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TitleAnalysis and Optimization of Systems: State and Frequency Domain Approaches for Infinite-Dimensional Systems: Proceedings of the 10th International Conference Sophia-Antipolis, France, June 9–12, 1992
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LanguageEnglish
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Total Pages660
Document Text Contents
Page 1

Lecture Notes
in Control and Information Sciences 185

Editors: M. Thoma and W. Wyner

Page 2

R.E Curtain (Editor)
A. Bensoussan,
J.L. Lions (Honorary Eds.)

Analysis and Optimization
of Systems: State and Frequency
Domain Approaches for Infinite-
Dimensional Systems
Proceedings of the 10th Intemational Conference
Sophia-Antipolis, France, June 9-12, 1992

Springer-Verlag
Berlin Heidelberg New York
London Paris Tokyo
HongKong Barcelona Budapest

Page 330

319

Hence Xel is an eigenvector of H, and so it equals ~ , for some n E Z. From the

definition of 7, and ~'~ we conclude that ~', = Xei = XT, . I

From the above proof we see that Theorem 2.5 is still valid if we instead of Assump-
tion 1.4, assume that H has only eigenvalues.

3 S e l f - a d j o i n t , l X T o n n e g a t i v e a n d S t a b i l i z i n g S o l u t i o n s

In the theory of A R E one is interested in self-adjoint solutions and especially in non-
negative and stabilizing solutions. These solutions can be characterize in terms of the
eigenvectors and eigenvalues of H too. We shall give the characterization of self-adjoint
solutions in the following theorem and the characterization of nonnegative solutions in
Theorem 3.2. In Theorem 3.3 we shall give necessary conditions under which the solutions
are stabilizable.

T h e o r e m 3.1 Suppose that Assumptions 1.1 and 1.4 hold. I f the index set J C Z is such
that {7~, n E J} is a Riesz basis for Z and )q ~ --~j for all i , j E J, then the operator X
defined by

x7~ = ¢. / o r n ~ J (28)

is a self-adjoint solution of the ARE.

P r o o f That X is a solution of the A R E follows from Theorem 2.4. So it remains to show
that X is self-adjoint. Let J denote the operator on g $ Z defined by

(0i
It is easy to see that

g ' J + J H = 0 (30)

on D(H).
We shall show that (Xr/~, 7m ) - (7~,Xrhn) = 0 for all n 6 O. Since {7, ,n E J} is a Riesz
basis for Z and X 6 / : ( Z ) we may then conclude that X is self-adjoint.

(xT. , 7..) - (7. ,x7~.) = (6., 7..) - (7., 6.,)

-7, . ' ¢,~ )
= (J~ . , ~. .)

i
- .~. +----~ [(2,,J¢,,, ~,.) + (J¢. ,~. .~, . )] (31)

i
- - - [ ( J H ~ , , e~, ) + ( J ¢ , , H~ , , , ) ]

1
- - - ( [ J H + H * J ] ~ . , ~ ) = 0 by (30).

U

Page 331

320

T h e o r e m 3.2 Suppose that Assumptions 1.I and 1.4 hold. Denote by J := {n E Z I
Re(l , ) < 0}. If {r/,, n E J} is a Riesz basis for Z, then the operator X defined by

Xr/, = ~, for n 6 J (32)

is a nonnegative self-adjoint solution of the ARE.

P r o o f From Theorem 3.1 we have that X is a self-adjoint solution of the ARE, so we
only have to show that X is nonnegative. Therefore define

:= span{@.}, (z,, z2)~ := (zx, Z2)zezl~. (33)
-EJ

Now it follows from the Assumptions that {(I),, n E J} is a Riesz basis for the Hilbert

( I ) By(32) w e h a v e t h a t l m ( S ) = Z . space Z. Define the operator S on Z as S := X "

From Theorem 2.4 we have that X E £(Z), and thus S E £(Z, Z). Introduce the operator
U(/): Z - 4 2 by

U(t)r/, = eX"'(I',. (34)

Let /1 : D(H) C Z, --* Z be given by H =/-/12" Then it follows immediately that

= - uCOr/., uco)r/. =

Note that /~ is a Riesz-spectral operator. Usin G the Assumptions we get that H generates
an exponentially stable Co-semigroup T(t) on Z. For every t _> 0 there holds T(t) 6 £(Z).
It is easy to see that

U(t) = T(t)S (36)
on span{r/,}.

nEJ
From the boundedness and linearity of S and T(t), the unicity of the bounded linear
extension of U(t) to Z and by (36) we obtain that

U(t) = TCt)S (37)
on the whole Hilbert space Z.
We introduce the operator L : Z (9 Z ---, Z (9 Z, which is given by

L = 0 0 "

For z 6 Z we have that

(Xz, z) = ( Z ] " - X r / - , Z . . r / . )
-EJ n~d

o ¢ . )
-EJ nEJ

= (~-'~ c~.L~. , ~ a , ,~ . ) (39)
ned nfiJ

=

Page 659

648

Pohjolainen S., Laaksonen M. "Optimal minimax controllers with a complex
Remez method", Systems Science XI, International Conference on Systems
Science, Wroclaw Poland, September 22-25, 1992.

[5] Tang, Pingtak Peter "Chebyshev Approximation on the Complex Plane", Ph.D.-
thesis, University of California, Berkeley 1987.

[6] Zemanian A.H. "Distribution Theory and Transform Analysis", Dover, N.Y.
1987.

Page 660

A U T H O R S I N D E X

ALPAY D. 563
AUDOUNET J. 436
AUGESTEIN D.R. 624
BARAS J.S. 624
BARATCHART L. 563
BARDOS C. 410
BASIN M.V. 336
BENSOUSSAN A. 184
BLAQUIERE A. 476
BONNET C. 574
BONTSEMA J. 388
CAIJ.IER F.M. 72
CONRAD F. 512
CURTAIN R.F. 140, 388
EL JAI A. 326
FABRE C. 524
FISHER S.M. 624
FLAMM D.S. 598
GEORGIOU T.T. 222
GOTOH Y. 401
HANNSGEN K.B. 551
INABA H. 290
KHAPALOV A. 489
KLIPEC K. 598
KRABS W. 447
KUIPER C.R. 314, 388
LAAKSONEN M. 636
LASIECKA I. 23
LAGNESE J.E. 423
LEBEAU G. 160
LEBLOND J. 563
LOGEMANN H. 102
MAIZI N. 585
MBODJE B. 436

MCMILLAN C. 459
MONTSENY G. 436
MORGOL O. 531
MORRIS K.A. 378
ORLOV Y.V. 336
OSINGA H.M. 388
OTSU-KA N. 290
PANDOLFI L. 543
PEDERSEN M. 467
POHJOLAINEN S. 636
PRITCHARD A.J. 1,326
PUEL J.P. 500, 524
RAO B. 512
REBARBER R. 366
SALLET G. 347
SIGAL-PAUCHARD M. 476
SKOGESTAD S. 610
SMITH.M.C. 222
STAFFANS O.J. 551
TANNENBAUM A. 242
TORAICHI K. 290
TOWNLEY S. 302
TRIGGIANI R. 459
VAN KEULEN B. 46
WANG Z.Q. 610
WHEELER R.L. 551
WlNKIN J. 72
XU C.Z. 347
YAMAMOTO Y. 401
YOUNG N.J. 199
ZUAZUA E. 500, 524
ZWART H.J. 279, 314, 401

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