##### 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

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