2.
ý
C
both
deÿned by (2.23) and (2.24) have the following prop-
LC
(Zÿ ; L2 ([0;
0,
1.
dt
2
Z
dt
dt:
36
Chapter 2.
Case 2:
0
< ÿ.
Z
þ
0
If
k
2
z
Zÿ ,
C T (t)z
then
k
2
Y
ÿ k k kk ÿ k k kk ÿ kk
dt
2
C
2
2
L
kC k z
0 (independent of
M2 >
CT (
L2 ([0;1);Y
(Zÿ ; L2 ([0;
)z
L2 ([0;
M
)
); Y )) for
z
Z
for any real
and
ÿ
ý.
Z
þ
2
k ÿ kk þ
0 ü
(A )
0
C
Case 1:
ý
û
2
L(Z;Y
0,
ÿ
)
0
T (t)
kk
ÿ
1
L1
L( Z 0 )
Z
þ
t
1
M
0.
>
>
0 we have
); Z 0 ) for
0
Z0
k k
k
0
C
0 ü
(A )
z Z
ÿ2ýt
2
dt
ÿ;
L2 ([0;
and
([0;
k
dt
K2 e
0
0
ý 0 ü ý ý(A ) T (t)0 C 0 C T (t)Aÿÿ z ý
0
=
ÿ
Z
ÿ2ýt
2
K2 e
0
1
So it follows that ) ; Y );
0
L(Y ;Z 0 )
0
T (t)
k
z
2
Z.
þ
0 Zÿÿ
for
ÿ >
ü
0
T (t) C C T (t)Zÿ
We now investigate the values of
(A0 )ü T (t)0 C 0 C T (t)Aÿÿ z
Z
Z
þ
); Y ) and moreover from the above we ob-
1 2
ÿ >
2
ÿ
þ ).
ÿ , for some
From property (4), for every
2.
Z
þ 2 1 ÿ kk ÿ 1 ý C 2L
Thus we have tain
2
z
M2
Z
) z
L(Z;Y
Z
2
) z
L(Z;Y
C
for some constant
Classes of well-posed linear systems
If
z
2
and
ý
C2 ý 1 . 2
0
Zÿü ,
for which
ÿ
Z,
dt
kk C
L(Z 0 )
L(Z;Y
)
ý ý ýT (t)Aÿÿ ý
kk z
L(Z )
ý ý ýT (t)Aÿÿ ý
L(Z )
Z dt
dt:
0. ÿ 1
ÿ1
If
z
2
Z,
Z þý ý ý(A0 )ü T (t)0 C 0 C T (t)Aÿÿ z ý 0
þ
0
Z0
dt
Z
ÿ k k kk ÿ k k kk ÿ kk C
C
M1
2
L(Z;Y
2
L(Z;Y
z
;
)
z
)
z
Z
Z
þ
t
0
Z
1
0
2
ÿ2ýt
K1 e
ü ÿÿ
2
ÿ2ýt
K1 e
ü ÿÿ
t
dt
dt
2.4.
37
The analytic class
for some constant
M1 >
Case 2:
ÿ 0.
þ <
0,
ý
0 (independent of
ÿ)
if
þ
ÿ
ý <
1.
ÿ 1
ÿ1
If
z
Z
þ
0
2 Z, ÿ
ÿ 0 þ ÿ ÿ(A ) T (t)0 C 0 C T (t)Aÿý z ÿ
Z
0
dt
ÿ kC k
) kz k
ÿ kC k2 (
) z
2
L(Z;Y
L Z;Y
ÿ for some constant
Case 3:
þ
M2 >
0 (independent of
Z
ÿ
Z
ÿ2üt
K1 K2 e
kk
Z
k k;
dt
ÿý
t
Z0 1
ÿ2üt
K1 K2 e
dt
ÿý
t
0
M2 z ÿ)
if
þý < 1, that is, if ý > þ1.
ý 0, ý > 0. ÿ 1
ÿ1
If
z
Z
þ
0
2 Z, ÿ
ÿ 0 þ ÿ ÿ(A ) T (t)0 C 0 C T (t)Aÿý z ÿ
Z
0
dt
) kz k
ÿ kC k2 (
) kz k
L(Z;Y
L Z;Y
ÿ for some constant
M3 >
Z
ÿ kC k
2
0 (independent of
k k;
if
þ <
1.
ÿ
0
ÿ2üt
K1 K2 e
Z0 1
Z
M3 z ÿ)
Z
dt
þ
t
ÿ2üt
K1 K2 e þ
t
dt
38
Chapter 2.
Classes of well-posed linear systems
ÿ 1
0
ÿ1
Case 4: If
2
z
ÿ <
0,
þ >
þ
0.
Z,
Z ÿÿ ÿ ÿ(A0 )þ T (t)0 C 0 C T (t)Aÿý z ÿ
Z
0
C
L(Z;Y
2
C
L(Z;Y
M4 z
for some constant
M4 >
0 (independent of
So (A0 )þ T (t)0 C 0 C T (t)Aÿý z
2
L1 ([0;
and using the above, we obtain for
k k
=
LC z Z 0 ÿ
= =
ÿ ÿ for some constant
3.
for
That þ >
C C 2 Lþ þ 0
1 2
M >
0
Zý ; Zþ
1 þ
ý ).
); Z ) for z
ÿ
þ
1 2
From 1,
2
ÿ <
ÿ <
C 2 Lþ
1 : 2 L2 ([0;
) z
Z
) z
Z
Z
ÿ
2
K2 e
0
1
0
ÿ2üt
dt
ÿ2üt
2
K2 e
dt
;
Z.
1,
If
z
þ >
2 þ
Zý ,
then
ý
A z
1,
2
Zý
ÿ ÿ ÿ 0 þZ 1 ÿ 0 0 ÿ(A ) T (t) C C T (t)zdtÿ ÿ ÿ ÿZ 1 0 ÿZ ÿ ÿ 0 þ 0 0 ÿ (A ) T (t) C C T (t)zdtÿ ÿ ÿ Z ÿZ0 1 ÿ ÿ ÿ 0 þ 0 0 ÿ ý ý ÿ ( A ) T (t) C C T (t)A A zdtÿ ÿ ÿ 0 Z Z 1 ÿ 0 þ ÿ 0 0 ÿ ý ý ÿ(A ) T (t) C C T (t)A A z ÿ dt 0 M
k k ý
A z
Z
=
M
kk z
þ;
Z
0. We depict this region (shaded) in Figure 2.5.
ý
for
þ >
þ
1 2
and
ÿ <
1 2
and is self-dual in
is a consequence of part 1 above.
and 0
1,
þ <
We only need to prove that the equality þ >
Z
ÿ k k kk ÿ k k kk ÿ kk
dt
2
1
); Y ); Zþ0
ý
for
ÿ <
LC
1 2
=
C C 2 Lþ 0
L 0
Z ý ; Zþ
ü
ý
Zý ; Z
0 ÿý
û
holds for
and it can be easily shown that
2.4.
39
The analytic class
ÿ 1
ÿ1
þ
0
LC .
Figure 2.5: The feasible region of parameter values for
for any
y
2
L2 ([0;
1
); Y )
C If
z
with
þ
Z
for
ÿ >
ÿ
þ
part 2 that
C þ z )(
z
ÿ <
0
=
CT (
Z
4.
C
L
C
0
þ
ÿ
Z ;Z
0
(Z; Y ) and so
71, Pazy [63]).
Since
þ
ÿ
for
L2 ([0;
1 2
þ <
0
ÿ
CA
ÿ ý 2L
ÿ >
1
1
0
0
<
1,
); Y )),
); Y );
2 ÿ ÿ ÿ 0
T (t) C C T (t)zdt
0
= 1,
0
1
T (t) C C T (t)zdt
0
C C 2 Lÿ 2L =
)z
Z
ÿ
Z :
(Zþ ; L2 ([0;
1 0
So
C2L þ 2
(C T ( )z ) =
1 2
0
T (t) C y (t)dt
0
)=
1 2
2
1 0
=
CC C þ ÿ ÿ þ ÿ ÿÿ þ 0
Now since
y
1 2 , then since
( and so
Z 0
1 2 and
Cz
þ <
1 2.
(Z; Y ) for
1 2
ÿ >
L
=
2
ý >
Z :
1, it follows from
>
ÿ 0
Z :
0 (using Lemma 6.3, page
is the inÿnitesimal generator of an exponentially
A
stable strongly continuous semigroup on the Hilbert space
Z,
it follows from
the standard theory for Lyapunov equations (see for example Theorem 4.1.23,
ÿ ÿ
page 140, Curtain and Zwart [27]) applied to þ ( Z,
A;
that there exists a unique self-adjoint solution
equation
h
Az1 ; Qz2
for all
z1
and
z2
in
iZ h +
D (A)
and
Z Qz
=
Qz1 ; Az2 Q
iZ
=
of the operator Lyapunov
ý z1 ; C A ý z2 ü
ÿ
CA
ÿ
is given by
1 0
0
ý
Q
T (t)
0 ÿ
(A )
ýC
0
ý ) with state space
ÿ
; CA
ý T (t)zdt;
ÿ
CA
Y
;
40
Chapter 2.
2
Z.
If
2
for all
z
and
D (A).
Consequently, we have
w1
hAA w1 ; QA w2 i ÿ
If
L
ÿÿ
>
ÿ
ÿ
0
þ
. If
w
ÿ
ÿ
Zÿ
ý
2 ih
Cw
0
2L
ÿ
0
Zÿ ;Zÿÿ
i
+1; Z
0
0 ÿÿ
Zÿÿ
=
þ
ÿ
and
0 ÿÿ
C
hw1 ; A L 0
Cw
2 ih
i
Aw
1 2 D (A
ÿ
hw1 ; A L 0
Cw
ý
Z
1
1
Z
. Since
ÿ
Zÿ ;
w
1
0
h
0 ÿÿ
0
Zÿ ;Zÿÿ
ü
h
we
i
0
Zÿÿ1 ;Z1ÿÿ
i
;
ÿ
ÿ
ü
2
ü
2
0
T (t) C C T (t)w dt
22Z
ÿ,
0 Z ÿÿ
1
h
0 Zÿÿ1 ;Z1 ÿÿ
h
0 Zÿ ;Zÿ ÿ
i
i
:
,
hAw1 ; T (t) C C T (t)w2 ih 0
0
Zÿ ;Z 0
ÿ þ 1 2 þZ1+ = D A1+ , +1 and since w2 2
0
0
ÿÿ i
denotes
ü
T (t) C C T (t)w dt
0
2
0
A LC
1 2 2 Z1+
2
1
0
0
Z
i
þ
2L
LC
Aw ;
=
1
0
(2.26)
w ;w
2
0
belong to
and so
T (t) C C T (t)w dt
Zÿ ; Zÿÿ
T (t) C C T (t)w
0 Zÿ ;Zÿÿ
0
2
:
Y
hþ; þih
0
and so
ý
2 ih
ÿ
A w
T (t) C C T (t)w dt
0
Aw ;
0
) and
If
1
part 2,
=
+1
þ
0, then for
ÿ >
0
ÿ
2L
A
=
Since
0
1
0
Z
1
0 ÿÿ
0 Zÿ ;Zÿÿ
0
Aw ;
1 belongs to D (A ). From 1+ ÿ Z , L w2 2 Z +1 ÿ Z ÿ
1
ý
with
Z
w ;A
ÿ
Aw Z
Zÿÿ ,
ÿ
and
Zÿ ; Zÿÿ
ÿ
0
and
=
A
ÿ
2 L
LC
1
ÿ
A w
h C w1 ; C w 2 i
=
Z
have
hw1 ; A L
then
2 Z1+ , then w 2 Z , since Z1+ ÿ Z .
the duality pairing between
0
ÿ,
ÿ
0, we have, using part 2, that
Zÿ ; Zÿÿ
since
Z
hQA w1 ; AA w2 i
+
Z
1+
belong to
w
Classes of well-posed linear systems
0
0
Zÿ ;Zÿÿ
i dt:
Consequently,
hw1 ; A L 0
Z =
Z0
= since (A0 )ÿ
1
1
û
1
Aw ; (A
û
ÿ
0
Z; Zÿÿ
þ
)
T (t)
0
ý
ÿ
1
0
0 ÿÿ
(A )
Z
2 ih
0
ÿÿ
0
C CA
1 2 Z.
0 Zÿ ;Zÿ ÿ
0
C CA
ÿÿ
ÿ
ÿ
2
T (t)A w
ú
T (t) ÿ
Z
0
Z
0 Zÿ ;Zÿ ÿ
i dt
dt
As a result we obtain that
i 0 ÿÿ
(A )
:
ú
2 h
T (t)A w
1
hAA w1 ; QA w2 i ÿ
0 ÿÿ
(A )
AA w
Cw
AA w ;
i
ÿ
and 0
=
0 ÿ
1
ÿ
hw1 ; A L
=
0 Zÿ ;Zÿ ÿ
AA w ; T (t)
0
2L
2 ih
Cw
0
ÿÿ
C CA
ÿ
2
ü
T (t)A w dt Z
(2.27)
2.4.
41
The analytic class
Similarly, it can be shown that for
hw1 ; L
C Aw
2 ih
ÿ
L
Zÿ ;
0 Zÿÿ
ÿÿ i
Zÿ ;Z
(Here, using part 2, we consider
þ
and
w1
Z
1+
ÿ,
ÿ
ÿ
2L
LC
in
we have
hQA w1 ; AA w2 i
=
0
2
w
ÿ
ÿ1; Z
0 ÿÿ
Zÿ
.)
þ
for
ÿ >
So by substituting (2.27) and (2.28) in (2.26), we see that
hz1 ; A L z2 ih 0
for
z
and
1
C C
z
i + hz1 ; L
2L
Zÿ ;
0
1
z
C
and
z
2
But since
i + hz1 ; L
2 ih
C Az
1+
0; we have
LC
0 Zÿ ;Zÿ ÿ
0
Zÿ ;Zÿÿ
Zÿ ,
LC A
2
satisÿes
i = hz1 ; C C z2 ih
is dense in
ÿ
(2.28)
:
0
0
Zÿ ;Zÿÿ
Z
, it follows that
0 Zÿ ;Zÿ ÿ
in
2 ih
C Az
ÿ ).
ÿ
0 Zÿÿ
hz1 ; A L z2 ih holds for
0
Zÿ ;Zÿÿ
2 ÿin Z1+ þ(ÿ Z
and 0
C
Z
and
0
A LC , LC A
i = hz1 ; C C z2 ih 0
i
0 Zÿ ;Zÿ ÿ
i
Zÿ .
Remark: We note that part 1 of Lemma 2.4.2 shows that C is an admissible
observation operator for the semigroup þ >
þ 21 .
fT (t)g 0 tþ
on the Hilbert space
Zþ
for any
Theorem 2.4.3 Let þ
þ
2R
2R
satisfying
þ 21
satisfy
þ 12
< þ <
for
þA; B; C ) is a well-posed
We now show that the system þ given by the triple ( linear system on
Zþ ,
1 2 + ý.
< þ <
1 2
+
ý.
If A, B and C satisfy
the assumptions A1, A2 and A3 listed at the beginning of this section, then the system given by the triple
þA; B; C )
(
is a well-posed linear system with input
space U , state space Zþ and output space Y , and has a transfer function given by G(s)
=
C (sI
+ A)ÿ1 B .
Proof From part 1 of Theorem 2.4.1 and part 1 of Theorem 2.4.2, it follows that B
and
C
are admissible input and output operators for the semigroup
with state space s
2 ü(þA).
1.
Let
L (Z
s
Zþ .
2 ü(þA).
We show that ý(s) =
Firstly, if
2
U,
+1 ), it follows that (sI + A)
ý ; Zý
it follows that (sI + A)ÿ1 Bu 2.
u
Furthermore, we have for
kýuk
Y
2 Z.
u
then
Bu
2
1 Bu 2 Z +1 .
ÿ
Hence
2 U,
C (sI
ý ý ý ýC (sI + A)ÿ1 BuýY
Zý
1B
ÿ
A)
fT (t)g 0 tþ
is well-deÿned for
and since (sI +
Furthermore, since
+ A)ÿ1 Bu
ý
C (sI
+
2 Y.
12
ÿ
A)
ý >
þ1,
42
Chapter 2.
Classes of well-posed linear systems
ÿ ÿ ÿ(sI + A)ÿ1 Buÿ Zÿ+1 ÿ ÿ ÿ 1ÿ ÿ (sI + A) Zÿ+1 ;Y ) L(Z ;Z ) kBukZÿ ÿ ÿ ÿ ÿ+1 ÿ 1 ÿ(sI + A) ÿ Zÿ+1 ;Y ) L(Zÿ ;Zÿ+1 ) kB kL(U;Zÿ ) kukU :
ÿ kC kL( ÿ kC kL( ÿ kC kL(
Z
Thus for each
s
2 ÿ(þA), ÿ(s) 2 L(U; Y ).
ÿ ÿ ÿC (sI + A)ÿ1 B ÿ
3.
ÿ+1 ;Y )
L(U;Y ) ÿ kC kL(Zÿ+1 ;Y )
In fact we have shown that
ÿ ÿ ÿ(sI + A)ÿ1 ÿ
L(Zÿ ;Zÿ+1 ) kB kL(U;Zÿ ) :
(2.29)
Using the resolvent identity, it is easy to check that for
we have ÿ(s)
s
and
þA; B; C )
ý
posed by showing that the map ÿ( ) is bounded in the half-plane
j
C
Re(s)
>
in
ÿ(
þA),
þ ÿ(þ ) = þ(s þ þ )C (sI + A)ÿ1 (þ + A)ÿ1 B:
Finally we show that the system þ given by the triple (
4.
þ
g
is well-
C+ 0 :=
fs 2
0 . In order to do this, we use the exponential stability of the
generator of the semigroup and (2.29): We have for all
s
2 C +0
ÿ ÿ ÿ ÿ1 ÿ ÿ(sI + A)ÿ1 ÿ ÿ ÿ L(Zÿ ;Zÿ+1 ) ÿ A(sI + A) L(Zÿ ) ÿ ÿ1 ÿ = ÿI þ s(sI + A) ÿ L(Zÿ ) ÿ ÿ ÿ ÿ kI kL(Zÿ ) + jsj (sI + A)ÿ1 ÿL(Zÿ )
for some
M >
ÿ
1+
jsj
=
1 + M;
M
jsj
(using Theorem 5.2.(c), Pazy [63])
0. This completes the proof.
Remark: A more general result with unbounded C can be proven in a similar
manner. We quote the following from Staýans [77]:
þA generate an exponentially stable analytic strongly confT (t)g þ0 in Z , B 2 L (U; Z B ), C 2 L (Z C ; Y ) and D 2 ÿ ý < ý + 1. Fix any ü satisfying ý þ 12 < ü < ý + 12 .
Theorem 2.4.4 Let tinuous semigroup
L(U; Y ), where ý Then þA, B , C
B
X
=
t
C
ÿ
ÿ
B
B
C
generate a regular well-posed system on
Zþ . The transfer function is given by G(s)
=
C (sI
(U; Zþ ; Y ),
+ A)ÿ1 B + D ,
where and it
satisÿes
jslim j!1 G(s) = D: 2C +0
s
In the sequel, the operators
LB LC
and
LC LB
play an important role, so we col-
lect some properties here which readily follow from Lemmas 2.4.1.2 and 2.4.2.2.
2.4.
Corollary 2.4.5
1. 2.
43
The analytic class
LB and LC deÿned by (2.16) and (2.24), respectively, satisfy:
LB LC 2 L (Zÿ ) for any ÿ satisfying ÿ1 < ÿ < 1 + þ. L
C LB
2 Lÿ
Z
þ 0
þ
for any
satisfying
ÿ
ÿ
ý <
1 + 2þ,
Lemma 2.4.2 states that
ÿ
1
< þ,
C
L
1.
Since
satisfying
ÿ
(1 + þ)
< ý <
From part 2 of Lemma 2.4.1, we know that
Proof
ý <
ý
1+
2 Lÿ
ÿ <
ÿ
þ, ý >
Z ;Z
þ 0
þ
ÿ
(1 +
for
ý, ÿ
B
L
þ).
1.
2 Lÿ þ ÿ
ÿ
0
Z ;Z
þ
for
ý, ÿ
Furthermore, part 2 of
satisfying
ý
ÿ <
1,
ÿ >
ÿ
1,
the intersection of the feasible regions depicted in Figures 2.4
and 2.5 is nonempty and is sketched in Figure 2.6. ý 1
ÿ1
þ
þ
1+þ 1 + 2þ
0
1
ÿ
ÿ(1 + 2þ) ÿ(1 + þ) ÿ1 Figure 2.6: The intersection of feasible regions.
Hence the claims in parts 1 and 2 follow. We now examine the Hankel operator associated with
h(
þ
)=
þ
CT (
)B . Through-
out the remainder of this section, we assume that the input and output spaces are ÿnite-dimensional: say, U = C m and Y = C p . Lemma 2.4.6 If A, B , and C satisfy the assumptions A1, A2 and A3 listed at the beginning of this section with U = C m and Y = C p , then:
þ
þ 2
1 1
!
2
1
1. The impulse response h( ) = C T ( )B L1 ([0; ); C pÿm ) and so it has a well-deÿned compact Hankel operator þ : L2 ([0; ); C m ) L2 ([0; ); C p ) given by
Z
(þu)(t) =
ÿ C2L
2. If
1 2
1
0
h(t
+ ü )u(ü )dü
1 2 + þ, then þ = (Zÿ ; L2 ([0; ); C p )). < ÿ <
1
t
ý
0; for
u
CB, where B 2 L
L2 ([0;
(L2 ([0;
1
m ); C ):
1
(2.30)
); C m ); Zÿ ) and
44
Chapter 2.
Proof
We have
1.
k
C T (t)Bu
and since
þ
k ÿ ÿ ÿ ÿ
k k k k
Cp
C
C C
C
k k k k
C
L(Z; p )
C
L(Z;C p )
A
A
h(
ÿÿ
T (t)A
ÿÿ
T (t) L(Z ) A
Ke
L(Z; p )
1, it follows that
ÿ <
k
T (t)Bu Z
k k k k k k kk ý2 1 C
L(Z; p )
C
k k k
Classes of well-posed linear systems
ÿ
ÿþt
A
ÿÿ
t
)
ÿ
Bu Z
C
u
2.
From part 1 of Lemma 2.4.1 it follows that
1
B 2 CB
( ( But
C T (t)
2L
u)) (t)
=
C2L
C T (t)
B
(Zý ; L2 ([0;
u
=
C T (t)
B2L 1 C
CB
Cm
Cm ;
1
2
1
T (þ )Bu(þ )dþ :
0
(Zý ; C p ) and so we have (see for example Theorem A.5.23, page
628, Curtain and Zwart [27]) ( (
u
(L2 ([0; ); C m ); Zý ) and p )). Consequently, if u
);
Z
kk
pþm ). Compactness follows
);
from Lemma 8.2.4 (page 399, Curtain and Zwart [27]).
from part 1 of Lemma 2.4.2, L2 ([0; ); C m ), u Zý , and
C
L( m ;Z )
B
L( m ;Z )
B
L1 ([0;
ÿ
u)) (t)
Z Z
=
1
C T (t)T (þ )Bu(þ )dþ
=
Z
0
=
1
C T (t
+ þ )Bu(þ )dþ
0 1
h(t
+ þ )u(þ )dþ = (ÿu)(t):
0
Remark: In fact, ÿ is nuclear, that is
P
1
k=1 ýk
Chapter 3). We will þrst show that
LB LC
2L
<
1
(see Theorem 3.2.3 in
(Zý ) is compact for all
satisfying
ü
þ
1
< ü <
1 + ÿ. This was shown in Curtain and Ichikawa [14] under the extra assumption that
T (t)
was compact for each
since we assume that
B
t >
0. Here we do not need this assumption,
has þnite rank.
Lemma 2.4.7 Under the assumptions A1, A2 and A3, with U =
C p,
LB LC
2L
(Zý )
The map
Proof
t
7!
0
0
0
1 ! Cÿ
is compact. Let
operators from (0;
)
0
0
Zü
Zü ; Zý
þ
to
0
T (t)BB T (t)
ÿ
in the norm topology, and since T (t)BB T (t)
þ 1 ! Lÿ
is compact for all ü satisfying
Zý .
C
B 0
: (0;
þ
)
1
< ü <
0
Zü ; Zý
t
=
and Y
1 + ÿ.
Zü ; Zý
is compact, for each
Cm
þ
is continuous
greater than zero
denote the Banach space of compact
Thus we have that the map
t
7!
0
0
T (t)BB T (t)
:
is continuous in the norm topology. Moreover, as in the
2.4.
45
The analytic class
proof of part 3 of Lemma 2.4.1, it can be shown that for þ
ÿ
ÿ <
1 + 2ý,
þ <
1 + ý and
1
Z 0
k
ÿ(1 +
ÿ >
0
k
0
T (t)BB T (t)
ý),
ÿ
and
satisfying
þ
we have
C(Zÿ0 ;Zþ ) dt
<
1
:
2
R1
ÿ
þ
0 0 C Zÿ0 ; Zþ (see for 0 T (t)BB T (t) dt example, Thomas [81] or Hille and Phillips [46]). Proceeding as in the proof
Consequently, the Bochner integral
of Corollary 2.4.5.1, it follows that satisfying
ÿ
1
< þ <
2L
Next we show that the spectra of ÿÿ ÿ are identical for all
2L
LB LC
1 + ý.
þ
satisfying
ÿ
1
< þ <
(Zþ ) and is compact for all
(L2 ([0; 1 + ý.
1
); C m )) and
LB LC
2L
þ
(Zþ )
C m and Y = p C , the nonzero Hankel singular values are equal to the square roots of the
Lemma 2.4.8 Under the assumptions A1, A2 and A3 with U =
2L
nonzero eigenvalues of LB LC
(Zþ )
ÿ
1
for þ satisfying
ÿ
< þ <
1 + ý.
1 + ý. From part 3 of Lemma 2 1 1 for þ < 2 + ý and ÿ > 2.4.1, we know that LB = Zÿ ; Zþ ý. 2 ÿ þ 1 0 0 Furthermore, from part 3 of Lemma 2.4.2, LC = Zþ ; Z ÿ for þ > 2 1 and ÿ < 2 .
Proof
First we prove the result forÿ
BB 2 L 0
Let us choose
þ
and
ÿ
1 2
ÿ ÿ ÿ
< þ <
þ
0
CC2L
satisfying
ÿ ÿ 1 2
ý < ÿ <
1 2
ÿ
;
1
< þ <
2
(We remark that such a choice is possible, since LB LC
=
0
=
2
ý >
BB C C B C 2 L 0
1
ÿ
ÿ
ÿ
1.) Consequently,
(Zþ ) ;
and owing to the compactness of ÿ (and hence of ÿÿ ), 1 1 for 2 < þ < 2 + ý.
ÿ ÿ
+ ý:
2L
LB LC
6
1 1 2 < þ < 2 + ý and suppose that ü = 0 is an eigenvalue of and v a corresponding eigenvector; LB LC v = üv . Then Let
ÿ
ÿÿ
C CBB C C C C 6 v
0
=
0
v
=
and so ü is an eigenvalue of ÿÿÿ , since 0, a contradiction).
Conversely, suppose that vector
y.
û
Then LB LC LB
C
LB LC v
v
=
C
(üv ) =
ü
C
(Zþ ) is compact
LB LC
2L
(Zþ )
v;
= 0 (for otherwise LB LC v =
LB
CC 0
v
=
is an eigenvalue of ÿÿÿ , with corresponding eigen-
0
y
=
LB LC
BB C 0
0
y
=
LB LC
B
ÿ
ÿ
y;
46
Chapter 2.
and
LB LC LB C y = BB C CBB C y = Bÿ 0
So
ÿ
0
is an eigenvalue of
0
0
0
B LC , since
L
0, a contradiction).
ÿ
B 6
2L
(Zÿ ) is compact for
ÿ
1
B LC
2L
B
=
y
satisfying
þ
ÿ
ÿ (ÿy ) =
B
ÿ
ÿ
ÿ
ÿ
1
< þ <
1+
y:
CB
ÿÿ y =
ý.
Since
1 + ý, it has a pure point spectrum.
< þ <
Moreover, by Lemmas 2.4.1 and 2.4.2, L
ÿ
ÿÿ
ÿÿ y = 0 (for otherwise ÿÿÿ y =
Finally, we extend the result to all
B LC
L
Classes of well-posed linear systems
ÿ
(Zÿ ; Zÿ +þ ) for
ÿ
1
1
< þ <
þ
1+ý
ÿ
ü;
min 1 + ý; 2 . From this it follows that the nonzero point spectra of LB LC (Zÿ ) and LB LC (Zÿ +þ ) are the same
for any for all
ü
þ
satisfying 0
2ÿ (
1; 1 + ý
ÿ
2L
< ü <
ü ).
ÿ1
ÿ 21
Hence the nonzero point spectrum of so
B LC ) = û (ÿ
û (L
Let
û1
ÿ
ÿ) for all
þ þ þ û2
thermore, let
:::
þ
+ÿ
1 2
B LC
ÿ
L
satisfying
1
2L
1+ÿ
2L
(Zÿ ) is independent of
þ
and
1 + ý.
< þ <
0 denote the Hankel singular values of the system. Fur-
ûl+1 < û < ûl . Then we have the following useful lemma which will be used in the next section. û
be such that
Under the assumptions A1, A2 and A3 with U = C m and Y = 1 ÿ (L2 ([0; ); C m ) has a spectrum ûl+1 < û < ûl , then J := I ý2 ÿ ÿ û (J ) contained in ( ; Æ) (Æ; ) for some Æ > 0, and û (J ) ( ; Æ) consists of exactly l negative eigenvalues.
C p , if
Proof
J
1
\ ÿ1 ÿ
ÿÿ ÿ is compact and has a pure point spectrum with 0 as the accumu-
lation point. Considering the resolvent that
2L
ÿ ÿ1 ÿ [ 1
Lemma 2.4.9
ýü
I
ÿý
1
ÿ
û
2ÿ ÿ
ÿ
úI
úþ1
, it is easy to see
has a spectrum which is a shifted version of the spectrum of ÿ
Finally, since ÿ ÿ has a pure point spectrum mulation point, and since
ûl+1 < û < ûl , J
ÿ
þ
2 2 û1 ; û 2 ; : : :
has exactly
l
1
ÿ
ý2 ÿ
ÿ.
with 0 as the accu-
negative eigenvalues.
Analytic semigroups are generated by parabolic and some hyperbolic partial diþerential equations. For a change, we give a diþerent example of an analytic system from the class of fractional transfer functions.
Example 2.4.10
(The fractional transfer function
s m , where
1 (1+ )
0
< m <
1.)
2.4.
Let
47
The analytic class
ÿ
2R
ÿ
satisfy
1
ÿ Z
=
2
f
ÿ Z 1 1j
1(< 0). We denote by
< ÿ < m
L2 (0;
It is easy to see that operator by
x
7!
f1 ; f2
0
Z
1 : 1+x
Z
=
k
RA f
k
2 Z
1
Z =
ÿ
1
ÿ
(1 +
RA ,
It is immediate that and
f (x)
dx <
1
;
f1 (x)f2 (x)dx:
RA
:
Z
for all
2 x)
j j þ f (x)
2
1
Z
dx
RA
which we denote by
D (A)
A
1 1+x
f (x)
!
f
be the multiplication
Z
2
Z:
(Z ), since
(1 + x)
algebraic inverse of
of
ÿ
j j
þ
2
(1 + x)
ÿ
ÿ
0
j j f (x)
2
dx
kk
=
f
2
Z
:
1
is injective, since if 0 = (RA f ) (x) = 1+x f (x) for almost all ), then f (x) = 0 for almost all x (0; ). Thus one can deÿne an
2 1 (0;
ÿ
is a Hilbert space. Let
0
Furthermore, x
2L
RA
(1 + x)
0
(RA f ) (x) = The operator
1
Z
i
ÿ
)
with the inner product
h
ÿ (1 + x)
the vector space
Z
A,
= ran (RA ) and
as follows:
ARA f
=
f:
is well-deÿned. The operator A is closed, since the graphs
are related by the following isomorphism:
RA
I: Z
A
2 1
ý ! ý Z
Z
I (f1 ; f2 ) = (f2 ; f1 )
Z;
2 ý
for all (f1 ; f2 )
Z
Z;
We have D (A)
= ran (RA ) =
Moreover, if g
2
D (A),
f
RA f
2 j 2 g f2 j Z
f
Z
ü ü2 g
(1 + )g ( )
1 C 1û C 1 1 ü ÿÿ ü 2 1 f g 2N û C 1 ! ü ÿÿ ü ü ÿ ÿ üü ÿ ÿ ÿ!1 ü ÿ ÿ ü 1 ÿ ÿ ÿ! ÿÿ 2 C
then (1 + )
2 f( )
a sequence
fn
L2 (0;
Hence we have
). Since
00 (0;
n
(1 + )
Z.
g
fn (
(1+ )
)
Z
2
Z
:
(Ag )(x) = (1+x)g (x). The set D (A) contains continuous
functions compactly supported in (0;
is
=
Z
2
), that is,
00 (0;
) such that fn (
)
(1 + )
f,
L2 (0;
fn (
(1+ )
(1 + )
fn
)
2
where
00 (0;
)
) is dense in
)
2
(1 + )
00 (0;
D (A).
L2 (0;
If
f
2
Z,
), there exists
2 f ( ) in L2 (0;
1
), that
2 f ( ):
). So
D (A)
is dense in
Chapter 2.
48
A = RAÿ :
It is clear that R
h
R
Classes of well-posed linear systems
indeed, for any f and g in Z , we have
1
Z
i
A f; g Z
=
(1 + x)
0
Z
þÿ
1
=
(1 + x)
h
=
0 f; R
1 1+x
þÿ
f (x)g(x)dx
ÿ
f (x)
i
þ
g(x)
dx
1+x
Ag Z :
Thus from Lemma A.3.65.(c) (page 603, Curtain and Zwart [27]), we have
þ üÿ Aÿ ÿ ÿ A
ÿþ
1
(A )
=
2
ý
A Aÿ f = f for all f D(A ), ran (RA ) = D(A). Furthermore, if f
So R
1
A
2
= R
A R
A:
= R
2
ÿ
f = f for all f
ÿ
Z and D(A ) =
D(A) = D(A ), then
A Aÿ f ) = (ARA ) (Aÿ f ) = Aÿ f
Af = A (R
2
and so A is self-adjoint. If f
h
Af; f
iZ
D(A), then we have
1
Z
=
0
Z =
Z
ÿ
0
1 1
(1 + x)
þÿ
(1 + x)f (x)f (x)dx
þÿ
(1 + x) f (x)
(1 + x)
(1 + x)
k kZ 0
=
2
f
j j
þÿ j
f (x)
j
2
2
dx
dx
:
þ
A is a nonnegative self-adjoint operator and so using Example 1.25 (page 493,
f gtý
Kato [49]) it follows that group
T (t)
0.
þ
The resolvent of ÿ(
A),
þ
þ
A. It can be shown that ÿ(
þ
1
(þI + A)
þ
2
The point spectrum of Af = þf , f
þf (x).
A is the inÿnitesimal generator of an analytic semi-
6
þ
6
þ
f (x)
A) =
{(
for all
{({(
þ1 þ 2 ;
f
2
Z:
þ1 þ ;
1] and for þ
þ1 þ þ
1]) = (
Af )(x) = þf (x), that is,
;
1]. Let
(1+x)f (x) =
But since x = þ + 1 almost everywhere, f (x) = 0 almost everywhere.
The semigroup e
þ
A is empty. ý(
þ1 þ
(continuous) spectrum is (
x tf (x).
(1+ )
þ +1+x
D(A), and f = 0. So (
Thus f = 0 (a contradiction!).
þ
1
f (x) =
A) =
f gtý T (t)
So the point spectrum of A is empty, and its
;
1], see Figure 2.7.
0 . Given t
ÿ
ÿ
ÿ
0, deÿne T (t)
2L
The operator T (t) is well-deÿned since for all f
k ÿ kZ T (t)f
2
1
Z
=
(1 + x)
0
þÿ j ÿ
ÿ
(Z ) by (T (t)f ) (x) =
(T (t)f ) (x)
j
2
dx
2
Z , we have
2.4.
49
The analytic class
C
2i i
ÿ3
ÿ2 ÿ1
1
2
3
0
ÿi ÿ2i Figure 2.7:
ÿ (A)
1
Z =
ÿ2
e
f þ gý
Next we prove that
þ
T
1. (T (0)f ) (x) = 2. For any
f
2
(t)
f (x)
Z,
kk
for all
f
2
Z
Z,
= = = =
þ(
þ
+ t) =
þ(
þ )T (t)
T
(t)f
T
þ (0) =
T
ÿ ÿ ÿ ( þ(
(1+x)(þ +t)
e
(1+x)þ
e
(1+x)þ
e
þ)
T
Choose a choose a
M >
nuity.
&
0
2 þ( )
ÿ ÿ ÿe
ÿ
x
[0;M ]
T
t f
k
(1 + x)
(1+x)t
ÿ k f
2
dx
ÿ
ÿ
ÿ ÿ
1ÿ
<
2
= 0, for all
f
þ 0
2
f (x)
(1+x)t
i f (x)
þ
(T (t)f ) (x)
þ
t
and
6 2 ÿ ÿ ÿÿ j j R1 j j
(1+x)t
<
t < Æ
1
M
2
f
f (x)
f (x)
2
Z.
2
(1 + x)
ÿ
ÿ
j j f (x)
2 dx
We have
dx:
dx <
implies that ý
Z.
þ.
= 0. Let 0 =
t
e
RM
Z:
I.
ÿ
e
ÿ ÿÿÿ ÿ
0 small enough so that 0
Æ >
0
Z
dx
(T (t)f )) (x):
0 large enough such that 0
sup
So limt
=
h
for all nonnegative
1
Z
2
f
2
f (x)
2
(2.31)
3. Finally we prove the strong continuity at
kþ ÿ k
ÿ
f (x)
:
and so
we have
þ
T
j j ÿj j x)t
ÿ
(1 + x)
2
f
;
ÿ eÿ2(1+
0
(T (þ + t)f ) (x)
Thus
ÿ1 ÿ1].
=(
is a strongly continuous semigroup on
0
t
t
2t
e
1
Z
ÿ
=
ÿc (A)
(1 + x)
0
þ
=
ý
2
. Next,
:
This proves the strong conti-
50
Chapter 2.
Hence
fTÿ t g þ ( )
t
0
Classes of well-posed linear systems
is a strongly continuous semigroup on
fTÿ t g ÿA; D A fTÿ t g þ þAÿ; D Aÿ f 2Z
Next we prove that the generator of the inÿnitesimal generator of
L2 (0; 1).
( )
( )
0
t
t 0 by ( (page 24, Curtain and Zwart [27]), for all
is (
( )). Let us denote
(
)). From Lemma 2.1.11
we have
Z 1 ý (ÿI ÿ Aÿ ) f = eý Tÿ (t)fdt 1
ÿt
0
ÿ > !0ÿ , the growth bound of the semigroup fTÿ(t)g þ0 . ÿ 2 C such that Re(ÿ)>maxfÿ1; !0ÿg we obtain
for Re( ) for
t
ÿ
Z
ÿI ÿ Aÿ )ý1 f (x) þ
(
0
(
Tÿ(t)f ) (x)dt
1
f (x) ÿÿ + 1 + x ý1 þ (ÿI + A) f (x):
= = From this it follows easily that
1 ý e
ÿt
=
Consequently,
D(A) = D(Aÿ ) and Aÿ = ÿA.
Since the semigroup is determined by its inÿnitesimal generator (see for example, Theorem 5.5, page 21, Pazy [63]), it follows that the semigroup is in fact the semigroup
fTÿ t g þ kT t k þ eý fT t g þ ( )
From (2.31) it follows that is less than
ÿ
t
0.
t
( )
and so the growth bound of
1. Hence the semigroup
( )
t
fT t g þ ( )
t
0
fT t g þ ( )
t
0
0 is exponentially stable.
Remarks:
fT t g þ LZÿ
1. Since the semigroup
LZ (
( )
) is continuous in the
t
(
is analytic, the map
0
)
T (ý)
2.1.1.(iv), page 35, Lunardi [53]). This can also be seen directly: 1 e
t0
x
0
First we show that for sup
2 1
x
(0;
)
þ, ý
both greater than zero,
ý ý ýþ(1+x) ýe
; 1)
: (0
!
topology (see for example, Proposition
ÿ eý
ý (1+x)
ý ý ý<
jþ ÿ ýj : fþ; ýg
min
2.4.
51
The analytic class
1 !R
7!
From the mean value theorem applied to the function ÿ (0;
)
, we have
ÿþ(1+x) ÿ eÿý(1+x)
e
7!
2 ÿü
for some ü x
(min
(x + 1)e
e
ÿ f g f g x 1 !R þ
þ; ý
(1+ )
ü (1+x)
; max
þ; ý
)
ÿü (1+x);
ÿü (1+x) <
ü
1
and so (1 + x)e
). It is easy to see that the function
[ü (1 + x)]
: we have
+ : : : > ü (1 + x);
. Finally, using the fact that ü > min
t!0 k
Thus for every t0 > 0, lim
topology.
1
2
2!
we have the desired result.
T (t + t0 )
f gtÿ
2. However, the semigroup
ü
is bounded by
= 1 + ü (1 + x) +
T (t)
:
= (x + 1)e
ý
: (0;
ÿÿ(1+x)
e
ÿ
k
T (t0 )
f g þ; ý
,
= 0.
0 is not continuous at t = 0 in the norm
Indeed, if it were, then its inÿnitesimal generator A would be
bounded (see for example Theorem 1.2, page 2, Pazy [63]); but we know that A is unbounded, since D(A) is not closed.
ÿ
2
ÿ
We note that A h(x) = (1 + x) h(x), h Z
ÿ
as described earlier.
Deÿne B :
1 p
Z
C
!
þ q
Z
by (Bu)(x) = x
x (1 + x) dx =
0 it is easy to see that B Let C : Z
!C
ÿ ÿ ÿ
þ(p)þ(1
2L C (
p
þ(
þ
sin(mû)
q)
û
ÿþ
A
, and for each ÿ
2R
if
p >
ÿ
1 and
p+q <
x
f (x)dx
for all f
0
2
(
ÿ
all
the
conditions
in
ÿ
1;
Theorem
2.4.3
are
satisÿed
ý
and
2L so
A; B; C ) deÿnes a regular well-posed linear system on Z , where
sm
1 1 + þ, and it realizes the transfer function G(s) = 2 (1+ ) G(s)
= =
=
=
þ 1 þm 2 Z 1 þm
C (sI + A) sin(mû) û
Z
B
x
0
m
;
s+1+x
dx
1
(Z;
ÿ
the
C ). triple
1 < ý < 2
: Indeed we have
þ þ m2
(s + 1 + x)
x
û 1
1
0
sin(mû)
(1 + s)
(2.32)
Z:
From (2.32) and the Cauchy-Schwarz inequality it follows that C Thus
, we deÿne
).
1 þm 2
Z
ÿ
u. Using
q)
;Z
be given by
Cf =
þ m2
D
x
dx
52
Chapter 2.
where we choose the following branch: for 1 (1 + s)m
=
ÿ
m[log
e
1+s
+i
s
Classes of well-posed linear systems
2 C n ÿ1 ÿ ÿ (
Arg (1+s)]
;
1],
;
ÿ <
Arg (1 + s)
< ÿ
ÿ
(see for instance pages 187-188, Lang [51]).
2.5
Transfer function algebras
In this section, we introduce some function spaces which will be used in the sequel. Throughout this book, we use the following notation. For
r
Figure 2.8) C+ r
C
r
Figure 2.8: The half-planes
Cÿ = r
1.
f 2C j f 2C j
(see
C+ r
C
r
C+ = r
2R
g g
s
Re(s)
> r ;
s
Re(s)
< r ;
+ C+ and C r . r
C+ r =
f 2C j 2C j s
Re(s)
s
Re(s)
Cÿ = f r
þg ýg
r ; r :
1 (C pþm ) denotes the set of complex püm matrix-valued functions deÿned c
H
in the closed right half-plane, which are bounded and analytic in and continuous in
C+ 0 .
The set
H
1 (C ), c
C+ 0 ,
with point-wise addition and
2
multiplication, is a commutative ring with identity. Note that if G c H (C p m ), then it does not necessarily follow that it has a limit at inÿnity.
1 þ
So for instance, 2.
1 l (C pþm )
H
c
û
X(
;
ÿ
e
s
is an element belonging to
denotes the set of complex
üm
c
matrix-valued functions
) of a complex variable deÿned in the closed right half-plane with a
decomposition
X
=
ý+
G
F,
open right half-plane, and
F
ý is the matrix transfer function of to l, with all its poles in the pþm ). ( C 1
where
a system of MacMillan degree
3.
p
1 (C ).
H
G
at most equal
2
H
c
1 l (C pþm ) denotes the set of complex p ü m matrix-valued functions (û) of a complex variable deÿned in the closed right half-plane with a decomposition = ý + , where ý is the matrix transfer function of H
c
;[
]
X
X
G
F
G
2.5.
53
Transfer function algebras
a system of MacMillan degree right half-plane, and
S1
4.
F
to
2 H1 (C pÿm ). equal
l,
l
with all its
denotes the set of complex-valued functions
nonzero limit at inÿnity in
C+ 0,
g
2 H1 (C ) c
R
denotes the class of proper, rational functions
cients such that
g
C+ 0.
has no poles in
that have a
C+ 0 , and the zeros
ÿnitely many zeros in
are all contained in the open right half-plane. 5.
poles in the open
c
with complex coeÆ-
g
R1 denotes the class of proper, rational functions g with complex coeÆ-
6.
cients such that
g
C+ 0 , and has a nonzero limit at inÿnity.
has no poles in
MH1 denotes the set of matrices (of any size) with each entry in H1 (C ). Similarly, we use MR, MH1 þ , and so on. c
7.
c
c
Suppose
and
M
N
belong to
Then the pair (M; N ) is
;
MH1 c
and have the same number of columns. over
right coprime
MH1 c
if there exist
MH1 such that the following Bezout identity holds: c
XM
Suppose that prime over call this a
G
MH1 .
If
M
for all
s
H
c
M; N )
;
M
right coprime factorization
G
G
and
Y
in
2 C +0 :
1 [l] (C pÿm ) and that the pair ( is such that det( ) 2 S1 and = of over M 1 .
belongs to
c
ÿYN =I
X
H
NM
is right co-
ý1, then we
c
Next we prove a few elementary facts concerning elements from the above classes of transfer functions, which are used in the proofs in the sequel. Lemma 2.5.1 If f H
1 l (C ) c
.
;
Proof
f g
2
1 (C )
H
c
2 S1
has at most
has a Laurent expansion around each zero f (s) g (s)
P
and g
XX r
=
mi
=1 j=1
i
ai;j
(s
pi
þ
M
g
has a nonzero limit at inÿnity in
such that
jsj > M
ÿÿ ÿÿ ÿÿ ÿ
implies that
g (s)
0
f g
2
So
m1 ; : : : ; mr
C+ 0 , given
ÿÿ ÿÿ ÿÿ ÿ
ÿ slim !1 g(s) + s2C
g.
zeros, then
+ ÿ(s);
ÿ p i )j
where p1 ; : : : ; pr are the zeros of g with multiplicities r ( i=1 mi l), and ÿ is analytic in C + 0. 1. Since
of
l
< þ:
þ >
respectively
0, there exists an
Chapter 2.
54
jj
1
Thus s
g(
)
jj g
is bounded for
s
> M.
Classes of well-posed linear systems
2
Since f
H
1 c
ÿÿP P ÿþ
j ÿj
,
f( )
is bounded for
ÿÿ ýÿ
> M . Moreover, choosing M large enough to include all the zeros pi 's
in the open disk for
jj s
fj j
s < M
r i=1
, we have that
> M . Thus ÿ is bounded for ÿ0 :=
f
s
g
mi j=1 (
2 jj j C+ 0
þ
ai;j
pi )j
s > M
þ 2 C j j j þC
2. Let þ := min Re(p1 ); : : : ; Re(pr ) . Since ÿ is analytic in
j ÿj
2
it is continuous in the compact set ÿ1 := and
ÿ( )
is bounded for s
s
ÿ1 .
M
+ 0
s
is bounded
.
ý ý
+ 0 , in particular M; Re(s)
ÿ
2
C
i
ÿ2
ÿ1 ÿ0
ÿ
0
iM Figure 2.9: The sets ÿ0 , ÿ1 and ÿ2 .
2
!
þ 2C jj j
j j
3. Finally consider the set ÿ2 :=
2.9). We prove that inf s ÿ2 g(s) in ÿ2 such that g(sn )
!
+ 0
s
ý
f g ! 2
(see Figure sn
f g f g
0. Consider the compact set ÿ2 , the closure of ÿ2 .
0. But since g
j j
ÿ
2
> 0. If not, there exists a sequence
2
There exists a convergent subsequence ÿ2 and g(snk )
s < M; Re(s) <
1C
H
c
(
snk
sn
of
such that snk
s0
), it follows that g(s0 ) = 0, which
P1
is a contradiction, since the zeros of g are contained in the complement of
ÿÿ ÿÿ ÿ ÿ
2
ÿ2 . Thus inf s ÿ2 g(s) f (s) g (s)
C+ 0
= ÿ0
[ [
Finally, since f
ÿ1
j ÿj ÿ( )
p1 ; : : : ; pr
C+ 0
gü{
ÿ2 ,
r i=1
<
1
7!
ÿ(s) =
n
\ 2C j
Thus ÿ is continuous in
s
C+ 0
+ 0
c
ÿÿ ÿ
. Hence
mi j=1 (
þ
ai;j
pi )j
is bounded in ÿ2 .
ÿ
and g are in H
s
is continuous in
ÿÿ
1
g (s)
ÿ2 , it follows from the above that ÿ( ) is bounded in
+ contained in C 0 , it follows that
this lemma.
f
is bounded in ÿ2 , and since
is bounded in ÿ2 . Thus
Since
2 ÿP
> 0, which implies that sups ÿ 2
C+ 0 .
, and the zeros of g, namely p1 ; : : : ; pr are
f (s) g(s)
ûXX û
Re(s)
and so ÿ
mi
r
ai;j
i=1 j=1
þ 2
o
ÿ
2
.
1C
H
c
(
(s
j
pi )
Moreover, ÿ is analytic in ).
C+ 0 .
This completes the proof of
2.5.
Lemma 2.5.2 If K
=
ization K
C+ 0
55
Transfer function algebras
NM
þ1
2 H1 [l] (C pÿm ), c
;
then there exists a right coprime factor-
, where M is rational,
C+ 0.
and they are all contained in
ý
det(M )
2 R1
has exactly
l
zeros in
ý is the matrix transfer function of a system of þ ý
Proof Let K = G + F , where G
MacMillan degree equal to l , with all its l poles in the open right half-plane, and c F H (C p m ). Let G = N1 M 1 be a right coprime factorization of G over
2 1 ÿ ý MR (which exists by Lemma A.7.37, Curtain and Zwart [27]). We claim that þ1 is a right coprime factorization over MH1 for K . M is K = (F M + N1 )M + square with det(M ) 2 R1 ÿ S1 having exactly l zeros in C 0 (this follows for c
example, from page 287, Rugh [71] and the fact that
l),
C+ 0 and
and they are all contained in
N
:=
FM
ý has MacMillan degree 2 M 1 by the ring
G
+ N1
H
c
1 (C ). Since ( 1 ) is a rational coprime pair over MR, there ý , ý 2 MR such that ý þ ý 1 = . Deÿning and in M 1 by = ý+ ý , = ý , we see that þ =[ ý+ ý ] þ ý[ + 1] = and so ( ) is a right coprime pair over M 1 .
properties of exist
X
X
H
c
N ;M
Y
X M
X
Y F
Y
XM
Y N
I
X
Y
YN
X
Y F
M
Y
FM
H
N
I
c
Lemma 2.5.3 If (N; M ) is a right coprime factorization of K and V
2 1 ÿ H
(C m m ) is invertible as an element of
also a right coprime factorization of K .
MH1 , c
Moreover, any two right coprime factorizations of K
2 H1 [l] (C pÿm ), c
;
then
2
H
(N V ; M V )
1 [l] (C pÿm ) c
;
unique up to a common right multiplication by an invertible element in
Proof
These proofs are analogous to parts
b
and
c
7.2.8, pages 353-354, Curtain and Zwart [27].
Lemma 2.5.4 If
1. M and N belong to
MH1 , c
2. M and N have the same number of columns, 3. M is a square matrix with 4.
det(M )
then N M
has
l
c
;
det(M )
2 S1
and
zeros in the open right half-plane,
þ1 2 1 l (C pÿm ) H
c
Y
N; M
c
H
.
is
are
MH1 . c
of the proof of Theorem
56
Chapter 2.
Proof
The inverse of the square matrix
1.
M
where adj(M )
2 MH1 , c
ÿ1 =
1 det(M )
Classes of well-posed linear systems
M
is given by
adj(M );
since its components are sums and products of the
1ÿ(C1 ) by assumption. Thus from Lemma ÿ1 can 2.5.1, we have that all the entries in belong to 1 l (C ) and so be written as the sum of a proper rational matrix þ and a matrix 2 M 1. ÿ1 2 1 k (C pým ), for some nonnegative integer k. In the remainder Thus
components of
M,
which are all in
H
c
NM
H
c
NM
;
G
NM
H
c
;
k ÿ l,
of the proof, we will show that
F
H
c
hence proving the claim.
ÿ1 2 1 k (C pým ), it follows from Lemma 2.5.2 that it has a right coprime factorization þ þÿ1 , where þ is rational, det ( þ ) 2 R1 has exactly k zeros in C and they are all contained in C . Since ( þ þ ) is right coprime, there exist and in M 1 such that þþ þ= Upon post multiplication by þÿ , we obtain þ þ þÿ = þÿ ÿ , we have þÿ = þ 2 M 1 . Consequently Using þ þÿ = 2 1 (C ). It is now easy to see that k ÿ l. ÿ Since
2.
NM
H
c
;
N M
M
+ 0
X
Y
H
c
XM
1
M
XM
1
N M
det(M ) det(M )
H
NM
YN
I:
M
1
YN M
1
M ;N
1
M
M
M
1
M
XM
M:
YN
H
c
c
2 H1 l(C pým ), c
ý
Lemma 2.5.5 If K then K1 K K2
2 H1 c
K1
;
pÿ mÿ ). ;l (C
From Lemma 2.5.2,
Proof
M
+ 0
K
=
NM
2 H1 (C pÿ ýp ) and K2 2 H1 (C mýmÿ ), c
c
ÿ , where 1
N, M
c
S1 has, say k (ÿ l) zeros in the open right half-plane. K1 K K2
=
1 det(M )
K1 N
adj(M )
2 MH1 and det(M ) 2 Thus
K2
and proceeding as in the proof of Lemma 2.5.4, we get that c c p mÿ ). (C pÿ mÿ ) H ;l (C ÿ ;k
ý ý 1
1
H
ý
Lemma 2.5.6 If K = N M
1 ý
H
c
ÿ 2 1 l (C pým ) 1
H
c
;[
]
, N
2
K1 K K2
1 (C pým )
H
c
(C m m ), with (M; N ) right coprime, then det(M ) has exactly
l
and M
2 2
+ zeros in C 0 ,
and the zeros are all contained in the open right half-plane.
From Lemma 2.5.2, there exists another coprime factorization of
=
þ þÿ , where þ 2 1 (C pým ), þ 2 1 (C mým ), with ( þ þ ) right coprime,
Proof N M
1
N
H
c
M
H
c
M ;N
K
2.5.
57
Transfer function algebras
Mÿ is rational and det(Mÿ ) 2 R1 has exactly l zeros in C +0 , and the zeros are all contained in C + 0 . From Lemma 2.5.3, M = Mÿ U , where U is an invertible c . From I = UU þ1 we have that [det(U )] ÿdet þU þ1 ýü = 1, and element in MH1 hence det(U (s)) 6= 0 for s 2 C + 0 . Since det(M ) = det (Mÿ ) det(U ), it follows that det(M ) has exactly l zeros in C + 0 , and the zeros are all contained in the open right half-plane.
The following technical lemma will be used in the characterization of solutions.
ÿ) 2 MH1c ;ý, then given any ÿ > 0, there exists a Æ > 0 þ þ þ Æ, we have kK (þ + iÿ)k1 þ kK (iÿ)k1 + ÿ; where k ÿ k1 denotes the L1 ýnorm (for a ÿxed þ 2 R, kK (þ + iÿ)k1 = sup!2R kK (þ + i! )k).
If K ( such that whenever 0 Lemma 2.5.7
In order to prove the above Lemma 2.5.7, we will use the following generalization of the so-called \3 lines theorem" to the matrix case. The statement and proof of the 3 lines theorem in the scalar case can be found, for instance, in Theorem 12.8, pages 257-258 of Rudin [70].
ý
ý
Lemma 2.5.8
Suppose
fþ + i! j a < þ < b; ! 2 Rg, ÿ = fþ + i! j a þ þ þ b; ! 2 Rg, 2. K : ÿ ! C ü is continuous on ÿ, 1.
ÿ=
p m
3.
K
is analytic in
ÿ,
kK (s)k þ B for all s 2 ÿ and for some ÿxed B < 1. If M (þ ) = supfkK (þ + i! )k j ! 2 Rg for a þ þ þ b, then we have M (þ )bþa þ M (a)bþÿ M (b)ÿ þa for all a < þ < b: 4.
Proof
1.
prove that function
We assume þrst that
M (a) = M (b) = 1. In this case, we have to For each ÿ > 0, we deþne an auxiliary
kK (s)k þ 1 for all s 2 ÿ.
ý a) for all s 2 ÿ: Since Re(1 + ÿ(þ + i! ý a)) = 1 + ÿ(þ ý a) ü 1 in ÿ, we have khþ(s)K (s)k = jhþ(s)j kK (s)k þ 1 for all s 2 @ ÿ; hþ(s) =
1 1 + ÿ(s
(2.33)
(2.34)
58
Chapter 2.
j
Classes of well-posed linear systems
i! ÿ a)j þ ÿj!j, so that B (2.35) for all s = þ + i! 2 ÿ n R: khÿ(s)K (s)k = jhÿ (s)j kK (s)k ý ÿj!j B Let R be the rectangle cut oþ from ÿ by the lines y = ü ÿ (see Figure 2.10). ÿþ
the boundary of ÿ. Also, 1 + ( +
ÿ
R
B ÿ
b
a
ÿ Bÿ
Figure 2.10: The rectangle
R.
khÿK (s)k ý 1 on @R, the boundary of the rectangle khÿK (s)k ý 1 on R, by the maximum modulus theorem (see for
By (2.34) and (2.35),
R.
Hence
example, Theorem 3.18.4, page 115, Hille and Phillips [46]). But (2.35) shows that
khÿK (s)k ý 1 on the rest of ÿ. Thus khÿK (s)k ý 1 for all s 2 ÿ and all If we ýx s 2 ÿ and then let ÿ tend to zero, we obtain the desired result.
ÿ > 0. 2.
We now turn to the general case with
M (a) 6= 0.
bÿs
Let
sÿa
f (s) = M (a) bÿa M (b) bÿa ; where for M > 0 and s complex, M s is deýned by M s = es log M ; M
and log
is real. Then
1.
f
is analytic in
2.
f
has no zeros,
C,
1
4.
f is bounded in ÿ, jf (a + i!)j = M (a),
5.
jf (b + i!)j = M (b),
3.
and so f1
hence
K satisýes our previous assumptions.
Thus
ÿ ÿ ÿ 1 K (s)ÿ ý 1 in ÿ, and ÿ f (s) ÿ
bÿÿ
ÿ ÿa
kK (þ + i!)k ý jf (þ + i!)j = M (a) bÿa M (b) bÿa :
2.5.
Hence 3.
59
Transfer function algebras
M (ÿ )bÿa ÿ M (a)bÿÿ M (b)ÿ ÿa .
Finally, we show that if
M (a) = 0, then K
2 6 E11 = 6 64
þ
We have, for positive , sup
fkþE
!2R and sup
0
0 . . .
0 . . .
0
0
kg
11 + K (a + i! )
fkþE
! 2R
1
11 + K (b + i! )
::: :::
is identically zero in ÿ. Let 0
:::
0 . . .
= sup
3 7 7 7 5
0
ÿ
p m
fkþE
kg þkE k
11 + 0
! 2R
kg ÿ !2RfþkE k kK b sup
:
11 +
=
( +
11
(
> 0);
i!)kg = þkE11 k + M (b):
Thus from the above, we obtain
kþE
11 + K (ÿ + i! )
This is true for all
þ > 0.
kK ÿ
Consequently
K
b ÿ 11 b a (
b ÿ b a
Passing the limit as
( +
of
k ÿ þ ÿÿ kE k ÿÿ þkE k
ÿ ÿa 11 + M (b)) bÿa
:
þ tends to zero, we have
i!)k ÿ 0 kE11 k bÿa M (b) bÿa bÿÿ
ÿ ÿa
=0
:
K (ÿ + i!) = 0 for all ! 2 R and a ÿ ÿ < b. Finally the continuity K (b + i!) = 0 for all ! 2 R. Thus K is zero in ÿ.
yields that also
Proof (of Lemma 2.5.7.) Let K (s) = G(s)+ F (s) where G is the matrix rational
transfer operator of a system of MacMillan degree, say
l,
with all its poles in
c . Let the poles of G be contained in F 2 MH1 + the half-plane C r for some r > 0. Consider the function s 7! K (s) deþned for s belonging to the inþnite strip ÿ := fs 2 C j 0 ÿ Re(s) ÿ rg. Clearly, K (þ) is the open right half-plane and
continuous in ÿ and analytic in the interior of ÿ. ÿ
C
0
ÿ
r
Using the triangle inequality, it is easy to see that
K (þ) is bounded in ÿ:
60
Chapter 2.
1. 2.
Classes of well-posed linear systems
s 7! G(s) is bounded in ÿ (since all its poles are in C +r ) and s 7! F (s) is bounded in ÿ (in fact, in C + ). 0
ÿ > 0,
For any
M (ÿ ) = sup!2RfkK (ÿ + i!)kg.
deþne
Using Lemma 2.5.8, we
M (ÿ ) ÿ [kK (iþ)k1]1ÿ r M (r) r : If kK (iþ)k1 = 0, then we have M (ÿ ) = 0, and the result follows trivially. now consider the case when kK (iþ)k1 6= 0: we have obtain
ÿ
ÿ
M (ÿ ) ÿ kK (iþ)k1 h
! kKMi rk1
i ÿr
ÿ
M (r) kK (iþ)k1
þ ÿr
We
:
Æ such that 0 < Æ < r, and for any ÿ satisfying 0 ÿ ÿ ÿ Æ, we have kK (ÿ + iþ)k1 ÿ kK (iþ)k1 + þ. Since limÿ
( ) ( )
0
Lemma 2.5.9
If
= 1, there exists a
2 H1c C ý
F
( p m ) and lim!
!ü1 F (i!) exists (say, F1 ), then
!1 F (s)
lim
s
s2C + 0 exists and equals
Proof
F1 .
It follows from Theorem 5.18 (page 96, Rosenblum and Rovnyak [69])
that
F (ÿ + i!) = Let
þ > 0.
satisfying
ÿ ý
1 ÿ1 ÿ
Z
First we choose an
j!j > R
1
2
+(
R>0
t ý !)2
F (it)dt; ÿ > 0:
(large enough) such that for all
!
in
, we have
kF i! ý F1 k < þ (
)
2
We have
kýF Zÿ 1i! ý F1 k ÿ ýýý ýÿ ÿ1 ÿ t ý ! ýZ ÿ ýý R ÿ ý ýý ÿR ÿ t ý ! ( +
)
1
2
+(
1
2
+(
Z ÿ R
ÿ kF iþ k1 ý 2
( )
)2
ÿR ÿ
ý ý
F (it) ý F1 ) dtýý
( )2
F (it) ý
(
1
2
+(
ý ý þÿZ R 1 F1 ) dtýý + dt 2 ý 2 ý ÿR ÿ + (t ! )2
t ý !)2
dt +
þ
2
þ
ý
;
1
R
2.5.
61
Transfer function algebras
1 ÿ 1 dt = 1: 2 þ ÿ1 ÿ + (t ÿ !)2
Z
where we have used
We note that
ÿ þ
Z R
1
ÿR ÿ 2 + (t ÿ !)2
dt =
and so it is clear that there exists a
jÿ
+
i!j > R, we have
ÿ þ Consequently, for all
kF ÿ
( +
Z R
1
i!)
Arctan
þ
Rþ (> R)
ÿR ÿ 2 + (t ÿ !)2
s 2 C +0
ÿ
1
dt <
Rÿ ÿ 2 + ! 2 ÿ R2 2
such that all
ÿ + i!
satisfying
ý
4
kF iþ k1 : ( )
jsj > Rþ ÿ F1 k ý kF iþ k1 kF ýiþ k1 with
þ
, we have
2
( )
4
( )
+
ý 2
=
ý
and this completes the proof.
Lemma 2.5.10
1.
If
c (C kýk ) F (þ) 2 H1
2. for every
! 2 R, F (i!) is invertible
!ü1 F (i!) exists (say, F1 ) and F1 is invertible c (C kýk ). then F (þ)ÿ1 2 H1 ;û 3. lim!
Proof
From Lemma 2.5.9 above, we have
!1 F (s) = F1 :
lim
s
We prove that
s 7! det(F (s))
s2C + 0
has a ÿnite number of zeros, which are all conCk k C is a continuous
þ
tained in the open right half-plane. Since det( ) : function, it follows that
!1+
lim det(
s
s2C 0
Consequently, there exists a every
s
2 C +0
with
jsj > R
ý
!
F (s)) = det(F1 ) 6= 0:
R>
0 large enough such that det(
. The function
s
7!
det(
F (s))
F (s))
6
= 0 for
is analytic in þ =
Chapter 2.
62
fs 2 C +0 j jsj < Rg to the continuity of
.
s
Classes of well-posed linear systems
But it is not identically 0 in ÿ (for otherwise, owing
7!
F (s)) in ÿ = fs 2 C +0 j jsj ÿ Rg we would ! 2 [þiR; iR], a contradiction!). Thus the zeros
det(
F (i!)) = 0 for all s 7! det(F (s)) have no accumulation point in ÿ. But there are no zeros on + [þiR; iR] [ fs 2 C 0 j jsj = Rg, and so the zeros of s 7! det(F (s)) have no
have det( of
accumulation point in ÿ. Since the set ÿ is compact, it now follows that there are only þnitely many zeros. Furthermore, they are all contained in the result follows using Lemma 2.5.4.
C+ 0.
Finally,
Chapter 3
Compactness and nuclearity of Hankel operators In this chapter, w e examine the relationships betw een the exponential (or strong) stability of certain classes of well-posed linear systems and the compactness and nuclearit y properties of the Hank el operator. New suÆcient conditions for nuclearit y are giv en for exponentially stable, regular, well-posed linear systems with an analytic semigroup. In the last section of this chapter w e will derive a bound on the
L1 ÿerror of a sub-optimal Hankel norm approximant of
a system with a nuclear Hankel operator.
We consider the Hankel operator of stable matrix-valued functions
1 (Cppÿm ). The Hankel operator with symbol
H
H2 (
C ) given b y
G f = ÿ(þG fþ )
H
where þ
G
is the multiplication map on
for
G
2
f
is the operator
2
G : H2 (C m ) !
H
C m );
H2 (
G
(3.1)
R; C m ) induced b y
L2 (i
G
(see Theo-
rem A.6.26, page 647, Curtain and Zwart [27 ]), ÿ is the orthogonal projection operator from L2 (iR; C p ) onto H2 (C p ) and f (s) := f ( s).
þ
ÿ
Practical control design is typically based on a reduced-order model of the original system. Many design methodologies utilize a rational approximation of a stable transfer function in the
1 ÿnorm (for example, see Chapter 9, pages
L
457-563, Curtain and Zwart [27 ]). For this to be possible the Hankel operator with symbol G H (C p m ) should be compact. We quote the follo wing
2
1 ÿ
criterion (see for instance Corollary 4.10, page 46, Partington [62]):
þþ 2 ()
1 ÿ
H
2 C
1
ÿ
L (iR; C p m ) determines a compact Hankel (C p m ) + 0 (iR; C p m ), where 0 (iR; C p m ) denotes the
Theorem 3.0.11 (Hartman) G operator iÿ G
ÿ
C
ÿ
A. Sasane: Hankel Norm Approximation for Infinite-Dimensional Systems, LNCIS 277, pp. 63−83, 2002. Springer-Verlag Berlin Heidelberg 2002
64
Chapter 3.
Compactness and nuclearity of Hankel operators
p ÿ m complex matrix-valued functions deÿned on þ 1.
space of continuous a (unique) limit at
R,
i
with
i
Since most models of inÿnite-dimensional systems are obtained, not as transfer functions, but as realizations, we are interested in deducing the properties of the Hankel operator from properties of the realization. To do this we introduce the time-domain Hankel operator, which is deÿned in terms of h, the inverseG. If h L1 ([0; ); C p m ) or L2 ([0; ); C p m ), we deÿne m the time-domain Hankel operator þh : L2 ([0; ); C ) L2 ([0; ); C p ) by
2
Laplace transform of
(þh u)(t) = for all
u
2
L2 ([0;
1
1
1
Z
0
h (t
1
ÿ
1
+ s)u(s)ds;
); C m ). In the case that
h
2
L1 ([0;
ÿ
! 1 ý 1 Cÿ 0;
t
);
(3.2)
p m ), it is well-known
that þh is compact (see for example Lemma 8.2.4, page 399, Curtain and Zwart [27]) and so þh has countably many singular values (square roots of the eigen-
þ
values of þh þh )
ý ý ýÿ 1 C
0 and these are also called the Hankel singular p m ), then (3.2) may not be in L ([0; ); C p ). If, 2 p m however, we also assume that G L (iR; C ), then (3.2) always deÿnes þ as a bounded operator from L ([0; ); C m ) to L ([0; ); C p ) (see Proposition 8,
values of
G.
If
2
ÿ1
h
ÿ2
1
:::
L2 ([0;
);
2 1 1
ÿ
1
h
2 2 page 224, Keulen [83]). In either of these cases, þh is isomorphic to HG under the Laplace (or Fourier) transform (see Lemma 8.2.3, page 397, Curtain and Zwart [27] and Keulen [83]).
Let us consider two classes of systems; the ÿrst was the subject of the book [27] by Curtain and Zwart and the second was the main topic of the recent book by Oostveen [61]. Class 1. ý(A; B; C ) where A is the inÿnitesimal generator of an exponentially
stable strongly continuous semigroup B (C m ; X ) and C (X; C p ).
2L
2L
f gtý T (t)
0 on a separable Hilbert space
X,
For this class, it is well-known and easy to verify that the impulse response is h(t) = C T (t)B , h L1 ([0; ); C p m ), and the Hankel operator is compact.
2
Class 2. ý(A
ü
the Hilbert space
1
þ
þ
), where ( m ; X ).
2L C
BB ; B; B X, B
ÿ
A
generates a contraction semigroup on
þ
ü
2
þü
1 ÿ
1B For this class, it is known that G(s) = B (sI A + BB ) H (C m m ), m m m m h L2 ([0; ); C ), but h L1 ([0; ); C ) in general (see Curtain and Zwart [28]). So by Proposition 8 (page 224, Keulen [83]) it follows that the
2
1
ÿ
62
1
ÿ
time-domain Hankel operator is well-deÿned by (3.2).
A
ü
BB
þ in general does
not generate an exponentially stable semigroup, but under mild conditions it does generate a strongly stable semigroup t
!1
for all
x
2
X ).
f B gtý T
(t)
0 (that is,
T
B (t)x
!
0 as
65
Since the Hankel operator of Class 1 is compact, one wonders if relaxing exponential stability to strong stability is still suÆcient for compactness. Sur-
h 2
i h
i
2
h
ÿ
prisingly, the reverse statement holds if A and A is,
f
x
x; Ax
g
+
Ax; x
ÿ
D(A )).
0
TB (t) tþ
f
= 0 for all x
g
D(A) and
i h
ÿ
+
In section 3.1 we prove that if A and A
and
ÿ
TB (t)
0
tþ
i
are skew-symmetric (that
x; A x
ÿ
ÿ
A x; x
= 0 for all
are skew-symmetric,
are strongly stable but not exponentially stable,
and X is inÿnite-dimensional, then þh will never be compact!
f
g
On the other
hand, by means of an example, we show that systems from Class 2 can have a compact Hankel operator even if
0
TB (t) tþ
is not strongly stable. In other
words, the stability of the semigroup says little about the compactness of the Hankel operator.
ÿ
While compactness of the Hankel operator is essential to obtain a rational approximation in the H1
norm, the property of nuclearity is desirable, because
P =1
1
it gives a-priori error bounds in terms of the singular values. We recall that a compact Hankel operator is
nuclear
if
1
k
ÿk <
, where ÿk 's are the singular
values in decreasing order. (For background material about nuclearity, we refer
2
1
\
1
to Chapter 1 of Partington [62].) h
1
L ([0;
integer
l,
);
C pým )
2
L ([0;
);
In Glover et al.
C pým ),
[37] it was shown that if
and þh is nuclear, then for a given
there exists a truncated balanced realization Gb;l and an optimal
0
Hankel-norm approximation G ;l , both of order
k ÿ k ÿ
k þ ý P = +1 k þ P = +1
G
Gb;l
G
G ;l 1
0
l,
1
2
1
k l
1
such that
ÿk ;
and
ÿk :
k l
In the last section of this chapter we will also ÿnd a bound on the L1
ÿ
error of
any sub-optimal Hankel norm approximant of a transfer function for which the Hankel operator is nuclear. There exist various known conditions for nuclearity in terms of the transfer functions.
Theorem 3.0.12 (Coifman and Rochberg [11].)
Z Z
is nuclear iÿ
1
ü1
1
0
k
00
G (x + iy)
k
If
G
dxdy <
2
1
:
C pým ), then
H1 (
HG
(3.3)
The Hankel operator given by + possesses on C 0 an expansion of the form
Theorem 3.0.13 (Coifman and Rochberg [11])
(3.2) is nuclear iÿ
G
X 1
G(s) =
=1
n
ÿ
1
s
an
Gn
(3.4)
66
Chapter 3.
where Gn 2 C pÿm and
an
Compactness and nuclearity of Hankel operators
2 C þ are such that X1 kG k 1 jRe j 0
n
n=1
<
(an )
(3.5)
:
These theorems made it possible to obtain suÆcient conditions for the nuclearity of transfer functions of delay systems (see Zwart et al. [94] and Glover et al. [38]). See also the recent results on fractional transfer functions in Bonnet and Partington [9]. In this chapter we give suÆcient conditions for nuclearity in terms of the state-space realizations. Our new results on nuclearity in Section 3.2 are 1. Class 1 has a nuclear Hankel operator (Theorem 3.2.1). 2. A class of regular well-posed linear systems with an exponentially stable analytic semigroup and unbounded
B
and C has a nuclear Hankel operator
(Theorem 3.2.3). Analytic semigroups are generated by parabolic partial diÿerential operators and hyperbolic partial diÿerential operators with structural damping (see Pazy [63]). Consequently, Theorem 3.2.3 has important consequences for model reduction of distributed systems with an analytic semigroup with unbounded sensing and control.
3.1
Compactness of Hankel operators
In the introduction to this chapter, we recalled that linear systems with bounded, þnite-rank input and output operators, and an exponentially stable semigroup always have a compact Hankel operator. One might hope that this would also be the case if we only have strong stability. However, by means of two examples, we show that the property of strong stability does not imply the compactness of the Hankel operator. Let us consider Class 2. This class arises by stabilizing the open-loop system 1 B , via the static output B (sI A)
ý
feedback
u
=
ÿ
y,
which results in the closed-loop system ý(A G(s)
=
G0 (s) (I
+ G0 (s))
ÿ
þ
ý
ý ) with transfer function ý )þ1 +
ý
ý(A; B; B ) with transfer function G0 (s) =
ÿ
BB ; B; B
þ1 = ý ( B
sI
ÿ
A
BB
B:
This closed-loop system has several nice properties (see Curtain and Zwart [28]):
3.1.
67
Compactness of Hankel operators
u
y
ÿ
ÿ(A; B; B )
ÿ1
ÿ ÿ generates a contraction semigroup f R1 ÿ B ( ) k2 ÿ 12 k k2 . C2. 0 k R1 ÿ B ( )ÿ k2 ÿ 21 k k2 . C3. 0 k C4. ( )= ÿ( þ + ÿ)ý1 2 1 (L( )).
C1. A
BB
T
B T
t x
B T
t
G s
B
dt
x
sI
g
B (t) tþ0 .
x
x
dt
A
BB
B
H
U
C2 and C3 show that the system has an impulse response L2 ([0;
1 L );
ý
h(
) =
ÿ
(U )) and bounded observability and controllability maps
deÿned as follows: 1.
B
:
L2 ([0;
1 ! Z 1 B ); U )
=
u
2.
C
:
X
!
L2 ([0;
C
(
1
X
0
=
and
is deÿned by
B (t)Bu(t)dt
T
for all
); Y ) is deÿned by
x)(t)
ý 2 B
B ( )B
C
B T
ÿ
B T
B (t)x
for all
t
u
ü
2
L2 ([0;
0 and all
1 x
); U ):
(3.6)
2
(3.7)
X:
f B gtþ f B gtþ f B ÿgtþ
The semigroup
0 is not necessarily strongly stable. SuÆcient conand T (t) 0 0 to be strongly stable can be found in Arendt and Batty [3]: ditions for
T
T
(t )
(t )
N1. The intersection of the spectrum of A with the imaginary axis is at most
countable. OR N2.
f 2 j x
X
ÿ
B T ( t) x
k
ÿ
T (t) x
k kk k =
x
=
T (t)x
k8 ü g fg t
0
=
0 .
ÿ ( ) = 0 can be replaced by ÿ ( )ÿ = 0, and ÿ ) is approximately controllable or observable, then N2 holds.
Note that in N2, þ(A; B; B
= 0;
B T t x
The Hankel operator ýh is equal to
B T t
CB
and the the observability Gramian Lyapunov equations:
L
x
. The controllability Gramian
C
=
CÿC
L
B
=
if
BBÿ
always satisfy their respective
68
Chapter 3.
N3. (A N4. (A
ÿ
ÿ
BB BB
Compactness and nuclearity of Hankel operators
ÿ ) B + B ( ÿ ÿ )ÿ ÿ )ÿ C + C ( ÿ ÿ ) L
x
L
L
x
A
L
BB
A
BB
f B ÿ gtþ f B gtÿþ
x
=
x
=
ÿ ÿ
ÿ
ÿ
BB x
for all
x
BB x
for all
x
2 2
D (A). D (A
ÿ ).
0 is strongly stable, N3 has the unique solution LB and if 0 is strongly stable, N4 has the unique solution LC . Suppose now that A and A are skew-symmetric and TB (t) t 0 and TB (t) t 0 are strongly sta1 ble, but not exponentially stable. Then N3 and N4 have the unique solution 2 I and we now prove that ÿh will not be compact. We have If
T
(t)
T
( t)
f
ÿ
ÿh ÿh =
gþ
f
ÿg þ
BÿC ÿCB Bÿ C B Bÿ B =
1
=
L
2
ÿ
I
=
1 2
BÿB
:
If ÿh is compact, then ÿh ÿh is also compact and it follows from the above
that
B
B
must be compact (the compactness of
follows, for example, from
BBÿ
Theorem 6.4.(c), page 131, Weidmann [85]). Consequently we obtain = 1 LB = IX must be compact, a contradiction, since we assume that X is an 2 inþnite-dimensional. So if A and A are skew-symmetric and TB (t) t 0 and
f
T
ÿ f gþ ÿ B ( ) gtþ0 are strongly stable but not exponentially stable, then ÿh can never t
be compact.
Many examples of partial diýerential equation systems of the Class 2 structure with a skew-symmetric operator
can be found in Oostveen [60]. Here we
A
give a simple example which can be readily analyzed and which captures the salient features of the partial diýerential equation examples. Example 3.1.1 Let A : D (A) (
A
with
2 66 66 66 =6 66 66 64
( D (A)
=
f gþ
x
2
0
þ
N ))
`2 (
!
N)
`2 (
be the operator given by
ÿ
1
0 0
2
ÿ
2
0 0
3
ÿ
3
0 ..
N)
`2 (
3 77 77 77 77 77 77 5
1
j
1ÿ X k=1
j ÿ kh
ý1 ij
x; e2k
2
+
jkh
x; e2k
;
.
ij
2
þ
)
<
1
;
n 1 denotes the standard orthonormal basis for `2 (N ). We þrst show is closed and densely deþned. Clearly, all elements x in `2 (N ) with x; en = 0 for all suÆciently large n lie in D (A) and form a dense set in `2 (N ). So A is densely deþned.
where that
h i
A
en
3.1.
m
69
Compactness of Hankel operators
Let
that
f gÿ
!
xm m 1 be a sequence in D (A) and let xm x0 and Axm . Since the sequence Axm m 1 is bounded, there exists a M
!1
f gÿ
X1 ÿjÿ h k
=1
k
Consequently, for any
X ÿjÿ h N
k
=1
2 2 ÿ1 ij jkh N2N
xm ; e
+
k
2 2 ij k
,
2kÿ1
xm ; e
k
xm ; e
ij2 jkh +
xm ; e
2k
ij2
þ
þ
< M;
for all
m
þ
1:
< M;
for all
m
þ
1:
Owing to the continuity of the inner product and the fact that
X ÿjÿ h 0
obtain
N
=1
k
x ;e
k
N
X1 ÿjÿ h 0
Since the choice of
n p1
0
x
2
+
=1
k
2kÿ1
x ;e
x ;e
k
þ
D (A)
with
ij2 jkh 0 2 ij2 ý +
0 = y0
Ax
x ;e
and so
In fact it can be easily checked that
o
þ
ij2 jkh 0 2 ij2 ý
was arbitrary, it follows that
k
Consequently,
2kÿ1
A
k
A
xm
!
>
!
0
as
0,
we
y
0 such
x
M:
M:
is closed.
is a Riesz spectral operator with
fü g 2
ni n N (see Figure 3.1) and the corresponding (orthogonal) Riesz basis of eigenvectors the (totally disconnected) set of simple unstable eigenvalues
2 (en
ü
+1)
ien
2N.
n
C
3i 2i i
ÿ2
ÿ1
0
1
2
ÿi ÿ2i ÿ3i Figure 3.1: Since
Ax
+
þ
A x
= 0 for all
x
2
D (A)
ÿ (A).
=
D (A
þ ),
it follows that
A
is the
inÿnitesimal generator of a contraction strongly continuous semigroup on the
70
Chapter 3.
Hilbert space `2 (N ). We have 0
2 66 66 66 ÿ 1 A =6 66 66 64
ÿ
0
ÿ(A)
and
1
0
ÿ
1 2 0
0 1 2
0 1 3
ÿ
1 3 0 ..
where
0 1
3 7 7 7 7 7 7 7 2 L(`2(N )): 7 7 7 7 7 5
1
It can be easily seen that
2 66 66 66 66 Rn := 6 66 66 66 4
2
Compactness and nuclearity of Hankel operators
Rn
!
A
.
ÿ1 as n ! 1 in the uniform operator topology,
3 7 7 7 7 7 7 7 7 2 L(`2 (N)): 7 7 7 7 7 7 7 5
ÿ
1
0 ..
. 0 1 n
ÿ
1
n 0
0
0
0
0 ..
.
ÿ
has ÿnite rank 2n, A 1 is the uniform limit of a sequence of compact operators and so A 1 is compact (see for example Theorem A.3.22.e, Since each
Rn
ÿ
page 587, Curtain and Zwart [27]). From Theorem 6.29 (page 187, Kato [49]) if follows that Let
B
2L C (
has compact resolvent.
A
N )) be deÿned by
; `2 (
2 6 6 6 6 6 B = 6 6 6 6 4
1 0 1 2 0 1 3 0 . . .
3 7 7 7 7 7 : 7 7 7 7 5
It follows from Theorem 4.2.3 (page 164, Curtain and Zwart [27]) that þ(A; B;
ÿ
ÿ ; ÿ; B ÿ ) is approximately
) is approximately controllable. Dually, þ(A
observable. Consequently, from Lemma 2.2.6 (page 23, Oostveen [60]) it follows that
B
:=
f B gtþ T
A
(t)
A
ÿ
0 and
ÿ and Aÿ
f B ÿgtþ B BB T
(t)
=
ÿ ÿ BB ÿ generate strongly stable semigroups
A
0 , respectively, on `2 (N ).
3.1.
71
Compactness of Hankel operators
ÿ generates a strongly stable semigroup, but the state linear system ÿ ÿ(A ÿ BB ; B; B ÿ ) has a time-domain Hankel operator þh that is bounded, but Thus
A
ÿ
BB
ÿ
not compact.
The following example shows that a system with the Class 2 structure can have
fTB (t)gtþ0 is not strongly stable.
a compact Hankel operator although
Example 3.1.2 We start by showing that the transfer function G(s)
=
1+
p1 2 s
+1
;
(with an appropriate choice of an analytic branch of the complex square root function) has a realization of the form ÿ(A
ÿ BB ÿ ; B; B ÿ ).
We do this by
interpreting it as the closed-loop system formed by applying the static output feedback to the system with transfer function G0 (s)
=
1 : 2+1
p
s
Following Baras and Brockett [7], we show that
ÿ
ÿ(A; B; B ), where be given by
A
G0 (s) has a realization generates a contraction semigroup. Let A (`2 (Z))
2 66 6 16 6 A = 26 66 4
It is easy to see that the spectrum of [
A
A
..
.
..
.
..
2L
.
0
ÿ1
1
0
ÿ1
1
0 ..
is bounded and that
A
.
..
.
..
.
3 7 7 7 7 7 : 7 7 7 5
ÿ
+ A = 0. It can be shown that
(see Figure 3.2) is the set
ÿi; i] = fs 2 C j Re(s) = 0
and
ÿ 1 þ Im(s) þ 1g:
(See Problem 241, page 135, and Problem 67, page 36, Halmos, [44].) Let
B
2 L(C ; `2 (Z)) be deýned as
2 6 6 6 B = 6 6 6 4
. . . 0 1 0 . . .
3 7 7 7 7 : 7 7 5
72
Chapter 3.
Compactness and nuclearity of Hankel operators
2i
C
i
ÿ2
ÿ1
0
1
2
ÿi ÿ2i Figure 3.2:
ÿ(A) = ÿc (A) = [ÿi; i].
A; B; B ÿ ) realizes the transfer function ps12 +1 . We know that s 7! B ÿ (sI ÿ A)þ1 B is analytic in C n [ÿi; i]. Moreover for ! > 1, þi it can be checked easily that B ÿ (i! ÿ A)þ1 B = p 2 ! þ1 , as follows. Let L2 (T) be the the space of square integrable functions on the unit circle and let L' be Claim: ÿ(
the multiplication operator corresponding to the bounded function
'(z ) = ÿ z þ1 + i! + z: 1
1
2
2
From Exercise 241 (page 135, Halmos [44]), it follows that the bilaterally inþnite matrix
2
i!I ÿ
6 6 6 6 A=6 6 6 6 4
..
.
..
.
..
3 .
i! ÿ 12
1 2 i! 12 1 2 i!
ÿ
..
.
..
.
..
.
7 7 7 7 7 7 7 7 5
L' on L2 (T) L T
is the Laurent matrix corresponding to the multiplication operator
with respect to the familiar standard orthonormal basis in 2 ( ), namely n þ 1 ( ) = . Thus ( ) corresponds to multiplication by n n2
fe
z
z
1
g
'(z )
i!I ÿ A
Z
= =
=
1
ÿ 12 zþ1 i! 12 z ÿ 1 þÿz 1 ýü i! ÿ 2i! z " û ú ÿ z i! i! z +
+
1
1
1
+
1+
Since we want to þnd out
1
2
+
1
ù +
1
i!
2
û
ÿz
+
1
z
#
úø2 +
::: :
(3.8)
B ÿ (i!I ÿ A)þ1 B , we are actually þnding the (0; 0)th
3.1.
73
Compactness of Hankel operators
i!I ÿ A)ÿ1 , which is the coeÆcient of z 0 in (3.8): 1 1 2n 1 n 2n : (ÿ1) n i! n=0 2i! But for ! > 1, we have
ÿþ
Xÿ þ
entry of (
p!ÿ2 iÿ
1
namely,
ÿ ÿ þÿ i! !2 ÿ X1 ÿ 12 þ ÿ ÿ 2 þ i! =0 ! ÿ þ X1 2 ÿ 2 ÿÿ 21 þ ÿ i! =0 i! ÿ 1 þ ÿ þ; 2 ÿ2 ÿ 1
=
1
=
n
n
1
=
n
2
1)
n
(
2
n
(
1
n
(
1
n
It is easy to see that
1 2
1
1
n 1) =
n
n n 1) 2
n
(
n 1)
:
2n
n
ÿi and so it follows that for ! > 1, B þ (i! ÿ A)ÿ1 B = p 2 ! ÿ1 . I
i i
II
Figure 3.3: The Riemann surface of the map
s! 7 ps12+1 .
s! 7 ps12 +1 . We shall take two copies of the sÿplane, with cuts along the segment [ÿi; i]. The formation of an analytic branch of the map s 7! ps12 +1 takes place in each of these planes. Now we shall join the edges of our cuts cross-wise with the help of two segments [ÿi; i], the inner points of Consider the map
which will be considered to be diÿerent, although geometrically they coincide.
s 7! ps12 +1 (see Figure 3.3). It is double-sheeted, with two branch points on i and ÿi. The function s 7! p 12 s +1 is
This is the Riemann surface of the function
single-valued and continuous on this surface. We consider the analytic branch
i! 7! p!ÿ2iÿ1 for ! > 1. and this analytic branch match on fi! j ! > 1g, the claim follows. of
s 7! ps12 +1
corresponding to the plane I;
Since
G0 (s)
A ÿ BB þ ; B; B þ ) realizes the transfer function G(s) = 1+p1s2 +1 . Since G(s) = 1+p1s2 +1 , it can be easily checked that G(s) 2 H1 (C ). Moreover, ! 7! G(i!) is continuous and has the limit 0 as ! ! þ1. So it follows from Thus þ(
74
Chapter 3.
Theorem 3.0.11, that
G
Compactness and nuclearity of Hankel operators
determines a compact Hankel operator. However, our
earlier discussion shows that in this case
A
stable semigroup.
3.2
ÿ
BB
ÿ cannot generate a strongly ÿ
Nuclearity of Hankel operators
In this section we study the nuclearity property of the Hankel operator of a system in terms of a given realization (A; B; C ). First we show that an exponentially stable state linear system with bounded inputs and outputs and ÿnite-dimensional input and output spaces is nuclear.
f gþ
Theorem 3.2.1 Let A be the inÿnitesimal generator of an exponentially stable
2L C
2L
strongly continuous semigroup B
( m ; X ),
ÿ
C
(X; C p ).
T (t) t
C
1. The observability operator CT (
1
)x
is Hilbert-Schmidt.
2. The controllability operator
R
C
Proof
=
1.
CÿC
, L
Deÿne
BBÿ
B=
C
k :
X
C
!
and
:
!
X
B
:
f
e1 ; : : : ; e p
jC
g
þh =
L2 ([0;
L2 ([0;
CB
j
),
1
); C m )
1; : : : ;
C T (t)x; ek
jh þ k kk þ kk R1 j ýÿt k ÿkj 1 =
=
x
C e
M e
C ÿ
(
x)(
deÿned by
X
B
) =
u
=
C
by
ÿ
i
C p . We have
ÿ ÿ ij ÿ ÿ kk þ k ( ) ýÿt k ÿ k
T t
!
deÿned by
x; T (t) C ek ;
x; T (t) C ek
x
); C p )
are all nuclear.
is the standard basis for
( k x)(t)
1
1 k 2 f pg h i h ÿ
L2 (0;
( k x)(t) = where
on the separable Hilbert space X , with
T (t)Bu(t)dt is Hilbert-Schmidt.
0 3. L
0
Then
x
kk
ÿk k ÿ k k kk
T (t)
C
e
;
2 C dt < . We will now use the following result which is 0 M e an adaptation of Theorem 6.12 (page 140, Weidmann [85]): and
1
Theorem 3.2.2 Let L2 (0;
).
K be a bounded linear
2 1
2 2H
If there exists a function ÿ
for almost all t
(0;
),
and all v
1
jK
jþ
operator from a Hilbert space
L2 (0;
, then
K
)
such that
(
v )(t)
is Hilbert-Schmidt.
H
kk
into
ÿ(t) v
3.2.
75
Nuclearity of Hankel operators
C
Applying this result we obtain that
X1 kC
f g
arbitrary orthonormal basis
j=1
k xj
k
of
xk
1
2
L2 (0;
k is Hilbert-Schmidt. Consequently, for an
X,
) <
1
X X1 kC
and
p
k xj
k=1 j=1
X1 kC
Thus
and so
Bÿ 2.
C
k
xj
j=1
2
k
for all
1
2
L2 (0;
L2 ([0;
1
k
) <
);C p ) <
2f
1; : : : ; p
1 1
;
From Theorem 6.9 (Weidmann [85]), it follows that
Bÿ
;
:
is Hilbert-Schmidt.
is Hilbert-Schmidt. But
g
B
is Hilbert-Schmidt iÿ
is Hilbert-Schmidt by applying the þrst part of
the lemma to the dual system ý(Aÿ ; C ÿ ; B ÿ ).
BB
3.
Using Theorem 7.10(b) (Weidmann [85]), we obtain that
ÿ
and üh =
CB
LC
are all nuclear.
We remark that the Hilbert-Schmidt property of mortier [29] (page 24, Proposition 1.0.2).
C
=
C ÿC
,
LB
=
was already shown in Du-
We now show that a class of regular linear systems with an exponentially stable analytic semigroup and unbounded
B
and
C
has a nuclear Hankel operator. We
use the notation introduced in Section 2.4. Theorem 3.2.3 Let
1.
ÿ
2. B
2L 2L þ
T (t) t
(C m ; Z
ÿ
0 on Z ,
B ),
and
(ZÿC ; C p ),
3. C
where ÿB
1.
f gþ
A generate an exponentially stable, analytic, strongly continuous semi-
group
ÿ
ÿC < ÿB
A, B , C
space C (sI
Cm
+
+ 1.
If þ satisÿes ÿC
ÿ
1 < þ < ÿB 2
1 + 2 ,
then
generate a regular linear system with state space Zþ , input
p
and output space C . The transfer function is given by G(s) 1 A) B , and it satisÿes
ý
jslim j!1 kG(s)k = 0: 2C +0
s
=
76
Chapter 3.
C2L
ÿ
3. h(
1
CT (
)B
2.
L1 ([0;
p m ) and the Hankel operator ÿ satisÿes h ); m ); Zÿ ) and (Zÿ ; L2 ([0; ); C p )).
);
(L2 ([0;
, where
Furthermore,
Proof
1
ÿ 2 1 Cÿ CB B2L 1 C
) :=
ÿh =
(Zÿ ; L2 ([0;
B2LL
( 2 ([0; ); C m ); Zÿ ) and the observability p ); C )) are Hilbert-Schmidt operators.
2. The controllability map map
Compactness and nuclearity of Hankel operators
ÿh
C2L
is nuclear.
1
This part follows from Staþans [79] (page 251) or Staþans [77].
1.
p = 1.
Just as in Theorem 3.2.1, it is suÆcient to prove this for the case
part 1 above, we know that C is an admissible observation operator for with state space every
z
2
C,
Zþ
jC (
and so
Zÿ
we have
j
C2L
j þ k k þ k k
z )(t)
=
=
C
C
= If
ý ü jC j
ÿC
and
t
7!
z )(t)
2 ý ý jC j
ÿýt e ÿC ÿþ
t
C
kL kL kL kL
(ZÿC ;C )
L2 ([0;
);
ü
þ > ÿC
Case 2:
If
C
(ZÿC ;C ) K1 þ > ÿC
and
t
7!
ý
e
ýt
0, then we have
þ <
(
2
z )(t)
L2 ([0;
Hilbert-Schmidt for all
k kL þ k kL 1 C =
);
þ
Since
B
ý
2
L
(Z )
ý Cý
ýt
e t
ý
ÿ
1 . 2
2L
L
kkþ z
Z
k k kk
(Z )
kkþ z Z
;
So by Theorem 6.12 (page 140,
ÿ þC ýÿ ÿ ÿA T (t)ÿ
C
(ZÿC ;C ) K2 e
ý
ýt
kkþ z
Z
L
(Z )
þ
satisfying
kkþ z Z
;
). So by Theorem 3.2.2, it follows that that
satisfying
ÿC < þ .
C
1 B
Z
u
= 0
ÿ
1
0
(ZþC ; C ), for
is Hilbert-Schmidt for all
is Hilbert-Schmidt for all
f gþ 2 1
is an admissible control operator for (L ([0; ); C m ); Z ) and for
L
ÿ ÿ
T (t)
þ
(ZÿC ;C )
1 . 2
is an element in
ÿ
T (t)A z
C
T (t) t
Z ÿ ÿ ÿ L(Z ) A z Z ÿ ÿ T (t)ÿ z Zþ : ÿ
C
From the above cases, it follows that þ > ÿC
C
Z
ÿ
ÿ þC ýÿ ÿ ÿA T (t)ÿ
(ZÿC ;C )
), for
k ÿC
Cý Cý Cý
þ
C
1 . 2
ÿC
); C )). Since
T (t)z
C
k kL þ k kL 1 C =
1
ÿ (ZÿC ;C ) A ÿ þ ÿ (ZÿC ;C ) A ÿ þ ÿ (Zÿ ;C ) A
Weidmann [85]), it follows that that ÿC
kÿ
0, then we have
þ
(
j
C T (t)z C
Case 1:
(Zÿ ; L2 ([0;
From
f gþ
T (t) t u
0
is
satisfying
with state space ); C m ),
L2 ([0;
T (t)Bu(t)dt:
þ
C
Zÿ ,
B
3.2.
77
Nuclearity of Hankel operators
B 2 Lÿ B ÿ 0
Thus the dual operator
0
Zÿÿ ; L2 ([0;
0
(
z)
()=
0
1 ÿ
); C m )
B T(
Proceeding as above, it can be shown that Hilbert-Schmidt.
B
that 3.
If
T (t)Bu
2C 2
C
But since
Zþ
B
C T (t)Bu
B 2 Lÿ 0
þ
ÿB
0, it follows that
ÿC
Cp
C
m
ÿB <
L2 ([0;
( (
;C p )
(
L Z
u))(t)
A
ý
B T (t)ý
ÿþ
L(Z )
ÿüt
e
K1 ;C p ) þ t
(
L Z
C ÿþB
1, we obtain that
(L2 ([0;
u
0
Zÿÿ ; L2 ([0;
1
); C m )
þ
is
2\
C
We know that sequently, if
0
Zÿÿ ,
and for t > 0, T (t)Bu ý 2RZý . Consequently p for every t > 0. Moreover,
Cp
C T (t)B
Finally, since
2
z:
2C
k þ k k ÿC ýý C ý ü k k ÿ þ k k ÿC ý B2L 1 C 2 1 C B 2 CB B 2L C
ÿC
C T (t)
2
Bu
and
Zþ
z
Thus, using Proposition 2 (page 261, Aubin [4]) it follows
m , then
C T (t)Bu
But
and for
is also Hilbert-Schmidt.
u
k
0
)
þ
h(
m ); Z ) and ÿ
);
); m ),
=
C T (t)
Zÿ ,
u
=
u
and
ÿ C2L )=
Z
k k B
L
(C m ;ZÿC )
k k ÿ 2
: (C m ;ZÿC )
B
CT (
kk u
Cm :
L
)B
L1 ([0;
(Zÿ ; L2 ([0;
1
1
); C pþm ).
); C p )). Con-
1
C T (t)
T (þ )Bu(þ )dþ :
0
p ) and so we have (see for example Theorem A.5.23, page
(Zÿ ;
628, Curtain and Zwart [27])
CB
( (
u))(t)
Z
1
=
C T (t)T (þ )Bu(þ )dþ
Z0 1 =
C T (t
+ þ )Bu(þ )dþ
Z0 1 =
h(t
+ þ )u(þ )dþ
0
Furthermore,
B
and
C
=
(ÿh u)(t):
are Hilbert-Schmidt operators, and so it follows from
Theorem 7.10 (b) (page 175, Weidmann [85]) that ÿh =
CB
is nuclear.
As explained in the introduction, analytic semigroups are generated by parabolic and some hyperbolic partial diþerential equations and so the above theorem is relevant to many distributed parameter systems with unbounded sensing and control. For Example 2.4.10 considered earlier in Section 2.4 of Chapter 2, the transfer function
G(s)
= (1+1s)m has a nuclear Hankel operator.
The analyticity assumption in Theorem 3.2.3 is crucial. In general, if generates an exponentially stable semigroup and if
B
or
C
A
is unbounded, the
78
Chapter 3.
Compactness and nuclearity of Hankel operators
Hankel operator will not be nuclear:
to show this, we revisit Example 2.3.4
given in Chapter 2.
Example 2.3.4 (continued) We note that the semigroup is not analytic, but it is exponentially stable, and B is bounded, but C is unbounded. The Hankel singular values are ÿk =
p 12
ÿk +1
, where the þk 's are the roots of the
transcendental equation
tan(þý ) =
ÿþ(3 ÿ þ2 ) 2 1 ÿ 3þ
(3.9)
(see for example, Theorem 8.2.10, pages 402-403, Curtain and Zwart [27]).
10
5
-7.5
-5
-2.5
5
2.5
7.5
-5
-10 Figure 3.4: Roots of the transcendental equation.
þ
For the sake of simplicity, we assume ý = 1. Because of the periodicity of tan( ),
ÿ
þ
its monotonicity in each periodic interval of the type
(2
k ÿ 1)
ü 2
k + 1)
; (2
and the monotonicity of the function f (x) =
ü 2
;
k 2 Z;
ÿx(3ÿx2 ) , for jxj > p3, the positive 1ÿ3x2
roots of the transcendental equation above satisfy 1 < þk <
kü
for
k
> 2 (see
Figure 3.4). Thus we obtain
Hence
P
ÿk =
p
1
2
þk + 1
>
p
ÿk diverges, and so ÿ
1
2
2
=
þk + þk
h
p1
2þk
>
p1
2ü
k
for
k > 2:
is not nuclear. However, it is Hilbert-Schmidt.
This follows from Exercise 8.9 (Curtain and Zwart [27]).
ÿ
Finally, we make the point that exponential stability is not a necessary condition for nuclearity: in the following example, the Hankel operator is nuclear, although the semigroup is not exponentially stable.
3.2.
79
Nuclearity of Hankel operators
Example 3.2.4 We construct the example below following Ober [58].
f gÿ
Let
k 1 be a decreasing bounded sequence of distinct positive numbers converging to 0. Consider the system ÿ(A; B; B ), with
ÿ
ÿk
ÿ
A
=
ÿk ÿl
þ
kl (ÿk + ÿl )
2 66 B = 6 4
and
þ
ÿ
ÿ1
1
ÿ2
2
ÿ3
3 . . .
3 7 7 7 5
2L C (
1
1 k;l<
N )) :
; `2 (
Then 1.
2L 2
A (`2 (N )) is Hilbert-Schmidt. This follows from the proof of Proposition 3.(i) (page 304, Ober [58] and Theorem 6.22, Weidmann [85]). Thus
0
þ (A)
ý
ü
and this rules out exponential stability.
However, from the
proof of Proposition 3.(iii) (page 304, [58]), it follows that the semigroup e
At
t
ý0 is strongly stable.
2. The Lyapunov equation Aþ
2 66 þ0 := 6 4
has a solution
+ þA =
ÿ1
0
0 0 . . .
ÿ
ÿ
(3.10)
BB :
0
:::
ÿ2
0
:::
0 . . .
ÿ3
:::
. . .
..
3 7 7 : 7 5
.
This is proved in Proposition 3 (page 304, [58]), and so it follows from Theorem 3.1 (page 10, Hansen and Weiss [45]) that admissible control operator. Since
A
=
ÿ
A
B
is an inýnite-time
generates a strongly stable
semigroup, it follows from [45] (page 10) that (3.10) has the unique solution
2 6 BBÿ = þ0 = 664
ÿ1
0
0
:::
0 0 . . .
ÿ2
0
:::
0 . . .
ÿ3
:::
. . .
..
3 7 7 ; 7 5
.
which is clearly compact. Hence it follows from Theorem 6.4.(c) (page 131, Weidmann [85]) that
B
is compact.
3. The time domain Hankel operator1 üh = with
þk
(üh ) =
ÿk
for all
B B
k
2N
.
BÿB 2 L
(L2 (0;
1
)) is compact,
1 The equality ÿ = ÿ follows from the remark after the proof of Corollary 4.4, page h 276, Ober [59]. That ÿk (ÿh ) = þk is established in Corollary 2 (page 304, [58]).
80
Chapter 3.
Compactness and nuclearity of Hankel operators
Finally, upon choosing the sequence
f g 2N
k obtain nuclearity: for example, we can take
Conclusions:
ÿk
such that 1 ÿk = 2 . k
P 2N k
ÿk <
1
, we can
ÿ
There already exist necessary and suÆcient conditions for the
compactness and nuclearity of the Hankel operator in terms of the transfer function. However, in most applications, the model is given in terms of a triple (A; B; C ) of operators. compact
Example 3.1.2
nuclear Example 3.2.4
A
þ exponentially stable
alytic semigroup
2 L(C m ; XÿB ) 2 L(XÿC ; C p ) ÿB ÿ ÿC < ÿB + 1
B C
Example 2.4.10 A
þ exponentially stable
emigroup Example 2.3.4
Example 3.1.1
A B
and
C
bounded
B
or
C
unbounded
þ strongly stable
emigroup
Figure 3.5: Overview of the results. We have given new suÆcient conditions for nuclearity for two classes of realizations: an exponentially stable realization with bounded
B
and
C
(The-
orem 3.2.1) and an exponentially stable analytic realization (Theorem 3.2.3). On the other hand, by means of examples, we have shown that the state space properties of exponential and strong stability of the semigroup have little to do with the compactness and nuclearity of the Hankel operator. Figure 3.5 gives a concise overview of the results shown in the previous sections of this chapter.
3.3
L1ÿerror
of sub-optimal Hankel norm ap-
proximants
In this section, we prove that a sub-optimal Hankel norm approximant is also a good approximation with respect to the
1 ÿnorm under the assumption that
L
the Hankel operator is nuclear. To prove Theorem 3.3.1 below, we will use the following results:
L1 ÿerror of sub-optimal Hankel norm approximants
3.3.
2
P12
P2. If
K1 ; K2
P1. If
ÿk
k=1
H
space
1
kk
ÿ
); C p m ) and ÿh is nuclear, then
([0;
are compact operators from the Hilbert space 2,
ÿ
1
h L1 ([0; );C pÿm ) (see for example, Theorem 2.1, page 866, Glover et al. [37]).
L1
h
81
then
ÿj+k
þ
1
(K 1 + K 2 )
ÿ
ÿj
(K1 ) +
ÿk
H
1
(K2 ) for all
to the Hilbert
j
and
k
in
N
(see for example, Theorem 7.7, page 171, Weidmann [85]). Theorem 3.3.1 Suppose that
2
1. h 2.
ÿh
L1
([0;
1
ÿ
); C p m ) ,
and let G denote the Laplace transform of h,
is a nuclear Hankel operator,
3. ÿl+1 (G) < ÿ < ÿl (G), and
þý 2
4. K (
)
H
1
;l
ÿ
(C p m )
k ý
If K( G
ý
)
=
ý (þý)
G(i
1 (C pÿm )
H
k ý G(i
ÿk
k
ÿ.
where
l;
(HG+Gÿ )
2N
and
;
ý k1 ÿ
ý
) + G (i )
We know that
Proof
for all
);
with MacMillan degree at most
)
then
ÿk
HG+Gÿ
k k =
4l
ÿl (G)
HG+K
X1
+2
ÿk (G):
k=l+1
kÿk
G
+K
k1 ÿ
ÿ < ÿl ;
(3.11)
. Moreover from P2. above, we obtain ÿk+l
since
ý
F(
) + K (i )
is the rational transfer function of a ÿnite-dimensional system
þý 2
F(
ý (ý ) +
G
ý k1 ÿ
is a solution to the sub-optimal Hankel norm ap-
proximation problem, that is,
(HG+Gÿ )
ÿ
ÿk
(HG ) + ÿl+1 (HGÿ ) =
ÿk
(HG ) ;
(3.12)
ý has MacMillan degree at most equal to l. Thus we have 9 ( > l( ) ÿ) ÿ = . . using (3 11) . > ; l( l( ) ÿ) ÿ 9 > l ( l( ) ÿ) ÿ = (3 11) and (3 12) both apply, . . . > ; but (3 11) is sharper. ( ) ÿ l l l( ) ÿ ) ( ) ÿ ( )
G
ÿ1 HG+G
ÿ
G
:
HG+G
ÿ
G
ÿ +1 HG+G
ÿ
ÿ
G
:
:
:
ÿ +
HG+G
ÿl+l+1 HG+Gÿ
ÿ
. . .
G
ÿl+1 G
(3:11) and (3:12) both apply, but (3:12) is sharper.
82
Chapter 3.
Compactness and nuclearity of Hankel operators
X1
Finally, employing the estimate from P1. above, we have
k ÿ G(i
where
ÿ k1 þ k
ÿ
) + G (i )
h
ÿk
+h
1
L1 ([0;
ÿ
);C p m )
þ
2
ÿk
k=1
(HG+Gÿ ) ;
ÿ denotes the inverse Laplace transform of ÿ . Thus we have 1 k ( ÿ) + ÿ( ÿ)k1 þ 4l l ( ) + 2 k( )
X
h
G
G i
G
i
ÿ
G
ÿ
G :
k=l+1
The error bound in Theorem 3.3.1 is like the one in Theorem 6.4 (page 889, Glover et al. [37]).
X1
Finally we investigate the nature of the error bound
El
:= 4l
ÿl (G)
+2
ÿk (G):
k=l+1
!
!1
In Theorem 3.3.3 below, we prove that El 0 as l . This shows that the sub-optimal Hankel norm approximants are also good approximations with respect to the
L
1ýnorm. In order to prove Theorem 3.3.3 below, we need the
following lemma.
Lemma 3.3.2 If a1
Proof
We have
ü ü ü a2
:::
1 X ü ak
0
an+1
and
P1
n=1 an
+ : : : + a2n
k=n+1
!1 n
Thus limn
a2n
!1
1 X ü ak
an+2
a2n
!1
limn
+ : : : + a2n+1
!1 n
a2n+1
, then
ün ü a2n
!1 n
limn
an
= 0.
0:
= 0:
k=n+1
Thus limn
1
= 0, and so lim 2n n
Furthermore,
<
ün
!1 2n
a2n+1 = 0, and so limn = 0, we obtain
lim (2n + 1) n
!1
a2n+1
= 0:
(3.13)
a2n+1
ü
a2n+1
0:
=
0.
But since
(3.14)
3.3.
L1 ÿerror of sub-optimal Hankel norm approximants
83
From (3.13) and (3.14) above, the result follows. Remark: We remark that the assumption a1
ÿ 2 ÿÿ þ h ÿ n2 a
:::
t
Consider for example the lacunary series in which the other terms are zero: then 1
but
fn g an
12
+0+0+
1
22
+0+0+0+0+
has the constant subsequence
G
Theorem 3.3.3 If H
That is, El
Proof
!
Since
0
= 4l
as
G
H
l
ýn2 1 ü n2
+0+:::
þl (G)
+2
1 X
= +1
þk (G) < ÿ
0,
for all
k l
!1
1
<
1
;
.
is nuclear, then given any ÿ >
such that El
1
32
0 is important.
term is n2 , and all
there exists a
l
N
2N
ÿN
:
.
is nuclear,
P
þk (G)
converges, and so
P1= +1
!
þk (G) 0. k l Hence the result
!1 l þl (G) = 0.
From Lemma 3.3.2 above, it follows that liml follows.
Chapter 4
Characterization of all solutions In this chapter, w echaracterize all solutions to the sub-optimal Hankel norm approximation problem in terms of a solution to a key J ÿspectral factorization problem. All the proofs in this chapter are based on purely \frequency domain" techniques. In Chapter 5 we will give formulas for a J ÿspectral factor in terms of the state space parameters A, B and C which satisÿes the assumptions in this chapter, hence solving the sub-optimal Hankel norm approximation problem for the Pritc hard-Salamonclass and the analytic class of inÿnite-dimensional systems, respectively.
4.1
The sub-optimal Hankel norm approximation problem
Although it is known (see Adamjan et al. [2]) that if inf
K (þ )2H1 l (Cpÿm ) ;
k ( ÿ) + ( ÿ)k1 = G i
ÿ2
( )
G i
K i
inf c
K (þ )2H1 l (Cpÿm ) ;
;
ÿl+1 ;
ÿ 2 C (R C
in this section, we will prove by simpler methods that if G(i ) limit 0 at , then
þ1
1 (R C pÿm ) then
L
k ( ÿ) + ( ÿ)k1 = G i
K i
;
ÿ
p m)
with
ÿl+1 :
In order to do this, we will start by ÿrst proving the following simple result. A. Sasane: Hankel Norm Approximation for Infinite-Dimensional Systems, LNCIS 277, pp. 85−99, 2002. Springer-Verlag Berlin Heidelberg 2002
86
Chapter 4.
H H !H
Lemma 4.1.1 Let S :
H !H 1
Characterization of all solutions
H
1 to the Hilbert space
If S
2 is an arbitrary bounded linear map of rank
ÿ
:
1
ÿl+1 .
ÿ ÿ ÿ k þ ÿk ÿ
2 be a compact, bounded operator from the
Hilbert space
f g
2 with singular values ÿ1
l,
ÿ2
then
0.
:::
S
S
Proof If ÿl+1 = 0, then the inequality is trivially satisÿed. Let us assume that
ÿ
ÿ
ÿl+1 > 0. Suppose that
wk
H
k k
f
wk
1
= 1. Deÿne vk =
g
k
ÿk S
ÿ ÿ
ify that the vk 's are orthonormal and S vk = ÿk wk . Let þ
2 onto span w1 ; w2 ; : : : ; wl+1 ; then
jection from
ÿ ÿ
Consider the following restriction of þ S :
f
ÿ ÿ
: span v1 ; v2 ; :::; vl+1
þ S
P
which has rank at most z =
l+1 k=1 ak vk ,
l,
P j j l+1 k=1
X
=
ak
ÿ
X
2
þ Sz
l+1
=
Thus
k þ ÿk ÿ S
S
k=1
ü 2 G
with symbol G, namely H
inf c
þü 2 k ü )
ü ÿü 1c C pþm ü k1 ÿ k (
G(i ) + K (i )
ÿ ÿ
ker (þ S ),
kk z
.
= 1. If
= 1, and
þ
2 2
ÿk ak
ÿ ÿ
þ S z
ÿ
k k
2 ÿl+1
1 R C pþm
L
(
;
X 2
=
l+1 k=1
ÿ
þ Sz
2
k
2
2
ak = ÿl+1 :
) be such that the Hankel operator
;l
ÿ
(C p
m)
k ü
ü k1 ÿ
ü
). Thus,
ÿ
G+G
l
G(i ) + K (i )
ÿ +1 :
ÿ þü
Let K ( ) = G ( ) + F ( ), where G ( H
2
g
S
, is compact. Then
K (ý )2H1
F(
span w1 ; w2 ; :::; wl+1
S
ÿl+1 .
Theorem 4.1.2 Let G(i )
Proof
2
S )
ak ÿk wk ;
k=1
k þ ÿk ÿ k S
f
þ ÿ k ý k þ ÿk
l+1
ÿ
þ Sz
S
g!
wk . It is easy to ver-
be the orthogonal pro-
þ (S
and hence there exists a z
then we have
ÿ
is an orthonormal sequence of eigenvectors of the
2
operator S S : S S wk = ÿk wk ,
+F
kH k =
) has MacMillan degree
ÿ kH
G +G
=
k
ý
l, and
G + HGþ k ÿ ÿl+1 (G):
H
The ÿrst equality follows easily from Lemma 8.1.2.c and Examples 8.1.3 (page 388, Curtain and Zwart [27]).
The last inequality is a consequence of Lemma
4.1.1 above and the classical result that the rank of the Hankel matrix of the transfer function of a ÿnite-dimensional system is equal to its MacMillan degree (see for instance Kalman et al. [48]).
4.1.
The sub-optimal Hankel norm approximation problem
ÿ 2C R C
Theorem 4.1.3 If G(i )
(
ÿ
p m ) with limit
;
þ 2 1 Cÿ
K(
Proof
1.
inf
H c ;l ( p
)
m)
k ÿ
0 at
ÿ k1 ý
G(i ) + K (i )
87
þ1
, then
i
ÿl+1 :
From Hartman's theorem (see for instance Corollary 4.10, page
46, Partington [62]), it follows that the Hankel operator corresponding to G is compact. Consequently, from Theorem 4.1.2 above, we have
þ 2 1 Cÿ
K(
2.
inf
H c ;l ( p
)
k ÿ
ÿ k1 ü
G(i ) + K (i )
m)
We now prove the reverse inequality.
f g 2N
ÿl+1 :
From Theorem A.6.11 (page 640,
Curtain and Zwart [27]), we have the existence of an approximating sequence Gn
n
2N k ÿû
üN
of rational transfer matrices with no poles on the imaginary axis for
G. Given any þ > 0, choose
N
large enough such that for all
G(i )
ÿ k1
Gn (i )
<
þ 3
n
,
:
(4.1)
Let s
where
s
ÿ2
Gn ( )
1 C pÿm
H
c
(
u
Gn = Gn +
ûÿ 2
u
) and
Gn (
)
H
Gn ;
1 C pÿm c
(
). From Lemma 8.1.2.c and
Example 8.1.3 (page 388, Curtain and Zwart [27]), it follows that s
ÿl+1 ( Gn )
We know that if
K1
the Hilbert space in
N
n
and
K2
ÿl+1 (Hs Gn )
=
ÿl+1 (Hs Gn + Hu Gn )
=
ÿl+1 (Hs Gn +u Gn )
=
ÿl+1 (HGn ):
ý
are compact operators from the Hilbert space
þ
2 , then ÿj+k
K
1( 1 +
K2 )
K
K
ÿj ( 1 ) + ÿk ( 2 ) for all
j
H
1 to
and
k
(see for example, Theorem 7.7, page 171, Weidmann [85]). Applying this
with
for
H
=
ü
K1
= HG ;
K2
= HGn
þ
j = l + 1; k = 1;
G;
N we have
s
ÿl+1 ( Gn )
=
ý ý ý
ÿl+1 (Gn )
û k û k1
ÿl+1 (G) + ÿ1 (Gn ÿl+1 (G) + ÿl+1 (G) +
ûÿ 2 k ÿ
Gn
þ
3
G)
G
(from Lemma 8.1.2, page 388, [27])
:
(4.2)
1 l C pÿm
In Glover [35] (Theorem 3.1), it was shown that given a stable rational matrix s
Gn , there exists a Kn ( s
)
H
c
;
(
) such that
ÿ k1 ý
Gn (i ) + Kn (i )
s
ÿl+1 ( Gn ) +
þ 3
:
(4.3)
88
Chapter 4.
þÿ
ÿÿ
2
)
that Kn (
H
k ÿ
1plÿmC pÿm c
þÿ 2 1 C ÿ þ ÿ k1 k ÿ þ ü k ÿþ ü ü
We note that since Kn ( u
) +
Gn (
)
c
H
( ;l
u
G(i ) + Kn (i )
; (
Gn (i )
Characterization of all solutions
þÿ 2 1 C ÿ nýN ÿ ÿ ÿ þ ÿ k1 ÿ k1 k ÿ ÿ k1 u
) and
Gn (
)
H
c
). Furthermore for
=
G(i )
G(i )
ÿ
Gn (i )
ÿ
+ þl+1 (G) +
3
s
+
s
ÿ 3
ÿ
it follows
, we have u
Gn (i ) + Kn (i ) + Gn (i )
+ þl+1 ( Gn ) +
3
p m ),
(
Gn (i )
Gn (i ) + Kn (i )
(using (4.1) and (4.3))
3
ÿ
+
(using (4.2))
3
þl+1 (G) + ÿ:
=
But since the choice of ÿ > 0 was arbitrary, the claim follows.
From the results shown in Theorems 4.1.2 and 4.1.3, we have the following:
ÿ 2C R C
Theorem 4.1.4 If G(i )
(
ÿ
p m ) with limit
;
þ 2 1 Cÿ
K(
)
inf
H
c
;l (
p
m)
k ÿ
û1
0 at
ÿ k1
, then
= þl+1 :
G(i ) + K (i )
In light of the above theorem, we are now ready to state the sub-optimal Hankel
norm approximation problem that we consider:
Let
ÿ 2C R C ÿ k ÿ
G(i )
(
p m)
;
þl+1 < þ < þl ;
If
þÿ 2 ÿ k1 ü
with limit
then ÿnd
such that
K(
0 at
)
1
H
G(i ) + K (i )
c
û 1ÿ Cp
;l (
:
m)
þ:
K is then called a solution of the sub-optimal Hankel norm approximation problem.
þÿ 2
In the following corollary, we show that any solution K0 (
problem in
þÿ 2 ÿ üÿ
ÿ
Corollary 4.1.5 If K (
)
1 l C pÿm ü þÿ
H
c
;
) is such that
(
þÿ 2
þl , then K ( ) = G ( ) + F ( ), where G (
l
ÿ
1
H
p m ). (C ;[l]
Proof degree
r
ÿ
ÿ
ü
k ÿ
r
þÿ 2 1 ýk ÿ and all
þl > þ
c
(
;
)
)
H
c
)
1 C pÿm
ÿ
H
c
(
ÿ
;l
ül ÿ k1 ý
G(i ) + K (i )
r
þ <
, with all
), that is, K (
)
þÿ þÿ 2 1 C ÿ k ÿ ÿ k1 ý lür r l ü
) has MacMillan
poles in the open right half-plane, and F1 (
p m ), (C
ÿ k1 ü l þÿ 2
G(i ) + K (i )
Suppose that K ( ) = G 1 ( ) + F1 ( ), where G 1 (
Since K (
1 l C pÿm
H
) has MacMillan degree
poles in the open right half-plane and F (
c
)
l unstable poles) of the sub-optimal Hankel norm approximation fact has an unstable rational part of MacMillan degree exactly l!
(with at most
)
H
c
(
p m ).
. From Theorem 4.1.2, it follows that
þ 2 1 Cÿ
K(
Thus þl > þr+1 , which implies that
)
inf
H c ;r ( p
l < r + 1,
m)
G(i ) + K (i )
and so
. Hence
þr+1 :
=
.
4.2.
4.2
89
Main assumptions and a few useful consequences
Main assumptions and a few useful consequences
In this section, we state the assumptions S1-S5 under which we solve the suboptimal Hankel norm approximation problem. We will also prove a few elementary consequences of these assumptions which will be used in the sequel. We make the following assumptions:
S1. S2.
G(iÿ) 2 C (R; C pÿm ) with limit 0 at þi1. ÿl+1 < ÿ < ÿl , where ÿk 's denote the Hankel singular values of G.
S3. There exists a ÿ(
ý
=
S4. ÿ(
ýÿ
)
Ip 0 G(i!)þ Im ý þ Ip ÿ(i! )
V (ýÿ) s 2 C +0 .
0
2
ÿp
(p+m)
üý
exists a
0
!!ý1 ÿ(i! ) =
1
Ip
üý
!2R
Ip G(i!) 0 Im
m
ü
i!):
c H þ1
ÿ
(4.4)
C (p+m)
ÿp
( +m)
such that ÿ(
þ
ýs V ýs
, that is, there
)
(
) =
Ip+m
ü
0
ÿIm
0
2
ÿ(
m
such that for
ýÿ I
0
ýI
þ
0
ü
ÿ H c C (p+m)ÿ(p+m) ý
( +m)
Ip
) is invertible as an element of
for all
S5. lim
ýÿ 2 H1c ÿC
.
Remarks:
1. From Hartman's theorem (see for example Corollary 4.10, page 46, Partington [62]) and S1, we have that the Hankel operator with symbol
G is
compact.
2. From S5 and Lemma 2.5.9, it follows that
!1+
ýs
lim ÿ(
s
s2C 0
ý
) =
Ip 0
ü
ÿIm : 0
iÿ) at þ1. Indeed, if there exists a iÿ) has a limit, say ÿ1 at þ1, then
3. We remark that S5 is only a normalization condition. All we actually need is the existence of a unique limit of ÿ( ÿ0 satisfying S3 and S4 such that ÿ0 (
90
Chapter 4.
Characterization of all solutions
it can be easily seen that ÿ deþned by
ÿ
s
Ip
ÿ( ) = satisþes S3, S4
and
þ
0
ÿIm
0
ÿ1 ÿ0 (s)
1
for all
ÿ
s 2 Cÿ 0:
S5.
Lemma 4.2.1 If the assumptions S1-S5 hold, then
1.
lim s
!1 V (ÿs) =
ÿ
Ip
2C +0
2.
5.
6.
.
ÿ
V22 (i!) is invertible for all ! 2 R,
3. There exists a 4.
1 Im
0
s
þ
0
and
ýý ý V22 (iþ)ÿ1 V21 (iþ)ý1 < 1.
Kþ(ÿþ) 2 H1 l (C pým ) such that kG(iþ) + Kþ (iþ)k1 ý ÿ. c
V22 (ÿþ)ÿ1 2 H1 l (C pýp ). c
;
;
K0 (ÿþ) := V12 (ÿþ)V22 (ÿþ)ÿ1 2 H1 [l] (C pým ) K0 (iþ)k1 ý ÿ. c
;
kG(iþ) +
and it satisÿes
V12 (ÿþ); V22 (ÿþ)) is a right coprime factorization over MH1 of K0 (ÿþ) 2 H1 [l] (C pým ). c
(
c
;
7.
det(
C+ 0.
V22 (ÿþ))
Proof 1. From S5,
ÿ
V (ÿs) ÿ I0p 10I m
þ =
üÿ
V (ÿ s )
2.
Ip
0
þ
ÿIm
0
ÿ
and the fact that
l
has no zeros on the imaginary axis, and exactly
ûÿ
ÿ ÿ(ÿs)
Ip 0
0
zeros in
1 Im
þ
;
ÿ
V (ÿþ) 2 MH1 , the result follows. c
ÿ satisþes S3, and so taking inverses, we obtain
V (i!)
ÿ = for all
! 2 R.
ÿ
Ip 0
Ip G(i!) 0 ÿIm
þ
þ ÿIm V (i!) 0
þÿ1 ÿ
Ip 0
0
ÿ 12 Im
þÿ
ÿ
Ip 0 G(i!)þ ÿIm
þÿ1
:
(4.5)
! 2 R:
(4.6)
; ÿblock of the above yields
Considering the (2 2)
V21 (i!)V21 (i!)þ ÿ V22 (i!)V22 (i!)þ = ÿ 2 Im ; ÿ 1
where
4.2.
91
Main assumptions and a few useful consequences
Thus for
u 2 Cm
we have
kV22 (i!)ÿ uk2 = kV21 (i!)ÿ uk2 + 2 kuk2: ÿ 1
V22 (i!)ÿ u = 0, then u = 0. So it follows that V22 (i!)ÿ is invertible for all ! 2 R, or equivalently, V22 (i! ) is invertible for all ! 2 R. ÿ ÿ2 ÿ2 1 ÿ From (4.6), we have ÿV22 (i! )þ1 V21 (i! )uÿ ÿ kuk2 = ÿ ÿ 2 ÿV22 (i! )þ1 uÿ . Let M > 0 be such that kV22 (i! )k þ M for all ! 2 R, we obtain kuk2 þ ÿ ÿ ÿ ÿ kV22 (i!)k2 ÿV22 (i!)þ1 uÿ2 þ M 2 ÿV22 (i!)þ1 uÿ2 . Hence Thus if
ÿ ÿ ÿV22 (i! )þ1 V21 (i! )ÿ2 þ 1 ÿ and so we have 3.
From S3 it
ÿ ÿ ÿV22 (iý)þ1 V21 (iý)ÿ
G(iý) 2 C (R; C pým )
<1
! 2 R;
for all
1 < 1. and lim!
4.1.4, the result follows. 4.
1
ÿ2 M 2
!ü1 G(i!) = 0.
So from Theorem
From (1), we know that 1
!1+ V22 (ÿs) = ÿ Im :
lim
s
s2 C 0
Thus applying Lemma 2.5.10 to c (C k k ).
H 1;û ý
V22 (ÿý),
V22 (ÿý)þ1 2
we obtain that
c (C pým ) satisfy kG(iý) + K (iý)k þ ÿ and suppose it Kÿ (ÿý) 2 H1 ÿ 1 ; [ l] þ 1 c , where N and M has the coprime factorization Kÿ (ÿý) = NM over MH1 + c are in MH1 , M is rational, and det(M ) 2 R1 has exactly l zeros in C 0 and Let
none on the imaginary axis. Deÿne
ÿ
U1 (ÿý) U2 (ÿý)
þ
ÿ
= =
ÿý)N (ý) + þ12 (ÿý)M (ý) þ21 (ÿý)N (ý) + þ22 (ÿý)M (ý) ÿ þ K (ÿý) ÿ þ(ÿý) M (ý): þ11 (
U1 (i!) U2 (i!)
For
! 2 R,
þ
ÿ
= þ(
ÿý)
ÿ
= þ(
i!) I0p G(Ii!) m
þþ1 ÿ
ÿ
þ
þ
G(i!) + Kÿ (i!) M (ÿi!); ! 2 R: Im
note that
U1 (i!)ÿ U1 (i!) ÿ U2(i!)ÿ U2 (i!) =
N (ý) M (ý)
(4.7)
Im
We have
ÿ
þ
U1 (i!) U2 (i!)
þÿ ÿ
Ip 0
0
ÿIm
þÿ
þ
U1 (i!) U2 (i!) ;
92
Chapter 4.
Characterization of all solutions
and appealing to S3, that
ÿ
Ip 0 G(i!)ÿ Im
þþ1
ÿ
i!)ÿ I0p
ÿ(
þ
0
ÿIm
ÿ
i!) I0p G(Ii!) m
ÿ(
þþ1 ÿ =
Ip 0
0
þ
ÿÿ2 Im
:
Thus
U1 (i!)ÿ U1 (i!) ÿ U2 (i!)ÿ U2 (i!) ÿ þÿ ÿ G (i! ) + Kÿ (i! ) Ip ÿ = M (ÿi! ) Im 0
Hence for all
u 2 Cm
and all
! 2 R,
þÿ
0
ÿÿ2 Im
þ
G(i!) + Kÿ (i!) M (ÿi!): Im (4.8)
we have from equation (4.8) that
kU1 i! uk2ÿkU2 i! uk2 k G i! Kÿ i! M ÿi! uk2ÿÿ2kM ÿi! uk2 þ kG iý Kÿ iý k1 þ ÿ kU1 i! uk þ kU2 i! uk: (
)
(
)
=
(
(
)+
(
))
(
)
(
)
0
;
(4.9)
since
( )+
( )
. So (
From (4.7) we obtain
V (ÿý)
ÿ
)
U1 (ÿý) U2 (ÿý)
(
þ
ÿ =
)
(4.10)
Kÿ(ÿý) M (ý); Im þ
(4.11)
and so
V21 (ÿý)U1 (ÿý) + V22 (ÿý)U2 (ÿý) = M (ý): (4.12) We claim that ker(U2 (ÿi! )) = f0g for all ! 2 R. Suppose on the contrary that there exists 0 6= u0 2 C m and a !0 2 R such that U2 (ÿi!0 )u0 = 0. Then from (4.10) and (4.12), we obtain M (ÿi!0 )u0 = 0, which implies that u0 = 0, a contradiction.
From (4.9), we deduce that
ý ý ýU1 (i! )U2 (i! )þ1 y ý2
and so
U1 (iý)U2 (iý)þ1 2 L1 (R; C pým )
þ kyk2
for all
ý
! 2 R;
ý
ý ý satisþes ýU1 (i )U2 (i )þ1 ý
1 þ 1.
c U2 (ÿý) 2 MH1 . We know that ÿ21 (ÿý) is strictly proper and + both ÿ22 (ÿý) and M (ý) are proper with invertible limits at inþnity in C 0 . Thus
Consider
from (4.7) we see that
!1+ U2 (ÿs)
lim
s
2
exists and is invertible
s C0
Thus it follows from the proof of Lemma 2.5.10 that þnitely many zeros in
:
(4.13)
s 7! det(U2 (ÿs)) has only
+ C+ 0 , and they are all contained in C 0 .
4.2.
93
Main assumptions and a few useful consequences
V22 (ÿÿ)), det(M (ÿÿ)) and det (U2 (þÿÿ)) are contained in some ÿ1 V21 (iÿ)ÿ < 1, there exists a Since ÿV22 (iÿ) 1 ÿ ÿ > 0. ÿ ÿ1 V21 (iÿ)ÿ1 = 1 þ r. It follows from Lemma 2.5.7 0 < r < 1 such that ÿV22 (iÿ) a Æ1 > 0 such that Æ1 < ÿ and for any þ satisfying 0 < þ < Æ1 , ÿthat there exists ÿ ÿV22 (þþ þ iÿ)ÿ1 V21 (þþ þ iÿ)ÿ ý 1 þ r . Similarly it follows from Lemma 2.5.7 2 1 that there exists a Æ2 > 0 such that Æ2 < ÿ and for any þ satisfying 0 < þ < Æ2 , r kU1(þþ þ iÿ)U2(þþ þ iÿ)k1 ý 1 + 1 þ4 r = 1 þ1 r : 4 4 Let Æ := min fÆ1 ; Æ2 g, and ÿx a þ satisfying 0 < þ < Æ . Deÿne ý(ü; s) = det (üV21 (þþ þ s)U1 (þþ þ s) + V22 (þþ þ s)U2 (þþ þ s)) ; where ü 2 [0; 1]. The zeros of det(
half-plane
C+ ÿ ,
where
a. We know that
ý(0; ÿ) ý(1; ÿ)
V22 (þþ þ ÿ)U2 (þþ þ ÿ)) and = det (V21 (þþ þ ÿ)U1 (þþ þ ÿ) + V22 (þþ þ ÿ)U2 (þþ þ ÿ)) + . are meromorphic (in fact analytic!) in C ÿþ=2 + b. ý(0; ÿ) has a nonzero limit at inÿnity in C 0 : det(V22 (þÿ)) has a nonzero limit + + at inÿnity in C 0 and det (U2 (þÿ)) has a nonzero limit at inÿnity in C 0 (see =
det (
(4.13)).
ý(1; ÿ) has a nonzero limit at inÿnity in C +0 , U1 (þÿ) is proper in C +0 , and the above.
since
V21 (þÿ)
is strictly proper,
ü; s) 7! ý(ü; s) : [0; 1] ü iR ! C is a continuous function, and ý(0; i!) = det (V22 (þþ þ i!)U2 (þþ þ i!)) = det(V22 (þþ þ i! )) det (U2 (þþ þ i! )) ; and ý(1; i!) = det (V21 (þþ þ i!)U1 (þþ þ i!) + V22 (þþ þ i!)U2 (þþ þ i!)) :
c. (
d. We have
ý(ü; i!) = det(V22 (þþ þ i!)) det (U2 (þþ þ i!)) ÿ ÿ1 V21 (þþ þ i!)U1 (þþ þ i!)U2(þþ þ i!)ÿ1 þ det I + üV22 (þþ þ i! ) 6= 0;
since
ý ý
þþ þ iÿ ÿ1V21 þþ þ iÿ Uý1 þýþ þ iÿ U2 þþ þ iÿ ÿ1ýý1 ý þþ þ iÿ ÿ1V21 þþ þ iÿ ý1 ýU1 þþ þ iÿ U2 þþ þ iÿ ÿ1 ý1 þ þr < ; 4
ý ýüV22 ( ý 1 ýV22 ( h ri 1
2
)
(
)
1
1
(
1
)
)
(
)
(
(
)
)
(
)
94
Chapter 4.
Characterization of all solutions
V22 (ÿÿ ÿ i!)) 6= 0 and det (U2 (ÿÿ ÿ i!)) 6= 0.
det(
þ(ý; 1) = 6 0, since V21 (ÿþ) is strictly proper, U1(ÿþ) is proper + det(V22 (ÿþ)) det (U2 (ÿþ)) has a nonzero limit at inÿnity in C 0 .
e.
in
C+ 0 , and
Thus the assumptions in Lemma A.1.18 (Curtain and Zwart [27], page 570) are satisÿed by
þ(1; þ)
þ,
and hence it follows that the Nyquist indices of
are the same.
þ(0; þ)
and
Consequently, the number of zeros are the same (the
þ(0; þ), þ(1; þ) are analytic in C +ÿ Æ ) and so the 2 + sum of the number of zeros of s 7! det(V22 (ÿÿ ÿ s)) in C 0 plus the number of zeros of s ! 7 det (U2(ÿÿ ÿ s)) in C +0 equals the number of zeros of s! 7 det (V21 (ÿ+ÿ ÿ s)U1(ÿÿ ÿ s) + V22 (ÿÿ ÿ s)U2(ÿÿ ÿ s)) (= det(M (ÿ + s), using 4.12) in C 0 . In particular, we obtain that the number of zeros of s 7! det (V22 (ÿÿ ÿ s)) in + C 0 is less than or equal to l. But since the choice of ÿ can be made arbitrarily + small, it follows that s 7! det (V22 (ÿs)) has at most l zeros in C 0 . Thus m ÿ m V22 (ÿþ) 2 H1 l (C ). number of poles is zero, as
c
;
K0(ÿþ) := V12 (ÿþ)V22 (ÿþ)þ1 . From Lemma 2.5.5 and part 4 above, it follows that K0 (ÿþ) is an element in H1 l (C pÿm ). We have
5.
Deÿne
c
ÿ
G(i!) + K0 (i!) Im
þ
= =
with
! 2 R,
;
ÿ
þÿ
þ
Ip G(i!) K0 (i!) 0 Im Im þ ÿ þ ÿ Ip G(i!) V (i!) 0 0 Im V22 (i!)þ1 ;
and so we have (for notational convenience, we restrict writing out
the argument
i!)
G + K0)ý (G + K0) ÿ ü2Im þý ÿ þÿ þ G + K0 I 0 G + K0 p = Im 0 ÿüþ2Imÿ Im þý ÿ ÿ þ þÿ þ ÿ ý 0 0 I I 0 I p G p p G ý = 0 Im 0 ÿ ü 2 Im 0 Im V V22þ1 V22þ1 V þ þý ÿ þÿ ÿ 0 0 Ip 0 = þ 1 þ 1 0 ÿIm V22 ; V22 ÿ
(
V . Thus it follows that K0 (i!))uk2 ÿ ü2 kuk2 = ÿ ýV22 (i!)þ1 uý2 for all u 2 C m and all ! 2 R. Hence it follows that kG(iþ) + K0 (iþ)k1 ý ü . where we have used S3 and the ýdeÿnition of ý
k G i! (
(
)+
Thus from Corollary 4.1.5, it follows that
K0 (ÿþ) 2 H1 [l](C pÿm ). c
;
4.3.
95
All solutions to the sub-optimal Hankel norm approximation problem
V ÿÿ ; V22 þÿ þÿ 2 H1 ÿ V12 þÿ V22 þÿ MH1c 22 þs V22 þs þ þ 21 þs V12 þs I; þ 21 þÿ MH1c 22 þÿ V12 þÿ ; V22 þÿ V12 þÿ V22 þÿ þ1 V22 þÿ + We now prove that ( 12 ( ) c (C p m ): of 0 ( ) ;[l]
(
ÿ
(
MH1c
6.
K
)) is a right coprime factorization over
(
) and
(
) belong to
. More-
over,
where ÿ (
7.
(
(
)
(
)
) and
(
ÿ
(
(
)
ÿ
(
) belong to
From Lemma 2.5.6, it follows that det(
All
(
)=
.
Hence it follows that
)) is a right coprime factorization of
axis, and exactly
4.3
))
l
zeros in
(
(
)
(
)
.
)) has no zeros on the imaginary
C0 .
solutions
to
the
sub-optimal
Hankel
norm approximation problem In this section we obtain a nice parameterization of all solutions to the suboptimal Hankel norm approximation problem under the assumptions S1-S5 listed in the previous section.
kQ iÿ k1 ý
K (þÿ) := R1 (þÿ)R2 (þÿ)þ1
Theorem 4.3.1 Suppose that S1-S5 hold.
( )
then
and
R1 (þÿ) R2 (þÿ)
ÿ
þ
þ1 := ÿ(þÿ)
Q(þÿ)
, where
ÿ
2
c (C pÿm ) H1
Q(þÿ) ; Im
satisÿes
þ
(4.14)
c (C pÿm ) and kG(iÿ) + K (iÿ)k ý ÿ . K (þÿ) 2 H1 1 ;[l]
Proof in
1,
If
C+ 0,
Step 1: We show that
det(
V21 (þÿ)Q(þÿ) + V22 (þÿ)) has exactly l zeros
and
they are contained in the open right half-plane.
By Lemma 4.2.1.7, we know that
s 7! det(V22 (þs)) has no zeros on the imagiþ > 0 such that all its
C+ 0 . So there exists an zeros are contained in the half-plane C + ÿ. nary axis, and exactly
l
zeros in
c (C pÿm ). Thus, we V22 (þÿ)þ1 V21 (þÿ) 2 H1 ;l V22 (þÿ)þ1 V21 (þÿ), that there exists a Æ , 0 < Æ < þ, such that whenever 0 ý ý ý Æ , ý ý þ1 sup ýV22 (þý þ i! ) V21 (þý þ i! )ý < 1: From Lemma 2.5.5, it follows that
have from Lemma 4.2.1.2 and Lemma 2.5.7 applied to
Fix such a
ý > 0.
!2R
Consider
ü(û; s) := det (ûV21 (þý þ s)Q(þý þ s) + V22 (þý þ s)) ;
96
Chapter 4.
ÿ + C ÿ) ÿ2
where in
2
;
Characterization of all solutions
þ(0; ÿ) and þ(1; ÿ) are meromorphic (actually analytic + + containing C 0 with nonzero limits at inÿnity in C 0 .
[0 1]. The maps
on an open set
We have: 1. (
ÿ; s) 7! þ(ÿ; s) : [0; 1] þ iR ! C
is a continuous function.
þ(0; i!) = det(V22 (ýý ý i!)), and þ(1; i!) = det(V21 (ýý ý i!)Q(ýý ý i!) + V22 (ýý ý i!)). Im + ÿV22 (ýý ý i!)ÿ1 V21 (ýý ý i!)Q(ýý ý i!) is invertible, since
2.
3.
ÿ ÿÿV22 ( ý ÿ ÿ ÿV22 ( ý
ü
<
:
ý ý i! ÿ1V21 ýý ý i! Qÿ ýý ý i! ÿÿ ý ý i! ÿ1V21 ýý ý i! ÿ kQ ýý ý i! k )
(
)
(
)
(
)
)
(
)
1
s 7! det(V22 (ýý ý s)) has no zeros on the imaginary axis, þ(ÿ; i!) is nonzero for all ÿ 2 [0; 1] and ! 2 R.
Moreover, since it follows that
þ(ÿ; 1) is nonzero for all ÿ 2 [0; 1], since
4.
!1+
lim det(
2C 0
s s
ÿV21 (ýs ý ý )Q(ýs ý ý ) + V22 (ýs ý ý )) =
1
üm
6
=0
:
So the assumptions of Lemma A.1.18 (Curtain and Zwart [27], page 570) are satisÿed by
þ
and so the Nyquist indices of
þ(0; ÿ)
and
þ(1; ÿ)
are the same.
Consequently, the number of zeros are the same (the number of poles in each case is zero, since
þ(0; ÿ), þ(1; ÿ)
are analytic in
C + ÿ ). We already know from
ÿ2
ÿ = 0, det(V22 (ýÿýý )) has l zeros in the closed right half-plane C +0 . Thus the number of zeros of þ(1; ÿ) is l. But since ý can be chosen arbitrarily + small, det(V21 (ýÿ)Q(ýÿ) + V22 (ýÿ)) has exactly l zeros in C 0 . Finally, det(V21 (ýÿ)Q(ýÿ) + V22 (ýÿ)) þhas no zeros on the imaginary axis, since ÿ det I + V22 (ýi! )ÿ1 V21 (ýi! )Q(ýi! ) = 6 0 for all ! 2 R. that when
Step 2: We show that K (ýÿ) V22 (ýÿ))ÿ1 2 H1 l (C pþm ). c
V11 (ýÿ)Q(ýÿ) + V12 (ýÿ))(V21 (ýÿ)Q(ýÿ) +
:= (
;
V21 (ýÿ)Q(ýÿ) + V22 (ýÿ)) has a nonzero limit at inÿnity in C +0 , since Q(ýÿ) 2 MH1 is proper, V21 (ýÿ) is strictly proper, V22 (ýÿ) is proper + + in C 0 and det(V22 (ýÿ)) has a nonzero limit at inÿnity in C 0 . So by Lemma 2.5.4 and using Step 1 above, it follows that K (ýÿ) = ÿ1 is a well-deÿned element (V11 (ýÿ)Q(ýÿ) + V12 (ýÿ)) (V21 (ýÿ)Q(ýÿ) + V22 (ýÿ)) of H1 l (C pþm ). det(
c
c
;
4.3.
97
All solutions to the sub-optimal Hankel norm approximation problem
Step 3: We show that
kG(iÿ) + K (iÿ)k1 þ ÿ.
that det(V22 ÿWe know ÿ(ýÿ)) ÿV22 (i! )ÿ1 V21 (i! )Q(i! )ÿ < 1
has
no
zeros
! 2 R.
on
the
imaginary
axis
and
R2 (i!) = V21 (i!)Q(i!) + V22 (i!) is invertible for every ! 2 R. Consequently with s = i!, for all ! 2 R, for all
Thus
G + K )þ (G + K ) ý ÿ2 Im ý ü ý üý ü ý þ ü ÿ þ þ þ Ip G þ Ip 0 I Q p G ÿ1 ÿ 1 þ Q Im V = (R2 ) 0 Im 0 ýÿ 2 Im 0 Im V Im R2 ÿ1 þ þ ÿ1 = (R2 ) (Q Q ý Im )R2 ; (
where we have used S3. Thus
Step 4: We show that
kG(iÿ) + K (iÿ)k1 þ ÿ.
K (ýÿ) 2 H1 [l] (C pým ). c
;
Finally, from Corollary 4.1.5, and Steps 2 and 3 above, it follows that c (C p m ). ;[l]
ý
H1
Next we show that any solution
K (ýÿ) 2
K (ýÿ) 2 H1 [l](C pým ) to the sub-optimal Hanc
;
kel norm approximation problem has the form (4.14).
K (ýÿ) 2 H1 [l] (C pým ) kG(iÿ) + K (iÿ)k1 þ ÿ, then K (ýÿ) = R1 (ýÿ)R2 (ýÿ)ÿ1 , where Theorem 4.3.2 Suppose that S1-S5 hold.
ý
for some
R1 (ýÿ) R2 (ýÿ)
ü
ý
= ÿ(
c
If
ýÿ)ÿ1 Q(Iýÿ) m
;
and
ü
Q(ýÿ) 2 H1 (C pým ) satisfying kQ(iÿ)k1 þ 1. c
K0 (ýÿ) 2 H1 [l](C pým ) satisfy kG(iÿ) + K0 (iÿ)k1 þ ÿ and suppose it has the coprime factorization K0 (ýÿ) = NM ÿ1 over MH1 , where N and M + are in MH1 , M is rational, and det(M ) 2 R1 has exactly l zeros in C 0 and none on the imaginary axis. Deþne U1 and U2 as in the proof of Lemma 4.2.1.4.
Proof
c
Let
;
c
c
From (4.7) we obtain
ý
K0 (ýÿ) Im
ü
ý
ü
ý
ü
U1 (ýÿ) M (ÿ)ÿ1 = V (ýÿ) U1 (ýÿ) M (ÿ)ÿ1 ýÿ)ÿ1 U U2 (ýÿ) 2 (ýÿ)
= ÿ(
(4.15)
and so
K0 (ýÿ) Im
= =
V11 (ýÿ)U1 (ýÿ) + V12 (ýÿ)U2 (ýÿ)] M (ÿ)ÿ1 ÿ1 [V21 (ýÿ)U1 (ýÿ) + V22 (ýÿ)U2 (ýÿ)] M (ÿ) : [
98
Chapter 4.
Characterization of all solutions
Thus
ÿ
0 (ÿ )
K
þÿ þÿ 12 þÿ 11 þÿ þÿ 12 þÿ 1 þÿ 2 þÿ ÿ1 1 þÿ 2 þÿ
= =
where
Q(
þÿ
) :=
)Q(
)+V
(
)) U2 (
(V
)Q(
)+V
(
)) (V
þ
RM
and
M
SN
=
I.
(
(
U
)U (
Next, we show that Since
U
(
)
)M ( )
(
.
) and
U
(
) are right coprime over
Now it is readily veriÿed using
þÿ 1 þÿ þÿ 1 þÿ
) + V12 (
)U (
þÿ 2 þÿ þÿ 2 þÿ
) + V22 (
)U (
) + V22 (
)Q(
are right coprime, there exist
N
11 ( 21 (
V V
þÿ ÿ ÿ1 21 þÿ þÿ
(V11 (
V
and
þÿ þÿ þÿ ÿ ÿ
R
(
S
)þ(
)U (
)
=
K(
)U (
)
=
M(
in
)=
þÿ
))
M M
1. 1 such that
H
H
c
c
that
I
)M ( ) =
ÿ1 ;
N(
ÿ
)
(4.16)
):
(4.17)
Consider now
þ ÿ 12 þÿ ÿ ÿ 21 þÿ 1 þÿ ÿ ÿþ ÿ ÿ
[
S(
)V
= R( )[V
(
)U (
= R ( )M ( ) =
þÿ 2 þÿ þ ÿ 11 þÿ þ ÿ 22 þÿ 2 þÿ þ ÿ 11 þÿ 1 þÿ
) + R( )V22 (
(
S(
)+ V
)] U (
(
)
)U (
)]
S(
(
) [V
(
)
R(
)U (
þÿ 1 þÿ 12 þÿ 2 þÿ
)V21 (
)+ V
(
)] U (
)
)U (
)]
)N ( )
I:
þÿ
1(
Thus
U
) and
2(
U
þÿ
) are right coprime over
þÿ 2
Finally, we show that has no zeros in
C+ 0.
Q(
)
H
det (V
(
ÿ >
1. c
c
det (U (
s)U
(
+
s))
ÿ
+V
det(V
s
s))
ÿ
s)
ÿ
, that is,
det (U (
s
s
!7 2 þ U
(
s)
Proceeding as in the proof of Lemma 4.2.1.4, we have
s
ÿ
using (4.17)) in of zeros of
H
!7 22 þ þ + 7! 2 þ þ C0 21 þ þ 1 þ þ 22 þ þ 2 þ þ l den l den C+ 0 + !7 þÿ C0 2þ þ
number of zeros of
7!
M
1 (C pÿm ), by showing that
that the sum of the number of zeros of s
[S ( )V
(
(Q) =
in
ÿ
s))
in
C+ 0 plus the
equals the number of zeros of
in
s)U
ÿ
(
(
s))
ÿ
, where
. Thus
(= det(M (ÿ + s),
(Q) denotes the number
Q(
C+ . But ÿ H (C p m ).
) has no poles in
0 can be chosen arbitrarily small, which implies that
Q(
þÿ 2 )
1 ÿ c
Consequently, our main result is the following:
Theorem 4.3.3 Suppose that S1-S5 hold.
k ÿ G(i
ÿ k1 ý
) + K (i )
for some Q(
þÿ 2 )
þÿ 1 þÿ 2 þÿ
þ iÿ K (
ÿ
R
(
)
R
(
)
1 (C pÿm )
H
c
)=
þ
þÿ ÿ þÿ þ1 þÿ k ÿ k1 ý )R
= þ(
)
2 (þÿ)þ1
1(
R
satisfying
Q(i
þÿ 2
Then K (
, where
Q(
)
Im
)
)
1.
þ
1 [l](C pÿm )
H
c
;
and
4.4.
99
A comparison with existing results
4.4
A comparison with existing results
It is appropriate to compare the results in Theorem 4.1.4 and Theorem 4.3.3 of this chapter with a corollary to the abstract results from Ball and Helton [5] as stated in Curtain and Ran [17]. Note that for an easier comparison we have chosen the version which is closer to our main result. Denote by ^(C p m ) ^ h the set h L1 ([0; ); C p m ) . ^l (C p m ) denotes the set of complex p m
n
j 2
1
matrix-valued functions
ÿ
þ
X(
o
A
A
ÿ
ÿ
ÿ
) of a complex variable deÿned in the closed right
half-plane with a decomposition
X
=
þ+
G
F,
þ is the matrix transfer
where
G
function of a system of MacMillan degree at most equal to l, with all its poles ^(C p m ). The following result follows in the open right half-plane, and F
2A
from Ball and Helton [5].
Theorem 4.4.1 Let G G. Then
2A
^
and let ÿk 's denote the Hankel singular values of
k þ þ k1 ýþ 2 A C ÿ ýþ 2 k þ þ k1 ü ýþ ýþ ýþ ý ÿ ýþ þ ýþ ý ÿ ýþ þ ýþ inf
K (ý )2A^ (C ÿ ) p
l
A
ÿ
m
G(i
If ÿl+1 < ÿ < ÿl , then there exists a
^ (C p m ) l
satisÿes
G(i
ýþ 2 Aÿ A
(4.4).
) + K (i ) R1 (
)
R2 (
)
) + K (i )
þ(
ÿ iþ K (
= þ(
=
)
1
)=
Q(
R1 (
such that K ( ) ) 1 , where
)R2 (
)
Im
k ýþ k1 ü ýþ 2 A C
ÿ
ÿl+1 :
^( (p+m) (p+m) )
)
^(C p m ) satisfying Q(i ) ) ^(C (p+m) (p+m) ) such that þ( ) 1
for some Q( element in
ÿ
ýþ
1. Moreover, þ( ) is any ^( (p+m) (p+m) ) and it satisÿes
ÿ
Notice that our result is for a more general class of transfer functions G in ( ; C p m ) with limit 0 at i , although we cannot exclude the possibility
CR
û1
ÿ
that our result could be deduced from the abstract results of Ball and Helton [5].
Further comparison reveals that our Theorem 4.3.3 only gives suÆcient
conditions, whereas Theorem 4.4.1 gives necessary and suÆcient conditions. ^ the necessity part is fairly easy to show using Theorem For the class G
2A
2.3 of Iftime and Zwart [47] (see also the elementary proof of part
)
b.
c.
of
Theorem 3.1 of [47]). This elementary proof relies basically on the existence of a spectral factorization, which is well-known for the Wiener class (see Clancey and Gohberg [10]). The major diýerence between Theorem 4.4.1 and our main result is that we have used elementary transparent proofs in contrast to the abstract methods in Ball and Helton [5] which subsequently need to be translated to this problem.
Chapter 5
State space solutions In this chapter, w e will sho w that for the Pritc hard-Salamon class and the analytic class, there do exist
J ÿspectral
factors for eac h class satisfying the
assumptions in Chapter 4, hence solving the sub-optimal Hankel norm approximation problem for these classes of inÿnite-dimensional systems. We will give an explicit formula for a
J ÿspectral
factor þ in terms of the state-space pa-
rameters (A; B; C ) of the original system and we will chec k that it has all the properties S1-5 required in Chapter 4.
J ÿspectral
5.1
factorization for the Pritchard-
Salamon class In this section, we show the existence of a solution to the key
J
ÿspectral fac-
torization problem (satisfying S1-S5 in Chapter 4) for the smooth Pritc hardSalamon class of exponentially stable inÿnite-dimensional systems with ÿnitedimensional input and output spaces, in terms of the system parameters C.
A, B ,
We list the assumptions below:
P1. ý(A; B; C ) is an exponentially stable, smooth Pritchard-Salamon system. P2.
U
P3.
ÿl
=
C m,
Y
=
C p.
+1 < ÿ < ÿl .
In Curtain and Ran [17] the sub-optimal Hankel norm problem was solv ed for the
A. Sasane: Hankel Norm Approximation for Infinite-Dimensional Systems, LNCIS 277, pp. 101−108, 2002. Springer-Verlag Berlin Heidelberg 2002
102
Chapter 5.
Pritchard-Salamon class by ÿnding a solution to the problem considered in S3 with
G(s)
=
C (sI
ÿ
A)
State space solutions
J ÿspectral
ÿ1 B .
factorization
However, there the
starting point was to quote a result from Ball and Helton [5], which states that the sub-optimal Hankel norm approximation problem is equivalent to solving the
J
ÿspectral factorization problem in S3. Then a solution is constructed from
a given realization of
G.
However, if one looks for the result quoted from [5],
one realizes that this is not an obvious corollary of the very abstract and general theory in [5]. This motivated the self-contained proofs of the sub-optimal Hankel norm approximation problem in Chapter 4. Here we use just the results from Chapter 4 and verify that all the assumptions are satisÿed. The sub-optimal Nehari problem is a special case of the sub-optimal Hankel K( ) H (C pÿm ) satisfying G(i ) +
ÿÿ 2 k ÿ ÿ)k þ ÿ for a given ÿ > kþk. In the case that ÿ > kþk, a solution ý to the J ýspectral factorization problem S3 was constructed in Curtain and Zwart norm approximation problem; one seeks K (i
1
1
[26] for the Pritchard-Salamon class.
ÿl+1 < ÿ < ÿl , but The proof of the following
For the sub-optimal Hankel norm problem we have nonetheless, the ý factor is precisely the same.
lemma is analogous to that of Lemma 2.11 in Curtain and Zwart [26]:
Lemma 5.1.1 Suppose that ü(A; B; C ) satisÿes P1-P3 and let G(s) = C (sI þ
A)
1B.
ý
Then
ÿ 0
ü
A ;
þ
C
ý
CB
0
L
ü
1
ý 2 ÿ
;
ýC LB ÿB
û
ü
ÿ 0
N ;
0
Ip
0
0
ÿIm
ûú
is an exponentially stable, smooth Pritchard-Salamon system and its transfer operator is
ý(
ü
ÿ=
ù
I
û
0
Ip
ý(s) =
N
ýÿ), where
0
+
ÿIm
ø ý ÿ12 LB LC 1 þ
ÿ
0
ü
A ;N
ÿ 0
þ
1
ü
2 ÿ
ÿB
and s
C
ýC LB þ
1þ
ÿ (sI + A )
N
0
2 C0 .
1 ÿ LC B
0
û 0
0 þ
C
0
CB
L
ý
(5.1)
;
Moreover,
ý
;
1 ÿ
2
ü
ýC LB ÿB
û ü ;
0
Ip
0
0
ÿIm
ûú
is an exponentially stable, smooth Pritchard-Salamon system and its transfer operator is V
ü V
(s) =
Ip
0
ýÿ), where
(
0
1 ÿ Im
û
1
ý 2 ÿ
ü
ýC LB B
0
û
1N þ ÿ
0 þ
(sI + A )
0
C
0
1 ÿ LC B
ý
;
s
2 C0 :
Furthermore, 1.
ýÿ) 2 MH c
ý(
1
is invertible in
MH c , and ý(ýÿ) 1 þ
1
is equal to V
ýÿ).
(
þ
J ÿspectral factorization for
5.1.
2. ÿ given by (5.1) is a solution of the in S3.
!1+ ÿ(ÿs) =
ÿ
3. lim s
2C 0
s
Ip
0
0
ÿIm
103
the Pritchard-Salamon class
J ÿspectral
factorization problem (4.4)
þ
.
Remarks:
1. Note that while it is algebraically straightforward to verify that ÿ(s) solves the
J
ÿspectral factorization problem, it is important to justify the well-
posedness of each term at all steps. Use is made of the Lyapunov equations from Lemma 2.3.1, but to make sense of all terms, it was necessary to make use of the smoothing properties of
and
LB
LC
from Lemma 2.3.1.
Of course, these smoothing properties are only known to hold for smooth Pritchard-Salamon systems. 2. If
jjþjj, it was shown in Curtain and Zwart [26] that ÿ11 (ÿþ) belongs to MH1 , is invertible in MH1 and ÿ11 (ÿþ)ÿ1 is the transfer operator of an exponentially stable smooth
ÿ >
(a) (b) (c)
c
c
Pritchard-Salamon system.
In the case that
ÿ11 (ÿþ)ÿ1
ÿl
+1 < ÿ < ÿl , ÿ11 (ÿþ)ÿ1 still exists, but it is not stable:
is the sum of a stable part in
rational part with at most
l
1 (C pþp )
H
unstable poles.
c
and an antistable
Finally we give the main result of this section.
ÿ1
Suppose that ý(A; B; C ) satisÿes P1-P3 and let Let ÿ be given by (5.1). Then we have:
Theorem 5.1.2 A)
B.
ÿþ) 2 H1 l (C pþm ) ÿ1 , where R1 (ÿþ)R2 (ÿþ) c
K(
;
ÿ
for some Proof
satisÿes
1 (ÿþ) R2 (ÿþ) R
þ
kG(iþ)
+
ÿ1 = ÿ(ÿþ)
þ)k1 ý
K (i
ÿ
Q(
ÿþ)
ÿ
G(s)
iþ
=
C (sI
ÿþ)
K(
ÿ =
þ
Im
ÿþ) 2 H1 (C pþm ) satisfying kQ(iþ)k1 ý 1.
Q(
c
The assumptions S1-S5 in Chapter 4 hold. S1 is satisüed since
G
is the
transfer operator of an exponentially stable smooth Pritchard-Salamon system (see 2.12) and S2 holds. Finally, Lemma 5.1.1 shows that S3-S5. Now the result follows from Theorem 4.3.3. Example 2.3.4 (continued) All the results in this section apply to the Example
2.3.4 considered earlier in Chapter 2.
ÿ
104
Chapter 5.
J ÿspectral
5.2
factorization
for
State space solutions
the
analytic
class In this section, we show the existence of a solution to the key
J ÿspectral
fac-
torization problem satisfying S1-S5 in Chapter 4 for the exponentially stable analytic class of inÿnite-dimensional systems discussed in Section 2.4 of Chapter 2 with ÿnite-dimensional input and output spaces: throughout this chapter,
ÿ
we consider the class of systems described by the triple ( A; B; C ) satisfying m p (see page 26). We list all the assumptions U = C , Y = C
A1, A2, A3 with below:
A1.
ÿ
A
is the inÿnitesimal generator of an exponentially stable analytic
semigroup
fT (t)gtÿ0 on the Hilbert space Z .
A2.
B
2 L(C m ; Zÿ ), where ÿ is a ÿxed number in (ÿ1; 0].
A3.
C
2 L(Z; C p ).
A4.
þl
+1 < þ < þl .
We construct a solution þ to the
J
ÿspectral factorization problem (??) which
has the properties S3-S5. The spectral factor þ is the same one as in Curtain and Ichikawa [14] (in which they consider the Nehari problem for a similar class with some diýerences; see also Table 2.1), but for the sub-optimal Hankel norm
approximation problem we have þl+1 < þ < þl . Extensive use of the results of Section 2.4 is made to show that all the conditions in Chapter 4 are satisÿed. The candidate solution is described by the quadruple
ÿ
ÿA ; 0
þ
C
0
L
ý
CB
1
;
þ
ü
2
CL
B
ÿþB
û
ü
þ 0
N ;
0
ûú
Ip
0
0
þIm
:
First we show that this deÿnes a regular linear system (see Section 2.1), and that its transfer function is invertible.
ÿA; B; C ) satisfy A1-A4 and let G(s) = C (sI +
Lemma 5.2.1 Let the triple (
1 B . Let ý be a ÿxed number satisfying ÿ1 < ý < ÿ. ÿý ÿ 21 < ü < 21 . Then A)
ÿ
ü1 = and
ü2 =
ÿ
ÿA ; 0
þ
C
ÿ
ÿA ; Nþ 0
0
0
þ
C
L
C
0
B
ý
;
1 þ
2
ü
CL
B
ÿþB
0
Choose any
û
ü
satisfying
ú þ 0
N ;
0
;
ü û ú 1 LC B ý ; 1 C LB ; 0 þ 2 ÿB þ 0
(5.2)
(5.3)
J ÿspectral factorization for
5.2.
105
the analytic class
C p+m ,
ÿ 0
are regular linear systems with state space Z , input space p+m . Denote the transfer function of ÿ by space C 1
ÿ) =
G1 (
ÿ
1 ÿ
þ
B
CL
þÿB
2
N
0
þ (ÿI + A )
0 ÿ1
0
ý
C
ü
CB
0
L
2 Hc
and output
û
C (p+m)þ(p+m)
1
ú :
(5.4) Then we have
!1 G1 (s) = 0:
lim
s
(5.5)
s C+ 2
Furthermore, if
þs) :=
ÿ
þ(
then
c
H1
ù
0
0
ÿIm
þÿ
þ( ) is an element in ø C (p+m)þ(p+m) and its inverse
V
and G2
C (p+m)þ(p+m) ÿ
1
G2 (s)
for all s
Proof
þÿ) =
=
ÿ
(
ù
2 Hc
1
ø
+ G1 (s)
c
H1
for all s
ù (p+m)þ(p+m) ø C ,
is given by V
þ
0
Ip
it
þÿ), where
B
is
þB
invertible
over
(
þ
0
(5.6)
þ G2 (ÿ);
1
þ Im
0
2 C +0 ;
ÿ2 :
denotes the transfer function of
CL
2 ÿ
þ
Ip
0 ÿ1
(sI + A )
N
þ 0
ý
C
0
þ LC B 1
ü
(5.7)
2 C +0 .
BûNþ , úwe ýrst note by appealing to Lemma 2.4.8 that Nþ 2 L (Zý ) and Nþ 2 L Z ý are well-deýned for any ù ø þ satisfying þ1 < þ < ý. From Lemma 2.4.1.2, it follows that LB 2 L Z ÿ ; Z ú û for ü < 1 + 2ý, and so LB Nþ 2 L Z ý ; Z for þ < 1 + 2ý. Here we remark that if þ satisýes þ1 < þ < ý, then in particular þ < 1+2ý. Since B 2 L (C m ; Zý ) for ú û þ1 < þ < ý, we have B Nþ 2 L Z ý ; C m . Furthermore, if þ1 < þ < ý, then using Lemma 2.4.2.2, we obtain that LC 2 L (Zý ; Z ), and so LC B 2 L (C m ; Z ). To make sense of the terms
0
þ
B N
0
and
0
L
0 ÿ
0
0 ÿ
0
0
Consequently, if
ý
C
0
We now ýx a possible, since
L
ü
0 ÿ
0
0 ÿ
þ1 < þ < ý, then we have:
CB
ü
2
ù ø L C p+m ; Z ;
satisfying
þ1 < þ .
þþ þ 21
ÿ
CL
B
þÿB
< ü <
1 . 2
þ
þ 2L
N
0
0
ù
0
ý
Zÿ ;
C p+m
þ1 < ý < þ
1 2
;
Case 2 :
ý
=
:
We remark that such a choice is
The freedom in the choice of
ü
is depicted in Figure 5.1,
where we have one of the possible regions depending on the value of
Case 1 :
ø
þ
1 2
;
Case 3 :
þ
1 2
< ý
ý.
ý 0:
106
Chapter 5.
ÿ
State space solutions
ÿ
ÿ
1 1 2
ÿý ÿ 12
ÿ1 ý
þ
0
þ
0
Case 1
þ
0
Case 2
Case 3
Figure 5.1: The possible cases.
B
From Theorem 2.4.4 with
C
= 0 and
ÿ
ÿ
=
ÿ
(which satisÿes the condition
þ
B ÿ ÿC < ÿB + 1), it follows that þ1 is a regular linear system on Zÿ with a transfer function G1 (þ) given by (5.4), which satisÿes lim s!1 G1 (s) = 0. 0
ÿ
s C+ 2
Similarly, it can be seen, again using Theorem 2.4.4, that þ2 is a regular linear
system on
ÿ
Z
0
with a transfer function
Finally we have to show that ý(
ýþ)V (ýþ) = Ip+m = V (ýþ)ý(ýþ).
ýs)V (ýs) ý
ÿ
ý( =
ÿ
1
2 ý 1
1
ý 4 ý
ÿ
ý
C
CL
1 þ LC B
0
Im
0
B z ý LC BB z 0
= =
in
0
ý 1.
Zÿ ÿ
þ
ý ý A ) 1 Nþ
ÿ
0
C
ý
1ý
0 ÿ
0
0 ÿ
CL
B
ýýB
0
C
= z
0
C CL
2
0
C
C
ü
1 þ LC B
0
0
1 þ LC B
ü
0
1 þ LC B
ü
1 þ LC B
0
þ
0
holds 0
1N
þ (sI + A )
Furthermore, it can be veriÿed that for
C CL
1ý
0 ÿ
þ (sI + A )
N
0
Consider
þ
þ
(sI
ü
0
0 ÿ
þ
0
Ip
(sI + A )
0
B
ýýB B
þ
B
ýýB
ÿ
CL
0
N
0
CL
ýýB We have
þ
B
ýýB
ý 2 ý ÿ
CL
2 (þ) given by (5.7).
G
ü
:
B ý LC BB :
ý,
Zÿ
0
where
ý1
< þ < ÿ,
there
C LB z ý LC LB A z 1 (sI + A ) z ý ý2 (sI + A ) (N ) 1 z 2 ý (Nþ ) þ 0
0
A L
0
ÿ
0
0
0
ÿ
This can be checked by using the Lyapunov equations (2.17) and
J ÿspectral factorization for
5.2.
107
the analytic class
(2.25), and Lemmas 2.4.1.2 and 2.4.2.2. Consequently, we obtain
ÿ
ÿ(
s )V
ÿ
ÿ )ÿ
(
s
0
þ
0
Ip
Im
ÿ
1
þ
C LB
ÿ
2 ÿ
=
ÿB
ÿ
ÿ 12 ÿ ÿ ÿ 14 ÿ
0
Nÿ
0
C LB ÿB
ÿ
ÿB
ÿ
h
ÿ
2 (N
(sI
ÿ
0
ÿ
)
1N
0 ÿ
(sI + A )
0
ý
1L
0
CB
ÿ
C
0
1L
(sI + A )
0
ý
ÿ
ü
1
0
1N
CB
ÿ
1 (sI + A ) ÿ ÿ2 (sI + A ) (N
)
ü
0 ÿ
0
Nÿ
0
C
ÿ
þ
ÿ
0 ÿ
A
þ
0
C LB
1ý
0 ÿ
(sI + A )
C
0
1L
0
CB
ÿ
ü
0
ÿ
)
1i
ÿ
= 0:
Remark: In several places in this section we need to verify various identities
which, taken algebraically, are easy to verify. However, when dealing with unbounded operators, it is necessary to check at every step that all the terms are well-deþned and in the correct spaces. We do this by making extensive use of Lemmas 2.4.1, 2.4.2 and Corollary 2.4.5. We have made this explicit in the þrst part of this proof of Lemma 5.2.1.
ÿ
Suppose that the triple ( A; B; C ) satisÿes A1-A4 and let ÿ1 A) B. If ÿ is given by (5.6), then ÿ is a solution to the spectral factorization problem (4.4) in S3.
Lemma 5.2.2 G(s) J
ÿ
=
Proof
C (sI
+
This is proved by a tedious but straightforward substitution, again
verifying that all the terms are well-deþned at each step. For a step by step calculation bounded Remark:
stable:
B
and
C,
In the case that
ÿ11 (þÿ) 1 ÿ
see Curtain and Zwart [27].
ÿl
+1
1
ÿÿ)
ÿ11 (
< ÿ < ÿl ,
ÿ
exists, but it is not c pþp ) and H1 (C
can be written as the sum of a stable part in
an antistable rational part with at most
l
unstable poles.
Finally we give the main result of this section.
þ
Suppose that ( A; B; C ) is a triple satisfying the assumptions A1-A4 listed at the beginning of this section and let G(s) = C (sI + A)ÿ1 B . Let ÿ be given by (5.6). Theorem 5.2.3
þÿ) 2 H l (C p m ) 1 , where R1 (þÿ)R2 (þÿ) K(
c
1; ÿ
þ
ÿ
satisÿes
1 (þÿ) R2 (þÿ) R
kG(iÿ)
þ
+
þÿ) 1
= ÿ(
ÿ
ÿ)k
K (i
ÿ
þÿ)
Q(
Im
ý
1
þ
ÿ
iþ
þÿ)
K(
=
108
Chapter 5.
for some
Proof
Q(ÿÿ) 2 H1c (C pÿm )
satisfying
State space solutions
kQ(iÿ)k1 þ 1.
The assumptions S1-S5 in Chapter 4 hold. S1 is satisÿed (see Theorem
2.4.4) and S2 holds. That S3, S4 and S5 hold follows from Lemma 5.2.1 and Lemma 5.2.2. The result now follows from Theorem 4.3.3.
Example 2.4.10 (continued) We remark that all the results in this section
apply to the Example 2.4.10 considered earlier in Chapter 2.
ÿ
Chapter 6 The non-exponentially stable case
In the previous chapters the sub-optimal Hankel norm approximation problem w as solv ed for arious v classes of inÿnite-dimensional systems under the assumption that
A generates an exp onentially stable strongly contin uous semigroup.
How ever, there exists an important class of systems with a transfer function (C p m ), for which does not generate an exponentially stable semi-
G 2 H1 ÿ
A
group (see Examples 3.1.1, 3.1.2 and 3.2.4). In [37], approximating solutions to the optimal Hankel norm approximation problem were obtained without assuming exponential stabilit y,but only for the case that the Hankel operator is n uclear, which is a strong assumption.
In this chapter, w e will solv e the
sub-optimal Hankel norm approximation problem for non-exponentially stable inÿnite-dimensional systems in terms of a solution to the sub-optimal Hankel norm approximation problem for an exponentially stable system which is obtained by shifting the generator of the semigroup of the original system. The outline is as follows. In section 6.1 we will prov e a useful lemma which will be used later and in section 6.2 we will introduce our class of terms of assumptions on
systems in
h and its Fourier transform, that is, without the use
of a state-space realization. We will prov e a crucial conv ergence result for this class and then prov e our main Theorem 6.2.4. Finally in the last section 6.3, we will apply the results from the previous section and the results from Chapter 5 to obtain state-space solutions to the sub-optimal Hankel norm approximation problem for non-exponentially stable inÿnite-dimensional state-linear systems and give an illustrative example.
A. Sasane: Hankel Norm Approximation for Infinite-Dimensional Systems, LNCIS 277, pp. 109−117, 2002. Springer-Verlag Berlin Heidelberg 2002
110
Chapter 6.
6.1
The non-exponentially stable case
A few useful results
First we will prove a few elementary results which will be used later in this chapter. The key to the proof of the new result (Theorem 6.2.4) is Corollary 6.1.2 which is an easy consequence of the following lemma.
G 2 H1 (C pÿm ) and ! 7! G(i!) : R ! C pÿm is uniformly ÿ > 0, there exists a Æ > 0 such that sup kG(i! ) ÿ G(þ + i! )k < ÿ for all þ satisfying 0 þ þ þ Æ:
Lemma 6.1.1 If
continuous, then given any
!2R Proof
It follows from Theorem 5.18 (page 96, M. Rosenblum and J. Rovnyak
[69]) that
G(þ + i!) = Since for
þ > 0,
Choose a
ü > 0 such that
þ ý
1 G(it) dt; þ1 (t ÿ !)2 + þ 2
Z
þ > 0:
1 1 dt = 1; þ1 (t ÿ !)2 + þ 2 and we have ÿ Z 1 ÿ ÿþ G(it) ÿ kG(i!) ÿ G(þ + i!)k = ÿ ý þ1 (t ÿ !)2 + þ 2 dt ÿ G(i!)ÿÿÿ ÿ ÿ Z 1 ÿþ G(it) ÿ G(i!) ÿÿ ÿ dt : = ÿ ý þ1 (t ÿ !)2 + þ 2 ÿ þ ý
Z
kG it ÿ G i! k < ÿ ( )
Now choose a
Thus
(
)
2
for all
t
Æ > 0 such that for any þ ÿ Z ÿþ ÿ ÿ ÿ ý Rn[!þÿ;!+ÿ] (t
ÿ
and
!
jt ÿ !j < ü:
satisfying
satisfying 0
ÿ ÿ 1 ÿ< dt ÿ !)2 + þ 2 ÿ
þþþÆ
, we have
ÿ
4
kG iý k1 : ( )
kG i! ÿ G þ i! k ÿ ÿ Z þ ýþ !þÿ kGt itÿ !ÿ2G i!þ 2 k dt ýþ Rn[!þÿ;!+ÿ] kGt itÿ !ÿ2G i!þ 2 k dt Z Z þ þ þ ! +ÿ 2 þ ý !þÿ t ÿ ! 2 þ 2 dt ý Rn[!þÿ;!+ÿ] t ÿkG! i2ý k1þ 2 dt þ ÿ kG iý k1 kG ÿiý k1 ÿ: (
=
)
( +
)
ÿ Z 1 ÿ ÿþ G(it) G(i!) ÿÿ ÿ dt ÿý (t ! )2 + þ 2 ÿ þ1 Z !+ÿ ( )
(
(
2
1+2
(
) +
( )
)
) +
4
( )
+
(
2
+
( )
(
=
(
) +
( )
) +
)
6.2.
111
The class of systems and the main result
Since the choice of
! is arbitrary, this completes the proof.
G(iÿ) : R ! C pÿm has limits Gþ1 at þ1, then given any ÿ > 0, there exists sup! 2R kG(i! ) ý G(þ + i! )k < ÿ whenever 0 ü þ ü Æ . G 2 H1 (C pÿm )
Corollary 6.1.2 If
Proof
Given any
and
sup
!2(ý1;M1 ]
[
Æ>
0
such that
ÿ > 0, there exist M1 > 0 and M2 > 0 such that sup kG(i!) ý G1 k < ÿ ; and !2[M2 ;1)
Moreover, since
is continuous and a
G(iÿ)
:
R
kG i! ý Gý1 k (
)
(6.1)
2
ÿ
<
2
:
(6.2)
!C ÿ
p m is continuous, it is uniformly continuous in
M1 ý 1; M2 + 1], and so given any ÿ > 0, there exists a Æ such that 1 > Æ > 0 !1 and !2 lie in [M1 ý 1; M2 + 1] and j!1 ý !2 j < Æ,
and whenever
kG i! ý G i! k < ÿ: (
1)
(
2)
(6.3)
Thus it follows from (6.1), (6.2) and (6.3) that whenever
R and
j! ý ! j < Æ 1
2
, then
kG i! ý G i! k < ÿ (
1)
(
2)
!1 and !2 belong to G(iÿ) : R ! C pÿm is
. Hence
uniformly continuous, and so the result follows from Lemma 6.1.1.
6.2
The class of systems and the main result
The speciÿc class of systems we consider in this chapter is deÿned below:
B1.
1. 2.
h 2 L2 ([0; 1); C pÿm ). G(iÿ) 2 C (R; C pÿm ) and has a unique limit G1 at þ1, where G := F (h), where F denotes the Fourier(-Plancherel) transform.
B2. There exists a 1.
G ( þ + ÿ)
Æ > 0, such that for every þ
2 H1 Cÿ ÿ ÿ ý û ý û ::: ÿ Gþ ÿ ý <ý<ýÿ ÿ ÿ ÿG þ iÿ K ÿ iÿ ÿ ü ý 1
singular values [ ] l+1
satisfying 0
< þ ü Æ,
( p m ) has a compact Hankel operator þ[ÿ ] , with [ ] 1
[ ] 2
, and
[ ]
l , then ( + ) has a solution to the sub-optimal [ÿ ] Hankel norm approximation problem: ( ) ;l (C p m ) such [ ] . that ( + )+ ( )
2. if
K
ýÿ 2 H1
ÿ
112
Chapter 6.
We remark that 2 of B1 implies that
The non-exponentially stable case
G(iÿ) 2 L1 (R; C pÿm ).
Consequently, from
Proposition 8 (page 224, Keulen [83]) it follows that ÿ is a bounded operator, (C p m ) and ( [ ( )]) ( ) = [ (ÿ )] ( ) for almost all R, where
G 2 H1 ÿ HG F u i! F u i! HG is the frequency domain Hankel operator with symbol G.
!2
Furthermore, we
note that from 2, by applying Theorem 3.0.11 (in Chapter 3) we obtain that
the Hankel operator with symbol
G is compact.
We will now prove a few properties of the class of systems that we consider which will be used in the proving our main Theorem 6.2.4. If 0
<ÿþÆ
, then the c time-domain Hankel operator corresponding to ( + ) (C p m ), that is, ÿ with kernel [ÿ ] ( ) = () ) C p m )), is denoted by ÿ[ÿ ] , and the lth 1 ([0 [ÿ ] Hankel singular value is denoted by l . We note that since ) C p m) 2 ([0 ÿ and ), it follows from the Cauchy-Schwarz inequality that [ÿ ] is 2 (0 integrable.
ÿ
h
eþ
eþ
ÿ 2L
h
2 L ;1
þ
;1 ;
ÿ 2 H1 h2L
Gÿ
ÿ
ÿ
;1 ;
h
ÿ
Lemma 6.2.1 If an inÿnite-dimensional system satisÿes the assumptions B1 and B2, then
1.
ÿ
ÿ[ÿ ]
þü
ÿ[ÿ ]
2. Given any
1.
Let
operator by eþ Proof
ÿt
! ü l2N
ÿ ÿ
þl[ ] ! þl ÿ
ÿ ! 0.
as
L2 ([0; 1); C m ) ! L2 ([0; 1); C m ) If u 2 L2 ([0; 1); C m ), then
S1[ ] ÿ
:
,
ÿ ! 0,
uniformly as
:
h
i
S1[ ]u (t) = eþ u(t)
eþ
ÿt
strongly as
Proof
for all
t ý 0:
ÿ
h
S1[ ] = S1[ ]
ÿ ! 0.
ÿ
6 u2L
Let 0 =
2 ([0;
ÿ
iü
1 ;C )
m strongly as
m ). Given any
1
Z
!I
( )
2
sup
2[0
t
;M
1
h
ÿ
iü
!
ý > 0, choose a M > 0 such that 2
implies that
ÿt
]
ÿ
2
Æ > 0 such that 0
ÿ
ÿ ! 0, and S2[ ] = S2[ ]
ku t k dt < ý : þÿ<Æ ý þ ý ýe ü ý < p ý kuk :
M
Now choose a
ÿ
ÿ
Sublemma 6.2.2
Ip
ÿt
S2[ ] : L2([0; 1); C p ) ! L2 ([0; 1); C p ) be the multiplication oper[ ] [ ] . Then it can be checked that ÿ[ ] = S2 ÿS1 .
Similarly, let ator by
ÿ
be the multiplication
2
2
6.2.
113
The class of systems and the main result
Thus
ÿ ÿ2 ÿ [ÿ ] ÿ ÿS1 u ÿ uÿ Z 1 ÿþ ÿ ÿ eÿÿ t ÿ 1ý u(t)ÿ2 dt
=
0
"
ÿ
sup t2[0;M ]
ÿ Hence
[ÿ ]
=
strongly as
þ
Since ÿ[ÿ ]
þ
ý
þ
h
ÿÿt
1
+
2
S1
[ÿ ]
i
!
þ
Im
0
2
2
u(t)
2
ÿ
dt
2
Z dt
!
þ
ÿÿ t
M
=
2
ü üe
1
+
2
ÿ
+
strongly as
[ÿ ]
i
þ
S1
þ üü k k 1
2
u(t)
2
dt
2
ÿ :
[ÿ ]
0. Similarly,
ÿ
[ÿ ]
[ÿ ] [ÿ ] [ÿ ] ÿþ S2 S2 ÿS1 , we have
=
h
[ÿ ] [ÿ ] [ÿ ] þ ÿ S2 S2 ÿ 1
S
1
K
k k
M
=
S2
h
[ÿ ]
i
!
þ
S2
Ip
0 strongly.
ÿ[ÿ ] =
[ÿ ]
#2 ü Z ü 1
k k ÿ
S1
!
þ
u(t)
M
h
ÿ
Deþning
Z
2
ÿ
ü üe
= [ÿþ ÿ] 2 , we have that
K
þ
i
þ
ÿ ÿ
[ÿ ] [ÿ ] þ [ÿ ] 1 + S1 ÿ ÿS1 :
(6.4)
S
K2
is compact, since
is compact and
K
is
self-adjoint (using Exercise 18, page 127, Gohberg and Goldberg [40]). We will use the following (Exercise 6:60 , page 136, Weidmann [85]).
LV W
!
Sublemma 6.2.3 Let
(
;
)
and that Tn
uniformly.
[ÿ ]
S1
!Ki h
K
and
U V W ,
,
2LU V
be Hilbert spaces.
T strongly. If
S
(
;
)
Suppose that Tn ,
is compact, then Tn
uniformly, using the Sublemmas 6.2.2 and 6.2.3. But
[ÿ ]
S1
þ
K
!K
þ
(=
K)
[ÿ ]
S1
uniformly, and so
[ÿ ] þ ÿ ÿS1
!
KS1[ÿ ]
!K
þ
ÿ ÿ uniformly as
KS1[ÿ ] =
S
h
!
2
T
[ÿ ]
S1
T
K
S
i
þ
,
uniformly. Consequently, þ
!
0:
(6.5)
!
[ÿ ] Using the Sublemmas 6.2.2 and 6.2.3, we obtain that S2 ÿ ÿ uniformly. Thus h iþ [ÿ ] [ÿ ] [ÿ ] [ÿ ] þ þ þ ÿ S2 = S2 ÿ ÿ uniformly. As a result, ÿ S2 S2 ÿ ÿþ ÿ uniformly, ÿ ÿ ÿ [ÿ ] ÿ 1, we have and since ÿS1 ÿ
! ÿ
[ÿ ]
S1
h
þ
ÿ
[ÿ ] [ÿ ] S2 ÿ
S2
þ
þ
ÿ ÿ
i
[ÿ ]
S1
! þ
0 uniformly as
From (6.4), (6.6) and (6.5), it follows that ÿ[ÿ ] 2. Since
þ
ÿ
ý [ÿ ]
þ
ÿ[ÿ ]
!
ÿþ ÿ uniformly as
þ
ý
þ
!
ÿ[ÿ ] 0,
! [ÿ ]
ýl
!
þ
!
0:
(6.6)
ÿþ ÿ uniformly as þ
!
Corollary 4.(a), page 1090, Dunford and Schwartz [30]).
ýl
as
þ
!
!
0.
0 (using
114
Chapter 6.
The non-exponentially stable case
Finally, we will prove our main result about the existence of solutions to the sub-optimal Hankel norm approximation problem.
Theorem 6.2.4 If an inÿnite-dimensional system satisÿes the assumptions B1
ÿl+1 <ÿÿ < ÿl , then there ÿ exists a þ0 > 0 such that for < þ < þ0 , ÿG(iÿ) + K [ÿ ](iÿ)ÿ1 þ ÿ.
and B2, and if satisfying
1. Let
Proof
0
0
þ þ < Æ1 ,
ý = þÿþ2l+1 > 0. sup
! 2R
Choose a
every
þ
Æ1 > 0 small enough so that whenever
kG(i!) ý G (þ + i!)k < ý:
This can be done, owing to assumption B1 and Corollary 6.1.2. 2. Next choose a 6.2.1.2)
3. Let
Æ2 > 0, such that whenever 0 þ þ < Æ2 , we have (see Lemma ] < ÿl[ÿ+1
ÿl+1 + ÿ 2
< ÿl[ÿ ] :
þ0 = min fÆ1 ; Æ2 g and consider any þ satisfying 0 < þ < þ0 . c (C pþm ) such that K [ÿ ](ýÿ) 2 H1 ;[l]
From B3, we
have the existence of a
ÿ ÿ ÿG(i! + þ ) + K [ÿ ] (i! )ÿ þ ÿ + ÿl+1 : ÿ
sup ÿ
! 2R
2
Thus,
ÿ ÿ ÿG(i! ) + K [ÿ ] (i! )ÿ ÿ ÿ
ÿ ÿ ÿG(i! ) ý G(i! + þ ) + G(i! + þ ) + K [ÿ ] (i! )ÿ ÿ ÿ ÿ ÿ þ kG(i!) ý G(i! + þ )k + ÿÿG(i! + þ ) + K [ÿ ](i!)ÿÿ =
þ =
ÿ ý ÿl+1
ÿ:
2
+
ÿl+1 + ÿ 2
This completes the proof.
6.3
State-space solutions
We now apply the results from the previous section and from Sections 5.1 and 5.2 of Chapter 5 to obtain explicit formulae for the solutions of the sub-optimal Hankel norm approximation problem in the case when
A
exponentially stable. We consider the following two classes:
is not necessarily
6.3.
115
State-space solutions
Class
A:
The triple (A; B; C ) satisÿes the following assumptions:
1. þ(A; B; C ) is a smooth Pritchard-Salamon system with input space C m and output space C p , 2. The impulse response 3. For every
ÿ >
0,
A
ÿ
h ÿI
satisÿes B1, is the inÿnitesimal generator of an exponen-
tially stable, strongly continuous semigroup.
Remarks:
1. We note that assumption 3 is satisÿed if either of the following is true:
A1 A2
: þ (A)
\ C +0
is empty and
assumption1 .
: A
A
satisÿes the spectrum determined growth
generates a contraction semigroup2 .
We note that from Exercise 2.4 (page 81, Curtain and Zwart [27]), it follows that if
A
generates the semigroup
fT (t)gtÿ0, then Aÿÿ ÿ I is theþ in-
ÿnitesimal generator of the strongly continuous semigroup
e
ÿt T (t)
ÿ
t 0 þ
fT (t)gt 0 and ÿÿ . Thus if either A1 or A2 is satisÿed, then for every ÿ > 0, A ÿ ÿ I ÿ þ
with a growth bound equal to the sum of the growth bound of generates the exponentially stable semigroup
2. If the triple (A; B; C ) belongs to Class
ÿt T (t)
ÿ
e
þ
t 0. þ
A, then the system þ (A ÿ ÿ I ; B; C )
is an exponentially stable smooth Pritchard-Salamon system. We denote
B[ÿ ], the observability map [ ÿ ] [ ÿ ] C , the Hankel operator by ý , andh the lth Hankeli singular value [ÿ ] . If þ[ÿ ] < þ < þl[ÿ ] , let Nþ[ÿ ] := I ÿ þ 2 L[ÿ ] L[ÿ ] 1 , where L[ÿ ] þ l+1 B B C ý [ÿ ] ü [ÿ ] ý [ÿ ] ü [ ÿ ] [ ÿ ] B B and LC := C C . the controllability map of this system by
ÿ
0
A
ÿ
:=
0
1 is said to satisfy the spectrum determined the semigroup, 0 equals sup 2 ( ) Re( )
! ÿ þA 2 That is, kT (t)k ÿ 1 for all t þ 0
by by
ÿ
growth assumption
if the growth bound of
116
Chapter 6.
Class
B:
The triple (A; B; C ) satisÿes the following assumptions:
ÿ
1.
The non-exponentially stable case
A
is the inÿnitesimal generator of an analytic semigroup Z, B (C m ; Z ), where 1 < ÿ
2L
on the Hilbert space C (Z; C p ),
2L
ÿ
ÿ
2. The well-posed linear system3 given by ( response
h
3. For every
ÿA; B; C )
fT (t)gtÿ0 þ 0, and
has an impulse
that satisÿes B1,
þ >
ÿA ÿ þ I is the inÿnitesimal generator of an exponen-
0,
tially stable, strongly continuous semigroup.
Remarks:
1. We note that assumption 3 is satisÿed if either of the following is true:
B1
: ý(
ÿA) \ C +0
is empty and
ÿA
satisÿes the spectrum determined
growth assumption.
B2
:
ÿA generates a contraction semigroup.
We note that from Exercise 2.4 (page 81, Curtain and Zwart [27]), it follows that
ÿÿA ÿ þ I is the þ inÿnitesimal generator of the strongly continuous
semigroup
þt T (t)
ÿ
e
t 0
with a growth bound equal to the sum of the
fT (t)gt 0 and ÿþ . Thus if either B1 or B2 is satisÿed, then for every þ > 0, ÿA ÿ þ I generates the exponentially stable semigroup ÿ þ growth bound of
þt T (t)
ÿ
e
t 0. þ
þ
þ
Furthermore, from Corollary 2.2 (page 81, Pazy [63]) we
obtain that in fact
ÿA ÿ þ I
generates an
B,
2. If the triple (A; B; C ) belongs to Class system given by the triple (
semigroup.
analytic
then it is easy to see that the
ÿA ÿ þ I ; B; C ) is a regular well-posed linear
system satisfying the assumptions A1, A2, A3 listed in Section 2.4 of
B[þ ], the [ þ ] [ þ ] th observability map by C , the Hankel operator by þ , and the l Hankel [þ ] [þ ]i 1 , [þ ] [þ ] h [þ ] singular value by ý [þ ] . If ýl+1 < ý < ýl , let Ný := I ÿ ý 2 LB LC ý ü [þ ] ý ü [þ ] where LB := B [þ ] B [þ ] and LC := C [þ ] C [þ ] . Chapter 2. We denote the controllability map of this system by
ÿ
ÿ
0
0
Theorem 6.3.1 Suppose that the triple (A; B; C ) is in Class C (sI
ÿ A) 1 B . ÿ
+1 < ý < ýl .
Let ýl
then there exists a Æ >
0
K
where
"
ÿý) 2 H
If Q(
1
such that for every þ satisfying
[þ ](ÿý) = R[þ ](ÿý) 1
h
[þ ] 2 (ÿý)
R
[þ ] (ÿý) # h i 1û [ þ ] 1 = ý (ÿý) [þ ] R2 (ÿý) R
(C pým ),
ÿ
Q
i 1
(
0
and
and G(s)
;
ÿ ý ÿþ )
ú
=
kQ(iý)k þ 1,
< þ < Æ,
ÿ
Im
A
1
6.3.
and
[ÿ ]
ÿ
117
State-space solutions
ÿ
ÿ
( )=
þ
Ip
0
0
ÿIm
is such that K
[ÿ ]
þÿ 2
(
ÿ
ÿ2 +ÿ
)
þh
[ÿ ]
CL
B
ÿB
l] (C
c
H
þ
1;[
[ÿ ]
i
0
(
Nþ
0
ÿ
I
+A
0
þI)
ý ýG(i ) + K [ÿ ] (i )
ÿ
pþm ) and
h
þ ÿ ýý ý ÿ1
1
Theorem 6.3.2 Suppose that the triple (A; B; C ) is in Class ÿ1 pþm ), and C (sI + A) B . Let ÿl+1 < ÿ < ÿl . If Q( ) H1 (C
þÿ 2
0
then there exists a Æ >
"
where
[ÿ ]
[ÿ ]
and
ÿ
ÿ
( )=
0
0
ÿIm
is such that K
[ÿ ]
)
R2
(
)
+ÿ
þÿ 2
(
)
)=
#
þÿ þÿ
(
þ
Ip
þÿ
(
R1
[ÿ ]
0
such that for every þ satisfying
K
[ÿ ]
0
[ÿ ]
L
C
i B
ÿ.
This is a consequence of Theorem 5.1.2 and Theorem 6.2.4.
Proof
ÿ
C
ÿ2
[ÿ ]
R1
h
[ÿ ]
= ÿ
ÿ
þ
CL
l] (C
c
1;[
)
þÿ
(
)
þh
[ÿ ]
B
ÿB
H
þÿ
(
h
i
ÿ1
pþm ) and
þÿ
(
ÿ
)
Q
and G(s)
k ÿk ý Q(i
)
1
< þ < Æ,
ÿ1
;
þÿþ
(
þ)
þ
[ÿ ]
i
h
ÿþ þ ÿ ÿ ýý ý 0
(
0
I
þI)
A
ý ýG(i ) + K [ÿ ] (i )
ÿ1
1
C
0
L
[ÿ ] C
i B
ÿ.
This is a consequence of Theorem 5.2.3 and Theorem 6.2.4.
Proof
Example 3.1.2 (continued) Example 3.1.2 in Chapter 3 þts in both classes
and
B
1.
= 1,
Im
Nþ
0
[ÿ ]
R2
i
B
A
considered above: A
þ
BB
ý
,
and
B
:=
C
B
ý
are all bounded, and so ý(A
smooth Pritchard-Salamon system with 2. Since
ü
A
ÿ
AÿBB t
e
3. Since
A
þû
+
BB
tü0 ý
A
ý
is bounded,
. Moreover,
B
W
=
V
þ
ý
BB ; B; C )
is a
= `2 (N ).
it generates the analytic semigroup ( m ; Zý ) with m = 1 and ý = 0.
2L C
= 0, it follows from Corollary 2.2.3 (page 33, Curtain and
Zwart [27]) that
A
generates a contraction semigroup. Thus from Lemma
2.2.6 (page 19, Oostveen [61]), it follows that the system given by the triple ý ý ý (AÿBB ÿ ) (A BB ; B; B ) is output stable, that is, the map z B e z
L
þ
(`2 (Z); L2(0;
ÿ
h(
) =
ý
B e
1
)). (AÿBB ÿ )
in Chapter 3 that
So in particular, with
2 ÿ
B
G(i
L2 (0;
1
z
=
B
7! 2
2
Z), we obtain
`2 (
). Furthermore, we have already veriþed
) is continuous with the unique limit 1 at
ü1
.
So the results in this section apply, solving the sub-optimal Hankel norm approximation problem for this non-exponentially stable example.
ÿ
Chapter 7
The case of regular linear systems In this c hapter, w e will solve the sub-optimal Hankel norm approximation problem for regular linear systems with generating operators (A; B; C ) satisfying some mild assumptions. In the special case of the sub-optimal Nehari problem, an explicit parameterization of all solutions is obtained in terms of the system parameters
A, B
and
C.
The limiting factor to extending the results from Chapter 5 to the general case of well-posed linear systems is the diÆculty in manipulating with the unbounded operators
B
and
C.
In general, it is not clear that the candidate
spectral factor given b y (5.1) is ev en well-posed (see Staÿans [78]). In this c hapter w e translate the problem to the analogous one for reciprocal systems which we deþne in Section 7.1. In Section 7.2, we will
solve the sub-
optimal Hankel norm approximation problem for regular linear systems. The sub-optimal Nehari problem is a special case in which ÿ > ÿ1 . In this special case, using the results from Curtain and Oostveen [16], we can give a parameterization of
7.1
all
solutions, and we do this in the last section of this chapter.
Reciprocal systems
recipr oal system
The
of the regular linear system with generating opera-
tors (A; B; C ) such that 0 of
ÿ
A),
ÿ
A
2
þ(A)
(here
þ(A)
denotes the resolvent set
is the regular linear system with the bounded generating operators þ 1 ; A 1 B; C A 1 . The key to our approach is the relationship betw een
ÿ
ÿ
ÿ
A. Sasane: Hankel Norm Approximation for Infinite-Dimensional Systems, LNCIS 277, pp. 119−125, 2002. Springer-Verlag Berlin Heidelberg 2002
120
Chapter 7.
The case of regular linear systems
ÿ
the transfer function of the original regular linear system G(s) = Cÿ (sI 1 B and that of its reciprocal system with the transfer function G (s) = r ÿ þ 1 sI 1 1 A 1 B . Note that A 1 , A 1 B , C A 1 are all bounded CA A ÿ 1þ operators and for all s ÿ(A) ÿ A there holds
ÿ
A)
ÿ
ÿ
ÿ ÿ
ÿ
Cÿ (sI
ÿ
2
ÿ
ÿ
\
ÿ
ÿ A)ÿ1 B
=
ÿCÿ Aÿ1 B ÿ C Aÿ1
G(s)
=
Gr
=
Gr
ý ü 1 s
ý
ÿ
ÿ Aÿ1
1 s
üÿ1 A
ÿ1 B;
(7.1)
ÿ Cÿ Aÿ1 B
ý ü 1 s
+ G(0):
(7.2)
1L
2
1L
2
From (7.2), it is clear that G H ( (U; Y )) iÿ Gr H ( (U; Y )). In addition, the reciprocal system has the same controllability and observability Gramians, which we prove in the following lemma.
Let (A; B; C ) be generating operators of a regular linear system with transfer function G. Suppose that 0 ÿ(A) and Gr is the transfer function of its reciprocal system with generating operators (A 1 ; A 1 B; C A 1 ). Then the following hold: Lemma 7.1.1
1.
2
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
is an inÿnite-time admissible observation operator for A iþ C A 1 is an inÿnite-time admissible observation operator for A 1 . If either C or 1 is inÿnite-time admissible (for the semigroups generated by A or CA 1 A , respectively), then the observability Gramians are identical.
ÿ
C
ÿ
ÿ
ÿ
ÿ
is an inÿnite-time admissible control operator for A iþ A 1 B is an inÿnite-time admissible observation operator for A 1 . If either C or A 1 B is inÿnite-time admissible (for the semigroups generated by A or A 1 , respectively), then the observability Gramians are identical.
2.
B
3.
G
ÿ
ÿ ÿ
2 H1 (L(U; Y )) iþ Gr 2 H1 (L(U; Y )).
Proof 1. From Hansen and Weiss [45] (see also Grabowski [41]), we know that C
is an inþnite-time admissible observation operator iÿ the Lyapunov equation
hAz1 ; LC z2 i + hLC z1 ; Az2 i = ÿ hC z1 ; C z2 i for all
z1
and
z2
in
D (A),
(7.3)
has a nonnegative deþnite solution
C
L
=
The equation (7.3) is clearly equivalent to the Lyapunov equation
û
C Aÿ1 x2
x1 ; L
ú
+
û
C Aÿ1 x1 ; x2
L
ú
=
ÿ
û
ÿ1
CA
ÿ1
x1 ; C A
x2
ú
L
þC þ 0. (7.4)
for all x1 and x2 in X , which establishes the equivalence. Moreover, the observability Gramians are the smallest positive solution and so the Gramians are identical.
7.2.
121
The sub-optimal Hankel norm approximation problem
2. This is dual to part 1 above. 3. This follows from (7.2).
The idea is then to translate the sub-optimal Hankel norm approximation problem for the regular linear system with transfer function with transfer function
Gr ,
the latter having
now elaborate on this. Clearly,
ÿÿ) 2 1 l (C pÿm ) iÿ
K(
are related by
H
;
Consequently we have
=
1
K
G(i
)
ÿ k1 k ÿ
) + K (i )
ÿk (G)
=
=
ÿk (Gr )
G
into one for the system
generating operators. We
1 l (C pÿm ), where
H
+ G(0); for all
s
k ÿ
orem 1.2.1, it follows that
7.2
þÿ 2
Kr (
ÿ þ
Kr (s)
bounded
;
s
2 C +0 ÿ k1
and
Kr
:
Gr (i ) + Kr (i ) for all k N.
2
K
. Thus from The-
The sub-optimal Hankel norm approximation problem
In this section we will solve the sub-optimal Hankel norm approximation problem for regular linear systems, by þrst translating the problem to its reciprocal system with bounded generating operators, albeit a nonexponentially stable semigroup. Subsequently we use Theorem 6.3.1 from Chapter 6 in order to obtain explicit formulae for solutions to the sub-optimal Hankel norm approximation problem for such systems. We will solve the sub-optimal Hankel norm approximation problem for the regular linear system on a Hilbert space
X
with generating operators (A; B; C )
under the following assumptions:
H1. H2. H3. H4. H5.
U
0
=
2
C m and
þ(A)
Y
and
=
C p.
\ C +0
ÿ (A)
is empty.
f g þ0 f g þ0
B
is an inþnite-time admissible control operator for
C
is an inþnite-time admissible observation operator for
ÿ
G(
) =
C
ÿþ
ÿ( I
unique limit at
ý1
A)
ý1
.
B
2
H
1 (C pÿm ),
G(i
ÿ 2C R C )
(
;
T (t) t
ÿ
.
T (t) t
.
p m ) and it has a
122
Chapter 7.
The case of regular linear systems
H1 ÿ H5 are satisÿed ÿ 3 of Class A on page ÿ ÿ1 ). Furthermore,
First of all, we will show that under if the assumptions by the original system (A; B; C ), then the assumptions 1 115 are satisÿed by the reciprocal system (A 1 ; A 1 B;
ÿ
ÿ
CA
from (7.2) and Theorem 3.0.11, the Hankel operator of the original system and that of the reciprocal system
H
Gr
H
Section 7.1, it follows from Theorem 1.2.1 that the Hankel singular values of and
G
are both compact. Also, as discussed in G
are equal. Applying Theorem 6.3.1, we give solutions to the sub-optimal
Gr
Hankel norm approximation problem for the reciprocal system and hence also to the original system. But ÿrst we will prove the following elementary result. Lemma 7.2.1 If
ÿ
ÿ
(A 1 ; A 1 B;
ÿ
(A; B; C )
H1 ÿ H5,
satisfy
ÿ1 ) satisÿes assumptions1
CA
1
ÿ3
then
the
of Class
reciprocal
A.
system
1 is obvious.
Proof
ÿ
ÿ
2 : Owing to the inÿnite-time input admissibility of C A 1 for the reciprocal 1 T ( )x system, it is clear that for all x X, CA L2 ([0; ); C p ). Conse1 1 p m quently, CA T ( )A B L ([0; ); C ). Furthermore, the continuity of
þ
ÿ
3 : It follows from
H2
Gr
ÿ
ÿ
2
2
2
1
ÿ
ÿ
þ 2
ÿ
on the imaginary axis follows from equation (7.2), and
H5.
and the spectral mapping theorem ÿ þfor the resolvent
þ1 \ C +0 is empty. þ1 is bounded, it satisÿes the spectrum determined growth
(see 1.13.(i), page 243, Engel and Nagel [31]) that Furthermore, since
1
ÿ
A
A
assumption (see for instance Corollary 2.4, page 252, Engel and Nagel [31]). Thus it is clear that the growth bound of the semigroup generated by nonpositive, and so
A
þ1 ÿ
þI
A
þ1 is
is the inÿnitesimal generator of an exponentially
stable, strongly continuous semigroup. We end this section by giving our main result. Theorem 7.2.2 Suppose that (A; B; C ) satisÿes
ÿþ 2
and Q(
1 (C pÿm )
)
H
that for every þ satisfying
0
ý ý
þ(s)
=
"
Ip
0
0
ÿIm
þ1
2 CA
1 see page 115
ü
)
ÿ
s)
s)
s)
:=
ÿ
1(
R
ü
ÿ
= þ(
1.
ÿ þ1 ÿÿ
2(
s)R
þ1
s)
ý
H1 ÿ H5.
Let ÿl
+1
< ÿ < ÿl
Then there exists a Æ >
Q
s)
(
0
such
;
s
ü
þ)
;
Im
+
# [þ ] h iý h ÿ þ B þý Nÿ[þ ] sI + Aþ1 ý þ 1 A B
ÿ1ÿ1ÿ ÿ
ÿ ÿ
1( R2 ( R
Q(i
< þ < Æ,
Kr (
where
k þ k1 ý
, and
L
ÿ
þI
iþ1h ÿ
þ1 þý
CA
[þ ] þ1 CA B
L
i ;
7.3.
123
The sub-optimal Nehari problem
ÿÿ) 2 1 l (C pÿm ) þ1 þ þ1 þ þ1 þ
is such that Kr (
H
[] ÿ
. Here LB
;
[] ÿ
and LC
denote the controllability
iþ1 þ 12 L[ ]L[ ] .
Gramian and the observability Gramian, respectively, of the exponentially stable system
ÿ
A
ÿI; A
B;
[] ÿ
, and Nþ
CA
more, deÿning
ý ü K (s)
=
Kr
1 s
h
:=
I
þ
ÿ
ÿ
B
C
Further-
þ G(0);
(7.5)
þÿ) 2 H1 l (C pÿm ) and kG(iÿ) + K (iÿ)k1 ý þ.
we have K (
;
Proof From Lemma 7.2.1 it follows that the reciprocal system þ1 þ þ1 ) satisÿes the assumptions 1 þ 3 of Class A. Let r denote ( þ1 þ1 þ þ1 ). Moreover, we have the transfer function of ( þ1 A
;A
B;
CA
G
A
(Gr ) =
þl
;A
B;
CA
+1 (G) = þl+1 (Gr ) :
þl (G) > þ > þl
Consequently, using Theorem 6.3.1, we have that
þÿ) 2 H1 l (C pÿm ) 2
Kr (
;
and
kGr (iÿ) + Kr(iÿ)k1 ý þ. Finally it is clear that K deÿned by (7.5) is such that pÿm ) and kG(iÿ) + K (iÿ)k = kG (iÿ) + K (iÿ)k K (þÿ) 2 H1 l (C r r 1 1 ý þ. ;
7.3
The sub-optimal Nehari problem
Finally, in this section we solve the sub-optimal Nehari problem for the regular linear system on a Hilbert space
X
with generating operators (A; B; C ) under
the following assumptions:
N1. = C m and = C p . N2. 0 2 ( ). N3. is an inÿnite-time admissible control operator for f ( )g ý0. N4. is an inÿnite-time admissible observation operator for f ( )g ý0. N5. (ÿ) = ÿ (ÿ þ )þ1 2 1(C pÿm ). U
Y
ý A
B
T t
C
G
t
T t
C
I
A
B
t
H
The sub-optimal Nehari problem can be thought of as a special case of the sub-optimal Hankel norm approximation problem with
l
= 0. So in principle,
the results of the previous section apply to this case. However, we do not do 2 Note that from Theorem 6.3.1, it follows that that
Kr (þ ) 2 H1 l (C pÿm ) ;
Kr (ÿ ) 2 H1 l (C pÿm ), but this implies c
;
124
Chapter 7.
that in this section.
The case of regular linear systems
This is because the Theorem 6.3.1, which was used in
the previous section, can be now replaced (in the
special
case of the Nehari
problem) by a more powerful result from Curtain and Oostveen [16], which will enable us to even obtain a
of
parameterization
all
solutions to the sub-optimal
Nehari problem. Also, we ÿnd that with this alternate approach we can solve the problem under weaker assumptions than those demanded in the previous
N2 versus H2, and N5 versus H5).
section (notice the diþerences in
However,
the broad approach in both sections remains the same: instead of looking at the original system, we translate the problem to the reciprocal system, solve it, and ÿnally retrieve solutions to the original problem. Suppose that
ÿ > ÿ1 (G)
=
ÿ r (þÿ) 2 1
ÿ)k1 < ÿ, then K
K (i
H
Kr (s)
Conversely, if K
) H (C p m ) satisÿes
ÿ (þÿ) 2 1 H
:=
ÿ þ 1
K
H
G
G i
i
K
i
< ÿ
+ G(0):
s
1 (C pÿm )
þÿ 2
Kr (
ÿÿ 2 1 (C pÿm ) satisÿes k ( ÿ) + k r ( ÿ) + r ( ÿ)k1 , where
(Gr ). If K ( ) (C p m ) satisÿes
ÿ1
satisÿes kGr (iÿ) + Kr (iÿ)k1 kG(iÿ) + K (iÿ)k1 < ÿ, where
ÿ þ
K (s)
:=
Kr
1 s
<
ÿ,
then
þ G(0):
So instead of solving the suboptimal Nehari problem for
G,
we solve the sub-
optimal Nehari problem for the reciprocal system with the bounded generating 1 . This system satisÿes all the conditions in the operators A 1 , A 1 B , CA
þ
þ
þ
þ
following result from Curtain and Oostveen [16], which we now recall. Theorem 7.3.1 Suppose that the triple (A; B; C ) satisÿes the following assumptions:
T1. A is the inÿnitesimal generator of a strongly continuous semigroup
2 L(C m ; X ) and C 2 L(X; C p ).
on the Hilbert space X , B
ÿ) = C (ÿiI þ A)þ1 B 2 L1(R; C pÿm ).
T2. G(i
T3. B is an inÿnite-time admissible control operator for
fT (t)gtý0.
T4. C is an inÿnite-time admissible observation operator for
Let ÿ > ÿ1 and let
ý
ý(s) = where L
B
ý
be given by
Ip
0
0
ÿIm
and L
C
fT (t)gtý0.
ü
ý +
þ ÿ12 C LB 1B
ÿ
ü
ü
ü
ü ]þ1 û ü
ÿ [sI + A
N
C
CB
L
ú
:
þ
denote the controllability Gramian and the observability
Gramian, respectively, of the system
(A; B; C ),
ÿ
and N
:=
û
I
þ ÿ12 LB LC
ú 1
.
7.3.
ÿÿ) 2 1 (C pÿm ) þ1 2 (ýÿ)
Then K ( R1 (
125
The sub-optimal Nehari problem
, where
ÿ
þ
ýs) R2 (ýs) R1 (
ýs) 2 H1 (C pÿm )
Q(
for some
kG(iÿ) + K (iÿ)k1 þ
satisÿes
H
ýs)R
= ÿ(
satisfying
N
ÿ
ýs)þ1
ÿ
ýs)
K(
iþ
=
þ
ýs)
Q(
Im
kQ(iÿ)k1 þ 1.
ýN
Indeed, under the assumptions 1 5, the reciprocal system has its transfer 1 are inþnite-time function in H (C p m ) and the operators A 1 B and CA
1 ÿ
admissible with the same Gramians
þ
B
and
L
þ
ý
C
as the original system. So in
L
light of what has been said above, upon applying Theorem 7.3.1 to the reciprocal system (A 1 ; A 1 B; C A 1 ), we obtain the following result:
þ
þ
þ
ý
Theorem 7.3.2 Suppose that (A; B; C ) satisÿes
ýÿ) 2 H1 (C pÿm ), and kQ(iÿ)k1 þ 1.
Q(
ÿ
ÿ(s) =
þ
ÿ
Ip
0
0
ÿIm
B
and L
+
C
where L
ý ÿ1ý2 C Aþ1üLB 1
ÿ
þ1 ý
A
B
þ
ý
N1 ý N5
Let ÿ > ÿ1 and
.
be given by
þ1 üý iþ1û ý þ1 üý
sI + A
CA
C Aþ1 B
L
ú
þ
;
denote the controllability Gramian and the observability
ýÿ) 2 H1 (C pÿm )
Gramian, respectively, of the system Then K (
ýh
ÿ
N
ÿ
Let
satisÿes
(A; B; C ),
and N
ÿ
:=
kG(iÿ) + K (iÿ)k1 þ ÿ
û
iþ
I
ý ÿ12 LB LC
ú 1
.
ù ø
K (s)
=
Kr
1 s
+ G(0);
ýs) = R1 (ýs)R2 (ýs)þ1 , and
where Kr (
ÿ
for some
ýs) R2 (ýs) R1 (
ýs) 2 H1 (C pÿm )
Q(
þ
þ1 = ÿ(ý ) s
satisfying
While it is tempting to try to write ÿ and
ÿ
Q
ýs)
(
þ
Im
;
kQ(iÿ)k1 þ 1. K
in terms of its reciprocal, we know
that this will not (in general) be well-deþned (see Staýans [78]). So we leave the explicit solution as it stands.
Chapter 8
Coda In this last chapter, w e discuss the approach follo w edin this book to solve the sub-optimal Hankel norm approximation problem for inÿnite-dimensional systems. In Chapter 4, we ga ve suÆcient frequency domain condi to ions
solve the
sub-optimal Hankel norm approximation problem. Subsequently in Chapter 5, w e solv ed the sub-optimal Hankel norm appro
ximation problem for the expo-
nentially stable smooth Pritchard-Salamon class and the class of exponentially stable analytic systems, by explicitly verifying the frequency domain conditions given in Chapter 4.
In Chapter 6, under certain conditions, w e ga v ea solu-
tion to the sub-optimal Hankel norm approximation problem in the case when the inÿnitesimal generator is not necessarily exponentially stable and ÿnally, in Chapter 7 w e solv ed the sub-optimal Hankel norm approximation problem in the case of regular linear systems. The main advan tage of our approac h in this book is that it is a self-contained, elementary solution to the sub-optimal Hankel norm approximation problem. Moreov er, our results in Chapter 4 hold for a class larger than the Wiener class. Although our results in Chapter 4 hold for a class larger than the Wiener class, unfortunately, except for the results in Chapters 6 and 7, most of our examples
h 2 L1 !
ha v e
One might conjecture that an approach similar to the one in this book w ould yield solutions to the sub-optimal Hank el norm approximation problem for more general classes of well-posed linear systems than the ones considered in this book. How ev er, w e suspect this task to be a formidable one, since ewneed the smoothness properties of the controllabilit y and observ abilit y Gramians to v erify the candidate spectral factor formulas given in this book.
A. Sasane: Hankel Norm Approximation for Infinite-Dimensional Systems, LNCIS 277, pp. 127−129, 2002. Springer-Verlag Berlin Heidelberg 2002
128
Chapter 8.
Coda
This book also gives alternate state-space formulae for the sub-optimal Hankel norm approximation problem for ÿnite-dimensional systems. We conclude this book by explicitly working out an elementary ÿnite-dimensional example, which elucidates the procedure involved and the theorems in Sections 5.1 and 5.2 of Chapter 5.
(An elementary ÿnite-dimensional example.) Consider the ra-
Example 8.0.3
tional function G(s)
=
1
ÿ
s
þ
; 1 2 + 2
which has the minimal (and hence controllable) state-space realization of MacMillan degree 2 given by
ý
ÿ 12
=
A
ü
1 1 2
ÿ
0
The impulse response is
h(t)
ý ;
=
=
B
Ce
0
ü and
1
A B = teÿ 12 t
t
for
and observability Gramians are given by
ý
B
=
L
respectively. We have L
B LC
ý
=
3
4
2
3
2
1
1
1
ü
and
ü ;
ÿ
C
L
(LB LC ) =
ý =
n
=
C
t
1
0
ú
:
ÿ 0 and its controllability
1
1
1
2
3+2
û
ü ;
p
2; 3
þ2
po 2
;
and so the Hankel singular values of the system are ÿ1
Let
l=1
and
ÿ
2
= 1
norm approximant of
ý
V
(s)
=
G
2
p
2+1
þ 1;
p
> ÿ2
þ
=
p
þ 1:
2
2 + 1 . Thus we seek a sub-optimal Hankel
of MacMillan degree at most 1. We have
ý ü 1 þC LB (sI + A ) 1 N û C 1 L B ú þ ÿ ÿ C 2 0 B ÿ ÿ Im ý ü ý üý ü 1ý 1 1 1 üý 1 0 þ 2 þ1 sþ 0 þ 1 2 2 2 þ 0 1 1 1 0 1 1 s þ þ 1 0 2 2 3 2 1 s 1 2 64 1 þ (s 12 )2 þ (s 121)2 7 5: s 2 2 1þ 2 1 1 s s ( ( ) ) Ip
0 1
ü
ÿp
=
0 ÿ
0
ÿ
=
ÿ
=
ÿ
ÿ
With
0
0
ÿ
ÿ
2
2
þs) = 0, we obtain
Q(
þs)
K0 (
=
þs)V22 (þs)
V12 (
1
ÿ
1 2
ü
Coda
129
=
s
ÿ
s
=
þ 1 2 + 2
ÿ
1þ
1 + 2
p12
ÿ
þ 1 2 + 2
+1
ih
p
G0 (s)
kG þ G0 kH ý 1.
i
1 1 + 2 + 2 1 + 1 1 þ 1 2 2 2 2 2 2 + 1 1 1 + 1 : s+ s+ 2 þ 2 2 2
þ
s
p p
p p
Thus one sub-optimal Hankel norm approximant of
and
#ÿ1
1 2
s
s
h
s
=
"
+1
=
p p
1 1 2 + 2 2 1 1 s þ 2 + 2
G
is
;
ÿ
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2
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Index ÿ
3
lines theorem, 57
G, 15 H2 (C k ), 4 HG , 4 H1 [l](C pÿm ), 52 H1 (C pÿm ), 52 H1 l (C pÿm ), 5 H1 (C pÿm ), 4 H1 (C pÿm ), 52 Xþ1 , 14 X1 , 14 Z , 25 c
;
c
;l
;
c
ÿ
ÿh , 7
bounded observation operator, 14
conjugate linear, 16 contraction semigroup, 115 controllability Gramian, 22, 26 controllability map, 22, 26
dual of an operator, 19 duality map, 16 duality pairing, 17
exponentially stable, 7 exponentially
þt , 14
C00 ; 1 C0 iR; C ÿ MH1 R R1 S1 ÿ+ (0
), 47
p m ),
(
stable
Pritchard-
Salamon system, 22
ý, 14
c
bounded control operator, 14
feedthrough operator, 16 63
, 53
, 53
, 53
generating operators of a well-posed linear system, 16 growth bound, 21
, 53
^, 7
Hankel, 2
Cr Cr
, 52
Hankel matrix, 1
, 52
Hankel norm, 8
þ
!0 , 21 C + , 52
þ r
Cr
, 52
ÿk (G), 5 ÿk (H ), 5 r(ÿ), 23
Hankel operator, 4, 22, 43 Hankel singular values, 5, 64 Hartman's theorem, 63 Hilbert matrix, 2
G
AAK theorem, 6 admissible control operator, 14 admissible observation operator, 14
impulse response, 7, 43 inünite-time admissible control operator, 15 inünite-time admissible observation operator, 15 inünitesimal
Bezout identity, 53
generator
of
a
well-
posed linear system, 15
140
Index
Schur, 2
input space, 15
semigroup of a well-posed linear sysLaurent matrix, 72
tem, 15
Lyapunov equation, 79
singular value, 5 skew-symmetric, 65
MacMillan degree, 5
smooth Pritchard-Salamon system,
model reduction, 8
21 Sobolev space, 24
Nehari, 1
solution to the sub-optimal Hankel
Nehari's theorem, 3
norm approximation prob-
nuclear, 65
lem, 6, 88 spectrum
observability Gramian, 22, 34 observability map, 22, 34 optimal Hankel norm approximant, 6
growth
as-
state-space, 16 strongly stable, 64 sub-optimal Nehari problem, 4
output space, 15
sub-optimal Hankel norm approxi-
output stable, 117
mant, 9 sub-optimal Hankel norm approxi-
pivot space, 17 Pritchard-Salamon admissible control operator, 20 Pritchard-Salamon
determined
sumption, 115
admissible
mation problem, 6, 88
time-domain Hankel operator, 7 ob-
transfer functions, 15
servation operator, 20 Pritchard-Salamon system, 21
unbounded control operator, 14
reciprocal system, 119
well-posed linear system, 15
regular well-posed linear system, 16 representation of the dual space, 17 right coprime, 53 right coprime factorization, 53
Standard notation Here we list some standard notation that we use in the thesis. Non standard symbols which are deÿned in the thesis can be looked up in the Index.
Set theory 1.
N:
2.
{ S:
f
the set of natural numbers complement of the set
g
1; 2; 3; : : : .
S.
Linear algebra Throughout this thesis, unless otherwise speciÿed, when considering a vector
C . In a matrix, blank entries are always to be
space, the underlying ÿeld is understood to be zeros.
V
1. dim( ), where
V
.
V
is a ÿnite-dimensional vector space: The dimension of
V !V
2. ker(T ) and ran(T ), where the vector space of
T,
T : 1 2 is a linear transformation from to the vector space 2 : the null space and the range
V
1
respectively.
V
Linear analysis 1.
LH H (
2 ),
1;
where
H
and
1
H
2
are Hilbert spaces: the Banach space of
all bounded linear operators with the norm 2.
ÿ(A),
þ (A),
p (A),
þ
linear operator, and
H
c (A), where
þ
A
:
kk T
D (A)
(
= sup(06=)x2H1
ÿH ! H )
H
, then
ÿ
A
is a closed
is a Hilbert space: the resolvent set, spectrum,
point spectrum and continuous spectrum, respectively, of dense in
kT xkH2 kxkH1 .
denotes the adjoint of
A.
A.
If
D (A)
is
142 3.
Standard notation
C
X ;X
X
( 1 2 ), where 1 continuous maps from
C
instead of 4.
`2 (N )
X1 ; C ).
(
`1 :
are topological spaces: the space of If
X2
=
X1 a b
n=1
ÿ
n n
respectively
C
(
X1 )
! 1 X ha; bi2 = anbn : n=ÿ1
the Banach space of summable complex sequences indexed by
kak1
=
1 X
n=1
the Banach space of complex sequences indexed by
kak1
N with
janj:
in absolute value with the norm
7.
C , we simply write
is
N (respectively Z), with the inner product
the norm
6.
X2 X2 .
`2 (Z)): the Hilbert space of square summable complex sequences
ha; bi2 `1 :
to
(
indexed by
5.
and
X1
= sup
n2N
N and bounded
jan j:
Lloc 2 (ÿ; X ), where ÿ(ÿ R) is measurable and X is a Banach space: ý þ measurable and : f : ÿ ! X j Rf iskfBochner-Lebesgue 2 K (! )k d! < 1 for any compact set K (ÿ ÿ)
þp<1 ! X j Rfÿ kf ! kpd! < 1
Lp (ÿ; X ), where
þ
f
:ÿ
with the norm
1
:
ý
is Bochner-Lebesgue measurable and
kf kp
=
( )
1 p ûp ÿ f (!) d! .If X
üR
k
k
is
C , we simply write
Lp (ÿ) instead of Lp (ÿ; C ) . If p = 2 and X is a Hilbert R space, then it is also a Hilbert space with the inner product hf; g i2 = ÿ hf (! ); g (! )id! . L1 (ÿ; X ):
þ
f :ÿ!Xj
with the norm 8.
is Bochner-Lebesgue measurable and
2ÿ kf (!)k < 1 kf k1 = esssup!2ÿkf (!)k.
H1 (X ), where X
(
f
f : C +0
ý
esssup!
is a Banach space:
!Xjf
is analytic and
)
sup
s2C + 0
kf s k < 1 ( )
:
