Differential and Algebraic Riccati Equations with Application to Boundary Point Control Problems: Continuous Theory and Approximation Theory (Lecture Notes in Control and Information Sciences 164)

Lecture Notes in Control and Information Sciences Edited by M.Thoma and A.Wyner

164 I. Lasiecka, R. Triggiani

Differential and Algebraic Riccati Equations with Application to Boundary/ Point Control Problems: Continuous Theory and Approximation Theory

Springer-Verlag Berlin Heidelberg NewYork London ParisTokyo Hong Kong Barcelona Budapest

Series Editors M. Thoma • A. Wyner Advisory Board L. D. Davisson • A. G. J. MacFarlane - H. Kwakernaak J. L. Massey • Ya Z. Tsypkin • A..1. Viterbi Authors Prof. Irena Lasiecka Prof. Roberto Triggiani Dept. of Applied Mathematics Thornton Hall University of Virginia Charlottesville, VA 22903 USA

ISBN 3 - 5 4 0 - 5 4 3 3 9 - 2

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Preface

T h e s e L e c T u r e NoTes collecT, and r a t h e r c o m p r e h e n s i v e a c c o u n t optimal

control wlth quadratic

equations

d y n a m i c s operator) continuous)

is at least

s e m i g r o u p and B

framework,

an u p d a t e d

of r e s u l t s c e n t e r e d on the T h e o r y of

cost

in a H i l b e r t space Y,

in a u n i f i e d

functionals

for a b s t r a c t

of the type y = Ay+Bu. the g e n e r a t o r of a s.c.

(control o p e r a t o r

(linear)

Here,

A (free

(strongly

is an u n b o u n d e d o p e r a t o r

w i t h a d e g r e e of u n b o u n d e d n e s s up to the d e g r e e of u n b o u n d e d n e s s Also,

u is the c o n t r o l

H i l b e r t space U.

function which

The p r e s e n t

is L 2 in time w i t h v a l u e s

Treatment

includes b o t h

well as The i n f i n i t e time h o r i z o n problems. a n a l y s i s of The c o r r e s p o n d i n g d i f f e r e n t i a l (operator)

equations,

which arise

in The

of the o p t i m a l s o l u t i o n p a i r {uO,y0}.

of A. in a

the finite as

It c u l m i n a t e s w i t h the and a l g e b r a i c

(pointwise)

Riccati

feedback synthesis

These Notes g i v e the m a i n

r e s u l t s of The t h e o r y - - w h i c h has r e a c h e d a c o n s i d e r a b l e d e g r e e of m a t u r i t y over The past

few y e a r s - - a s well as the authors'

p h i l o s o p h y of approach, f i f t e e n years. provided,

while,

which

is c o n t a i n e d

basic

in their w o r k of the past

O n l y some key p o i n t s of the T e c h n i c a l d e v e l o p m e n t s are for the most part,

r e f e r r e d to in the a p p r o p r i a t e

detailed Technical

literature.

p r o o f s are

Both continuous

T h e o r y as

well as n u m e r i c a l a p p r o x i m a t i o n T h e o r y for the R i c c a t i e q u a t i o n s are included. Essentially, abstract

two

(non-necessarily m u t u a l l y e x c l u s i v e )

c l a s s e s of

e q u a t i o n s are i d e n t i f i e d by means of a r e s p e c t i v e a b s t r a c t

assumption,

(H.1) and

(H.2)

below.

T h e s e a s s u m p t i o n s are,

n o t h i n g but p r o p e r t i e s w h i c h c a p t u r e d i s t i n c t i v e c o n c r e t e classes of p a r t i a l d i f f e r e n t i a l

in fact,

f e a t u r e s of the

equations

of interest.

IV

(I) First class. only p a r a b o l i c

This includes p a r a b o l i c - l i k e dynamics:

(or diffusion)

not

e q u a t i o n s but also wave and p l a t e

equations w i t h a high degree of internal damping.

All these equations

are identified by the p r o p e r t y that the free dynamics o p e r a t o r A g e n e r a t e s a s.c. analytic s e m i g r o u p on Y. (2) Second class. dynamics,

plate-like

This includes w a v e - l l k e

(hyperbolic)

(both h y p e r b o l i c or not) dynamics,

and S c h r O d i n g e r

equations.

These are identified by a d i s t i n c t i v e a b s t r a c t

regularity'

p r o p e r t y of the c o r r e s p o n d i n g free d y n a m i c s

duality,

'interior regularity'

an

'trace

(and, by

p r o p e r t y of the corresponding non-

h o m o g e n e o u s problem). In either case, already,

the control operator B may have,

as r e m a r k e d

a degree of u n b o u n d e d n e s s up to the degree of u n b o u n d e d n e s s of

the free dynamics operator A.

This framework captures,

a m o n g others,

m i x e d p r o b l e m s for partial d i f f e r e n t i a l equations. Special e m p h a s i s is paid to the following topics.

(i)

Abstract operator models for b o u n d a r y control problems.

(ii)

Identification of the space Y of optimal regularity of the solutions,

p a r t i c u l a r l y for the second class:

it is w i t h respect

to the norm of this space that then the s o l u t i o n y is p e n a l i z e d in the q u a d r a t i c cost functional.

(iii)

Identification of the r e g u l a r i t y p r o p e r t i e s of the optimal pair

{uo,y0} (iv)

Verification of the s o - c a l l e d

'finite cost condition'

(F.C.C.)

in the space Y (of optimal r e g u l a r i t y m e n t i o n e d in (ii)),

in the

case of the infinite time h o r i z o n p r o b l e m and r e l a t e d algebraic Riccati equations.

This is the p r o p e r t y w h e r e b y for e a c h

initial c o n d i t i o n in Y, there exlsts some u ~ L2(O,~;O)

such

that the c o r r e s p o n d i n g s o l u t i o n y • L 2 ( 0 , ~ ; Y ) so that the q u a d r a t i c cost functional ks finite.

V

(v)

Constructive

solution

variational

(Riccati operator)

of a Riccatl equation,

of a

whether

or algebraic.

differential

(vi)

a p p r o a c h to the issue of e x i s t e n c e

D e v e l o p m e n t of numerical a l g o r i t h m s w h i c h r e p r o d u c e n u m e r i c a l l y the key p r o p e r t i e s of the continuous problems.

(1) relies,

As to the abstract m o d e l i n g problem

in the p a r a b o l i c case,

as simplified,

on the ideas of lB.1, Sect.

and refined in [T.5],

of e l l i p t i c theory and i d e n t i f i c a t i o n domains of a p p r o p r i a t e

IT.6],

[Las.4],

[L-T.1].

an incipient [L-T.24]

4.12],

[L-T.5],

[W],

by means

[Gr.1] b e t w e e n S o b o l e v spaces and

fractional powers of the basic d i f f e r e n t i a l

o p e r a t o r A; and in the h y p e r b o l i c / p l a t e case, [T.2],

(i), the t r e a t m e n t

on the model

ideas of

These works b e n e f i t e d from and pushed further to use

idea of [Fa.1].

See Notes at the end of S e c t i o n 4 in

for more details on operator modeling.

These o p e r a t o r models

have been s u c c e s s f u l l y used by the authors in a large v a r i e t y of b o u n d a r y control p r o b l e m s equations;

(optimal q u a d r a t i c cost p r o b l e m s and Riccati

u n i f o r m stabilization;

spectral p r o p e r t i e s a s s i g n m e n t and

s t a b i l i l i z a t i o n via a f e e d b a c k operator, (i~)

etc.)..

As to the optimal r e g u l a r i t y p r o b l e m

drastic difference hyperbolic/plate

(ii), we point out a

in the role played by a b s t r a c t models

mixed problems

e q u a t i o n s on the one hand, time) on the other.

in

(second order in time) or S c h r ~ d i n g e r

and p a r a b o l i c mixed p r o b l e m s

In the latter case,

(first order in

the c o m b i n a t i o n of s e m i g r o u p

m e t h o d s w i t h elliptic theory and i d e n t i f i c a t i o n of d o m a i n s of appropriate provide

(or re-prove)

optimal r e g u l a r i t y results for p a r a b o l i c mixed

see e.g.,

[L-T.8].

problems,

the

fractional powers w i t h S o b o l e v spaces is s u f f i c i e n t to

'Hilbert theory'

different

[Las.4] and [L-T.4], of, say,

(energy) means.

[Lio-Mag.1],

Not so, however,

This theory includes

w h i c h ds o b t a i n e d by for h y p e r b o l i c / m i x e d

V!

problems.

Here,

regularity

theory comes

methods else

the first

(energy,

crucial

or m u l t i p l i e r

In the case operator

beginning

of hyperbolic/plate/Schr~dinger m i x e d problems,

abstract

data,

useful

duality

[L-T.I],

tools

only at a s u b s e q u e n t

or transposition,

and unified

in the

~lli)

we note

(H.2)

is most

readily

verified

controllability

property

alternatively, feedback

'velocity'

is v e r i f i e d

via

feedback].

conditions. equation

Their

methods

through

The r e g u l a r i t y

trace of the h o m o g e n e o u s

(parabollc-like feedback

is at most

finite

(~v),

dynamics),

stabilization, dimensional.

(wave/plate/Schr~dinger

with

regularity

more d e m a n d i n g and explicit, and exact

respectively,

solution

of p r o b l e m

v i a a s t u d y of the exact

the g e n e r a l l y

to an u p p e r b o u n d and,

than L 2

argument,

on the space of optimal

stabilization

class.

pair

input u s m o o t h e r

via u n i f o r m

space of the d y n a m i c s

the F.C.C.

of the s e c o n d

Cost Condition (F.C.C.) class

of

from e n e r g y

of the optimal

'boost-strap'

in the case of the second class

dynamics),

uniform

on a

property

can then be a b s t r a c t e d

properties

in the case of the first

as the u n s t a b l e Instead,

case,

As to the Finite

that

the F.C.C.

amount

trace property'

As to the r e g u l a r i t y

in the p a r a b o l i c (iv)

[or,

'abstract

properties

(for

after a key

been o b t a i n e d

these rely on regularity theory with

{uO,y°}, and,

'trace regularity'

etc.),

level

a trace r e g u l a r i t y

homogeneous p r o b l e m - - h a s

the c o r r e s p o n d i n g These

[L-T.2],

hyperbolic [L-L-T].

provide

control

form or

to bear on these

with second-order

preliminary regularity r e s u l t - - t y p i c a l l y

methods.

in d i f f e r e n t i a l

[Lio.2],

methods

higher/lower

either

form) w h i c h were b r o u g h t

only v e r y recently,

equations w i t h D i r i c h l e t

methods,

block of a

differential e q u a t i o n

from p u r e l y partial

in p s e u d o - d i f f e r e n t i a l

problems

step or b u i l d i n g

property

(il)

of

dissipative

controllability

a lower bound

respect

as in

issues

of a s u i t a b l e

to the initial

verification is obtained by partial d i f f e r e n t i a l

(energy/multipller m e t h o d s in d i f f e r e n t i a l

or

Vll

pseudo-differential hyperbolic

form

equations,

[L-T.25]

and,

on m i c r o - l o c a l

In the case of s e c o n d - o r d e r

a n al y s i s

[B-L-R]

to a c h i e v e

sharp

results. Another emerged

approach

on these (v)

Riccati

issues over

Riccati

mentioned

the o p t i m a l i t y operator,

and s u b s e q u e n t l y operator,

parabolic

was

that

introduced

[L-T.5],

abstr a c t

treatment

abstract

parabolic-like

a suitable provides algebraic

[L-T.6],

operator,

of the o p t i m i z a t i o n opera~or

generally

problems

in time,

differential verified

justified.

Riccati

Riccati

equation.

with a

(T < ~)

and since

in h y p e r b o l i c

based

and

in

In the case of T = ~, quadratic

cost

under theory

this time on the R i c c a t i stability,

stabilization

an additional

bonus

via a R l c c a t l

and v e l o c i t y

for s e c o n d - o r d e r

and need not be dissipative. part,

equations,

the c o r r e s p o n d i n g

Uniform

'bona fide'

for T = ~;

the optimal

acts on both p o s i t i o n

examples

to hold

[F-L-T]

[L-T.22].

operator,

candidate

(both T < ~ and T = ~) ; in an

condition',

theory.

is a

in [L-T.4]

which yields uniform

As a n integral of i l l u s t r a t i v e

[L-T.IO]

first,

data of the p r o b l e m

used by the authors

problems

feedback

an e x p l i c i t

in c o n n e c t i o n

control

for the second class

'detectability

another

to construct

(abstractly)

then it has been s y s t e m a t i c a l l y

has

of a

in two steps:

the c o r r e s p o n d i n g

problem with Dirichlet

probl e m s

this consists

this candidate

in fact,

literature

to the e x i s t e n c e

in terms of the original

one shows

w h i c h solves,

This a p p r o a c h

in (v),

A vast

five years.

approach

conditions

defined

in [Lit.].

the past

As to the c o n s t r u c t i v e

operator

one uses

is p r o p o s e d

true.

these N o t e s

c o n t a i n also a large c o l l e c t i o n

of b o u n d a r y / p o i n t where

all

This also

numerical

control

the r e q u i r e d applies

schemes.

Thus

problems

assumptions

to the n u m e r i c a l the a b s t r a c t

for partial are

indeed

analysis

theory

is

of

VIII

These Notes ratio

are a s u b s t a n t i a l

3 to I) of the authors'

outgrowth

review article

(approximately

in the

entitled:

A l g e b r a i c Riccati e q u a t i o n s a r i s i n g in b o u n d a r y / p o i n t control: A r e v i e w of t h e o r e t i c a l and numerical results. Part I: C o n t i n u o u s theory; Part II: A p p r o x i m a t i o n theory, in P e r s p e c t i v e s in Control Theory, P r o c e e d i n g s of the S i e l p i a Conference, S~elpia, Poland, 1988, Editors: B. 3akubczyk; K. Malanowski; and W. Respondek, Blrkhauser, Boston, 1990, pp. 175-235. Also, relevant

regularity

in the text, Dirichlet

the authors'

[L-T.24]

t h e o r y of s e c o n d - o r d e r

and b a s e d

case and on

on

[L-T.I],

[L-T°20],

These n o t e s are p r e s e n t l y F~nally,

reference

being e x p a n d e d

the a u t h o r s

hyperbolic

[L-T.2],

[L-T.21],

gratefully

provides

[Lio.l],

[L-T.23]

acknowledge

by the f o l l o w i n g

agencies

and i n s t i t u t i o n s

reported

in these Notes:

National

Science

Sciences:

Mathematical

and

Ricerche,

Italy;

Air Force Office

Information Scuola

Sciences;

Normale

equations [L-L-T]

in the N e u m a n n

financial

of S c i e n t i f i c

Superiore,

Pisa,

support

Division Research,

Nazionale Italy.

case.

book.

for r e s e a r c h

Foundation,

Consiglio

invoked

in the

into a s e l f - c o n t a i n e d

received

Mathematical

a r e v i e w of

delle

work of

Table

I n t r o d u c t i o n ; two abstract c l a s s e s ; s t a t e m e n t of m a i n problems . . . . . . . . . . . . . . . . . . . . . . . . . .

I.

PART 2.

3.

4.

of C o n t e n t s

I: C o n t i n u o u s

theory.

Case

1

T <

A b s t r a c t D i f f e r e n t i a l R i c c a t l E q u a t i o n for t h e f i r s t c l a s s s u b j e c t to t h e a n a l y t l c i t y a s s u m p t i o n (H.1) = (1.5) . . . . .

8

2.1.

The general

. . . . . . . . . . . . . . . . . . .

8

2.2.

The smoothing

. . . . . . . . . . . . . . . . . .

12

2.3.

Counterexamples

. . . . . . . . . . . . . . . . . . . .

17

case case

A b s t r a c t D i f f e r e n t i a l R i c c a t l E q u a t i o n for t h e s e c o n d c l a s s s u b j e c t to the t r a c e r e g u l a r i t y a s s u m p t i o n (H.2) = (I.6)

22

S.1.

Pointwise

22

3.2.

T h e D R E for s e c o n d - o r d e r h y p e r b o l i c e q u a t i o n s with Dirichlet control: existence and properties

24

3.3.

T h e DRE:

3.4.

Nonsmoothing

synthesis

existence case:

of o p t i m a l

pair

. . . . . . . . . .

and uniqueness

. . . . . . . . . . .

26

weaker

of s o l u t i o n

28

notions

.....

A b s t r a c t D i f f e r e n t i a l R l c c a t J E q u a t i o n for t h e s e c o n d c l a s s s u b j e c t to t h e r e g u l a r i t y a s s u m p t i o n (H.2R) = (1.8) . . . . .

31

4.1.

32

Theoretical

results:

Theorems

4.1 a n d

4.2

. . . . . . .

Case T = 5.

Abstract Algebraic Riccati Equations: Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.

5.2.

6.

5.2.

6.3.

7.

A l g e b r a i c R i c c a t i E q u a t i o n for t h e f i r s t c l a s s s u b j e c t to t h e a n a l y t i c i t y a s s u m p t i o n (H,I) = (1.5) . . . . . .

36

A l g e b r a i c R i c c a t i E q u a t i o n for t h e s e c o n d c l a s s s u b j e c t tO t h e t r a c e r e g u l a r i t y a s s u m p t i o n (H.2) = (1.6)

39

E x a m p l e s of p a r t i a l d i f f e r e n t i a l e q u a t i o n p r o b l e m s s a t i s f y i n g (H.I) . . . . . . . . . . . . . . . . . . . . . . 6.1.

35

51

C l a s s (H.1): H e a t e q u a t i o n w i t h D i r i c h l e t b o u n d a r y control . . . . . . . . . . . . . . . . . . . . . . . .

51

C l a s s (H.1): H e a t e q u a t i o n w i t h N e u m a n n b o u n d a r y control . . . . . . . . . . . . . . . . . . . . . . . .

53

C l a s s (H.I): S t r u c t u r a l l y d a m p e d p l a t e s w i t h p o i n t c o n t r o l or b o u n d a r y c o n t r o l . . . . . . . . . . . . . .

57

E x a m p l e s of p a r t i a l d i f f e r e n t i a l e q u a t i o n p r o b l e m s s a t i s f y i n g (H.2) . . . . . . . . . . . . . . . . . . . . . .

71

X

7.1.

C l a s s (H.2): S e c o n d o r d e r h y p e r b o l i c e q u a t i o n s w i t h Dirichlet boundary control . . . . . . . . . . . . . .

71

C l a s s (H.2): E u l e r - B e r n o u l l i equations with boundary control . . . . . . . . . . . . . . . . . . . . . . . .

75

C l a s s (H.2): S c h r ~ d i n g e r e q u a t i o n w i t h D i r l c h l e t boundary control . . . . . . . . . . . . . . . . . . .

83

7.4.

Class

85

7.5.

~ l a s s (H.2): K i r c h o f f p l a t e w i t h b o u n d a r y c o n t r o l in the b e n d i n g m o m e n t . . . . . . . . . . . . . . . . . .

88

C l a s s (H.2): A t w o - d i m e n s i o n a l p l a t e b o u n d a r y c o n t r o l as a b e n d i n g m o m e n t

91

7.2.

7.3.

7.6.

7.7.

9.

hyperbolic

systems

......

model with . . . . . . . . .

94

Class point

(H.2): K i r c h h o f f e q u a t i o n w i t h i n t e r i o r control . . . . . . . . . . . . . . . . . . . . .

97

E x a m p l e of a p a r t i a l d i f f e r e n t i a l e q u a t i o n p r o b l e m s a t i s f y i n g (H.2R) . . . . . . . . . . . . . . . . . . . . . .

I01

8.1.

PART

First-order

C l a s s (H.2): W a v e e q u a t i o n w i t h i n t e r i o r p o i n t control . . . . . . . . . . . . . . . . . . . . . . . . .

7.8.

8.

(H.2):

II:

B o u n d a r y c o n t r o l / b o u n d a r y o b s e r v a t i o n for h y p e r b o l i c m i x e d p r o b l e m s of N e u m a n n type. A p p l i c a t i o n of T h e o r e m s 4.1 a n d 4.2 . . . . . . . . . . . . . . . . . Approximation

Numerical Algebraic 9.1.

for t h e

107

(H.l)-class

. . . . . . . . . . .

110

assumptions

. . . . . . . . . . .

II0

9.1.1.

Approximation

9.1.2.

C o n s e q u e n c e s of a p p r o x i m a t i n g a s s u m p t i o n s on A and B . . . . . . . . . . . . . . . . . .

111

A p p r o x i m a t i o n of d y n a m i c s a n d of c o n t r o l problems. Related Riccati Equation ......

112

Main

113

9.1.3.

9.1.4. 9.2.

theory

a p p r o x i m a t i o n of t h e s o l u t i o n to t h e a b s t r a c t Riccatl Equation . . . . . . . . . . . . . . . . .

Approxlmat~on

101

Approximation

results

of a p p r o x i m a t i n g

for t h e

schemes

.....

(H.2)-class

. . . . . . . . . . .

121

assumptions

. . . . . . . . . . .

121

9.2.1.

Approximating

9.2.2.

A p p r o x i m a t i o n o f d y n a m i c s a n d of c o n t r o l problems. Related Riccati Equation ......

122

9.2.3.

Approximating

122

9.2.4.

Discussion

8.2.5.

Literature

results

. . . . . . . . . . . . .

o n the a s s u m p t i o n s

. . . . . . . . .

. . . . . . . . . . . . . . . . . .

125 126

XJ

10.

E x a m p l e s of n u m e r i c a l a p p r o x i m a t i o n f o r t h e c l a s s e s (H.1) a n d (H.2) . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.

10.2.

10.3.

10.4. 11.

C l a s s (H.I): H e a t e q u a t i o n w i t h D i r l c h l e t b o u n d a r y control . . . . . . . . . . . . . . . . . . . . . . . C l a s s (H.i): E x a m p l e 3.1

T h e structurally d a m p e d plate p r o b l e m i n . . . . . . . . . . . . . . . . . . . . .

126

126

124

C l a s s (H.2): T h e w a v e e q u a t i o n w i t h D i r i c h l e t b o u n d a r y control . . . . . . . . . . . . . . . . . . . . . . .

138

Class

.....

145

. . . . . . . . . . . . . . . . . . . . . . . . .

149

Conclusions

(H.2):

11.1.

Theoretical

11.2.

Numerical

FirSt-order

aspects

aspects

hyperbolic

systems

. . . . . . . . . . . . . . . . . .

149

. . . . . . . . . . . . . . . . . . .

150

1.

Introduction;

Problem

two a b s t r a c t

and prellmlnarv

problem:

Given

model.

the dynamical

classes:

Consider

the q u a d r a t i c

of m a i n

the following

problems

optimal

control

system

Yt = A y + B u ; minimize

statement

y(0)

= Y0 e Y

(1.1)

functional T

J(u,y)

ff,,Ry(t),[~+{[u(t),,~]dt+,,Gy(T)I,~

=

(1.2)

0 over all

u 6 L2(0.T;U).

the p a p e r ,

we shall

with

make

y solution

the

following

of

(i.I)

standing

due

to u.

Throughout

assumptions

on

(I.1),

(1.2):

(i)

Y. U.

Z. a n d W a r e H i / b e r t

A: Y m ~(A)

spaces;

~ Y is t h e g e n e r a t o r

of a s.c.

)]',

) with

semigroup

e At o n Y.

t > 0; (lii)

B:

U ~

[~(A

Y-topology.

A

the dual

being

of ~ ( A

the Y - a d J o i n t

respect

of A;

to t h e

more precisely,

it is

assumed-that (A)-qB s ~(U;Y) (iV)

the operator

for s o m e

constant

0 & ~ < 1;

(1.3)

G is b o u n d e d , G e Z(Y;W).

Instead,

the o p e r a t o r

in S e c t i o n which

4 where

follows

unbounded

In then take

Abstract

5.8 w i l l

document

finite

Our motivation

partial

structures'

Accordingly,

necessarily

R E Z(Y;Z),

that

the c a s e w h e r e easier

except

(The a n a l y s i s

than

the

R is case where

or i n f i n i t e .

If T = ~,

we shall

(1.2).

(in a n y s p a c e

'flexible

bounded:

to b e u n b o u n d e d .

~s q u a l i t a t i v e l y

T > O may be

G ~ 0 in

to,

be generally

be a 1 1 o w e d

a n d B is u n b o u n d e d . )

(1.2),

clas~es.

directed

below.

Theorem

a n d B is b o u n d e d

R is b o u n d e d

control

R will

it w i l l

(1.4)

mutually

differential

comes

equations

dimensions), problems;

from,

see

including

and

with

we shall

distinguish dynamics,

boundary

those

the e x a m p l e s

exclusive

is u l t i m a t e l y or point

that may arise

of S e c t i o n

two general for w h i c h

6 and

classes

different

in 7

of n o t treatments

must

be a p p l i e d

~n order

of the two classes ('regularity')

First

to capture

will be s i n g l e d

assumptions

The

~lass.

(H.1)

satisfies

Second

class

assumption: extension

(first

semigroup

continuous

is s t r i c t l y

form).

e

At

is a n a l y t i c

henceforth

Y ~ L2(0, T;U);

Each

on Y,

~ appearing

less than

The second

for any 0 < T < ~,

denoted

thereof.

by one of the two

the assumption:

t > O, and the constant (1.3)

properties

below.

first class

the s.c.

optimal

out and m o d e l l e d

class

the operator

in

(1.5)

I: ~ < 1. (first

form)

B*e A*t a d m i t s

by the same symbol

satisfies

the

a continuous At B *e

satisfying

:

i.e.,

T

flJB*eA

(H.2)

2

tytl dt _< CT~lyllY ,

y ~ Y.

(1.6)

0 In

(H.2) we have B

[~(A

)

¢ ~(~'(A ),U)

for the dual

(B v , U ) u = (v, Bu)y;

Second

of B, after

identifying

with ~(A)

class

(second

the assumption: continuous

B*e A t R * :

form).

The s e c o n d

for any 0 < T < ~,

extension

continuous

v ~ ~(A

), u ~ U.

class

(second

the o p e r a t o r

(denoted h e n c e f o r t h

(1.7)

form)

satisfies

B*e A tR* admits

by the same symbol)

a

satisfying

Z ~ L2(O, TIU) :

T

(H.2 R)

fllB*e A tR*=,~at

_< ca:llzH ~ ,

= - z,

(z.8)

0 where

R

from Z to Y is the adjoint While

and B,

conditions

instead

~n S e c t i o n

(H.I)

condition

4) o p e r a t o r

and

(H.2R)

guarantees optimal

the e x i s t e n c e

control

problem

only

the

the o p e r a t o r s

(possibly

A

unbounded:

out by the notation.

for either

of a u n i q u e (1.2)

involve

requires also

R, as p o i n t e d

W h e n T = ~, we shall need

of R.

(H.2)

optimal

class

an a s s u m p t i o n

pair {uO, y O} of the

that

3

Fin____/~t_eeCost Cond____~tio____nn: For every YO ~ Y" there exists u E L2(O,~;U ) such that the

(F.C.C.)

satisfies J(u,y(u))

R e m a r k 1.1. Conversely,

(1.9)

c o r r e s p o n d i n g functional in (1.2)

Condition condition

< ~.

(1.6) implies (1.8) if R is bounded, (1.8) implies (1.6) if R is an isomorphism.

W h e n T < ~, we introduce the (input-solution)

o p e r a t o r L and its

L2-ad~oint L : t

(Lul(t)

= ~ eA(t-rlBu(rldr,

(i.i0)

0

T (L v)(t)

=

f

(1.11)

B~e A (~-t)v(r)dr,

t

where

(a)

(Lu, V ) L 2 ( O , T ; y ) = (u,L V ) L 2 ( O , T ; U ). Condition

(H.2) = (1.6)

Then

is equivalent

to [L-T.2],

[L-T.3],

(i.i2)

L: continuous L2(0, T;U) ~ C([O,T];Y),

(i.i3)

L : continuous LI(0, T;Y) ~ L2(O,T:U);

(b)

Condition

(H.2R) = (1.8) is equivalent to

(i.i4)

RL: continuous L2(O,T;U) ~ C([0,T];Z), L'R*: where L R

(1.1s)

c o n t i n u o u s LI(O,T;Z) - L2(O,T;U),

means

(EL)

w h e n one of the factors is unbounded.

To fix our ideas at the outset, like b o u n d a r y problems;

•

the f~rst class covers p a r a b o l i c -

not only the usual heat e q u a t l o n s / d i f f u s i o n

equations,

but also w a v e - l i k e or plate-like problems w i t h high d e g r e e

of d a m p i n g

('structural damping'),

second class covers undamped,

see S e c t i o n 6.3 below.

or conservative,

with b o u n d a r y or point control;

e q u a t i o n problems.

(e.g., w i t h

or S c h r ~ d l n s e r

We shall refer to (H.2) = (I.S) as to an

trace theory property,

the

or m i l d l y damped

w a v e - l i k e or p l a t e - l i k e partial differential e q u a t i o n s viscous damping)

Instead,

'abstract'

for this is what it amounts to in partial

differential will

problems.

be a p p l i e d

problems control

and D i r i c h l e t the

second

molntwlse

= (1.8)

order e q u a t i o n s

svnthesls

and Riccatl

a f t e r we a s s e r t

with Neumann

eauatlons.

the e x i s t e n c e

pair ,-(uO(t,0;Y0),yO(t,0;Yo) of p r o b l e m

(Pi)

and u n i q u e n e s s

(1.1),

(in time)

feedback

(synthesis),

via a Riccatl

operator

P(t),

u O in terms of the o p t i m a l

solution

yO

u 0 ( t , 0 ; Y 0 ) = - B * P ( t ) y O ( t , 0 ; Y 0 ),

the o p e r a t o r

P(t)

P(t)

Riccatl

+ P ( t ) A + A*P(t)

(DRE),

of a n optimal

representation of the optimal

in 0 ~ t ~ T,

(1.16)

of an a p p r o p r i a t e formally written

+ R R - P(t)BB

P(t)

as

= 0,

(1,17)

0 ~ t < T;

and

to be p r o p e r l y

interpreted

control

such as g i v e n by

a.e.

is a s o l u t i o n

Equation

(and

are the

(1.2).

Polntwise

where

control

boundary

Qualitatively

~n this paper

Case T < ~:

Differential

class

trace.

informally), the m a i n p r o b l e m s of interest following,

of the s e c o n d

observation, the o p e r a t o r R being in

boundary

(Dirichlet)

(H.2R)

4) to some purely boundary optimal

(in S e c t i o n

for h y p e r b o l i c

this case

Feedback

The form

in a technical

sense,

described

below.

(P2)

Cas~ T = ~:

Pointwise

Riccatl

operator,

optimal

solution

uO(t,O;Yo) where

Equation

*

= -B P y

and to be p r o p e r l y

0

_ _(t,O;Vo), _

P is a s o l u t i o n

+

A

representation,

via a

u 0 in terms of the

a.e.

in t > O,

of an a p p r o p r i a t e

(1.18) Algebraic

f o r m a l l y written as

(ARE)

PA

below.

feedback

of the optimal control 0 y , such as given by

the o p e r a t o r

Riccati

(in time)

P

+

R

R

interpreted

-

PBB P = O, in a technical

(1.19) sense d e s c r i b e d

(P3)

Numerical

approximation

of the DRE and the A R E for the

computation of the Riccatl Diff~cultles

related

the case w h e r e as it a r i s e s

to

operators

B unbounded.

in both point boundary

equations.-

For B bounded,

The present

control

control

p r o b le m s

problems

see e.g.,

only on the natural

"regularity"

for the dynamics,

least at the o u t s e t - - o n R and G.

is n e c e s s a r y

Indeed,

these

two d y n a m i c a l

as they d i s p l a y

peculiarly

B contributes

study of the Riccatl T < ~ and by (1.2).

(1.18)

feedback

consequent inherent

Riccati

synthesis,

modelled

waves a n d p l a t e s classical

by a s s u m p t i o n

unbounded

problems:

arguments

difficulty

arises

regularity

and,

for T = ~,

p.d.e,

in s h o w i n g

(assumption

(H.2))

as well

properties,

which

the r e q u i r e d

of several

are n e e d e d

for (i.I),

levels.

(1) One

theory of e x i s t e n c e

the d i f f i c u l t y

More precisely,

the gain o p e r a t o r

equation.

or v a r i a t i o n

thereof,

(ii)

Cost C o n d i t i o n

(1.9),

regularity

level

of

of

particularly

in specific,

the Finite

are no longer

assumptions

"concrete" w h i c h were

These

techniques

results boundary

controllabillty/unlform

to v e r i f y

the

A second

only very recently. optimal

B P is

Thus,

of the a b s t r a c t

(H.2),

in the

cases of c o n s e r v a t i v e

waves/plates/Schr~dinger

as their exact

in the

(1.16)

problem

techniques are required,

to bear on these p r o b l e m s

succeeded

problems,

Here,

Finite

of the

that for the class of

must be devised.

by a s s u m p t i o n

by

overcome

see

interesting

in the v e r i f i c a t i o n

for the class m o d e l l e d problems.

(H.2),

w i t h B bounded,

and new a p p r o a c h e s

any

difficulties

control

and B P.

or S c h m b d i n g e r

available,

brought

w h i c h escape

at two general

we shall

in the most

techniques,

to the DRE and P to the ARE and the

B P(t)

interest,

exclusive)

from each class.

expressed

and w h i c h must

to the g a i n o p e r a t o r s

inherently

p.d.e,

P(t)

(not

and--at

The u n b o u n d e d n e s s

the optimal

are present

of the s o l u t i o n

case T = ~ of u l t i m a t e dynam i c s

properties

synthesls--as

(H.2}

(H.2R),

very different

of m a t h e m a t i c a l

for T = ~ --of

or else

is based

of the o b s e r v a t i o n

level, w h i c h is aimed at a general

is the a b s t r a c t and u n i q u e n e s s

the p a p e r

results

"unificatlon."

to a number

These difficulties

require

the more

(non m u t u a l l y

best

o n l y on and s u c h

differential

(H.I),

into two

different

and n o n - a r t i f i c i a l

above all,

Thus,

action

to extract

classes

focuses

or its v a r i a t i o n

a non-smoothlng

in order

and,

assumptions

This d i s t i n c t i o n

classes

operator

paper

as in (1.3},

for partial lB.1].

mutually exclusive)

meaningful

and P.

the input o p e r a t o r B is unbounded,

challenging

operators

P(t)

control

stabilization

Cost C o n d i t i o n

for

T h e s e regularity/exact c o n t r o l / a b i l i t y / u n l f o r m

these systems. stabilization

results will

each s p e c i f i c

dynamics.

he r e v i e w e d

We have a l r e a d y p o i n t e d typical

dynamics

plates,

and S c h r i d i n g e r

see C o r o l l a r i e s inapplicable B, w h i c h

modelled

however

required

these systems;

cost

e.g.,

problems,

the constant

Qvervlew.

(1.5)

[P-S].

paper

6.1,

(I.3).

problems

See R e m a r k

presents

state are,

subject

take G = 0 in this case; instance,

our p r i n c i p a l

Riccati

operator

the DRE

(class

(H.I)

in this case

satisfying

case

(H.I))

in Section

applied

to,

(H.1),

(Section

3.1.

boundary

all

the r e q u i r e d

For

the class

the Riccati

allow

(1.12)

(1.6),

setting

more

all

(point)

typically

2.1 for

(classes

(H.1) and

control

S to be s m o o t h i n g results

set-up

(1.6) will

results

of a

in S e c t i o n

regular

regular =

For

and u n i q u e n e s s

after a treatment

cases,

for the class

in R e m a r k 3.1).

the o p e r a t o r

(H.2)

(when

we may just as well

5 for the ARE

A more

assumption

In all

equations,

=

as d e s c r i b e d and

t a k e n to be non-smoothing.

of e x i s t e n c e

2.2).

3.2 and 3.3,

B is u n b o u n d e d

(continuity)

(H.2)

in this general

we shall

all

excluded

and r e a s o n a b l y

R o n the t r a j e c t o r y

in S e c t i o n s

Then,

likewise

in the area of the three

and c o r r e s p o n d i n g l y

genuine

differential verify

and

the r e g u l a r i t y

in S e c t i o n s

an u p d a t e d

see more on this results

are g i v e n

Subsequently,

for the class

property

6.3

5.1 below.

at first,

to a s s u m p t i o n

analytic

of Section

than ~: ~ < q < i.

for the case where

(In v i e w of the r e g u l a r i t y of d y n a m i c s

and most

by our r e s u l t s - - a r e

The observation, o p e r a t o r s

T < ~) G on the final

given

for

5.3 at the end of

and c h a l l e n g i n g

6.2,

is g r e a t e r

are c o v e r e d

The present

aforementioned

given

as

incompatible

see R e m a r k

distinctive

in S e c t i o n s

q in

the treatment of

(H.2)).

and

(on the o b s e r v a t i o n

w h i c h are

[P-S],

complete account of results a v a i l a b l e in

and

waves

dealing with unbounded

restrictions

etc.), of

rules out,

treatments

additional

in the most

such as those

these s y s t e m s - - w h i c h from

conclusion

other

the results

~nteresting

(conservative

5. In addition,

below,

This

conditions,

(H.2)

of

the g a i n o p e r a t o r B P is unbounded,

equations)

to these systems,

7 in the context

that for the most

by a s s u m p t i o n

5.4 and 5.5.

R, on the finite

Section

out

in S e c t i o n

will be

for d y n a m i c s likewise

be

of the n o n - s m o o t h i n g are

illustrated

by,

and

problems for partial

in any dimension,

w h e r e we shall

assumptions.

of systems

satisfying

theory has reached

the

(analyticity)

a considerable

level

assumption

of m a t u r i t y

and c o m p l e t e n e s s the a l g e b r a i c

the d i f f e r e n t i a l case w h e n T < ~, and

in b o t h cases,

case w h e n T = ~.

This is a l s o so, b e c a u s e all o t h e r

mathematical p r o b l e m s w h i c h i n t e r w e a v e w i t h the R i c c a t l t h e o r y (regularity, class

stabilization,

etc.) are a l s o well u n d e r s t o o d

for the

(H.1}. In contrast,

the s i t u a t i o n

h y p e r b o l i c equations,

is m o r e d e l i c a t e

p l a t e - l i k e equations,

that fall o u t s i d e the scope of a s s u m p t i o n theory is a l s o a v a i l a b l e

for the c l a s s e s of

S c h r S d i n g e r equations,

(H.1).

Here,

in b o t h cases T < ~ a n d T = ~,

etc.

a rich Riccatl to be sure.

But it is p a r t i c u l a r l y in the a l g e b r a i c c a s e w h e n T = ~ a n d

for the

class of d y n a m i c s s a t l s f y ~ n g the trace r e g u l a r i t y a s s u m p t i o n (H.2) =

(1.6)

that the c o r r e s p o n d i n g Riccatl

c o m p r e h e n s i v e and r e a s o n a b l y complete. present many

'concrete'

the a b s t r a c t

theory,

satisfied.

p.d.e,

t h e o r y m a y be considered

Equally

important,

there are at

p r o b l e m s w h i c h s e r v e as i l l u s t r a t i o n s of

w h e r e all of the r e q u i r e d a s s u m p t i o n s are

This latter s t e p is far f r o m b e i n g a t r i v i a l

one:

it

involves s u c h delicate mathematical q u e s t i o n s as (optimal/Sharp) r e g u l a r i t y of s o l u t i o n s

to m i x e d problems,

as well as exact

c o n t r o l l a b i l i t y / u n i f o r m s t a b i l i z a t i o n c o n c e p t s on s p a c e s of o p t i m a l

regularity.

The issues h a v e b e e n r e s o l v e d o n l y v e r y r e c e n t l y for m a n y

(but not all) equations, equations.

mixed problems

for h y p e r b o l i c e q u a t i o n s a n d p l a t e - l i k e

not n e c e s s a r i l y hyperbolic, Indeed,

as w e l l as for S c h r ~ d i n g e r

one m a y s a y that in o r d e r

p i c t u r e in the c a s e T = ~ for the class

to c o m p l e t e

(H.2),

the o v e r a l l

a major task which

is

left for further i n v e s t i g a t i o n ks not so m u c h to p u s h f u r t h e r the abstract Riccati few p h y s i c a l l y

theory,

but to r e m o v e the g a p e x i s t i n g at p r e s e n t

important

'concrete'

of r e g u l a r i t y w h e r e a s s u m p t i o n (smoother)

space where

p.d.e,

(H.2)

holds

the F i n i t e Cost C o n d i t i o n

controllability/uniform stabilization} This is a p u r e l y p.d.e,

problems between

= (1.6)

problem.

true,

in a

the space

a n d the

(exact

has b e e n a s c e r t a i n e d so far.

A m o n g the d y n a m i c s w h e r e

this

u n d e s i r a b l e gap e x i s t s we cite the w a v e e q u a t i o n w i t h N e u m a n n control, and the E u l e r - B e r n o u l l i s e c o n d a n d third). Riccati

equation with

On the other hand,

'high'

boundary operators

w h e n T < ~,

t h e o r y with R n o n - s m o o t h i n g that still

issues that m a k e it, overall, S e c t i o n 3.

(e.g.,

it is the a b s t r a c t

encounters some subtle

less s a t i s f y i n g t h a n the case T = ~,

see

It is p r e c i s e l y in o r d e r to c a p t u r e best r e s u l t s a l s o An

the c a s e of purely boundary h y p e r b o l i c p r o b l e m s , and b o u n d a r y observation,

that the v a r i a t i o n

with boundary control

(H.2R)

of the class

(H.2)

is introduced.

That a s s u m p t i o n

(H.2R) holds true,

along with the other

a b s t r a c t a s s u m p t i o n s of Section 4, in the context of the p u r e l y b o u n d a r y h y p e r b o l i c p r o b l e m of Neumann type,

is a c r i t i c a l

of the recent sharp r e g u l a r i t y theory for the d y n a m i c s

consequence

[L-T.20],

[L-T.23]. Part I of this paper deals with the c o n t i n u o u s case.

It is

f o l l o w e d by Part II w h i c h deals w i t h an a p p r o x i m a t i o n t h e o r y thereof.

A b s t r a c t D i f f e r e n t i a l R i c c a t i E q u a t i o n for the first class subject to the a n a l y t i c i t y a s s u m p t i o n (H.1} = (1.5)

2.

We shall first p r o v i d e

result under a minimal

a general

a s s u m p t i o n on G (Section 2.1), and then a more regular result when G is a s s u m e d to be a s m o o t h i n g o p e r a t o r O < T < ~ in this section,

we may a s s u m e

(modulo an innocuous translation) y: A -I E ~ ( ~ , well

and Chat

(Section 2.2).

~he

that

fractional

without

In any case, loss

A is b o u n d e d l y powers

of

since

genepality

inver~ibie

on

(-A) 8, 0 < e < I, a r e

defined.

2.1.

The qeneral case Complementing

(i.i0), we shall let L T be the

(unbounded)

operator T

(2.~)

LTU = f eA(T-t)Bu(t) dt 0 w i t h d e n s e l y d e f i n e d domain ~(LT)

= {u e L2(O,T;U):

LTU e Y}, which

d e s c r i b e s the map from the input u to the s o l u t i o n y(T) time t = T, w i t h YO = O.

* Its adjoint L T,

of

(I.I) at

(LTU, Y) Y = (U, LTY) L 2 ( 0 , T ; y )

ks the c l o s e d o p e r a t o r *

T h e o r e m 2.1. be c l o s e d

~

A

= B e

[L-T.4],

[L-T.22] Let the (densely defined)

(or elosable),

y,

O ~ t ~ T, y e y.

(2.2)

o p e r a t o r GL T

as an o p e r a t o r L 2 ( O , T ; U ) D ~(GLT) ~ W.

there exists a u n i q u e o p t i m a l (i.I),

(T-t)

{LTY}(t)

Then,

pair { u O ( t , O ; Y o ) , y O ( t , O : Y o )} of p r o b l e m

(1.2) with T < ~, e x p l i c i t l y g i v e n by

is introduced.

That a s s u m p t i o n

(H.2R) holds true,

along with the other

a b s t r a c t a s s u m p t i o n s of Section 4, in the context of the p u r e l y b o u n d a r y h y p e r b o l i c p r o b l e m of Neumann type,

is a c r i t i c a l

of the recent sharp r e g u l a r i t y theory for the d y n a m i c s

consequence

[L-T.20],

[L-T.23]. Part I of this paper deals with the c o n t i n u o u s case.

It is

f o l l o w e d by Part II w h i c h deals w i t h an a p p r o x i m a t i o n t h e o r y thereof.

A b s t r a c t D i f f e r e n t i a l R i c c a t i E q u a t i o n for the first class subject to the a n a l y t i c i t y a s s u m p t i o n (H.1} = (1.5)

2.

We shall first p r o v i d e

result under a minimal

a general

a s s u m p t i o n on G (Section 2.1), and then a more regular result when G is a s s u m e d to be a s m o o t h i n g o p e r a t o r O < T < ~ in this section,

(modulo an innocuous translation) y: A -I E ~ ( ~ , well

and Chat

(Section 2.2).

we may a s s u m e

~he

that

fractional

without

In any case, loss

A is b o u n d e d l y powers

of

since

genepality

inver~ibie

on

(-A) 8, 0 < e < I, a r e

defined.

2.1.

The qeneral case Complementing

(i.i0), we shall let L T be the

(unbounded)

operator T

(2.~)

LTU = f eA(T-t)Bu(t) dt 0 w i t h d e n s e l y d e f i n e d domain ~(LT)

= {u e L2(O,T;U):

LTU e Y}, which

d e s c r i b e s the map from the input u to the s o l u t i o n y(T) time t = T, w i t h YO = O.

* Its adjoint L T,

of

(I.I) at

(LTU, Y) Y = (U, LTY) L 2 ( 0 , T ; y )

ks the c l o s e d o p e r a t o r *

T h e o r e m 2.1. be c l o s e d

~

A

= B e

[L-T.4],

[L-T.22] Let the (densely defined)

(or elosable),

y,

O ~ t ~ T, y e y.

(2.2)

o p e r a t o r GL T

as an o p e r a t o r L 2 ( O , T ; U ) D ~(GLT) ~ W.

there exists a u n i q u e o p t i m a l (i.I),

(T-t)

{LTY}(t)

Then,

pair { u O ( t , O ; Y o ) , y O ( t , O : Y o )} of p r o b l e m

(1.2) with T < ~, e x p l i c i t l y g i v e n by

9

-uO(t,O;x)

-I = {AoT[LTG

y0(t,0;x)

Ge

AT

x+L R R(e

A

xl]}(t),

(2.3)

= eAtx+(gu 0) (t), 8

(2.4)

S

AOT = I+L R RL+LTG GLT, with L, L*, defined (2.2).

Moreover,

in (I.I0),

below)

explicitly

even more,

CT~e (T_t)~ "

(2.7)

~

CT~ (T_t)~

0 ~ O < 1;

0 < t < T;

(2.8) (2.9J

"

for any 0 < £ < T, B P(t)

~ Z(Y;C([0, T-s];Y));

for each YO e Y, the optimal pointwise

feedback

*

The following

control

(2.10)

u0(t,O;y 0 ) is given in

form by

uO(t,0;Y0 ) = -B P(t)¥

(vii)

0 _( t < T;

~ Z(Y;C([O,T-z];Y)),

IfB*P(t)lJ~(y;u)

(iv)

(vi)

(2.6)

for any 0 < a < T, (-A*)OP(t)

(v)

operator

for O ( 8 < I,

}I(-A*)eP(t)J]a(y) ~ (iii)

self-adjolnt

~n terms of the data in

P(t) e Z(Y;C([O,T];Y)); ~n fact,

by (2.1),

such that

(i) (ii)

and LT, L; defined

there exists a non-negatlve,

P(t) = P (t} ~ O, defined (x) = (2.20)

(1.111,

(2.5)

symmetric

0

(t,O;Yo),

relation

0 ~ t < T.

(2.11)

holds:

T

(P(t)x,y)y

-- f(Ry0(r,t;x),Ry0(y,t;y))zdr+(GyO(T,t;x),GyO(T,t;y))w t T (B P ( r ) y 0 (r,t:x),B

+ t

s

P(r)y0(r,t;y) )udr,

(2.12)

10 from which the optimal

cost of the optimal control

problem on

[~,T] initiating at the time r at the point x ~ Y is j(u0(.,r;x),yO(°,r:x)) (viii)

for O < t < T, P(t) satisfies Equation

= (P(r)x,X)y

(2.15)

the following Differential

for all x,y e ~((-A)~),

Riccatl

v ~ > O,

= -(Rx, Ry)z-(P(t)x, Ay)y-(P(t)Ax, y) Y

(P(t)x,y)y

+ (B*P(t)x,B*P(t)Y)u (ix)

;

The following regularity properties

.

(2.14)

hold true for the optimal

pair )IuO{.,T;x)IIL2(T,T;U)+IIyO(.,T;X)IIL2(T,T;U)

~ CTllXllY

llsyO(T,z;x)llW ~ CTl[xl{y;

(2.16)

CTvUXU Y

flu°( •, r ;x) ilCl( [?,T] ;U) liyO(-,r;x)llc([r,T];y)

~ CT~llxlly

IlyO( • , r ; x ) ] ] C2q_l+~([r,T];y) In (2.17}, number,

(2.18},

(2.17)

if 0 ~ ~ < ~;

~ CTqllxi]Y

if M ~ q < 1.

= {f(t) ¢ C([T,T);X):

[]fI[Cr([T,T];X )

for x e Y and for each r fixed,

optimal control uO(t,T;x) are r e s p e c t i v e l y differentiable aYO

(2.18b )

the Banach space defined by

= sup (T-t)rI1f(t)l[X < ~}. r£t
(2.18a)

if X is a HAlbert space and r any real

C r ( [ r , T ] ; X ) denotes

Cr([T,T];X)

(2.15)

(2.19)

O ~ r < T, the

and the optimal s o l u t i o n yO(t,T;X)

U-valued and Y-valued

functions w h i c h are

8u ° in t ~ (T,T) with ~ (t,r;x) ~ U,

(t,r;x) e Y.

In fact

functions uO(t,r;x)

these U-valued and Y-valued

and yO(t,r;x)

are analytic in t E (s,T)

if the operator A has compact resolvent

in Y.

11

(x)

The operator

P(%)

is g i v e n

( c o n s tructively)

by

T . . P ( t ) x = ~ e A (r-t)R*RyO[~,t;x)dT+eA (T-t) G * Gy 0 (T,t;x).

(2.20)

t

(xi)

If we d e f i n e

the e v o l u t i o n

~(t,r)x the follow~ng

operator

= yO(t,r;x),

weak convergence

(2.21)

r e s u l t s hold true:

l i m ( G ~ ( T , t ) X , z ) Z = (Gx, z), tTT

v x ~ X, v z ~ Z;

(2.22)

z

llm(P(t)x,y)y t~T

Remark

2.1.

With reference

closed operator

(GLT)

= (G Gx, y),

V x , y E y.

to the a s s u m p t i o n

|

of T h e o r e m

be elosable

W ~ ~((GLT)

L 2 ( O , T ; U ) D ~(GLT)

) ~ L2(O,T;U)

be densely

we have GL T

as an o p e r a t o r ~ W

(2.24)

defined

(-A*)P/2G * be densely an o p e r a t o r W ~ ~ ( ( - A for s o m e ~ > 2~-1 The e q u i v a l e n c e

2.1,

densely defined operator

be

densely defined as a n o p e r a t o r

LTG

(2.23)

defined as •

)p12 G * ) ~ Y

(2.25)

is a s t a n d a r d

result

we compute

[K.I,

from

p.

168].

To see

the

sufficient

condition,

(2.2),

{L;S*z}(t)

= B*e A ( T - t ) G * z = B*(-A*)-~(-A*)~-~/2e A ( T - t ) ( - A * I ~ / 2 G * z ,

(2.26) use

(1.3),

and notice

that

{ - A ' ) ~ - P / 2 e A (T-t) ~ ~ ( y ; L 2 ( O , T ; y ) )

for

2~-p < 1. We e m p h a s i z e

that c o n d i t i o n

not i n v o l v e B - - i s only sufficient be c l o s a b l e , example

which

in S e c t i o n

instead 2.3.

(2.25)

on

(-A)P/2G

for the u l t i m a t e

involves

B.

- - w h i c h does

requirement

that GL T

This will be s e e n in one

12

Remark

2.2.

assumption

2.2.

(i) A n e x a m p l e that

The s m o o t h i n q In this

we s h a l l

in a d d i t i o n

that G ~s a s m o o t h i n g (-A*)~G*G

(which

the D R E

is a u t o m a t i c a l l y stronger

(2.14)

2.3 w i l l

(closable)

show

cannot

that

the

be d i s p e n s e d

with.

case

subsection,

assume

accordingly,

in S e c t i o n

GL T be c l o s e d

e Z(Y),

satisfied results

ks u n i q u e

and

to GL T b e i n g operator

(closable),

in the s e n s e

that

for s o m e ~ > 2~-1

(2.27)

w i t h @ = 0 if 0 ~ ~ < ~).

follow. the

closed

In p a r t i c u l a r

limits

as

Then,

the s o l u t l o n

t~T of T h e o r e m

to

2.2 are

strong.

Theorem (i)

2.2.

[D-I],

(Regularlty

[L-T.22]

of o p t i m a l

Assume (2.27). pair)

For

Then:

x e Y and a n y a > O,

luO(',r;X){c1_~_¢([r,T];U)÷170(',r;×)IC([r,T];y) ~ CT~IXl Y, (2.28)

yO(T,.;x) from which

= @(T,')x

for a n y

(2.29)

in p a r t i c u l a r lim @ ( T , t ) x t~T

(ii)

E C((T,T];Y),

= x,

(2.30)

x e Y;

(-A *)Sp( t)x E

0 < 8 < I, & > O, x ~ Y, we h a v e

C e + 1 _ 2 ~ + a ( [ 0 , T];Y)

-< CT~

{ (-A* )ee(t)Iz(y)

1

(2.31)

;

1-e iT-ti (iii)

B*P(t)

E ~(Y;CI_7_a([0,T];U)),

CT

*

tB P(t)XlU ~ (iv) (v)

lim P(t)x tTT (uniqueness) Eq.

(2.20),

the s o l u t i o n of

1

z-~ (T-t)i-~-~ = G Gx,

P(t),

the D i f f e r e n t i a /

(2.32)

i.e.,

Ixl

Y; (2.33)

x E Y;

given Riccati

constructively Equation

by

(2.14)

a n d of

13 the terminal

condition

(2.29)

self-adjoint

operators

P(t)

,_ B P(t)x

Theorem

2,3.

is unique w i t h i n

the class

~C~([O,T];U)

if 0 ~ ~ < ~, where p = O,

(2.34)

LCI_~_~([O,T];U)

if

(2.35)

[F.1],

[L-T.22]

~ ~ I

Under

< 1,

the a s s u m p t i o n

(-A~)~G*G ~ ~(Y}, which

is stronger

regularity

(ii)

than a s s u m p t i o n

results

hold true,

(2.27)

B P(t)

2.3.

[L-T.22]

All

(or

the optimal general

control

incorporating

defined

the above

[L-T.4]).

strategy

singularity

< q, additional

x e Y;

CT ~ ~

control,

where

[Las.4]. identity

results

problem

an idea of

defined

[D-I.I]

(2.37)

the constant

that

P(t)

(2.39)

q in

(1.3)

(iii)

finally,

(see

constructive

and explicit

the spaces

parabollc boundary

z > 0 [T.5],

G was taken

[T.6],

to be the

case.

The

is explicit and c o n s t r u c t i v e 0 0 pair u ,y is c h a r a c t e r i z e d (see

(2.20)) in terms

the operator

Equation

describe

a non-smoothing

the

in [L-T.4]

hence u l t i m a t e l y

Riccati

follows

(from

while

second-order

is 7 = ~+~,

the o p e ra t o r

the optimal

is c o n s t r u c t e d

the problem;

which

at t = T via the B a n a c h

certainly

introduced

Differential

from

approach

d o m a i n ~ of R n with Dirlchlet

in [L-T.4],

(1) first,

evolution,

•

[L-T.4],

to q u a n t i t a t i v e l y

in terms of the data of the p r o b l e m

an o p e r a t o r

12.381

equation)

treated a general

on a bounded

approach

;

2.1 are taken

contribution

quantities

G = I (with Z = Y),

in the sense

of T h e o r e m

to the Riccatl

[L-T.4]

Moreover,

variational

e Z(Y:C[O,T]:U)).

of the original

in (2.19).

I .iT_t)S_q

They are proved by a v a r i a t i o n a l

of the various

equation

optimal

2q-i

for any 0 < 8 < ~,

(iii)

solely

since

~ CTl×ly ,

[l-A*lePltl[z(y)

Remark

(2.36)

namely

IuOt-,r;X) IC((T,T];U)

(i)

of

such that

(li) next,

in terms of original of the original

P(t)

is shown

and its limiting

approach

(2.3)-(2.5);

is used also

to s a t i s f y

condition

and

data of

as t~T.

for the results

the (This

of the

14

s u b s e q u e n t sections for the class p r e v i o u s l y studied control,

(H.2) = (1.6).)

The case G = 0 was

(also for the p a r a b o l i c p r o b l e m w i t h D i r i c h l e t

and also by abstract methods)

in [B.2].

The p r e s e n c e of the

p e n a l i z a t i o n operator G in (1.2) introduces additional g e n u i n e difficulties. compensate'

Qualitatively,

the a n a l y t i c i t y of e At tends

'to

the effects of the u n b o u n d e d n e s s of B on any interval of

the type [0, T-&l, non-smoothlng

V & > 0 small.

Instead,

the p r e s e n c e of a

o p e r a t o r G p r o d u c e s a s i n g u l a r i t y at t = T for J

" * {LTG Ge AT x}(t) = B'e A (T-t)G*GeAtx,

formula

w h i c h occurs in the explicit

(2.3) for the optimal uO(t,O;x).

This is r e f l e c t e d by the

q u a n t i t a t i v e s t a t e m e n t s of T h e o r e m 2.1: (2.17) for uO; and (2.18b) y 0 w h e n ~ ~ ~ < I, where the s i n g u l a r i t y is m e a s u r e d by the spaces

(2.19).

for

This s i n g u l a r i t y is p r o g r e s s i v e l y r e d u c e d in T h e o r e m 2.2 under

the s m o o t h i n g a s s u m p t i o n eliminated,

see (2.37),

if further s m o o t h i n g is imposed on G as in

(2.36) of T h e o r e m 2.3. statements

(2.9),

(2.27) on G (vacuous if 0 < ~ < ~) and finally Likewise,

(2.32), and

it is i n s t r u c t i v e to compare

(2.39) of i n c r e a s i n g r e g u l a r i t y for the

gain o p e r a t o r B*P(t) under p r o g r e s s i v e l y s t r o n g e r s m o o t h i n g assumptions on G.

The above considerations,

that for 0 ~ q < ~, surprising.

the optimal y

in p a r t i c u l a r (2.18a) and (2.28), 0 is in C([O,T];Y). This is not

show

In fact, s t a n d a r d r e g u l a r i t y p r o p e r t i e s on a n a l y t i c

s e m i g r o u p theory yield the w e l l - k n o w n result that if q < ~ in (1.3), then the operator L in (I.I0) is c o n t i n u o u s L2(O,T;U)) thus every solution

natural

s o l u t i o n of

and

(1.1) w i t h YO E Y --not only the optimal

yO --lles in C([O,T];Y)! 'cutting line'

~ C([O,T];Y)

Thus,

the value ~ = ~ gives the

in the range of values of ~, w h i c h c r u c i a l l y

bears on the d e g r e e of technical d i f f i c u l t i e s present in the analysis. The case q < ~ behaves like the outset

'B-bounded'

case and one has at the

the ~mportant p r o p e r t y that any s o l u t i o n y(t),

optimal s o l u t i o n yO(t,O;Y0),

more d e m a n d i n g if instead M ~ ~ < i. -uO(.,r;x)

•

•

= [Ir+LrR RLr]

-1

.~_~_

{urK ~e

We have, A(.-T)

for the optimal control p r o b l e m on [r,T],

from *

*

in p a r t i c u l a r the The s i t u a t i o n is

belongs to C([O,T];Y).

(2.3)-(2.5), 0

x + L r T G Gy (T,r;x)}

(2.40)

0 < r < T, w h e r e L T, LrT are

the o p e r a t o r s L in (l.lO) and L T in (2.1) s t a r t i n g now from r rather than O.

Crucial

to the proof of s t a t e m e n t s

for yO is the key p r o p e r t y that

[Ir+L?R RLr]

(2.17)

for u

0

and

(2.18b)

~ ~(C~([T,T];U))

with

15

uniform

b o u n d w h i c h may be taken

accomplished

via a b o o s t - s t r a p

independent

argument

of r.

This

is

from the a-prlorl

starting

L2-regularity and u s i n g the smoothing properties of regularity of the operators

L and L

in (2.5) satisfies

.

cruclal

except that the process -i /[rT G ~ ( C ~ _ ~ ( [ r , T ] ; U ) )

taken I n d e p e n d e n t A boost-strap

of r.

This

technique

that the o p e r a t o r A(~:U)

Similarly,

based

is b o u n d e d l y

on the interval

as those of P(t), class

(H.1)

in the sector This step

properties of {uO, yO},

The r e g u l a r i t y

analytic

set,

pair

see

(2.7),

= (1.5).

available

for the class

key fact

in e s t a b l i s h i n g

(2.8),

of a n a l y t i c l t y

is crucial

((ix)

see

in subsequent

well-posedness

of

2.1).

(ix) of T h e o r e m

are dlstinctlve sections.

on

to obtain

of T h e o r e m

They should he c o n t r a s t e d

(H.2)

w h i c h shows

in the space

on ~ and c o n t i n u o u s

[O,T].

of the optimal

argument.

in [L-T.4]

Invertible

functions w h i c h are analytic

the analyticity p r o p e r t i e s

as well

bound w h i c h may be

is also done by a b o o s t - s t r a p

~, w h e r e ~ is an open symmetric

2,2 in

(same as 2[OT

now at T rather than OJ

starts

with a u n i f o r m

is also behind the proof

[I+L R RL]

of U - v a l u e d

exp(At),

to the proof of T h e o r e m

case is the key fact that the o p e r a t o r A r T

the s m o o t h i n g

2.1,

of the

with

those

A common goal--a

of the Riccatl

equatlon--is

s

that the g a i n o p e r a t o r

clear w h e n B is unbounded: in the s m o o t h i n g

Remark

2.4.

appr o a c h

Another

approach,

in Remark

from a direct

to the optimal

proposed

in [F.I],

taken to be

global

'smoothing'

bound),

In these references, [Da-I~

and

(2.27)

the s o l u t i o n Moreover,

'direct'

a typical

of the v a r i a t i o n a l (as it proceeds

programming)

of a s s e r t i n g

assumption

G is

a unique

argument and

condition on G are

as tTT. (2.S6)

in which case existence and u n i q u e n e s s Riccati

the various

Equation

quantities

in

is

Here the o p e r a t o r

(by local c o n t r a c t i o n

smoothing

(2.39)

of the Riccatl

as well as for the limiting

for IF.I],

(2.27),

[DaP.1].

for both the purposes Equation

case and

facts.

p r o b l e m via dynamic

to the Differential

under

so-called

following

is not a-priorl

in the general

of this

in a sense a converse

2.3,

control

[D-I.I],

of the Rlccati

a-priori

(2.10)

which

study of the w e l l - p o s e d n e s s

Equa t i o n

solu t i o n

Eqns.

case are statements

described

reverse

be well defined,

B P(t)

for of

is asserted.

u0:y0

(_A*)Sp(t),

s

8 ~ ~; B P(t) stated

do not experience

in T h e o r e m

G = 0. Equation

2.3,

thereby e x t e n d i n g

In a m o r e recent work for e x i s t e n c e

singularity

IF.5],

at t = T any

longer

the theory a v a i l a b l e

as

for

the direct s t u d y of the Riccati

(not for uniqueness)

is carried

out

in the non-

16 smoothing

case for G.

(closable)--a [L-T.4],

Instead

natural

of a s s u m i n g

hypothesis

[L-T.22]--[F.5]

makes

that GL T is closed

on G in the v a r i a t i o n a l

approach

of

the following a s s u m p t i o n

on G, say in

G n E ~(Y,W)

such

the case G ~ Z(Y,W): there (a)

exists

a sequence

{G~Gn}

Gn(-A}$/2

~ ~(Y,W);

is a n o n d e c r e a s i n g

family of s e l f - a d j o i n t

operators w h i c h c o n v e r g e s m o n o t o n ~ c a l l y G G in the sense

to

that as n ~ ~:

IiGn xlt2 T

(GnGnX, X) Y

ltSxlI~ ,

(G*GX, X)y

v x E Y Under

this a s s u m p t i o n

solution which

P(t)

convergence

(2.23)

of T h e o r e m

approximation assumption

than

invo l v e s

B.

(i)

of a

(2.7),

among

obtains

also

uses

the in

monotonic 2.3

that

invoke B, o n l y A and G--is of T h e o r e m

2.1,

which

an a p p r o x i m a t i n g

by the two p r o p e r t i e s

Riccati

problem

condition

is h a n d l e d principle

G*G nn

and [F.5]

see in S e c t i o n

does not

3.2]

existence

on the left side),

(versus w e a k c o n v e r g e n c e

GL T c l o s a b l e

Thm.

(2.41)

.

argument

(a) and

(b) of

(2.41):

The a p p r o x i m a t i n g symmetric

(ii)

IF.5,

reflected

(2.6)

of the p o s t u l a t e d

in [F.5]--which

of

shows

(P(t)x,y)y

In addition,

We shall

the a s s u m p t i o n

on two steps

assumption

because (b).

3,2]

properties

of P(t) ~ G G as tTT

property

The proof

Thm.

(with ~ t

2.1 above.

2.1),

(2.41)

stronger

[F.5,

(2.14)

the r e g u l a r i t y

as in T h e o r e m

strong

based

(2.41),

of the DRE

satisfies

others,

that

there exists ~ > 2q-1 s u c h that each G n s a t i s f i e s the a s s u m p t i o n

(b)

of o p e r a t o r s

(-A)

by the general followed

solution

involving

under

n

~/2G*nGn (_A)~/2 strategy

by a-priori

operator

G

Pn(t)

~ ~(Y),

of a local

estimates

the a s s u m e d some ~ > 2~-I,

contraction

to y i e l d a global

of the DRE

(2.14)

with endpoint

at t = T;

the general

case u n d e r a s s u m p t i o n

approximating

(2.41)

from b e l o w s u c h Riccati

is then b a s e d

solutions

Pn(t).

on

17

The c o n d i t i o n in (a) on Gn(-A

*

p/2 ~ Z(Y,W) is implied by (2.27)

)

[F.I, Lemma 3.1] but does not imply (2.27),

Thus the a p p r o x i m a t i n g

problem in (i) is not fully covered by Theorem 2.2: the proof in [F.5] uses a r g u m e n t s w h i c h are new over those in IF.l],

[D-I],

based on a change of variable introduced in IDa-L-T]:

and which are

unlike

[F.I],

[D-I],

the more d e l i c a t e step in IF.5] is now local existence.

2.3.

Counterexamples It was i n d e p e n d e n t l y noted in [F.6] and [L-T.22; R e m a r k 5.1]

that s u i t a b l e o n e - d i m e n s i o n a l range

(finite range) o p e r a t o r s G furnish

examples w h i c h illustrate the sharpness and/or limitations of the theory p r e s e n t e d in Sections 2.1,

2.2.

2.3.1. C o u n t e ~ e x a m D l e to the existence of the optlmal control u O The example.

Consider,

say the heat equation defined on a (smooth)

bounded d o m a i n ~ c R n with L 2 ( 0 , T ; L 2 ( F ) ) - c o n t r o l boundary conditions, (6.1a).

as in Example 6.1, Eq.

Here Y = L2(n),

U = L2(F).

in the D i r i c h l e t

(6.1), w i t h c = 0 in exists # E Y,

There

l@I = I such

that T

flB*e A (T t'+t, dt _-

(2.42)

0

for then,

otherwise,

by transposition,

the map u ~ y(T)

(where YO = O)

would be c o n t i n u o u s L2(0, T;L2(F)) ~ L2(~ ) = Y, w h i c h is false even in the o n e - d i m e n s i o n a l

case,

e.g.,

[Lio.3; p. 217].

consider the a s s o c i a t e d optimal control problem R

=

0; Gy

=

(y,#)y#;

G*

=

G

Following

[F.6], we

(1.2) w i t h =

S

*S .

(2.43)

Note that we have by (2.1) and (2.43), T

$

G L T U = ()/ e A ( T - t ) B u ( t ) d t ' @ ) Y ~ = (u'B*eA (T-')#)L2(O,T;U) @

(2.44)

0 so that GL T is finite rank and u n b o u n d e d by (2.42), hence u n c l o s a b ] e

[K.i: p. 16e].

18 Claim.

There is no optimal

control

~n this case.

[L-T.22],

if an optimal

control uO(-,O;x)

satisfies

the present v e r s i o n of (2.3),

In fact,

following

= u 0 e L2(O,T;U ) exists,

i.e.,

0 " * 0 * * AT * A (T-t)~, -[U +LTG GLTU ] = LTG Ge x = (eATx,~)yB e where we have used (2.43),

(2.43) on G G and

it

(2.2) for L T.

(2.45)

Moreover,

by

(2.44),

•

L;G*GLTU

~

= LT{ILTU,~Iy@ } = (

T

eA(T-t)Bu(t)dt,~)yB

* e A (T-tl~

.

12.46)

0 Using -u

(2.46) 0

in (2.45) yields

= {(uO, B*eA

Since B*eA

)#)L2(O,T;UI+(eATx,#)y}B*eA

(T-t)@ ~ L2(O,T; U) by

u O ~ L2(O,T;U),

R e m a r k 2.5. possible

(T-.

(2.42),

a contradiction.

the following

in the present

optimal problem

that there exists a unique optimal

(ii)

that there exists P(t), such that identity differentiable, satisfied,

@ ~ ~((-A*) ~/2)

(2.13)

it is not to satisfy

@ E ~((-A*) p/2)

self-adjoint,

holds; (P(t)x,x)

P(t)x ~ ~((-A*) ~) and the DRE

is

12.14)

is

i

for all ~ > 2~-i,

(2.43) does not s a t i s f y a s s u m p t i o n

(2.42)

and hence

for ~ implies [F.5; Sect.

12.41) of IF.5].

that

3.1],

In fact,

G in if we had

we would obtain that

B*e A (T-t)# = B * ( _ A * ) - I ( _ A * ) ~ - P / 2 e A would belong to L2(O,T;U) * ~/2 ~((-A ) S

case,

(2.43)

0 ~ t ~ T, non-negatlve

We note that the choice

contradicting

11.2),

0 control u ;

that for every 0 ~ t < T and x ~ ~(A),

Remark 2.6.

that

three desirable properties:

(i)

(iii)

(2,47) yields

•

It is argued in [F.6] that,

for the c o r r e s p o n d i n g

then

(2.47)

(T-t)#.

(2.42).

) = {0},

(T-t)(_A*)P/2 #

(2.48)

by (1~3) and a n a l y t i c i t 7 with 2q-p < I, thus

We note that in this case we have v p > 2~-I.

|

lg

2.3.2.

Assumption

(2.25)

We s h a l l p r o v i d e vloIated,

negative,

self-adjoint,

corresponding eigenvalues sequences

T h i s is not s u r p r i s i n g

GL T - - d o e s not i n v o l v e B.

maintain

orthonormal

Let (en,

Let ~i"

integers

i = 1,2,

that e x h a u s t

~ntegers z: ~1 U ~2 = z; ~1 A ~2 = ~" ~2 = {n = 1 , 3 , 5 , - . . } .

Consider

Y = Y1 + Y2" Let ~ i

be the o r t h o g o n a l

e

~I = {n = 2,4,6,...}, decomposition

of Y

i = 1,2.

(2.49)

so that ~i c o m m u t e s

a n d Y. are i n v a r i a n t 1

under e

I

for all ~ > 2q-i,

~

~ l ( b , en)yI 2 =

At

-

ne~ I

l O,

(2.50)

n e ~2

so that b ~ ~((-At)P/2),

Next, w i t h U = Y = W, d e f i n e

unbounded o p e r a t o r s

One r e a d i l y o b t a i n s

the b o u n d e d

> 2ff-1.

operators

(2.51)

G ,G and the

Yi = ~i y e Yi;

Gy = (Yl,b)ya+Y2;

=

;

(-A)~y2

Y p

B ,B by

G*y = ( Y l , a ) y b + Y 2 ;

Ls72

all of the p o s i t i v e

Example:

in n E ~i s . t .

sequence

0

disjoint

b ~ Y1 by s e t t i n g

(b, e n ) y =

=

be the

be two infinite,

of Y onto Yi'

At

(We shall,

of A on Y w i t h

Yi = s p a n {e n, n e ~i} , projection

is

as c o n d i t i o n

n = 1,2,...}

the o r t h o g o n a l

with A, h e n c e w i t h the s e m i g r o u p Define a v e c t o r

resolvent.

b a s i s of e i g e n v e c t o r s

{ - p n }, P n > O.

(2.25)

Let the g e n e r a t o r A be

say with compact

the n o t a t i o n A=.)

of p o s i t i v e

for GL T to be c l o s e d

a c l a s s of e x a m p l e s w h e r e c o n d i t i o n

yet GL T Is closed.

(2.25)--unlike

however,

is o n l y s u f f i c i e n t

LB 72

O;

71

= ITly e Yl

1-A*)~72 ; 72

by (2.52),

(2.51)

a e Y.

that

.27

e Y2 n

(2.52)

.

~((-a) ~)

{2.53)

20 ~((-A

*)P/2G*

(-A *)~/2

) = ~((

G" y

_A*)Pl 2 )

n Y2;

= (-A *)~/2y2,

Thus, ~ ( ( - A " ) ~ / 2 G ~) is not dense

y

e

9((

-A*}~/2G * ) .

in Y2" and c o n d i t i o n

(2.25)

(2.54) is

violated. On the other hand, s i n c e B~lu(t) and e At • we obtain by (2.1) and

(2.52)

~ 0 and Y2 is invariant under A

p

T GLTU = G~ e A ( T - t ) B u ( t ) d t O T

T

Gf eA(T-tlB~lu(t)dt + G~ eA(T-t)B~lu(t)dt

=

0

0 T

T

= G~ eA(T-t)l-Al~g2u(t)dt 0

= (-A) 1 ~ eA(T-t)g2u(t)dt, 0

w h e r e in the last step we have u s e d in Y2"

Thus,

p. 164].

(2.52) on G, w i t h the integral term

GL T ~s a closed o p e r a t o r

boundedly invertible operator

(2.55)

(being the product of a closed,

(-A) ~ and of a b o u n d e d o p e r a t o r

Our claim is proved.

Note that,

by (2.2),

[K°I;

one likewise has

S

= * y}(t) = ( - A=) Y e A {LTG

(T-t) y2,

Y2

= ~2 y e Y2"

Z

2.3.3. V a ~ t i o n a l versus d~rect approach: A s s u m p t i o n direct a p p r o a c h fails, vex GL T is closed We have a l r e a d y noted that a s s u m p t i o n a p p r o a c h of IF.5; Thm. 3.2]

(2.41)

(2.41) of the

for the direct

in the case w h e r e G is non-smoothing,

involves only the o p e r a t o r s A and G, not B.

Instead,

the a s s u m p t i o n of

the v a r i a t i o n a l a p p r o a c h of T h e o r e m 2.1 that GL T be c l o s a b l e all the data of the problem:

G, A, and B.

Thus,

provide new classes of examples where a s s u m p t i o n is closed.

Thus,

of

is.

[L-T.22]

[F.5; Thm. 3.2]

involves

not surprisingly, (2.41) fails,

~s not applicable,

we

yet GL T

w h i l e Theorem 2.1

We return to the example of Section 2.3.2 and set

GlY = (Yl,V)YlV,

G 1 = G 1 = GIG I,

v ~ YI"

Ivl = i,

(2.56)

21 where we recall It follows G 1 in

that

the s u b s p a c e s

from an o b s e r v a t l o n (2.56)

satisfies

in [F.5;

assumption

*)P12)

•=~ v E ~ ( ( - A Next,

choose

(2.50),

GIy =

(2.51).

Thus,

Sect.

for

under A,

the o p e r a t o r

it follows

G = GI+I 2 does not s a t i s f y G 1 as ~n (2.58):

as seen

Remark

2.7.

in S e c t i o n

2.3.2,

B o t h approaches,

in [L-T.22]

can be e x t e n d e d

readily

Remar~u~.8.

we h a v e s e e n i n

(i)

~ as in

direct a p p r o a c h (see b e l o w

Suppose

the a s s u m p t i o n s ¢losable

[F.5]

(see R e m a r k

2.6)

on Yl"

(2.~8)

G,

(2.59)

on Y2"

[]

and

the v a r i a t i o n a l

G to be unbounded:

2.3.1

satisfies

in

on Y;

not be p u r s u e d

Section

neither

(2.57)

defined

the o p e r a t o r

in IF.5]

to a l l o w

2ff-1.

here.

that

the

assumption nor does

[]

operator (2.41)

it m a k e

G in of

the

GL T c l o s a b l e

(2.44)).

(li)

proved

of

Thls will

(2.42)

that

identity

the d i r e c t

p > 0.

with

=

>

G1 ,

(2.41)

(2.41)

p

vector

GL T is c l o s e d .

G e ~(~(-A)P,Y),

(2,43)

12

that

some

(normalized)

(Yl,b)Ylb does not s a t i s f y

Since YI are Invarlant

Yet,

3.1]

(2.41)

n Y1

v = b, with b 6 Yl the

satisfying

u n d e r A and e At

Yi are i n v a r l a n t

and

that G(-A) ~/2

of both a p p r o a c h e s

(li) a s s u m p t i o n

in IF.5;

Section

is c l o s a b l e

are satisfied;

(2.41)

3.4.1].

for some $ > 27-I.

holds

Statement

then GL T = G ( - A ) $ / 2 V T is the p r o d u c t

true. (1)

i.e.,

(I) GL T is

Statement follows

of a c l o s a b l e

Then

(ll)

at once,

operator

is since

and of a

T bounded

operator

VTU = ~(-A)PeA(T-t)(-A)-ffBu(t)dt, 0

that V T ~ ~(L2(0, T ; U ) , Y ).

[]

p = ~-~/2

< ~, so

22 3.

Abstract Differentlal Riccatl Euuatlons for the s e c o n d class subject to the trace reuularltv a s s u m D t l o n fH.2) = (1.61 We shall

first provide

(Section 3.1)

synthesis

of t h e optimal pair under

smoothing

required on the o b s e r v a t i o n

mlnlmal

assumptions

in the specific in the D i r i c h l e t

of smoothing

pointwise

synthesis

constructed) does,

lheorem

3.1.

[L-T.6,

R e ~(Y;Z}.

including

satisfy the DRE. of s m o o t h i n g

y

Finally,

by

on R, u n i q u e n e s s

of the

(Section 3.3).

[F-L-T,

Thm.

2.1]

Ricca~

We assume

(H.2) = (1.6) on the dynamics

the

and that,

moreover,

Then:

there is a unique s o l u t i o n pair of functions 0

the claim in the

Here the theory is

smooth initial data.

of ODtlmal Dalr and candidate

Thm.l.3],

regularity h y p o t h e s i s (1)

in fact,

~equirement

svnthesls

wlth n o

under some

e q u a t i o n s with control

(Section 3.2),

operator will also be claimed Polntwlse operator

(H.2),

Next,

operator P(t) w h i c h occurs

for a p p r o p r i a t e l y

imposing an a d d i t i o n a l

3.1.

operator R.

(in time)

on R, further results will be provided

b o u n d a r y conditions

(explicitly

Riccati

the sole a s s u m p t i o n

case of second order hyperbolic

that the

rather complete

the p o l n t w i s e

= yO(t,O;Yo),

(1.1),

0 <_ t _< T, of the optlmal

u 0 = uO(t,O;Yo ) and

control

problem

(1.2) which satisfy u

0

E L2(O,T;U),

yO e

C([0, T];Y);

(3.1}

s

(ii)

u 0 and yO are related by

(recall L, L

in (1.9), (1.10);

LT, L T in

(2.1}, (2.2)) uO(.,O;Y0)

= -L*R*R{yO(.,O;Y0)}-L;G=Gy0(T,O;Yo )

(3.2)

and e x p l i c i t l y given by (see IOT in (2.5)) -1 • * A. * • AT -uO(t,O;Yo ) = {J[oT[L R R[e yo]+LTG Ge yo]}(t), m s --1 A. = ~= 0 yO(t,O;Yo) = {[I+LL R R] "[e- Yo-LLTG Gy (T,O;Yo)]}(t)

[I+LL=R'R] -1 = I - L [ I + L = R ' R L ] - I L * R = R (iii)

there exists an operator P(t) ~ Z(Y),

~ Z(L2(O,T;Y)); given explicitly

(3.3) (3.4)

(3.5) by

23 T

$

s

P(t)x = ~ e A (r-t)R'Ry0(r,t;x)dr+eA

(T-t)G*GyO(T.t;x),

xE

Y

(3.6a)

t

(3.6b}

: continuous Y ~ C([O,T];Y);

(iv)

(pointwlse

feedback synthesis) "

uO(t,O;Yo ) =-B

P(t)y

0

(t,O;Yo)

a.e.

in [O,T];

(3.7)

T

(v)

= fiRy0iT, t

(P(t)x,y)y

t;x),RyO(y,t;y)zdT+iGy0(T,t;x),Gy0(T,t;Y))w

T

x , y e Y;

+ f(u0 (r, t ;x), u0 (r, t ;Y)udr,

(3.8)

t

Pit)

ivi)

= P (t)

.> 0,

0 _< t .< T;

(3.9)

(P(O)x, XJy = J ( u ° i . , o ; x ) , y ° ( . , o ; x ) ) .

(vii)

(viii) The operator

P(t),

if the dynamical ~tit)

0 ~ t < T, is an i s o m o r p h i s m

system

g: ~.e.,

on Y if and only

(in short the pair {A*,R'})

= A ~ ( t ) + R g(t),

is exactly c o n t r o l l a b l e L2-contro/s

i3.1o)

on Y over

~iO)

= O,

[O,T-t]

from the origin wlth

the totality of s o l u t i o n p o i n t s ~(T-t)

fllls all of Y as g runs over ali of L2(O,T-t;Y). Property

(vlil)

Remark 3.1. approach.

Is further p u r s u e d

in S e c t i o n 3.4.

Theorem 3.1 is proved in [L-T.6], defined operator P(t)

(iv) requires

acting on the optimal

that the of the DRE;

Note that the s y n t h e s i s

that the operator B P(t) trajectory.

by a variational

is a bonafide s o l u t i o n

see more on this in S e c t i o n 3.4 below. property

[F-L-T]

There is no claim in the above g e n e r a l i t y

constructively

•

Instead,

be well d e f i n e d as

the DRE w o u l d require that

s

B Pit) be well defined on, say, ~(A). regularlty p r o p e r t y addition cause

for the operator

L under a s s u m p t i o n

to (1.2) of a final state p e n a l i z a t i o n

now essential

situation of T h e o r e m y(t,r;x),

(1.12)

Because of the general

changes

to the analysis,

2.1 for t h e class

operator

in contrast

(H.I).

x ~ Y --not only the optimal s o l u t i o n

In fact,

(H.2),

G will not to the any s o l u t i o n

yO(t;r;x)

--Is

24 c o n t i n u o u s in t, r ~ t ~ T for r fixed. t ~ y0(T,t;x)

is a l s o c o n t i n u o u s

o p e r a t o r x ~ y0(t,r;x)) 3.2.

This then y i e l d s that the map

(using the e v o l u t i o n p r o p e r t i e s of the

and this fact is n e e d e d in (3.6a).

The DRE fo~ ~ e ~ o n d - o r d e r h y p e r b o l i c e q u a t i o n s w l t h D i r l c h l e t c o n t r o l L E X i s ~ e n c e ~Dd DroDertles In this s e c t i o n a m i n i m a l a s s u m p t i o n of s m o o t h i n g is i m p o s e d on

R, w h i c h will then y i e l d that the operator P(t) the DRE,

D i r i c h l e t control, shall s p e c i a l i z e

of w h i c h

(7.1)

the d y n a m i c s

wtt = -Aw + ADu; with y(t)

~n (3.6) d o e s s a t i s f y

in the case of s e c o n d - o r d e r h y p e r b o l i c e q u a t i o n s w i t h

= [w(t),wt(t)],

is a canonical example.

Thus,

we

(1.1) to

i.e.,

to A =

_

,

Bu =

ADu "

of the form that arises in m i x e d p r o b l e m s for

s e c o n d - o r d e r h y p e r b o l i c e q u a t i o n s on a bounded d o m a i n ~ c R n, w i t h

Dirlchlet control, such as (7.1).

In (3.11), A is (for s~mplicity)

a

p o s i t i v e s e l f - a d ~ o i n t o p e r a t o r on X = L2(~) w i t h compact resolvent,

D

the D l r i c h l e t o p e r a t o r in ~(U;X),

U = L2(F) , d e f i n e d by

t h r o u g h o u t a s s u m e d that the r e g u l a r i t y h y p o t h e s i s

(7.4).

It Is

(H.2) = (1.6) holds

true for A and B as in (3.11) on the space y m X x [~(A~)] " = L2(n)×H-I(Q),

w h e r e d u a l i t y is w i t h respect to the X-topology;

=

e

(3.12)

and moreover,

that

(3.131

Z(Y;Z).

R2 Thus.

Theorem

T h e o r e m 3.2.

3,1

holds

[L-T.6]

RIRI:

true.

Moreover,

(a) Assume,

in addition,

c o n t i n u o u s H~-2&(~)

that

~ ~(A ~-6) ~ ~(A~),

s

R2R2: c o n t i n u o u s H-~-26(n)

= [~(A ¼+6) ] " ~

[~(A ¼1 ] ",

(3.14a) (3.14b)

% w h e r e R 1 is the X - a d ~ o l n t of R 1 and R 2 is the [~(A~)]'-ad~olnt of

R2,

so that

(3.14a-b)

c o l l e c t i v e l y mean that R~R: c o n t i n u o u s

25

Yr ~ ~(A¼)x[~(A¼)]'" space

(of regular

where R" is the Y-adjolnt

data}

Yr is defined

Yr ~ ~(A¼-6)x[~(A¼+6)]" Then,

the following

regularity

holds

of R and where

the

by = H½-26(n)xH-~-26(n)" true for the optimal

(3.15) pair {uO,y0},

O for initial data YO = [Wo,W 1 ] m Yr: yO[wO, wt] u 0 e H~'~(X),

a fortJori

w 0 ~ C([O,T];~(A~-6))

u 0 ~ C([O,T];La(F));

(3.16)

n H~-2$(0, T;X);

(3.z7)

w 0t E C([O,T];[~(A~+6)] ". (b) Let now {RI,R2}

satisfy,

(3.18) in addition

to (3.13),

R~RI: ~ontinuous ~-261n} = ~(A ~'~) ~ ~(A~+~) = .~+2~In);

(3 ~gal

R~R2:

(3.19b)

(slightly

continuous

more

H-~-26(n)

restrictlve

= [~(A~+&)] " ~ [~(A~-6)] "

than

(3.14}).

Then

8

(bl)

(3.20)

B P(t) E ~(Yr;C([O,T];L2(F)). (Existence)

(b 2 )

by (3.6a)

The non-negatlve satisfies

self-adjoint

operator

P(t) defined

the DRE for all x,z E Yr" and O ~ t < T,

( P ( t ) X , Z ) y = - ( R x , R Z ) z - ( P ( t ) x , A Z ) y - ( P ( t ) A x , Z)y+(B P ( t ) x , B P ( t ) z ) U ;

I PIT)

= G*G.

(3.21)

Remark 3 . 2 . The above result was originally proved in [L-T.6] In the case of the wave equation (or second-order hyperbolic equations) with Dirlchlet

control,

by a combination

of abstract

methods

and p.d.e.

methods (once the regularity property (H.2) = (i.6) has been ascertained [L-T.2], [Lio.l], [L-L-T]. In this case we have equivalent

(with

norms)

X = L2(n);

~(A~)

= H~(~);

[~(A~)] " = H-I(~);

(3.22)

26

~{A~-6)

= H~-2~(~) ; ~(A~+6)

= .~+2~ "0

Y = L2(QlxH-I(Q):

Yr

H--Y~-2~(~ (n);

[~(A~+6)]

" =

= H~-2~(~)xH-~-26(~}'

A key issue in the proof of

(3.16)-(3.18)

);

(3.23) (3.241

is that

[IT+LTL;R*R]-I

Z(Y;L2(0, T;Yr)) w i t h u n i f o r m bound w h i c h may be taken i n d e p e n d e n t of r. Th~s is a c h i e v e d by c o m p a c t n e s s arguments. f l r s t - o r d e r h y p e r b o l i c systems,

A c o m p a n i o n paper for

see Section 7.4 below,

is [Ch-L].

The

c o m b i n a t i o n of a b s t r a c t and p.d.e, methods of these papers should be e x t e n d a b l e to, say, some f o u r t h - o r d e r operators A. 3.3.

DRE:

B

E x i s t e n c e and u n l q u e ~ e s s

Imposing a s t r o n g e r a s s u m p t i o n of s m o o t h i n g on the o p e r a t o r R yields also u n i q u e n e s s of the Riccati operator. (H.2) = (1.8) on the d y n a m i c s and R • L(Y;Z)

In a d d i t i o n to

on the observation,

we may

assume in this subsection: (A.1):

the map R*R eAtB can be e x t e n d e d as a map:

continuous

U ~ LI(O,T;Y): T

~

*

IIR Re

At

(3.25}

BU[]ydt < CT[lUJlu.

0

(A.2):

the map B"eA=tG" can be e x t e n d e d as a map c o n t i n u o u s Y , L (O,T;U): I

sup IB*eA tG'xlU O~t~T T h e o r e m 3.3.

[DaP-L-T.1]

~ c~lXly

,

x-

Y

Under the above assumptions,

(3.26)

there exists a

n o n - n e g a t l v e s e l f - a d j o i n t o p e r a t o r P(t) = P (t) ~ O, 0 ~ t ~ T, g i v e n 0 e x p l i c i t l y by the same formula (3.6a) (where y is the optlmal t r a j e c t o r y g u a r a n t e e d by T h e o r e m 3.X(i})

P(t)

(1)

S P(t)

(li) (ill)

B*P(t)eA(t-T)B

such that

(3.27a)

G ~(Y;C([O,T];Y);

(3.27b)

~ ~(Y;C([O,T];U);

e Z(U;L2(T,T;U ) uniformly

i n T,

27 T sup I~IB'P(t)eA(t-r)Bu~l~dt ~ CT~[U~I~ . O~r~T ? (iv)

(3.28)

The u n i q u e optimal pair { u 0 , y 0} s a t l s f l e s the p o l n t w l s e f e e d b a c k synthesis property

(3.7)

(except that it is now for all

t G [O,T]) as well as p r o p e r t i e s

(v) = (3.8) and

(vll) = (3.10)

for the optimal cost of T h e o r e m 3.1.

(v)

(Existence) For 0 ~ t < T, the o p e r a t o r P(t) s a t i s f i e s the DRE (3.21),

n o w for all x,z e ~(A), s c o n d i t i o n P(T) = G G. (vi)

(Uniqueness)

as well as the t e r m i n a l

The o p e r a t o r P(t) g i v e n by formula

u n i q u e s o l u t i o n of the DRE as in point

(3.6a) is the

(v) above,

w i t h i n the

class of n o n - n e g a t i v e s e l E - a d j o l n t o p e r a t o r s w h i c h s a t i s f y properties Remark 3.3. method

(1) = (3.26),

(ii) = (3.27),

(ill) = (3.28).

The above result was p r o v e d in [DaP-L-T]

by a

(from the DRE to the optimal control problem):

•

'direct'

this first

e s t a b l i s h e s w e l l - p o s e d n e s s of the DRE (3.21) and next constructs, dynamic programming, original DRE. achieved,

the optimal control problem w h i c h g e n e r a t e s

W e l l - p o s e d n e s s of the DRE

{3.21)

via the

(for all x,y e ~(A))

f o l l o w i n g the original s t r a t e g y in [DaP.

is

] for the

"B-bounded" case, by a local c o n t r a c t i o n a r g u m e n t near T, followed by global a-prLor~ bounds.

This s t r a t e g y e n c o u n t e r s a d d i t i o n a l

d i f f i c u l t i e s to be sure. used in [D-L-T]. T h e o r e m 3.3(vi))

This w a y

In particular,

a new change of v a r i a b l e is

(existence and) u n i q u e n e s s

is obtained,

technical

(in the sense of

at the price of the s m o o t h i n g a s s u m p t i o n

(A.I) = (3.25) on R, a q u a n t i t a t i v e s t a t e m e n t

thereof will be g i v e n in

Remark 3.4 in the case of the w a v e e q u a t i o n w i t h D i r l c h l e t control and in S e c t i o n 7.4 in the case of f l r s t - o r d e r h y p e r b o l i c systems. Instead, approach

the prior w o r k

[L-T.6],

[C-L.I]

followed a v a r i a t i o n a l

(from the optimal control p r o b l e m to the DRE)

leading to

T h e o r e m 3.2 w h i c h has a m a r k e d l y w e a k e r a s s u m p t i o n of s m o o t h i n g on R, but does not claim uniqueness, R e m a r k 3.4. (3.11),

i

(Wave e q u a t i o n w i t h D i r i c h l e t control)

the w a v e e q u a t i o n

Dirlchlet control,

We n o w r e t u r n to

(or a s e c o n d - o r d e r h y p e r b o l i c equation) w i t h

to be analyzed in m o r e d e t a i l s ~n S e c t i o n 7.I below.

The r e l e v a n t spaces are g i v e n by (3.22)-(3.24} the r e g u l a r i t y a s s u m p t i o n s

(3.14),

or

(3.1g),

of R e m a r k 3.2.

Plainly

of T h e o r e m 3.2 hold true

28

if RIR i has a "smoothing a c t i o n of the order of A -e" (3.19b)

is equivalent

In contrast, assumption

to A ~ + $ R ~ R 1 A-y4+6 ~ Z(X),

it can be s h o w n IDa-L-T]

X ~ L2(~)

say

in our case).

that T h e o r e m 3.3 r e q u i r e s for its

(A.I} = (3.25) to be s a t l s f i e d that R~RI~+¢

3.4.

(technically,

Z(L2(Q));

•

R;R2A~+e E Z(H-I(n)).

(3.29)

N o n - s m o o t h l n u case: W e a k e r n o t i o n s of s o l u t i o n For the class

(H.2) = (1.6) of dynamics,

a s s u m p t i o n that R 6 ~(Y,Z), s e l f - a d J o l n t o p e r a t o r P(t)

under the sole

T h e o r e m 3.1 p r o v i d e s the n o n - n e g a t 2 v e in (3.6) n e e d e d for the p o i n t w l s e s y n t h e s i s

(3.7),

as well as several of its properties.

absent

in the statement of T h e o r e m 3.1 is a c l a i m that in this

g e n e r a l i t y such P(t)

What

is a s o l u t i o n of the DRE.

a v a i l a b l e at present,

is c o n s p i c u o u s l y

No s u c h c l a i m is

the m o s t g e n e r a l s t a t e m e n t s

of e x i s t e n c e

being

the ones of T h e o r e m 3.2 for s e c o n d - o r d e r h y p e r b o l i c e q u a t i o n s w i t h D i r i c h l e t control

[L-T.6],

h y p e r b o l i c systems

the r e s u l t s of [Ch-L.I]

(see S e c t i o n 7.4),

also T h e o r e m 4.1 w h i c h follows).

for f i r s t - o r d e r

and T h e o r e m 3.3 IDa-L-T]

W h e n R is o n l y in ~(Y;Z),

(see

lack of

(proof of) r e g u l a r i t y p r o p e r t i e s of the g a i n o p e r a t o r B P(t) p r e v e n t s one from Justifying the formal steps leading to the d e s i r e d c o n c l u s i o n that s u c h operator P(t) s a t i s f i e s the DRE Note that, on [O,T],

(3.21)

for, say,

x,y ~ ~(A).

at least w h e n the pair {A*,R ~} is e x a c t l y controllable, the operator P(t),

say,

0 ~ t < T, Is an i s o m o r p h i s m on ¥

(Thm. 3.1(viii)) and h e n c e B P(t)

is b o u n d e d from Y to U if and o n l y if

so is B, the trivial case.

~n general,

Thus,

and it is an issue w h e t h e r e.g.

U n d e r these circumstances, c o n s t r u c t e d o p e r a t o r P(t)

B P(t) may be unbounded,

is even d e n s e l y defined. it is of interest to regard the

of T h e o r e m 3.1 as "solution" of the

c o r r e s p o n d i n g Riccati D i f f e r e n t l a l

Equation

(3.21) in a s u i t a b l y w e a k e r

sense.

V i s c o s i t y solution.

One a p p r o a c h to this may be g i v e n by s e e i n g such

P(t) as limit of a p p r o p r i a t e R i c c a t i problems,

o p e r a t o r s P (t) of r e g u l a r i z i n g

where all P (t) s a t i s f y the DRE

Recrularizinu problems.

(3.21).

We i n t r o d u c e a p a r a m e t e r

of r e g u l a r l z a t l o n

~0 ~ ~ > 0, ~ ~ O, and c o n s i d e r the family {R } of o b s e r v a t i o n

29

operators

satisfying Re

Z(Y;Z):

e

R strongly:

Re

(3.3o)

X ~ Y.

Rex ~ Rx,

T *

(A.1 e )

ilReRee

At

(3.31)

BUlIudt <. CT, elIullU.

0

Remark

3.5.

Such family always

Re = R E R ( , A )

exists and we may in fact take

in terms of the resolvent

llR:ReeAtBullu

=

~1

of A, so that

1 * )R *ReAtR(ZE A)Builu ~ ~-~ 1 CT,~llu[ iU • IIR(E,A

since IIR(~,A)BII ~ c e by (1.3) with ~ = 1, and

(3.31)

follows

at

[]

once.

the corresponding optimal control

With each R e we associate problem

(OCP)~

solutions by, say,

on [O,T],

T < ~, yleldlng

{u~(t,O;Yo),y~(t,O;Yo} Theorem

the unique pair of optimal

and the corresponding

3.I or Theorem

3.3.

Moreover,

operator

by Theorem

3.3, Pz(t)

is the unique solution of the corresponding DRE e as explained The desired

connection

between

the original

regularizing family of non-negative operators

Pc(t)

is the following

for numerical purposes.

in Theorem

3.1 a "viscosity"

Theorem 3,~.

[Las.5]

self-adjolnt

It makes

solution

there.

and the constructed bona fade Riccati

which was originally

result,

envisioned

P(t)

Pz(t)

the operator

of the DRE

We have the following

P(t)

of (3.6)

(3.21).

strong

limits

as e$O:

(3.32)

(J)

P(t)x = llm Pe(t)x,

x E y;

(il)

0 uO(.,O;Yo ) = llm uz(-,O;Yo)

in L2(0,T;Y);

(3.33)

(ili)

yO(.,O;Yo)

in C([O,T]:Y);

(3.34)

(iv)

31u0,701

= llm y~(',O;Yo) 0 0 = lim Jlue,ye).

[]

(3.35)

30 The Dual D l f f e r e n t l a l Riccatl E u u a t l o n IDDRA]. of p r o p e r t y

Under the c i r c u m s t a n c e s

(viii} of T h e o r e m 3.1 we have seen that the (candidate)

Riccatl o p e r a t o r P(t) g i v e n by 43.6) is an I s o m o r p h i s m on Y for e a c h t ~ [O,T) if and o n l y if the pa~r {A ,R } Is e x a c t l y c o n t r o l l a b l e on [O,T].

Thus,

in this case, we let Q(t) be the n o n - n e g a t i v e

s e l f - a d j o l n t o p e r a t o r on Y: Q(t) = p-14t), and we r e a d i l y v e r i f y that Q(t)

0 ~ t < T,

formally s a t i s f i e s the f o l l o w i n g Dual

D i f f e r e n t i a l Riccatl E q u a t i o n for all x,z e ~ ( A d

(3.36)

) and 0 ~ t < T:

(Q(t)x,z)y = - (B*x,B*z) U + (Q(t)x,A * Z)y

+ (Q(t)A*x,Z)y + 4RQ(t)x, RQ(t)Z)z. The DDRE

(3.37)

43.37) arises from a q u a d r a t i c cost optimal control problem,

w h i c h h o w e v e r r e q u i r e s that A be a s.c. group generator;

see S e c t i o n 5,

C o r o l l a r y 5.7 and ff. for m o r e d e t a i l s in the case T = ~. Following the o r i g i n a l DRE (~) the DRE

IF.3] one m a y introduce a w e a k e r n o t i o n of s o l u t i o n of 42.14) as follows.

One first assumes that the end point c o n d i t i o n P(T) = PT of

42.14) be a n o n - n e g a t l v e s e l f - a d j o l n t i s o m o r p h i s m on Y and

s t u d i e s the DDRE

(2.37)

Note that the DDRE

In Q(t) wlth e n d - c o n d l t l o n Q(T) = QT =

(2.37) has the quadratic term i n v o l v i n g the b o u n d e d

o p e r a t o r R, w h i l e the original DRE

(2.14) in P(t) has the q u a d r a t i c 8

term i n v o l v i n g the u n b o u n d e d o p e r a t o r B . easier to handle than the DRE

(2.14).

Thus the DDRE

In fact,

local c o n t r a c t i o n coupled by global a-priorl

(2.37)

the general s t r a t e g y of

bounds

[DaP.1]

y i e l d s a u n i q u e s o l u t i o n Q(t} of this dual p r o b l e m IF.3]. proves that Q(t)

is an i s o m o r p h i s m 0 ~ t ~ T.

is

readlly Next,

IF.3]

One can thus d e f l n e the

o p e r a t o r P(t) by P(t) = Q-1(t),

0 5 t 5 T.

(3.38)

T h e n IF.3] proves that such P(t) s a t i s f i e s the identltles (3.8) of T h e o r e m 3.1. (2.38) solves

There is no claim,

the original

DRE

however,

(3.6a) and

that P(t) d e f i n e d by

(2.14).

(il) The general case w h e r e PT is not an i s o m o r p h i s m is reduced step

(i) by an a p p r o x i m a t i o n argument.

More precisely,

[F.3]

31

approximates

the general PT by PT+~I, el0, w h i c h is an isomorphism.

Step (i) gives then o p e r a t o r s Pc(t) defined by Pz(tl = Q~i(tl satisfying identities an o p e r a t o r P(t)

(3.6a) and

= llm P~(t}

satisfies identities

(3.8).

By letting t$O,

IF.3] o b t a i n s

in 2(Yl u n i f o r m l y in [0, T}, w h i c h l i k e w i s e

(3.6a) and

such P(t) solves

the or~glnal

an Isomorphism.

Nevertheless,

(3.81.

DRE

A g a i n there is no clalm

(2.14), nor

that such

llmlt

that

P(t)

is

the o p e r a t o r P(t) c o n s t r u c t e d in this

fashion may be v i e w e d as a "weak notion" of s o l u t i o n of the original DRE

(2.14).

4.

A b s t r a c k D { f f e r e ~ t l a l Rio carl E q u a t i o n s for the s e c o n d class s u b j e c t to the r e a u l a r l t v a s s u m n t l o n s (H.2R) = (1.81

Orientation.

In the present s e c t i o n we shall adopt the v a r i a t i o n

(H.2R) = (1.8) of the r e g u l a r i t y a s s u m p t i o n addition, Thus,

In

we shall a l l o w the o b s e r v a t i o n o p e r a t o r R to be unbounded.

in the present setting,

the u n b o u n d e d c o e f f i c i e n t o p e r a t o r s A and

B give rise to a (posslblyl u n b o u n d e d (1.101

(H.2) = (1.6).

i n p u t - s o l u t l o n o p e r a t o r L in

from L 2 ( O , T ; U ) to L2(0, T;Y ) and, moreover,

o p e r a t o r R is likewise

(possibly) u n b o u n d e d

is not c o v e r e d by the one of S e c t i o n s 3.i,

the o b s e r v a t i o n

from Y to Z. 3.3 as

This s e t t i n g

(H.2) = (1.6) a s s u m e d

there a m o u n t s to L c o n t i n u o u s L 2 ( O , T ; U I ~ C([O,T];YI,

see

(1.121.

Yet

there are i m p o r t a n t and natural b o u n d a r y control p r o b l e m s - - s u c h as the N e u m a n n b o u n d a r y c o n t r o l / D i r l c h l e t b o u n d a r y o b s e r v a t i o n p r o b l e m for s e c o n d - o r d e r h y p e r b o l i c e q u a t i o n s d e s c r i b e d in the s u b s e q u e n t S e c t i o n 8 - - w h l c h give rise p r e c i s e l y to this sltuatlon.

It is,

in fact,

the

d e s i r e to cover this h y p e r b o l i c p r o b l e m that m o t i v a t e s the p ~ e s e n t s e c t i o n and p r o v i d e s a g u i d i n g example.

Technically,

the s e t t i n g of

S e c t i o n 3.3 is not I n c l u d e d in the present s e t t i n g elther, speaking, say that

u n l e s s R is an isomorphism; 'morally'

of S e c t i o n 3.3,

see R e m a r k 1.1.

strictly

(However, we may

the p r e s e n t level of g e n e r a l i t y includes the s e t t i n g

in the sense that

the f o l l o w i n g T h e o r e m

the t e c h n i q u e of proof of [L-T.IO] of

4.1 of this s e c t i o n w o u l d also a p p l y to the

s e t t i n g of S e c t i o n 3.3 and p r o d u c e Theorem 3.3). and in llne with the spirit of this paper, f r a m e w o r k is m o t i v a t e d by the

'concrete'

wave e q u a t i o n wlth L 2 ( ~ ) - N e u m a n n control,

As a l r e a d y remarked,

the present a b s t r a c t

q u a d r a t i c cost p r o b l e m for the w h i c h p e n a l i z e s also the

31

approximates

the general PT by PT+~I, el0, w h i c h is an isomorphism.

Step (i) gives then o p e r a t o r s Pc(t) defined by Pz(tl = Q~i(tl satisfying identities an o p e r a t o r P(t)

(3.6a) and

= llm P~(t}

satisfies identities

(3.8).

By letting t$O,

IF.3] o b t a i n s

in 2(Yl u n i f o r m l y in [0, T}, w h i c h l i k e w i s e

(3.6a) and

such P(t) solves

the or~glnal

an Isomorphism.

Nevertheless,

(3.81.

DRE

A g a i n there is no clalm

(2.14), nor

that such

llmlt

that

P(t)

is

the o p e r a t o r P(t) c o n s t r u c t e d in this

fashion may be v i e w e d as a "weak notion" of s o l u t i o n of the original DRE

(2.14).

4.

A b s t r a c k D { f f e r e ~ t l a l Rio carl E q u a t i o n s for the s e c o n d class s u b j e c t to the r e a u l a r l t v a s s u m n t l o n s (H.2R) = (1.81

Orientation.

In the present s e c t i o n we shall adopt the v a r i a t i o n

(H.2R) = (1.8) of the r e g u l a r i t y a s s u m p t i o n addition, Thus,

In

we shall a l l o w the o b s e r v a t i o n o p e r a t o r R to be unbounded.

in the present setting,

the u n b o u n d e d c o e f f i c i e n t o p e r a t o r s A and

B give rise to a (posslblyl u n b o u n d e d (1.101

(H.2) = (1.6).

i n p u t - s o l u t l o n o p e r a t o r L in

from L 2 ( O , T ; U ) to L2(0, T;Y ) and, moreover,

o p e r a t o r R is likewise

(possibly) u n b o u n d e d

is not c o v e r e d by the one of S e c t i o n s 3.i,

the o b s e r v a t i o n

from Y to Z. 3.3 as

This s e t t i n g

(H.2) = (1.6) a s s u m e d

there a m o u n t s to L c o n t i n u o u s L 2 ( O , T ; U I ~ C([O,T];YI,

see

(1.121.

Yet

there are i m p o r t a n t and natural b o u n d a r y control p r o b l e m s - - s u c h as the N e u m a n n b o u n d a r y c o n t r o l / D i r l c h l e t b o u n d a r y o b s e r v a t i o n p r o b l e m for s e c o n d - o r d e r h y p e r b o l i c e q u a t i o n s d e s c r i b e d in the s u b s e q u e n t S e c t i o n 8 - - w h l c h give rise p r e c i s e l y to this sltuatlon.

It is,

in fact,

the

d e s i r e to cover this h y p e r b o l i c p r o b l e m that m o t i v a t e s the p ~ e s e n t s e c t i o n and p r o v i d e s a g u i d i n g example.

Technically,

the s e t t i n g of

S e c t i o n 3.3 is not I n c l u d e d in the present s e t t i n g elther, speaking, say that

u n l e s s R is an isomorphism; 'morally'

of S e c t i o n 3.3,

see R e m a r k 1.1.

strictly

(However, we may

the p r e s e n t level of g e n e r a l i t y includes the s e t t i n g

in the sense that

the f o l l o w i n g T h e o r e m

the t e c h n i q u e of proof of [L-T.IO] of

4.1 of this s e c t i o n w o u l d also a p p l y to the

s e t t i n g of S e c t i o n 3.3 and p r o d u c e Theorem 3.3). and in llne with the spirit of this paper, f r a m e w o r k is m o t i v a t e d by the

'concrete'

wave e q u a t i o n wlth L 2 ( ~ ) - N e u m a n n control,

As a l r e a d y remarked,

the present a b s t r a c t

q u a d r a t i c cost p r o b l e m for the w h i c h p e n a l i z e s also the

32

L 2 ( Z ) - n o r m of the (Dirichlet)

trace of its solutions.

the o b s e r v a t i o n o p e r a t o r R is the D i r l c h l e t trace,

In this example,

and it happens

that

the c o m p o s i t i o n R L - - w h i c h gives the maps from the control space L2(O,T;L2(r))

to the space L 2 ( O , T ; L 2 ( F ) )

of the h y p e r b o l i c solutions--is,

of the D i r l c h l e t o b s e r v a t i o n s

in fact, bounded;

than e a c h of its c o m p o n e n t s v i e w e d separately. (Dirlchlet)

i.e.,

At is nicer

This says that the

trace of the s o l u t i o n b e h a v e s m o r e r e g u l a r l y than it could

be A n f e r r e d from l o o k i n g at the interior ~ e g u l a r i t y of the s o l u t l o n and a p p l y i n g trace t h e o r y

(even formally).

This p r o p e r t y is, in fact, a

d i s t A n c t i v e feature of w a v e s and plates problems, d i s c o v e r e d in recent years

as It has b e e n

in a v a r i e t y of situations.

S e c t i o n 4.1 will p r o v i d e the m a ~ o r t h e o r e t l c a l results and S e c t i o n 8 will be d e v o t e d to the b o u n d a r y c o n t r o l / b o u n d a r y o b s e r v a t i o n i11ustratlon.

We m u s t h a s t e n to point out that verification that all

r e q u i r e d a b s t r a c t a s s u m p t A o n s are s a t i s f i e d in the case of the wave e q u a t i o n p r o b l e m w i t h b o u n d a r y o b s e r v a t i o n is not a trivia/ task: it requires, [L-T.23]

in a critical way,

the sharp r e g u l a r l t y theory

that has become a v a i l a b l e o n l y v e r y recently,

r e g u l a r i t y results

[L-M],

[M.1],

[L-T.20],

w h i l e earlier

for s e c o n d - o r d e r h y p e r b o l i c mixed

p r o b l e m s of N e u m a n n type is i n a d e q u a t e and i n s u f f i c i e n t .

This will be

seen more t e c h n i c a l l y in S e c t i o n 8.1. 4.1.

T h e o r e t i c a l results:

Th~o~s

4.1 and 4 ~

The f o l l o w i n g a b s t r a c t a s s u m p t i o n s either capture,

or else p r o p e r l y contain,

Gas 4!.3) and

(H.2R)) will

i n t r i n s i c properties of the

h y p e r b o l i c p r o b l e m of S e c t i o n 4.2. Assumptions.

In a d d i t i o n to the s t a n d i n g a s s u m p t i o n s

41.3) and

(H.2R) = (1.8), we shall a s s u m e that G = 0 and that (h.O)

R ~ ~(~(A);Z);

e q u i v a l e n t l y RA -1 E Z{Y;Z)

(4.1)

(~(A) e n d o w e d w i t h norm JJY~[~(A) = [~Ay[Iy , e q u i v a l e n t to the g r a p h norm,

as A is a s s u m e d b o u n d e d l y i n v e r t l b l e w i t h o u t loss

of generality); (h.1)

the m a p R e A t B can be e x t e n d e d as a map: U ~ LI(O,T;Z):

continuous

33 T

~

llReAtBullzd't:

<_ CTt]UlJ u

,

u

•

U;

(4.2)

0

(h.2)

the map Re At can be extended

as a map:

sup llReAtxl]z ~ CT~IXllY ,

Theorem

4.1.

through

(h.2) = (4.3)

[L-T.IO]

Under

11.2),

given explicitly

to (i.3) and

14.3)

(h.0) = (4.1)

(H.2R)

= 11.8)),

{uO(t,O;Y0),yO(t, 0;Yo) } of problem by (see

11.9),

y0(t, 0;Y0)

=

[L'R*(ReA'yo)]}(t

).

{

'~(P(t)x,z)y P(T)

a solution

(1.1),

14.41

eAty0+(Lu0)(t);

(4.Sa)

Ry0(t,0;Y0 ) = {[I-RL(I+L*R*RL)'IL*R*][ReA'y0]}(t there exists

there

(1.I0)),

-uO(t, 0;y 0) = {[I+L R RL]

Moreover,

Y ~ L (O,T;Z):

x • Y.

the above assumptions

(in addlt~on

exists a unique solution

contlnuous

P(t) • Z(Y),

),

(4.5b)

0 ~ t ~ T of the DRE

= -(Rx, RZ)z-(P(t)Ax, z)y-(P(t)x, Az)y+(B P(t)x,B P(t)z) U , v

x,z • $(A),

= 0,

(4.6) which is, in fact, given constructively

by

T s P(t)x = ~ e A (r-t)R*Ry0(r,t;x)dr,

x •

y.

(4.7)

t

Such P(t)

enjoys

the

(i)

following P(t)

= P (r),

(ll)

e(t)

(lii)

B P(t)

(iv)

B*P(t)eA(t-r)B

properties:

(4.8)

0 ~ t ~ T;

e ~(Y;O([O, TJ;Y);

(4.9)

E ~(Y;C([O,T];U);

e ~(U;L2(r,T;U)) , u n i f o r m l y

14.1o) in r:

34 T

(4.11)

o_
(v)

(feedback

synthesis)

uO(t,O:Yo)

(vi)

(P(t)x,Z)y

the optimal

pair

is related

= -B P(t)yO(t,O;Yo) .

by

(4.12)

0 _< t <_ T;

T = ~(RyO(r,t;x),Ry0(r,t;Z))zdr t T + ~(B* P(r)yO(r,t;x) , B " P(r)yO(r,t;Z))udr, t

from which

the optimal

cost is

j(uO(-,r;yo),y0(.,t;y0 (vii)

(Uniqueness)

The operator

to enjoy properties A more regular state

)) = (P(r)Y0,Yo)y

~t, we shall

P(t)

(i) = {4.8)

(h.5),

through

postulate

the existence

(see

(4.13) solution

(iv) = (4.11).

•

result.

of a Hilbert

and topologically)

such that the following

hold true where

Yo e Y.

is the unique

case is given by the following

7L[O,T ] c L2(O,T;U ) (algebraically 7L[t,T ] c L2(t,T:U),

in (4.7)

,

To

space

with restriction

assumptions,

(h.3)

through

(1.19)): ?

(Ltu)(r)

= f eA(f-r)Bu(r)dr.

(4.14)

t (h.3) (h.4)

LO: continuous S

the map LOR

•

[Re

A o

~ C([O,T];Y);

] can be extended

L~R'[ReA']: (h.5)

~[O,T]

for each t ~ [O,T],

continuous

T < ~,

(4.15)

as a map: Y ~ ~[O,T];

(4.1e)

the map

LtR RL t is compact 7L[t,T ]

itself.

('4.17)

35 ~emark (h,2}

4.1. =

Assumptlon

(4.3)

and

The next

(h.4}

(H.2R)

theorem

=

=

(4.16)

(1,8)

gives

~s s t r o n g e r

comblned

regularity

than

[L-T.IO,

results

assumptions

Remark

for

the

6.2].

[]

optimal

pair

{u°l.,r:Y0),y°(.,r;Y0)}. Theorem

4.2.

(h.5}

(4.17).

=

[L-T.IO]

(1)

Assume

hypotheses

sup l l u 0 ( . , T : x ) l l x [ t O
5.

Assume,

in a d d i t i o n ,

of t h e s e

~bstract

In this situation

two

Almebralc

section

where

=

(4.16)

and

~ CTllxll Y

hypothesis

theorems

Riccatl

where

the s.c.

(4.18)

,T]

s u p ily°(.,r;x)il C ( [ r , T ] ; Y ) O~T~T Application

(h.4}

Then

will

(h.3)

& CT11xilY

be g i v e n

Eauatlons:

T = ~,

semlgroup

we

w i t h w 0 = llm[(inJIexp(At)i[)/t]

Z(Y),

so t h a t

"

8.

and unlaueness

the g e n e r a l

is g e n e r a l l y

> 0 as

(4.19)

•

in S e c t i o n

treat

Then

(4.15).

Existence

shall

exp(At)

i.e.,

=

t ~ + ~

unstable

on Y,

in the u n i f o r m

norm

(w0+a)t WO+Z.

We

= fixed

IleAtll ~ Me

then

consider

> w 0, so

-A is the g e n e r a t o r

,

v z > O,

throughout

the

that A h a s w e l l - d e f l n e d of an s.c.

t ~ 0, a n d M d e p e n d i n g

translation

semigroup

fractional e -~t

on

A = -A+wI, powers

on Y a n d

on Y s a t i s f y i n g

A

lie-Ate[ < Me -Wt,

t > O; ~ = W - W O - Z

> O.

Moreover,

G = 0 in

(1.2)

while

R is n o n - s m o o t h l n g R G ~(Y;Z). In this following

sectlon

abstract

we shall

(5.0)

then discuss

Algebraic R i c c a t i

the s o l v a b i l i t y

Equation

of

the

(ARE)

(Px, A y ) y + ( P A x , y)y+(Rx, R y ) z - ( B ' P x , B ' P Y ) u

= O;

v x,y e ~(A).

(5.1)

35 ~emark (h,2}

4.1. =

Assumptlon

(4.3)

and

The next

(h.4}

(H.2R)

theorem

=

=

(4.16)

(1,8)

gives

~s s t r o n g e r

comblned

regularity

than

[L-T.IO,

results

assumptions

Remark

for

the

6.2].

[]

optimal

pair

{u°l.,r:Y0),y°(.,r;Y0)}. Theorem

4.2.

(h.5}

(4.17).

=

[L-T.IO]

(1)

Assume

hypotheses

sup l l u 0 ( . , T : x ) l l x [ t O
5.

Assume,

in a d d i t i o n ,

of t h e s e

~bstract

In this situation

two

Almebralc

section

where

=

(4.16)

and

~ CTllxll Y

hypothesis

theorems

Riccatl

where

the s.c.

(4.18)

,T]

s u p ily°(.,r;x)il C ( [ r , T ] ; Y ) O~T~T Application

(h.4}

Then

will

(h.3)

& CT11xilY

be g i v e n

Eauatlons:

T = ~,

semlgroup

we

w i t h w 0 = llm[(inJIexp(At)i[)/t]

Z(Y),

so t h a t

"

8.

and unlaueness

the g e n e r a l

is g e n e r a l l y

> 0 as

(4.19)

•

in S e c t i o n

treat

Then

(4.15).

Existence

shall

exp(At)

i.e.,

=

t ~ + ~

unstable

on Y,

in the u n i f o r m

norm

(w0+a)t WO+Z.

We

= fixed

IleAtll ~ Me

then

consider

> w 0, so

-A is the g e n e r a t o r

,

v z > O,

throughout

the

that A h a s w e l l - d e f l n e d of an s.c.

t ~ 0, a n d M d e p e n d i n g

translation

semigroup

fractional e -~t

on

A = -A+wI, powers

on Y a n d

on Y s a t i s f y i n g

A

lie-Ate[ < Me -Wt,

t > O; ~ = W - W O - Z

> O.

Moreover,

G = 0 in

(1.2)

while

R is n o n - s m o o t h l n g R G ~(Y;Z). In this following

sectlon

abstract

we shall

(5.0)

then discuss

Algebraic R i c c a t i

the s o l v a b i l i t y

Equation

of

the

(ARE)

(Px, A y ) y + ( P A x , y)y+(Rx, R y ) z - ( B ' P x , B ' P Y ) u

= O;

v x,y e ~(A).

(5.1)

36

As is well possesses feedback

known,

a solution

P of this e q u a t i o n

certaln

regularity

properties)

operator

which occurs

optimal

control

obvious

difficulty

operator'

law.

(even w h e n the e x i s t e n c e The c r u x of the m a t t e r P possesses

in the r e p r e s e n t a t i o n

to the I n t e r p r e t a t i o n

of a "Riccatl

operator"

this:

'regularity'

P ~ ~(Y)

a proper d e f i n i t i o n

of the gain o p e r a t o r

operator

domain

received 5.1.

We b e g i n w l t h

more a t t e n t i o n

the first

class

I.

2,~

([D-I],

Existence.

(H.I)

=

exists

(1.5)

of t h e ARE (i)

[F.2],

and subject

(5.1)

^ *

(A)

l-ep

w h i c h will

B P, at least

('analytic')

class

subject

class

covered

on Y;

(1.18)

w h i c h has

to the

to the Finite

Cost C o n d i t i o n

definite

solution

(1.9),

there s ~ ~(Y)

0 ~ P = P

that:

e ~(Y)

thus,

a$ an

by h y p o t h e s i s

W • > O;

(5.2)

and indeed ~ m a y be taken & = 0 if the o r i g i n a l self-adjolnt

is asserted). the R i c c a t l

[L-T.7]).

non-negative

such

on Y

in the literature.

For the first

a self-ad~oint,

or normal

A is

or has a Ri e s z basis of e i g e n v e c t o r s

P is c o m p a c t

if A has compact

resolvent:

(ll)

B P • Z(Y,U);

(5.3)

(ill)

j(uO, y O) =

(5.4)

(iv)

u O ( t ; y O) = - B * p y O ( t ) y 0 ( t ; y 0 )

II.

Reuularitv

functions U-valued

(Py0,Y0)y;

of t h e o p t i m a l

y 0 ( t ; y O) and uO(t;Y0) functions,

the

'gain

in Y for the r e p r e s e n t a t i o n

A l a e b r a l c Riccat~ E q u a t i o n for the first a n a l v t i c i t v a s s u m p t i o n (H.I~ = (1,5)

Theore~

that

properties

guarantee

w i t h dense

of the

operator,

of the

To p r o v e

unbounded

to be m e a n i n g f u l .

(1.18)

need not be even d e n s e l y d e f i n e d

is t h e r e f o r e

certain

and it

the s o u g h t - a f t e r

In the case where B is an u n b o u n d e d

is r e l a t e d

B P, w h i c h a p c i o r i

operator

(if it e x i s t s

provides

for all 0 < t < ~.

pair.

For e a c h

are a n a l y t i c

a consequence

(5.5)

fixed Y0 e y,

in t as Y - v a l u e d

of the a n a l y t i c l t y

of the

the or

feedback

AFt semlgroup

e

below

in

(5.11)

Remark.

In the case w h e r e

equation

on a b o u n d e d

the f o l l o w i n g true

[L-T.7]:

and of

(I.I)

(5.3).

models

a second

order

d o m a i n ~ c R n with D i r i c h l e t

additional

regularity

properties

parabolic

boundary

control,

of the optimal

pair hold

3'7

[

e-~ty0(.;yO)

if Y0

e

L2(n )

e HI"2e'M-e(Q~),

Q® = (0,®)xO,

e'~tuO(.;y0) V e"

>

e H~ - 2 ~ ' ' ~ - ~ ' ( ~ ) , > O, Z ~ =

e

"e-WtyO(. ;yo)

e

(O,~)xF;

•

(5.7)

HZ'-2P'~-P(Qoo ), p

if YO

(5.6)

(5.8)

> O,

H~-P(n) =

e-WtuO(. ;yo) e H2-2p ", 1-p'(~), p" > p

III.

Uniqueness.

In a d d i t i o n

that the following so-ca/led

to the a s s u m p t i o n

~S.C.

semigroup

of part I, we assume

'detectability condition'

There exists K ~ Z(Z,Y) (D.C):

(5.9}

> O.

(D.C.) holds:

such that the

e (A+KR)t generated by A+KR

is e x p o n e n t i a l l y

(uniformly)

(5.10)

stable on Y.

Then

(a)

the solution P to the ARE

(5.1) is unique within the class of

non-negative

operators

regularity

self-adjolnt

requirement

the s.c.,

analytic

exponentially

semigroup

(uniformly)

e

, generated by Ap = A-BB P is

stable on Y:

Apt -Wpt lie ]l~(y) ~ Mpe , for some constants Mp, Wp > O.

R e m a r k $,0.

t > 0

constructive

(5.11)

B

If the original s e m A g r o u p exp(At)

i.e., ~0 < 0 for the constant above explicit,

which satisfy the

(5.3); Apt

(h)

in ~(Y),

(5,0),

is (uniformly)

stable,

then one can give an

formula for P in terms of the optimal dynamics,

whlch in turn is given e x p l i c i t l y

in terms of the data of the problem;

p r e c i s e l y as in the case T < ~, seen before in (2.20).

If instead,

~0 > O, then the explicit

formula

for P becomes a c t u a l l y an Identity

s a t i s f i e d by P; see e.g.,

[L-T.7;

Section 2].

i

38

As in the case T < ~ ,

two distinct,

yet complementary,

a p p r o a c h e s are a v a i l a b l e to prove T h e o r e m 5.1 uniqueness): so-ca/led

(1) a variational a p p r o a c h

'direct' a p p r o a c h

[D-I],

(existence and

[L-T.7],

IF.2].

[L-T.19],

and

(ii) a

The v a r i a t i o n a l a r g u m e n t in

[L-T.7] starts from the control p r o b l e m as the p r i m a r y issue and c o n s t r u c t s an e x p l i c i t c a n d i d a t e for the Riccatl o p e r a t o r

(in terms of

the data of the p r o b l e m w l t h the help of the optimal solution,

Remark 5.0), w h i c h is then s h o w n to s a t i s f y the ARE contrast,

the direct a p p r o a c h as in [D-I],

of w e l l - p o s e d n e s s

programming) task,

see

In

IF.2] takes the direct s t u d y

(existence and uniqueness)

object and only s u b s e q u e n t l y recovers

(5.1).

of the ARE as the p r i m a r y

the control p r o b l e m

w h i c h g e n e r a t e s the original ARE.

(via d y n a m i c

In c a r r y i n g out its

the direct m e t h o d begins a c t u a l l y w i t h a direct s t u d y of the

corresponding Differential

(or Integral)

optimal p r o b l e m over a finite Interval

Riccati E q u a t i o n of the

[O,T], T < ~, and operates a

limit process as T ~ ~ o__nnthe D i f f e r e n t i a l Riccati E q u a t i o n w i t h a classical approach, technical difficulties, to B P).

w h i c h now, however,

(in llne

has to o v e r c o m e n e w

p a r t i c u l a r l y the s t r o n g c o n v e r g e n c e of B PT(O)

In both approaches,

a key point c o n s i s t s in e s t a b l i s h i n g that

the gain operator B P (a priori

fact, a bounded operator;

see

not n e c e s s a r i l y well defined)

(5.3).

is, in

In the v a r i a t i o n a l approach,

this

latter p r o p e r t y is a c c o m p l i s h e d by using a n a l y t l c l t y of the free dynamics,

together w i t h a c e r t a i n

'bootstrap' a r g u m e n t b a s e d on the

Y o u n g i n e q u a l i t y to s h o w that the optimal pair is more regular, e - ~ t u O ( t ; Y o ) E C([O,~];U}

(A priori,

and e - W t y O ( t ; Y o ) ~ C([O,~];Y}

we only k n o w that u 0 ¢ L2(O,~;U),

u ~ L2(O,T;U)

while a general control

need Dot p r o d u c e in general a c o r r e s p o n d i n g s o l u t i o n

y e C([O,T];Y);

a counterexample being o b t a i n e d by a p a r a b o l i c e q u a t i o n

on n, w i t h D i r l c h l e t - b o u n d a r y control where U = L2(F), [Lio.3;

indeed

for Y0 E y.

p. 217].)

and Y = L2(~ )

All this leads to the r e g u l a r l t y p r o p e r t y

(5.2) via

the explicit representation of P in terms of the optimal s o l u t i o n R e m a r k 5.O), w h i c h in turn leads to p r o p e r t y of the direct a p p r o a c h

[D-I],

IF.2],

(5.3).

the b o u n d e d n e s s

Instead,

(see

in case

(5.3) of the gain

o p e r a t o r B'P is e s t a b l i s h e d by p r o v i n g first that the s o l u t i o n of the

corresponding D i f f e r e n t i a l Riccati E q u a t i o n for the p r o b l e m on [O,T] p o s s e s s e s the d e s i r e d r e g u l a r i t y properties, limit as T ~ ~.

This,

in turn,

a p p l i c a t i o n s of the Y o u n g ' s

and then by p a s s i n g to the

is a c c o m p l i s h e d in [F.2] by r e p e a t e d

i n e q u a l i t y to prove that the optimal

39

trajectory

is in C([O,T];Y)

study of the e v o l u t i o n

Remark

5.1.

As r e m a r k e d

it s h o u l d be n o t e d becomes

rather

appearing

that

in S e c t i o n the p r o o f

in a s s u m p t i o n

satisfies

in fact,

true

s~mpllfles

or even 7 = ~,

or has a R i e s z

standard

property

for YO E Y.

analytic

for the optimal

basis

estimates

of

give

function

y E C([O,T];Y),

Thus,

and

the c o n s t a n t

y to an L 2 ( O , ~ ; U } - c o n t r o l

in fact the r e g u l a r i t y

e - W t y ( t ; Y o ) ~ C([O,~];Y)

for the case T < ~,

to be ~ < ~,

or normal,

in this case,

2.4,

5.1 g r e a t l y

in case

can be t a ke n

that a n y s o l u t i o n

automatically

2, R e m a r k

of T h e o r e m

A is s e l f - a d j o l n t Indeed,

at the o u t s e t

(1.3)

by a direct

v i a a flxed p o i n t a r g u m e n t .

straightforward,

if the o p e r a t o r elgenvectors.

for any T > O; or in [D-I]

equation

V T > O;

such property h o l d s

yO in this case

(while

it is a

distinctive p r o p e r t y of yO to be p r o v e d w h e n ~ < q < 1, not s h a r e d by general s o l u t i o n s y to L 2 ( O , ~ ; U ) - c o n t r o l s consequence,

one o b t a i n s

the e x p l i c i t

representation

sections

6.1,

physically Theorem

6.2,

immediately

analytic

5.1 will apply,

cannot

cover

these cases,

Remark

5,~.

The

property stable used

that

as in

particularly operator

the former. attractive

provides,

the free d y n a m i c s with. 5.2.

and,

treatments

Our

o n distinctive,

such as the one

Here, in [PSI

(D.C.)

= (5.10)

guarantees

P to the ARE, but also the Apt semlgroup e is e x p o n e n t l a l l y

solution

feedback

indeed,

it is the latter p r o p e r t y that Apt The exponential d e c a y of e is a

feature

in applications,

constructlvely,

a stabilizing

y = A y w h i c h m a y be,

possibly,

is

for then

the Riccatl

feedback

operator

unstable

to b e g i n

of

m A l o e b r a l c R i c c a t l E a u a t l o n for the s e c o n d c l a s s ~trace' r e u u l a r l t v a s s u m D t l o n (H.2) = ~ I ~ ) The s t u d y of the A R E Is more

dynamics (H.I}

by u s l n g

solution,

~n fact ~ < 7 < 1.

assumption

of the R i c c a t l

(5.11);

where

As a

m

the r e s u l t i n g

to p r o v e

other

'detectability'

not o n l y u n i q u e n e s s

concentrate

problems

while

above).

then that B P is bounded,

of P in terms of the optimal

a n d 5.3 b e l o w will

relevant,

u, as d i s c u s s e d

subject

= (1.5).

to a s s u m p t i o n Indeed,

in this case,

the free d y n a m i c s

w h i c h will

operator

in fact,

B.

And,

complicated

(H.2)

= (1.6}, there

make up for

in most

subject

for the c l a s s

rather

than

the u n b o u n d e d n e s s

of

to a s s u m p t i o n

is no s m o o t h i n g

of the i n t e r e s t i n g

to the

effect

of

of the

situations,

the

40

gain operator below.

B=P is i n t r i n s i c a l l y unbounded,

d e s c r i b e d in s e c t i o n 5.1, w h e n a s s u m p t i o n Thus,

see C o r o l l a r i e s

This feature is in sharp c o n t r a s t w i t h the

b o u n d e d n e s s of B P for the class

for the class

(H.I) and u n b o u n d e d n e s s of B P

(H.2) in the most i n t e r e s t i n g s i t u a t i o n s is a

it should be noted that,

s e c t l o n 5.1, h y p o t h e s i s

implies by d u a l i t y the desired (see (1.12)), w h i c h under the

(H.I) = (1.5) case ~ < q < 1, ~s g e n e r a l l y false,

p r o v e d to be true, in s e c t i o n 5.1. Equation

On the other

in contrast w i t h the s i t u a t i o n of

(H.2) = (1.6)

r e g u l a r i t y u e L 2 ( O , T ; U ) ~ y e C([O,T];Y) hypo~hesls

however,

for the optimal pair

(u0,yO),

but can be

as r e m a r k e d

A rather c o m p l e t e theory for the A l g e b r a i c Riccati

under present a s s u m p t i o n s was first given in [L-T.6]

c a n o n i c a l case of the wave equation, equations,

hyperbolic

was treated,

however,

w ~ t h D i r l c h l e t control in L2(O,T;L2(F)) , w h i c h by a b s t r a c t o p e r a t o r - t h e o r e t i c methods.

c o m p l e m e n t e d by further results,

I.

([L-T.6],

EXistence.

[L-T.9],

[FLT]).

For the second class covered by h y p o t h e s i s

exists a self-ad~olnt,

(11

such

This and

in [FLT].

(H.2) = (1.6) and subject to the F i n l t e Cost C o n d i t i o n ARE ( 5 . I )

in the

or more g e n e r a l l y s e c o n d - o r d e r

t r e a t m e n t was later put fully on an a b s t r a c t space framework,

T h e o r e m 5.2

5.5

situation

(H.I) instead is in force.

d i s t i n g u i s h i n g feature that tells the two cases apart. hand,

5.4,

'analytic'

(1.9),

non-negative solutlon 0 ~ P = P

e ~(Y)

there of the

that:

P E ~(~(AI,~IApI) n Zl~(ApI,~IA

(5.12)

11,

w h e r e the o p e r a t o r

(5.13)

Ap = A-BB P g e n e r a t e s a s.c. s e m i g r o u p on Y; thus,

the ARE

(5.1) holds

true also for all x , y E ~(Ap);

(5.14)

(ii)

S e ~ Z(~(A),U)

(lii)

j(uO, y 0) = (PyO,Y0)y;

(5.15)

(iv)

UO(t)- = -e * Py 0 (t);

(s.16)

n ~(~(Ae);U);

w h e r e we w r i t e yO(t) = yO(t;yo), is u n d e r s t o o d a.e.

uO(t)

= uO(t;Yo),

In t if YO e Y; w h i l e instead,

and

(5,16)

if Y0 e ~(Ap},_

41

then (5.14)

implies y0(t;Yo) ¢ C ( [ O , T ] ; ~ ( A p ) } ,

u 0 ( t ; Y 0 ) 6 C([0, T];U)

II.

~Diqueness.

(5.16),

for any T > 0.

In a d d i t i o n to the a s s u m p t i o n of p a r t I, we a s s u m e

that the f o l l o w i n g (D.C):

and by

'detectability'

condition

(D.C.) h o l d s true:

T h e r e exists K: Z D ~(K) ~ Y d e n s e l y d e f i n e d s u c h that s

IIK x[lZ ~ C[[B*X[u+[~X[[y], V X G ~ ( B

) c Y,

(5.17)

so that the o p e r a t o r A K = A+KR

(interpreted as closed)

(5.18)

AKt is the g e n e r a t o r of a s.c. semigroup a

on Y, w h i c h is then a s s u m e d

to be e x p o n e n t l a l l y s t a b l e on Y:

ile f o r some MK, oR > 0. s u f f i c i e n t l y large,

AKt

t > 0,

and the d e t e c t a b l l l t y a s s u m p t i o n

with constant c

(5.17)-(5.19)

the s o l u t i o n P to the ARE

(5.1) is u n i q u e w i t h i n the class of w h i c h s a t i s f y the

r e g u l a r i t y properties (5.14); Apt the s.c. s e m i g r o u p e g e n e r a t e d by Ap in (5.13) exponentially

is

Then

n o n - n e g a t i v e self-ad~olnt operators in Z(Y)

(b)

(5.19)

(For R > O, we choose K = -c2R - I

a u t o m a t i c a l l y satisfied.) (a)

-~K t [[~(y) ~ MKe ,

(uniformly)

stable on Y,

is

l

The proof of T h e o r e m 5.2 is g i v e n in [FLT] and follows the a b s t r a c t t r e a t m e n t of the c a n o n i c a l case of s e c o n d o r d e r h y p e r b o l l c e q u a t i o n s w i t h D i r i o h l e t control approach.

[L-T.6].

The f o l l o w i n g comments,

It is b a s e d on a v a r i a t i o n a l

w h i c h c o n s t r a s t the technical

m e t h o d o l o g y a v a i l a b l e in the case of T h e o r e m 5.2 w i t h that a v a i l a b l e in the case of T h e o r e m 5.1, apply.

A m a i n d i f f e r e n c e between the two

cases is that, as p o l n t e d out in s e c t i o n 3, at p r e s e n t no D i f f e r e n t i a l R i c c a t i E q u a t i o n on [0,T] is a v a i l a b l e u n d e r the a s s u m p t i o n (H.2) = (1.6) w i t h no s m o o t h i n g of the o p e r a t o r R; i.e., only to a s s u m p t i o n

(5.0) of boundedness,

avallable under assumption

(H.1) = (1.5)

w l t h R subject

in c o n t r a s t w i t h the s i t u a t i o n in s e c t i o n 2.

Thus,

an

42

to the issue of existence

approach based

on the classical

Differential

Riccati

case of a s s u m p t i o n as d e s c r i b e d applied

Equation

(H.1)

in s e c t i o n

[L-T.6],

[FLT].

under assumption

(H.1),

obtained First,

= (1.5),

described finally,

the e x i s t e n c e (H.2)

45.12),

regularity

(5.14),

one v e r i f i e s

using

the a n a l y t i c

approach

a different

[D-I],

treatment

of

[L-T.7]

to the ARE is n o w

solution);

its e x p l i c i t

(i)

in terms of

of a solution,

properties

IF.2],

strategy is n o w

the following steps:

candidate

(the optimal

(unlike

case

is

as T ~ ~ on the

of a s o l u t i o n

through

an explicit

the n e c e s s a r y

in

in the direct

Therefore,

As in the a n a l y t l c

the d a t a of the p r o b l e m establishes

of the ARE w h i c h

process

is out of q u e s t i o n

5.1).

under assumption

one c o n s t r u c t s

of a s o l u t i o n

idea of a l i m i t i ng

(ii) next,

one

of s u c h a candidate, representation;

as

(lil)

that such candidate o p e r a t o r does s a t i s f y

the ARE

(5.1). As m e n t i o n e d to n o t i c e under

that,

in c o n t r a s t

the a n a l y t l c i t y

is n o w g e n e r a l l y Indeed,

Theorem

Let

assume

the s i t u a t i o n (H.1)

the h y p o t h e s e s (1.9) hold

the f o l l o w i n g

(We shall

say,

guaranteed

Qorollarv

and only

Then,

by T h e o r e m

5.4.

of T h e o r e m

5.3,

Under

that

assumption

(H.2)

(the i n t e r e s t i n g

=

which

(H.2)

= (1.6)

as in T h e o r e m

is the

for the d y n a m i c s

controllability is e x a c t l y

over

some

[0, T],

part

and the I.

(5.20)

v.

the pair {A ,R } is exactly

is an I s o m o r p h i s m

part

I,

the a s s u m p t i o n s

(H.2)

P to the ARE

= (1.6),

(5.1)

on Y.

(1.9),

•

and

(5.20)

The operator B: U m ~(B) ~ Y is b o u n d e d B P:

5.4,

(1.5),

In

condition:

T < ~,

operator

case),

5.2,

controllable

5.2,

w e have:

Corollary

5.1

B*P

situations).

result,

the s o l u t i o n

if the o p e r a t o r

From

interesting

the class of L2(0, T ; Z ) - c o n t r o l s

in short,

controllable.)

by T h e o r e m

the g a i n o p e r a t o r

true,

exact

In Y from the o r i g i n within

it is important

described

= (1.5),

from the next

the e q u a t i o n y = A * y + R * v (E.C.)

5.2,

3.1(viil).

Cost C o n d i t i o n

addition,

of s e c t i o n

(in the most

follows

of T h e o r e m

5.3.

with

assumption

unbounded

this p r o p e r t y

counterpart

Finite

in the i n t r o d u c t i o n

from its d o m a i n

we see that

for the s e c o n d

where moreover

the r e q u i r e m e n t

class

B is an u n b o u n d e d that

if

in Y ~ U is bounded.

subject

•

to

operator

the g a i n o p e r a t o r

B P be

43

bounded

runs

into conflict

controllability original times

free d y n a m i c s

problems,

holds

true,

stabilizable well-known

result

implies

or equivalently,

as

that

5.5.

K

of D. Russell

origin,

well

conservative

D.C.

the pair

Assume

the F i n i t e

(1973)

(5.17)-(5.19).

Assume,

group u n i f o r m l y

bounded

and

and

(ll)

the

(exponentially)

=

(5.17)).

Then,

a

for time-reverslble

[Ru.2]

(H.2)

controllable

(to the

Thus:

(1.6)

for the dynamics,

(1.9) and the D e t e c t a b i l i t y that

the

w a v e and plate

groups)

of

{A ,R } is e x a c t l y

further,

(1)

for n e g a t i v e

is u n i f o r m l y

from the origin}.

hypothesis

if

bound

of the pair

in the n o t a t i o n

Cost C o n d i t i o n

of exact

= (5.17)-(5.19)

then the pair {A*,R*}

systems

Corollary

group u n i f o r m l y

in fact u n i t a r y

condition

(by the o p e r a t o r

(5.20)

On the other hand,

the interesting which yield

desirable d e t e c t a b i l i t y {A,R}

the a s s u m p t i o n

e At is a s.c.

(the case of all

Schr~dinger

with

of the pair {AS, R*}.

the free d y n a m i c s

for n e g a t i v e

times.

B is b o u n d e d

if and only

as

Condition

e At is an s.c.

Then the c o n c l u s i o n

of

s

Corollary

Remark

5.4 applies:

5.3.

Algebraic

The

following

Riccatl

In addition,

reference

Equation

however,

under

[P-S]

[P-S]

also deals w i t h

the a b s t r a c t

makes

if B P is bounded.

hypothesis

the f o l l o w i n g

two

•

the

(H.2)

= (1.6).

further

assumptions: (1)

an a s s u m p t i o n

R e ~(Y,Z)

as e x p r e s s e d

of the s m o o t h n e s s

on the o b s e r v a t i o n

operator

by the r e q u i r e m e n t T

f IlReAtxll~dt ~ OT~lXl]~ 0 where

V is a space

Y, s u c h

strictly

that B ~ ~(U,V)

larger

(5.21)

than Y and w i t h w e a k e r

and e At generates

topology

than

semlgroup on both Y

a s.c.

and V. (ii) for all

The a s s u m p t i o n

initial

data

(This a s s u m p t i o n plates

the F i n i t e

is Dot true

assumptions,

Detectability P en~oys

Condition

[P-S]

in the cases

V" = dual

regularity

Condition

space

6.2,

6.3, and,

holds

true

V, not o n l y on Y.

of c o n s e r v a t i v e

existence

on V, u n i q u e n e s s

the f o l l o w i n g

P e Z(V,V'),

claims

Cost

larger

problems such as those of s e c t i o n s

the a b o v e

where

that

in the s t r i c t l y

and under

of the s o l u t i o n

waves

6.4).

and Under

the a d d i t i o n a l P of the ARE,

property:

of V w i t h respect

to Y-topology,

(5.22)

44

which

in turn implies

boundedness

of the gain o p e r a t o r (5.23)

B P e Z(Y,U). In v i e w of the a b o v e are the r e s u l t s 5.2,

first

operator

of

given

methods

in the c a n o n i c a l [L-T.6], and

cannot

cover

important

the a b s t r a c t

In addition, physically

6.1,

violated

6.2,

is b o u n d e d

of

regularity

[PS]

than

control)

setting

cannot

of

in [P-S]

and plates and R > 0, w h i c h for

in the first place.

cover

problems

the distinctive,

such as those of our

as the r e g u l a r i t y

required

by

[PSI

is

5.1).

(5.21)

result

where

in

[P-S]

holds

the gain o p e r a t o r B P

on the s m o o t h i n g suffices

of the

to a c h i e v e

this

true.

to the h y p o t h e s e s

property

waves

= (1.6)

in s i t u a t i o n s

the f o l l o w i n g

In a d d i t l o n

5.6.

(H.2)

by a b s t r a c t

the results

justification

then an a s s u m p t i o n

R much weaker

Indeed,

following

(5.23),

a main

(parabolic)

and 6.3 below,

is i n t e r e s t e d

as in

observation

Theorem

analytic

(~ < ~ < 1: see R e m a r k

If one

goal.

assumption

the results

relevant,

5.2,

class of c o n s e r v a t i v e

Theorem

equation

in a b s t r a c t

with Theorem

(point or b o u n d a r y

not only

by the e a r l i e r

case of the wave

the class w h i c h o f f e r s

introducing

section

in contrast

with B unbounded

iS p r e c i s e l y

we can say that:

and then fully cast

moreover,

problems

5.5,

on the ARE s u b s u m e d

[FLT];

the

5.4,

Corollaries

[P-S]

of T h e o r e m

5.2,

assume

the

on R:

o@

~

BUllydt _< Cllull u,

IIR Re

u e U

(5.2Ibis)

0

where

-A is the t r a n s l a t i o n

generates

~>o,

a n s.c.

e -~t u n i f o r m l y

above stable:

(5.0),

which

l[e-Atll < Me -~t,

t~o. Then

Theorem

5.2,

the o p e r a t o r satisfies

Hypothesis satisfies 3.3.

semigroup

of A i n t r o d u c e d

(5.2Ibis)

the D i f f e r e n t i a l

Then,

Theorem

P,

a limiting

in a d d i t i o n

(5.23):

to the p r o p e r t i e s

B P ~ ~(Y,U).

guarantees Riccatl

argument

that

Equation

guaranteed

the c o r r e s p o n d i n g

PT(0)

for all T > O, see S e c t i o n

on the formula

defining

P yields

5.6. Since

assumption,

in the t r e a t m e n t then c o n d i t i o n

of

(5.21)

[P-S],

we have B e ~(U,V)

in [P-S]

by

•

implies

by

a for~ioFi

that

45

T

I[[ReAtBu[[z2dt <. CT[[U[[~, 0 A c o m p a r i s o n b e t w e e n our c o n d i t i o n (5.21iris)

reveals that

(5.2Ibis)

U ~ U

(5.21trls)

(5.21bls) and c o n d i t i o n

is w e a k e r than (5.21tris)

on a few

grounds: (i)

(5.2Ibis)

r e q u i r e s o n l y an L 1 c o n d i t i o n on any finite T, while

(5.21trls) (ii)

condition

requires an L 2 c o n d i t i o n in time; (5.21bls) uses the s m o o t h i n g of R R ("twice" that of

the o r i g i n a l R), w h i l e c o n d i t i o n o n l y of R; in other words, (5.21iris) condition The next

requires

(5.21trls) n e e d s the s m o o t h i n g

even w i t h i n an L2-test,

"twice as much"

condition

s m o o t h i n g of R than

(5.21his) does. result is the c o u n t e r p a r t of the Dual D i f f e r e n t i a l

Riccati E q u a t i o n C o r o l l a r y 5.7.

(3.37)

for T < ~.

[F-L-T]

Under the a s s u m p t i o n s of T h e o r e m 5.3 w h i c h

g u a r a n t e e that the Riccati o p e r a t o r P is an i s o m o r p h i s m on Y, set Q = p-1E

Z(Y).

Then,

Equation

(DARE):

Q satisfies

the f o l l o w i n g Dual A l g e b r a i c Riccati

(AQx, y) y+(QA*X,y)y+(RQX, R Q y ) z - ( B "x,B "Y)U = 0, V x,y

Q ~ Z(~(A Equation

(5.1).

Dual ARE

(5.24)

Ap = A-BB P.

to the (original)

A comparison between

•

(5.25)

(5.1}

(5.24)

A l g e b r a i c Riccatl

(5.24) and

(5.1) r e v e a l s the

(Table 5.1).

Correspondence

O r i g i n a l ARE

) c y;

(5.24) will be h e n c e f o r t h r e f e r r e d to as Dual A l g e b r a i c

following correspondence

TABLE 5 , 1 .

) c ~(B

} ;~(Ap)) n Z ( ~ ( A p ) ; ~ ( A ) ) ,

Riccatl E q u a t i o n w i t h respect Equation

E ~(A

A -A

Between O r i g i n a l A -A

R B

and Dual ARE B R

B R

P q

Z u

46

Thus, (infinite

to the o r i g i n a l

horizon)

dynamics

control

dynamics

problem

and its c o r r e s p o n d i n g

TA B L E

5.2.

Original

Original dynamics

(1.1)

(1.2),

control

and to its c o r r e s p o n d i n g

there

corresponds

problem

indicate

the dual

below.

and Dual P r o b l e m

Problem

Dual

(i.i):

Problem

dynamics:

= A y + B u on Y;

= - A z + R v on Y;

Cost (1.2):

Cost: J(v,z)

0

From DARE

0

the c o r r e s p o n d e n c e

(5.24)

As a s s o c i a t e d

well-posedness,

however, -A) b e

(equlvalently, that A *

consequence

A) be

we shall

Lemma

5.2,

7.0]

we see p l a i n l y

z = -A*z+R*v,

the a d d i t i o n a l of a s.c.

the g e n e r a t o r

of this a s s u m p t i o n

IF-L-T,

L2(O,T;U)

requires

the g e n e r a t o r

(equlva~ently,

be s h o w n

of Table

to the d y n a m i c s

a n d of h y p o t h e s i s

discuss

the DARE u n d e r

a special

automatically

but

fulfilled

(H.2)

case

in the case of,

=

o n Y.

(I.6),

element

say,

As a

it m a y of results

that A he an s.c.

(this h y p o t h e s i s

-A

i.e.,

In the f o l l o w i n g

the a s s u m p t i o n

important

that

o n Y;

group

that B*z is a w e l l - d e f i n e d

for any T > O, when v e L 2 ( O , T ; Z ).

generator,

assumption

semigroup

o f a n s.c.

that the

whose

group

is

conservative

waves

and

equations).

plates

It was a l r e a d y below Theorem present

5.2

available

of the o p e r a t o r boundedness. operator

noted

in s e c t i o n

S as well

that no D i f f e r e n t i a l

Riccati

under

(H.2)

R,

the a s s u m p t i o n

i.e.,

with R subject

In such generality,

P of T h e o r e m

Equation

= (1.6)

on

it is still

[O,T]

is at

w i t h no s m o o t h i n g

o n l y to a s s u m p t i o n

5.2 can be i d e n t i f i e d Px = . J i m PT(O)x,

as in the p a r a g r a p h

true that

(5.0)

of

the Riccatl

b y the limit r e l a t i o n ,

x ~ Y,

(5.26)

TT= where

PT(t)

explicitly

~ ~(Y), defined

0 ~ t ~ T, in terms

is the o p e r a t o r

of the s y s t e m ' s

in

(3.6)

data via

which

is

the c o r r e s p o n d i n g

47 optimal s o l u t i o n of the q u a d r a t i c p r o b l e m over PT(t) does

[O,T]: moreover,

such

realize the p o l n t w i s e (a.e.) f e e d b a c k synthesis, u (t,O;y O) = -B P T ( t l y

(t,O;Yo),

a.e.

in [O,T],

(5.2?1

of the optimal pair {u~,y~} as seen in T h e o r e m 3.1 as well as some

relations typical of the Riccatl theory, n o t e d there and in However, what is m ~ s s ~ n g ~n s u c h g e n e r a l l t y for R ~ ~(Y,Z) is a claim that PT(t) s a t i s f l e s the D i f f e r e n t i a l Riccatl further

Section 3.4.

Equation.

In w h a t follows, we shall show that w h e n the d y n a m i c s is

tlme r e v e r s i b l e connection,

(A g e n e r a t e s an s.c. group),

under natural assumptions,

it is p o s s i b l e

to make a

b e t w e e n the a l g e b r a i c

Riccati

o p e r a t o r P and the s o l u t l o n to 'some' D i f f e r e n t i a l Riccatl Equation, indeed,

the Dual D i f f e r e n t i a l Riccati E q u a t i o n i n t r o d u c e d b e l o w in

(5.30). T h e o r e m 5.8.

([FLT],

assume h y p o t h e s i s

IF.3])

Let A be an s.c. g r o u p g e n e r a t o r on Y and

(H.2) = (1.6) for the dynamics.

Finally,

assume the

Finite Cost C o n d l t ~ o n for the Dual P r o b l e m in T a b l e 5.2:

For each Zo ~ Y, there exlsts v e L2(O,~;Z) such

I

that J(v,z)

~

{5.28)

~, w h e r e z is the s o l u t i o n due to v.J

Then: AS

= Q

{i) there exists an operator Q e ~(Y), s a t i s f i e s the DARE

(5.24);

(ii) if, An addition, {A,B}

>_ 0, w h i c h

(equivalently,

the palr {-A,B}

Is e x a c t l y c o n t r o l l a b l e over some interval

of L2(O,T;U)-controls,

then the DARE

(5.24) admits a unique solution,

^

g i v e n by Q,

the pair

[O,T] w i t h i n the class •

in the class of

all Q ~ ~(Y), s u c h that Q = Q

> 0;

(lil) Q is g i v e n by the s t r o n g limlt Qx = llm QT(O)x,

x

e

Y,

(5.29)

TTw h e r e QT(t) e ~(Y),

0 ~ t ~ T, is the u n i q u e s o l u t i o n of the f o l l o w i n g

Dual D i f f e r e n t i a l Riccatl E q u a t l o n

48

{ ~t

s

(OT(t)x'Y)Y = (QT(t)x,A y ) y + ( A X, Q T ( t ) y ) y s

+ (RQT(t)x, R Q T ( t ) y ) z - ( B x,B YIU

(5.30) for all x,y ~ ~ ( A

1,

QT(T) = O, u n i q u e n e s s b e i n g in the class of o p e r a t o r s Q(-) E Z ( Y ; C ( [ O , T ] ; Y ) that

(Q(t)x,y)

(iv) the pair {-A,B} c o n t r o l l a b l e on s o m e and o n l y ~f QT(0)

(equivalently,

the p a i r {A,B} is e x a c t l y

[O,T] w l t h i n the class of L 2 ( O , T ; U ) - c o n t r o l s ,

~f

is an i s o m o r p h i s m on Y, in w h i c h case Q is an

i s o m o r p h i s m on Y as well.

•

The proof of T h e o r e m 5.8 is in [FLT, T h e o r e m s where

such

is d l f f e r e n t i a b l e in t for e a c h x,y ~ ~(A'):

further results may be found.

a n a l y s i s of the DARE

(5.24),

2.6 and 2.7],

It s h o u l d be n o t e d that the

as well as its d e r i v a t i o n s t a r t i n g from

the Dual control p r o b l e m in Table 5.2, are a m u c h s i m p l e r a n a l y s i s of the original ARE o r l s l n a l control p r o b l e m (RQx, RQy) Z in (~.24)

task than the

(5.1) and its d e r i v a t i o n s t a r t i n g from the

{1.1),

(1.2).

Indeed,

the q u a d r a t i c

term

in the u n k n o w n Q occurs w i t h the b o u n d e d o p e r a t o r

R, w h i l e the q u a d r a t i c term (B Px, B PY)u in (5.1) in the u n k n o w n P s

occurs w i t h the u n b o u n d e d o p e r a t o r B .

The w e l l - p o s e d n e s s

(existence

and uniqueness)

of the DARE for Q can be r e a d i l y h a n d l e d by a r g u m e n t s

by now s t a n d a r d

(fixed point plus a priori

bounds)

IF.3],

[FLT],

f o l l o w i n g the o r i g i n a l t r e a t m e n t in [DaP]. The s a m e c o n s l d e r a t l o n s a p p l i e d to the o r i g i n a l ARE

(5.1) say

that the case of R u n b o u n d e d and B b o u n d e d is m u c h e a s i e r than the case of R b o u n d e d and B unbounded;

compare the term

(Rx, R y ) y w i t h the term

(B Px, B PY)u in (5.1}. T h e o r e m 5.8 states that the o p e r a t o r Q d e f i n e d by the limit (5.2g) of s o l u t i o n s

to the Dual D i f f e r e n t i a l Riccatl E q u a t i o n

s a t i s f i e s the same Dual A l g e b r a i c Riccati E q u a t i o n o p e r a t o r p-1 --if it e x l s t s ! - - w o u l d satisfy,

(5.24)

(5.30)

that the

see C o r o l l a r y 5.7.

under exact c o n t r o l l a b i l l t y of the pair {-A,B},

Since

the o p e r a t o r Q is an

i s o m o r p h i s m on Y, the q u e s t i o n arises as to w h e t h e r or w h e n the a n a l y s i s of the original control p r o b l e m leadin~ to the Riccati o p e r a t o r P, and the a n a l y s i s of the dual control p r o b l e m leadlng to the dual R1ccatl

operator Q merge,

i.e., m o r e p r e c i s e l y as to w h e t h e r or

49

when we have Q = p-l, negative,

see

or ~-1 = p.

Example below.

In general,

Indeed,

the a n s w e r is in the

the v e r y i d e n t i f i c a t i o n of P w i t h

~-1 r e q u i r e s that P be an i s o m o r p h i s m on Y.

It ~s most g r a t i f y i n g

t h e r e f o r e that the I d e n t l f l c a t l o n P = ~-1 holds t r u e generator)

controllable

on some [O,T], T < ~,

i.e., p r e c i s e l y the c o n d i t i o n s u n d e r

w h i c h Q and P are both Isomorphisme, 5.7(iv),

(when A is a g r o u p

p r o v i d e d that both pairs {-A,B} and { A * , R *} are e x a c t l y see T h e o r e m 5.3 and T h e o r e m

this is the content of T h e o r e m 5.9 below.

respectively:

s

Example.

[FLT, p.325]

Let R = 0, B e Z[U,Y},

exactly c o n t r o l l a b l e over some other hand,

[O,T].

Then,

the F i n i t e Cost C o n d i t i o n

(5.28)

s a t i s f i e d since -A

-A

stable,

trivially,

and {A,B}

P = 0.

On the

for the dual p r o b l e m is

is stable and

(Qx, y)y = ~ ( B * e -A tx, S*e-A ty)udt,

(5.31)

0

Moreover,

where Q s a t i s f i e s the DARE. T h e o r e m 5.8(iv): T h e o r e m 5.9.

^--1

Q

[FLT,

However,

E ~(Y).

Theorem 2.7]

Q is an i s o m o r p h i s m on Y, by q

~ p.

m

Let A g e n e r a t e an s.c. g r o u p on Y.

If b o t h pairs {A,B} and {A*,R*} are exactly c o n t r o l l a b l e o v e r some [O,T],

then p -1

= Q.

•

(5.32)

C o m b i n i n g Theorems 5.8 and 5.9, we o b t a i n C o r o l l a r y ~.10.

Under a s s u m p t i o n

(H.2),

let A g e n e r a t e an s.c. g r o u p

on Y, and let {A,B} and {A ,R } be both e x a c t l y c o n t r o l l a b l e on some [O,T].

Then Px = llm T T--

w h e r e QT(t),

(Olx,

x e y,

(5.331

0 ~ t ~ T, is the unique s o l u t i o n to the Dual D i f f e r e n t i a l

Riccati Equation

{5.30).

•

The a b o v e C o r o l l a r y 5.10 c h a r a c t e r i z e s a s t r o n g limit of solutions

to the DDRE

the R i c c a t i o p e r a t o r R as

(5.30),

as desired,

50 The accompanying diagram illustrates a few main points of the orlg~nal and dual problem, and their merging at the level of establishing Q defined by Q = p-1 coincides with Q defined by (5.29).

maU~maml~a

Oriulnal dvnami,cs

(A group generator)

= Ay+Bu on Y Orloina! OPC(~)

Dual

z = -A*z+R*u on Y OCP(~)

0

0 !

Starting from finite time problem on [O,T], under Finite Cost Condition for original OCP (~) I

Starting from finite time I problem [O,T], under Finite Cost Condition for dual OCP (~)

l

as T 7 ~

P = llm PT(O) strongly s.t. P satisfies original

I

as T T ~

Q = lira QT(O)(strong'l'~Y;I s.t. Q satisfies dual !

ARE (5.1)

DARE (5.24), QT(t) satisfies DDRE (5.30)

[ I

{ {A ,R } exactly controi fable

(-A*,S*} I~ detectable I

solution

detectable 1

(-A,B} e x a c t l y

controllable

unique solution

I ..... isomorphism on Y

I P isomorphism on Y and Q m p -I

l

Q = Q when ) .... {A',R ~} and {-A,B} exactly controllable

51 6.

E x a m p l e s of Dartlal d i f f e r e n t i a l e a u a t l o n p r o b l e m s s a t l s f v l n a (H.I} In this section,

we illustrate the a p p l i c a b i l i t y of T h e o r e m s 2.1

{T < ~) and 5.1

(T = ~) for the

{H.I) = (i,5}.

In passing,

which s a t i s f y b o t h

'analytic'

class s u b j e c t to h y p o t h e s i s

some p.d.e, p r o b l e m s will be e x h i b i t e d

{H.I) and .(H.2).

Obvious c a n d i d a t e s

analytic class are heat or d i f f u s i o n problems.

A few canonical

thereof will be treated in s e c t i o n s 6.1 and 6.2 below, [Las.4],

[L-T.7].

cases

following e.g.,

In s e c t l o n 6.3, we shall then a n a l y z e e x a m p l e s of

(structural damping),

plates w i t h a s t r o n g d e g r e e of d a m p i n g

may a r i s e in the s t u d y of flexible structures. sections 6.1,

for the

6.2,

7.1,

7.2,

s u c h as

All e x a m p l e s in

Y.3, and most of those in s e c t l o n 6.3, are

not c o v e r e d by other treatments such as the one in [PSI. 6.1.

C~ass

(H.I):

Heat e q u a t i o n w l t h D i r l c h l e t b o u p d a r y control

Let ~ c R n be an open bounded d o m a i n w i t h s u f f i c i e n t l y smooth boundary r.

In n, we c o n s i d e r the D i r i c h l e t m i x e d p r o b l e m for the heat

e q u a t i o n in the u n k n o w n y(t,x): Yt = d y + c 2 y

in (O,T]x~ ~ Q,

(6.1a)

y(0,-) = Y0

in n,

(6.1b)

l

L-/Ylx = u

in

(O,T]xr

~ ~,

with b o u n d a r y control u e L2(Z ) and YO • L2{O)" T = ~.

(6.1c) We e x p l i c i t l y c o n s i d e r

The cost functional w h i c h we w i s h to m i n i m i z e Is then

J(u,v) =

f

(fly(t) U ~2 {O)+IIu(t) HL2 (r) } dt.

(6.2)

0

Note that the a b o v e p r o b l e m class

(H.2).

y • H½*¼(Q),

In fact, with,

say,

(6.1),

(6.2} does not belong to the

Y0 = 0 and u • L2(~ ), we only have

but y does not b e l o n g to C([0, T];L2(n)),

even in

1-dimension.

Abstract setting

[B.I],

[W],

[Las.4],

[T.6].

To put p r o b l e m

(6.2) into the a b s t r a c t setting of the p r e c e d i n g sections,

(6.1),

we i n t r o d u c e

the o p e r a t o r Ah = dh+c2h;

~(A)

1 = H2(Q) 0 H0(O),

(6.3)

52

select

the spaces Z = Y = L2(Q);

U

(6.4)

L2(F),

=

and finally define the operators R

Bu = -ADIu; where D I (Dirichlet map) h =Dlg and by elliptic

=

(6.5)

I,

is defined by

iff

(d+c2)h = 0 in ~;

hit

=

(6.6)

g,

theory and [Gr],

DI: continuous

L2(F ) ~ HM(Q)

c

ADh = -dh, In (6.5) A is the isomorphic

HM-2z(Q) ~(AD)

m ~(A~-Z),

¥ Z > 0,

(6.7)

= H2(Q) N H~(Q),

extension of A in (6.3),

(6.8) from,

say,

n2(n) ~ [~(A)]"

Assumption

(1.3]:

(A)-~B e Z(U,Y).

Assumption

our present

case w i t h 7 = ~+~, v e > 0.

From

we have

(6.7),

AD: continuous

(6.5),

= [~(A~+¢)],

AssumDtlon

(H.I) = (1.5).

in

(6.9)

that our claim ~s verified,

i-VB = -AD~AD e Z(L2(F);L2(n))

semigroup

is satisfied

In fact, we may take A = A D.

L2(F) ~ [ ~ ( ~ + ¢ ) ] ,

and we then have with 7 = ~+~ via

(1.3)

= ~(u;Y).

(6.10)

The operator A in (6.3) generates an s.c.

e At , on L2(~) , which is m o r e o v e r analytic here for t > 0 (and

c o n t r a c t l o n after a suitable

translation

of the generator).

Finite Cost C o n d l t i o n

(I.~).

constant

only flnltely many e i g e n v a l u e s

c 2 in (6.3)}

multlplicity,

The generator A has

since its resolvent

Thus,

the s t a b i l i z a t i o n

[M-T,

Appendix],

large

of finite

is compact and e At is analytic.

theory as in [T.1],

etc. applles:

(for suitably

The problem

[T.5],

[L-T.1Y],

is s t a b i l i z a b l e

on L2(Q)

if

53 and only if i t s

p r o j e c t i o n onto the finite d i m e n s i o n a l u n s t a b l e

subspace is controllable.

In particular,

one may p r e s c r i b e the s t a b i l i z i n g

as s h o w n in fT.5],

[L-T.17],

feedback to be of the form

N

u(t) =n=l Z (Y(t)'Wn)L2 (n)gn

(6.11)

for s u i t a b l e v e c t o r s w k 6 L2(O) and gk e L2(F) and s u i t a b l e as d e s c r i b e d there,

in order to s t a b i l i z e

feedback s y s t e m in the n o r m of H~-~(O) Finite Cost C o n d i t i o n on L2(~)

Detectability Condition in our case R = I, see

Conclusion:

T = ~.

(minimal)

N

u n i f o r m l y the c o r r e s p o n d i n g

in fact.

Thus, a fortiori

the

~s satisfied.

(5,~O).

This is a u t o m a t i c a l l y s a t i s f i e d since

(6.5).

T h e o r e m 5.1 a p p l i e s to p r o b l e m

(6.1),

(6.2)

Theorem 2.1 applies to p r o b l e m

(6.1) and

[L-T.7].

~oncluslon:

T < ~.

T < ~ for a n y final operator.

state o p e r a t o r

See R e m a r k 2.1,

G that m a k e s GL a closed

in p a r t i c u l a r

(6.2) w i t h (closable)

the (only) s u f f i c i e n t

c o n d i t i o n (2.25).

R e m a r k 6.0.

The a b o v e a n a l y s i s applies,

y(t) p e n a l i z e d in L2(O,T;H~-~)),

w l t h no e s s e n t i a l

w i t h Y0 ~ HM-z(n)"

L2(O,T;L2(n)

as in (6.2) where then ~ = I-~/2.

6.2.

(H.1):

Class

change,

to

rather than in

i

Heat e u u a t l o n w l t h N e u m a n b o u n d a r y control

Now we consider p r o b l e m 87, ~-i~ u e L2{7 ) and cost f u n c t i o n a l

(6.1a-b) w i t h = u,

on ~,

(6.1c) r e p l a c e d now by (6.12)

for T = ~:

0

instead of

(6.2), w h e r e we take go ~ Hl(n)"

present p r o b l e m (H.2).

(6.1a-b),

This is so since,

(6.12),

(6.13),

We note a g a i n that the

does not b e l o n g to the class

say, w l t h 70 = 0 and %% ~ L2(• ), we have

54

y e H~'~(Q), b u t y d o e s not b e l o n g

~bstract

settlnu.

abstract

setting,

we s h a l l

and select

To put p r o b l e m we Introduce

consider

(6.1a-b),

~(A)

into the

as l i f t e d

(6 " lS~)

"

to

the s p a c e s a n d o p e r a t o r s

w l t h A In

(6.17)

= ~(A½);

U = L2(r),

(6.15a).

Here,

the i s o m o r p h i c

extension,

invertible o n L2(n)

h=

Ng

We h a v e f r o m e l l i p t i c N: c o n t i n u o u s

L2(F)

(1.3):

Neumann

theory

~ H~(~)

L2(O)

we a s s u m e B.C.,

a n d the N e u m a n n

(~,c2)h=

i~

(6.17)

say,

loss of general~ty,

without

(6.i6)

R = I,

a n elgenvalue of d w i t h h o m o g e n e o u s

present

(6.13)

~-~ I F = o) ah

= { h e H2(n),

B U = -ANu,

Assumption

(6.12),

the o p e r a t o r

Z = Y = HI(~)

boundedly

(6.14)

to C([0, T ] ; H I ( ~ ) ) .

Ah = dh+c2h; which

h e n c e y e C([0, T ] ; H ½ ( Q ) ) ,

~

[~(A)]"

that

of A in

-c 2 is not

so that A is

m a p N is well d e f i n e d

o inn~

~lr

= g"

by

(6.18)

and [Gr],

¢ H~-2e(~)

(A)-~B ~ ~ ( U ; Y ) .

c a s e w i t h T = ~+~, V ~ > 0.

= ~(A~-~),

Assumption In fact,

(1.3)

Y ~ > 0,

holds

w i t h ~ = ~+E,

(6.19)

true

in the

we n e e d to

s h o w that

A-tB equivalently

that

(see

A~A-~AN which

is p r e c i s e l y

E Z(U,Y)

= ~(L2lrl,~(A~)),

(6.20)

(6.17)), = A~-Y+~+¢A~-~N

true in v i e w of

~ Z(L2(F),L2(~)),

(6.19)

s i n c e ~-q43A+e = 0.

(6.21)

55

Assumption

(H.1) = (1.5]..

Since A d e f i n e d in (6.15a) g e n e r a t e s an s.c.

a n a l y t i c s e m l g r o u p on L2(~),

then its llftlng as in (6.15b) g e n e r a t e s

an s.c. a n a l y t i c s e m l g r o u p on ~(A~) ~lnl%eCost

Condition

(1.9).

= HI(~)

C o n s l d e r a t i o n s i m i l a r to those made for

the D i r l c h l e t case apply now; see e.g., s t a b i l i z a t i o n results in H~-Z(~) Cost C o n d i t i o n

(1.9} on HI(Q)

as desired.

[T.5],

In fact.

holds t r u e

[L-T.17]

Thus,

a

for u n i f o r m the Finite

fortlorl

for p r o b l e m

(6.1a-b),

(6.12),

(6.13). (5.10].

Detectabilltv Condltlon

With

R

=

I,

this is a u t o m a t i c a l l y

satisfied. Conclusion:

T = ~.

Theorem 5.1 applies to p r o b l e m

(6.1a-b),

(6.12),

T < ~.

T h e o r e m 2.1 a p p l i e s to p r o b l e m

(6.1a-b),

(6.12),

(6.13), Conclusion:

(6.13) w i t h any final state o p e r a t o r that m a k e s GL c l o s e d See R e m a r k 2.1,

in p a r t i c u l a r the (only) s u f f i c i e n t c o n d i t i o n

We also remark that the above a n a l y s i s applies, essential

change,

YO E H~-Z(n) = i-~/2.

R e m a r k 5.i.

(closable).

to y(t) p e n a l i z e d in L2(0, T;H~-~(~))

rather than In L 2 ( O , T ; H I ( n ) ) H e r e we can take G = ~ - ~ / 2

(2.25}.

w i t h no wlth

as in (6.13),

w i t h Z = L2(~).

where then •

The choice of the functional

:(uy)

=

_

(6.227

dr,

0 in p l a c e of

(6.13) c o n s i d e r a b l y s i m p l i f i e s the analysis,

since with

y = L2(~ ) one e a s i l y sees now that in this case we h a v e that a s s u m p t i o n (i.3) holds true w i t h ~ = ~+~

< ~.

Then,

easier p r o b l e m belongs also to the class and 5.2 are applicable,

R e m a r k 6.2. heat

This

Thus b o t h T h e o r e m s

5.1

but T h e o r e m 5.1 is p l a i n l y to be preferred.

•

H a v i n g solved in HI(Q)

equation problem

H~-¢(O),

R e m a r k 6.I applies.

(6.1a-b),

(H.2).

the q u a d r a t i c cost p r o b l e m for the

(6.12),

(6.13)

{indeed,

as r e m a r k e d in

if we llke}, we can then o b t a i n as a c o n s e q u e n c e a s o l u t i o n to

66

the "purely boundary"

quadratic

cost problem w h i c h p e n a l i z e s

the cost

functional

Irl[L2(r)+l[u(t)

} dt

(6.231

0 over a l l u E L2(O,~;L2(F)) E

YO

HI(~).

w l t h y s o l u t i o n to (6.1a-b),

Now we take Y = Hl(~),

Z = L2(F)

trace operator y ~ Ry = yIF: continuous retailed u n i f o r m s t a b i l i z a t i o n the c o r r e s p o n d i n g as in (6.11), ~n H~(U) (6.23)

Condition K

E

and R is the

HI(~) ~ H~(F)).

guarantees

in fact.

a

The p r e v i o u s l y

for the s o l u t i o n y in Hl(n)

Thus,

in order to satisfy

to the s t a b i l i z a t i o n

to obtain the required

Z(L2(F),HI(Q))

of

of the form

u n i f o r m d e c a y of y(t)ir

the required Finite Cost C o n d i t i o n

Moreover,

(5.10) we appeal [L-T.18],

exponential

Fortiori

and

(Dirlchlet)

feedback closed loop problem with u, say,

iS satisfied.

see also

results

(6.12),

(1.9)

for

the D e t e c t a b i l i t y

results as in [T.6],

"stabillzlng"

operator

in (5.10), w h i c h may be taken of the form N

K" = l l ( - [ F , W n ) L 2 ( F } g n for s u i t a b l e w n E L2(F } and gn ~ HI(~)" e q u a t i o n with homogeneous

If such K is added to the heat

b o u n d a r y conditions, N

Yt = (~+c2)y + ~ (Ylr'WnlL2(r}gn n=l

in Q,

(6.24c)

y(O,.)

in ~,

(6.24b)

Z,

(6.24c)

= Yo

8y X -= 0 ~-u

then u n d e r suitable Hl(n) will result L2(O,T;L2(F})

conditions

[T.6],

on Wn, g n, u n i f o r m decay in (at least)

as desired.

In (6.33) can be pushed,

through w i t h no essential take

in

YO ~ H~-~(Q)"

|

change,

The p e n a l i z a t i o n

y(t)l~ e

for the above a n a l y s i s

to y(t)l~

to go

E L2(O, T ;H I-~ (r)) ~f we

57 6.3,

C l a s s fH,~): S ~ u t t u r a l l y b o u n d a r y control

~ / L ~ .

damped Dlateswlth

The case u = ~ in [C-R],

D o l n t control o~

[C-T.I-2].

Consider

the

f o l l o w i n g model of a plate e q u a t i o n in the d e f l e c t i o n w(t,x),

where

p > 0 is any constant:

2

wtt+d w - p d w t = 6(x_x0)u(t)

in

W(O,.)

i n ~,

(6.25b)

in

(6.25c)

r

lLwI~

= WO:

~ ~wlZ ~

wt(o,-)

= w1

0

(O,T]xR

(6.25a)

= Q,

( O , T ] x F = Z,

w i t h load c o n c e n t r a t e d at the interior point x O of an o p e n bounded (smooth) d o m a i n ~ of R n, n ~ 3.

R e g u ] a r l t y results for p r o b l e m

and o t h e r p r o b l e m s of this type,

are g i v e n in [T.4].

these results,

3(u.w)

the c o s t

functional we wlsh to m l n l m i z e

" 0

Consistently with Is for T = ~:

2 +llw.(t)ll2L2(n)+llu(t)llL2(n)}

= [{}]W(t)ll2~ H-IA)

(6.25),

dr,

(6.26)

"

i(O) w h e r e {Wo,Wl} G [H2(~) n H 0 ]xL2(n).

A b s t r a c t settlnu.

To put p r o b l e m s

s e t t i n g of the p r e c e d i n g sections,

(6.25),

(6.26)

into the a b s t r a c t

we introduce the s t r i c t l y p o s i t i v e

definite operator A h = &2h; ~(A) = {h e H4(~]): hIF = ~ h I F = O}

(6.27)

and s e l e c t the spaces and o p e r a t o r s Y = ~(A~)xL2(f} ) = [H21~) • HI(~)]×L2(C~); U = ~I,

A =

10ii ;

-A

Bu =

-pA y,

to o b t a i n the a b s t r a c t model

(1.1),

i01

;

R = I

(6.28)

(6.29)

& (x-x O) u

(1.2).

We n e e d to v e r i f y a few

assumptions.

is easy to verify

(1.3):

assumption

(1.3) is s a t i s f i e d w i t h ~ = I.

r e q u i r e that

(-A)-~B 6 ~(U,Y).

It

As~mptlon

Indeed,

that

from ( 6 . 2 9 ) ,

we

58

° I

6(x_xOlu

~

°ul

• Y,

from (6.28), we require that A - ~ 6 ( x - x O) e L2(~),

i.e.,

I6.30)

or that

(#): 6 ( x - x O) E [~(A~)] ", the dual of ~(A ~) w i t h respect to L2(n). Since it is true that ~(A ~) c H2(~) (6.27) thus

[H2(~)] " c

for the fourth order o p e r a t o r A in

regardless of the p a r t i c u l a r b o u n d a r y condltlons), and

(in fact,

[~(A~)] ", then c o n d i t i o n

6 ( x - x O) e [H2(~}] ", i,e.,

(#) is s a t i s f i e d p r o v i d e d

p r o v l d e d H2(Q) c C(~}, w h i c h is indeed the

case by S o b o l e v e m b e d d i n g p r o v i d e d 2 > ~, or n < 4, as required. However, as--accordlng

the above result is not s u f f i c i e n t for our p u r p o s e s

to a s s u m p t i o n

take ~ < 1 in (1.3).

(H.1} = (1.S)--we need to s h o w that we can

As a m a t t e r of fact, we now show that a s s u m p t i o n

(1.3) holds true for any ~ > ~, w h i c h then for n ~ 3 yields ~ < 1 as To this end, we note that

desired.

(-A)-~B e ~(U,Y)

if and only If B ~ ~ ( U , [ ~ ( ( - A ~ } q ] "

w i t h d u a l i t y w l t h respect to Y.

But ~ ( ( - A

) = ~((-A)~):

(6.31)

th~s follows

since A ~s the d~rect sum of two normal o p e r a t o r s on Y, ~ i t h p o s s i b l y an a d d i t i o n a l [C-T.1],

f i n l t e - d l m e n s i o n a l component

[C-T.2,

L e m m a A.1,

(if I is an e i g e n v a l u e of A)

case via) w i t h ~ = ~].

Moreover,

[C-T.4,

w ~ t h ~ = ~], we h a v e ~((-A') ~) = ~((-A) ~) = ~(~+~/2)x~(A~/21,

0 < 7 < I

(6.32)

(the first c o m p o n e n t does not r e a l l y matter in the a r g u m e n t below). Thus,

from (6.32) and B as in (6.29),

it follows that

(6.31) holds

true, p r o v i d e d ~ ( x - x O) E [~(Aq/2)] " (duality w i t h respect to L2(~} ), w h e r e ~ ( A ~/2) c H2~(~),

and hence,

p r o v l d e d 5 ( x - x O} e [H2~(Q)] " c

[~(~7/2}],° i.e.,

But this in turn is the case, p r o v i d e d H2~(n) c C(~); n by S o b o / e v e m b e d d i n g p r o v i d e d 2q > ~, as desired. We conclude:

assumption 4

(1.3)

(-A)-~B e Z(U,Y)

holds ~ U ~

for p r o b l e m

(6.25) w ~ h

< ~ < i, n ~ 3.

~$sumptlon

tH.l) = (1.5), At

contraction semigroup e t > 0.

The o p e r a t o r A in (6.29) g e n e r a t e s an s.c. on Y, w h i c h m o r e o v e r is a n a l y t i c here for

(This Is a special case of a much more general

result

59 [C-T.I-2]).

This.

along w i t h the r e q u i r e m e n t ~ < 1 p r o v e d above

g u a r a n t e e s that p r o b l e m Remark 6.3.

(5.25) s a t i s f i e s a s s u m p t i o n

(H.I) = {1.5).

S i n c e the s e m i g r o u p e At is a n a l y t i c on Y and also

u n i f o r m l y stable

[C-T.2],

we have by the j u s t - v e r l f i e d p r o p e r t y

(1.3),

in the n o r m of ~(Y,U):

lIB*e A tll wlth'~

< ~ < I,

=

liB

n ~ 3.

"( - A * I - ~ I - A This

is

*

,'°'"tll

a sharp

estimate,

(the i n t e r e s t i n g cases) does not a l l o w t o (H.2) = (1.6) holds true. Finite Cost C o n d i t i o n

Instead,

~1.9).

(exponentially)

Condition

(1.9) holds true w i t h u ~ 0.

along w i t h

Then,

< t,

for

15.33)

n

2,3

=

c o n c l u d e that a s s u m p t i o n

stable in Y [C-T.2],

Suppose that instead of Eq.

(6.25b-c).

which

W i t h A as in (6.29),

wtt+(d2+kl)W-{&+k2)wt

0

(H.2) holds true o n l y for n = 1.

uniformly

R e m a R k 6.4.

°I l ,

=

the s e m i g r o u p e At is

and thus the Finite Cost

(5.25a),

= ~{x-xO)u(t)

one has in Q,

(6.34}

if 0 < kl+k 2 is s u f f i c i e n t l y large,

g e n e r a t o r A has f i n i t e l y m a n y u n s t a b l e elgenvalues Since e At is a n a l y t i c on Y, the usual t h e o r y

in {Re ~ > 0}.

[T.1] applies:

The

problem is s t a b l l l z a b l e on Y if [T.1] and only if [M-T, Appendix] projection onto the finlte-dlmenslonal

the

its

u n s t a b l e s u b s p a c e is

controllable. For instance,

if A1 .... 'AK are the u n s t a b l e e l g e n v a l u e s of A,

a s s u m e d for s i m p l i c i t y to be simple, corresponding elgenfunctions

and @1 ..... @K are the

in Y, then the n e c e s s a r y and s u f f i c i e n t

c o n d i t i o n for s t a b i l i z a t i o n is that ~ k ( X O) ~ O, k = 1 ..... K. If X 1 ..... XK are not simple,

determines

then their largest multiplicity

M

the s m a l l e s t number of scalar c o n t r o l s n e e d e d for the

s t a b l l I z a t l o n of (6.34), w h e r e n o w the right hand side is r e p l a c e d by M

$(x-xl)ui{t),

along with

{6.25b-c).

The n e c e s s a r y and s u f f i c i e n t

i=1 c o n d i t i o n fop stabilization [T.I].

is now a w e l l - k n o w n full rank c o n d i t i o n

60

Detectabllttv

Condition

Conclusion:

T = ~.

{5.10).

5.1

Theorem

n _< 3, and p r o v i d e s

existence

(5.1),

operator

w ~ t h Riccatl

(6.32),

is the direct

flnite-dimenslonal eigenvectors

characterizations

a p p l y as well

for n = i.)

Concl u s i o n :

T < ~.

(closable);

see R e m a r k

Remark

Theorem

on Y plus

A has a R i e s z see

(6.27),

to the ARE

in particular,

basis

for the

a

of for the

we have 2.2 w o u l d

for any G that m a k e s

in p a r t i c u l a r

above

possibly

(6.28)

(Note that T h e o r e m

5.1 a p p l i e s

2.1,

operators

Thus,

= v2(x01.

LV j

(6.25)-(6.26),

of the s o l u t i o n

(since A, as r e m a r k e d

= ~ ( A ) x ~ (A ~ ) ,

of these spaces.

is satisfied.

problem

in particular,

w h e r e ~(A)

where

to

P ~ Z(Y,~(A))

sum of two normal

B P ~ ~(Y;U),

condition

applies

and uniqueness

component,

on Y),

W i t h R = I, this

(only)

GL closed

sufficient

(2.25).

6.5.

Essentially

the same a n a l y s i s

w i t h minimal

changes

applies

also to p r o b l e m (6.25a-b), w i t h the B.C. (6.25c) r e p l a c e d n o w by 8w x s u 8~w~ ~-~I ~l~ ~ 0. The new d e f i n i t i o n of A incorporates, of course, thes e b o u n d a r y

conditions,

and it is still

operator

is p r e c i s e l y A ~,

before.

The m a i n d i f f e r e n c e

self-ad~olnt dimensional

# = [#i,#2],

to s t a b i l i z e

~

where

form

(6.29)

as

A is n o n - n e g a t i v e

functions.

one-

Thus,

the

with corresponding

#2 = O.

Then,

as the c o n d i t i o n

one m a y c h o o s e

L~(n)

the d a m p i n g

with correspondlng

by the c o n s t a n t

@i = const,

the system,

(With no harm,

Y = ~(A~)xL~(n), null

spanned

that

the same

the p r e s e n t

A has I = 0 as an e i g e n v a l u e

eigenfunctlon

satisfied.

is that

and has p = 0 as an e ~ g e n v a l u e eigenspace,

new operator

applies

true

so that A n o w has

Remark

6.4

@ ( x O) ~ 0 is

to w o r k on the s p a c e

is the q u o t i e n t

space

L2(n)/N(A),

the

s p a c e of A.) :

plate

The case a = I [C-T.1-2]. equation

in the d e f l e c t i o n

w(t,x)

The K e l v l n - V o i g t is

model

for a

61

'Wtt+A 2W+p,.',,2wt w(O,-)

= WO;

= 6(x-x0)u(t) wt(o,-)

aw{~+(i-~)siw

with

0 < M

boundary

< ~

the P o i s s o n

operators

-

in

B I and B 2 are

(6.35a) (6.35b)

In

(0, T ] x F

in

Z;

= Z;

(6.35c)

(6.35d)

and p > O any

zero

= q~

in Q;

= wI

o

modulus

(0,T];~

constant.

for n = I, a n d

The

[Lag.2]

for n = 2:

B1w = 2u iu 2Wxy-U 2lWyy-U 22Wxx; a

B2W where

again

Regularity with

x 0 is an results

these,

we

2

.2.

= ~-~[ ~ U l - U 2 ) W x y Inter/or

polnt

fo~ p r o b l e m

take

the

cost

+.

,

UlU2(Wyy-Wxx)

of the o p e n

(6.35)

are given

functional

to be

],

(6.36}

bounded in

the

~ ¢ R n,

[T,4]. same

as

n ~ 2,

Consistently (6.26)

wlth

{Wo, Wz} ~ H~(n)xn2(n). ~betract Ah

settlna.

= A2h,

~(A)

We

introduce

the n o n - n e g a t l v e

self-adJolnt

= 0; ~a ~ h

= {h ~ H4(fl): a h + ( 1 - B ) B l h l r

+

operator

(1-~)B2hlF

=

o}

(6.37) and select

the s p a c e s

and

operators

Y = ~(A~)xL2(f])

A =

; Bu =

the a b s t r a c t

Assummtlon verify require

(1.3J:

that

U = R I,

I: II I°I -

to o b t a i n

= H2(~)xL2(~]);

(-A)-~B

assumption

(1.1),

E ~(U,Y).

(I.3)

; R = I

(6.39)

6 ( x - x ° )u

-pA

model

(6.38)

(1.2).

Again,

ks s a t i s f i e d

it is s t r a i g h t f o r w a r d

wlth

~ = I:

From

(6.39),

to we

that

(-A)-IBu

=

= -

6 ( x - x ° )u

e Y,

(6.40)

i .e., . from (6.38) we require that A*6

(x-xO). The same argument below (6.30) then applies yielding that (6.40) holds true if n 5 3.

However, in order to verify assumption (H.l) = (1.5) which requires that 7 should be < 1, the most elementary way is to check that assumption (1.3) holds in fact true with I = 5. in fact rely on the direct computation of (-A)+

In this case, we can (for simplicity of

notation, we take henceforth p = 1)

(where the entries (1) = A*(~I+A')-~(I+A')

and ( 2 ) = -A'(PI+A'

not really count in the present analysis), and avoid the domain fractional powers as in [C-T.41. We need to compute

From (6.42), we then readily see that (-A)-'BU provided ( I ) :A-~S(X-XO) c L2(0).

E

Y =

But P(dX) = Ii2(n)

and, in fact,

2

only 2(dK) c H ( R ) suffices for the present analysis) so that condition ( # ) is satisfied provided 6(x-xO) E [H2(n)] ' (duality with respect to L2(n) )

;

i.e.,

provided 2 >

2

provided H (n) c ~ ( n ) ,i .e., by Sobolev embedding n z,

or n < 4, as desired.

We have shown:

Assumwtion (1.3)

x(U,Y) holds true for problem j6.35) with n ( 3, and 7 = H . Indeed, I = ?4 is not the least 7 for which assumption (1.3) holds true. To obtain the least I for which assumption (1.3) holds true, we proceed as in the case of problem (6.25) above, in the argument which begins with (6.31) and uses 7 the domains of fractional powers X((-A) ) . As below (6.31) and ff., we 7 need to show that ( # # ) : Bu E [2((-A) ) ] ' , duality with respect to Y. (-A)-'B

E

The above argument shows some 'leverage.'

But for 0 < 7 J K, we have from [C-T.4, with a = 11 that

Thus, from B as in (6.391, we see that condition (##I above holds true, 0 I provided 6(x-x ) c [ B ( A ) ]', duality with respect to L2(R); i.e., provided 6(x-x0 ) c [H47 (R)]', since P(A 7 ) c H4'(n) for the fourth-order

63

operator case,

in

(6.37),

provided

(-A)-~B ~ Z(U,Y) n

i.e.,

p r o v i d e d H4q(O) c C(~), w h i c h in turn is the 1 n or ~ ~ ~ > ~. We conclude: A s s u m p t i o n (1.3)

4~ > ~, holds

true

for ~ r o b l e m

(6,$5)

n 1 ~ < ~ ~ ~,

p~ovlded

<_ 3 .

Ass1~mnt~on

(H.1)

contraction t > 0.

Th~s,

=

(1.5).

semlgroup

The o p e r a t o r

case of a much more

a l o n g w i t h the r e q u i r e m e n t

problem

Remark

(6.25)

6.6.

satisfies

property

q < i proved

assumption

S i n c e the s e m i g r o u p

~ust-verified

(6.39)

generates

e At on Y, w h i c h m o r e o v e r is a n a l y t i c

~s a speclal

(This

A in

(1.3),

(H.I)

general above

an s.c.

here

result

for

[C-T.2].)

guarantees

that

= (1.5).

e At is a n a l y t i c

on Y, w e have

by the

~n the n o r m of Z(Y,U) s

IJB*e A tll = IIBS(-A*)-~(1A*l~e A tll < o[~ 1, n 1 w i t h ~ < T < ~,

uniformly (H.2)

=

Finite

stable

(1.6)

Cost

n ~ 3, w h e r e we can take all [C-T.2].

holds

Thus,

Condition

(1.9).

(exponentially)

nullspace

of A [C-T.2],

and

satisfied

% = O, R e m a r k

6.4 applies

Detectabili~

~ondition

T = ~.

for n ~ 3, we o b t a i n

With A as in (6.39),

stable

automatically

(The o r e m

t > 0 as e

At

is also

that

assumption

as well.

uniformly

Cgncluslon:

o < t,

thus

the f i n l t e - d i m e n s i o n a l

the F i n i t e

Cost C o n d i t i o n

to p r o b l e m

(5.10).

Theorem

For

(1.9)

At

~s

is

the e l g e n v a l u e

(6.35).

W i t h R = I, this is satisfied.

5.1 a p p l l e s

5.2 w o u l d a l s o a p p l y

e

in Y/N(A),

on this space w i t h u m 0. also

the s e m l g r o u p

to p r o b l e m

for n ~ 3, but

(5.35)

for n ~ 3.

the c o n c l u s l o n s

of T h e o r e m

5.1 are stronger.)

Conclusion: a n y final 2.1,

T < ~.

in p a r t i c u l a r

ExamPle consider

Theorem

state operator

6.3.

the

2.1 also a p p l i e s

G that makes

(only)

to

GL c l o s e d

(6.35)

for n ~ 3 for

(closable);

sufficient condition

(2.25).

(A s t r u c t u r a l l y damped p l a t e w i t h b o u n d a r y

the p l a t e p r o b l e m

see R e m a r k

control.)

We

64

which

wtt+~2w-pdw t = 0

in (O,T]x~ ~ Q,

(6.44a)

w(O,-)

in n,

(6.44b)

wl~ m 0

in (O,T]xF E Z,

(6.44c)

~wl~ ~ u

in ~,

(6.44d)

= WO; wt(O,. ) = w I

is the same model as the one in (6.25),

upon by a b o u n d a r y control u E L2(O,T;L2(F)) point control as in (6.25a). (6.26)

we introduce

Then,

Following

the Green map G 2 defined by

if A ~s the same operator defined

(6.44)

J as in

in the L2(r)-norm.

Y = G2v~=~ {d2y = 0 in n; ylr = 0; ~Ylr

straightforward

that it is acted

We take the same functional

except that now u is p e n a l i z e d

[L-T.14],

except

m L2(~) , rather than a

in (6.27),

to see that the abstract

= v}.

it is rather

representation

of problem

is g i v e n by the e q u a t i o n w t t + A w + p A M w t = AG2u.

(Indeed, =

0

in

¢6.451

Q;

problem

(6.44)

(w-G2v)Iz

can be rewritten

(6.46)

first as w t t + d 2 ( w - G 2 u ) - p d w t

= ~(w-G2u) I~ = 0 by (6.45);

hence abstractly by

wtt+A(w-G2u)+pA~wt = 0 because of the B.C. since now A~h = -dh, ~ ( A ~) = H2(~) N H~(~). original A in (6.27),

From here, as usual,

follows by e x t e n d i n g

by i s o m o r p h i s m

[~(A)]"

It can be shown

expressed

in terms of the Dirichlet G 2 = -A-½D,

(6.46)

[L-T.14]

the

to, say, A: L2(Q )

that the Green map G 2 can be map D defined below,

as follows:

where y = Dv¢=, {by = 0 in n; yIF = v } ,

(6.47)

where D satisfies D: c o n t i n u o u s

Abstract

L2(F) ~ H~(~) c H~-2e(~)

settinu.

Thus,

m ~(A~-~/2),

• > O.

(6.46)

(6.46) becomes the abstract e q u a t i o n w t t+Aw+pA~wt

= -A ~Du,

(6.49)

65

or

dIwI=AIwl wt

on t h e s p a c e s

~Ssu/qption that

; A =

wt

Y = ~(AY=)xL2(tl);

tl.3|.

assumption

-pA y~

; Bu =

-A y'

(6.50)

Du

U = L2(F).

(-A)-~'B e Z ( U , Y ) .

(1.3)

I° 1

11

-

is s a t i s f i e d

Again,

wlth

it is e l e m e n t a r y

~ = 1:

Indeed,

to v e r i f y

from

(6.50)

we

require

(-AI-IBu

=

= 0

-A ~

~ Y = $

,

Du (e.51)

which

certainly

value

q = ~

IT.4],

holds

fails:

we o b t a i n

true

from

(6.48)

{6.48).

We may also

computations

verify

(as ~n

that

(6.41))

the

or

from

(say w i t h p = i)

(-A)-F'Bu and

by

from direct

we see

~

that

A- ~

A~,Dul

(-A)-~Bu

in

~

IA¼D u

(6.52)

(6.52)

,

fails

by ~

+ z12,

to be

in Y. w e have that:

Indeed, ~

<~

AssumDtio~

fI.3)

holds

The a b o v e of

power

claim

the p r e c e d i n g

[C-T.4,

c a n be v e r l f l e d two e x a m p l e s ,

by a n a r g u m e n t

based

(only

the s e c o n d condition

3 = ~ +~,

Assumption

component Bu e

provided,

respect

of

to the

fractional

w i t h G = ~]

usual

satisfied

similar

o n the d o m a i n

• ((-A) 7) = ~(A~+~/2)×$~(A'f/2),

and

for all

< 1.

ones

with

true

from

to L2(n);

Y z, a n d

(H.ll

in our

the

(6.50)

and

from

claim

is p r o v e d .

This

contraction,

(6.48)

A~Du

(6.53) whereby

respect

to Y,

the

is

E [ ~ ( A ~ / 2 ) ] ", d u a l i t y

provided

Is s a t i s f i e d analytic

argument),

with

(6.53),

i.e.,

ffi (1.5).

e At is a n s.c.

is n e e d e d

[ ~ ( ( - A ) ~ ) ] ", d u a l i t y

~ .( ~ < I

3

m

~ + ~ _< ~ or

as ~ < I w a s

semigroup

on Y

proved

above

[C-T.1-2].

66

Remark

6.7.

Since ~ < ~, a c o m p u t a t i o n

assumption

Finite

(H.2)

= (1.6)

Cost C o n d i t i o n

as i n

(6.33)

shows

that

is not satisfied.

tl.91.

unlformly

(exponentially)

Condition

(1.9)

W i t h A as in

stable

on Y

is a u t o m a t i c a l l y

(6.50),

[C-~.2],

satisfied

the s e m ~ g r o u p

and thus

with u m 0

e At is

the F i n i t e

Cost

(see also R e m a r k

3.4).

C onc l u s i o n :

T = ~.

for a n y n.

(Theorem

Conclusion:

T < ~.

closed

(closable);

condition

Remark

Theorem

to p r o b l e m

Theorem

2.1 a p p l i e s

see R e m a r k

2.1,

w i t h any G s u c h

in p a r t i c u l a r

the

that GL is

sufficient

(only)

(2.25).

A s i m i l a r a n a l y s i s a p p l i e s if the B.C. (6.44c-d) 8w 8~w e by ~-QIZ m O, u~-~--iX = u; refer to R e m a r k 6.5.

So far, we have damping

operator

differential

considered

is equal

operator,

Example 6.2.

¢ = ~

In the next

which we now complement only

Example

6.5.

in E x a m p l e s

of d a m p e d

example,

we r e t u r n

with boundary

sense

the

elastic

and a = 1 in

of b o u n d a r y

to the same

conditions

in a technical

operator,

choice

are

plates where

of the o r i g i n a l

6.1 and 6.3,

to the special

'comparable,'

of the e l a s t i c

examples

to the u - t h p o w e r

This was due

conditions.

operator

(6.44)

2.2 is not a p p l i c a b l e . )

6.8.

replaced

power

5.I w i t h R = I a p p l i e s

Eq.

that m a k e [C-T.1-2],

(6.25a), the d a m p i n g to the ~ - t h

~ = M.

On s o m e s m o o t h ~,

d i m ~ = n ~ 3, c o n s i d e r

the plate model

w l t h p > 0 any constant: { wtt+d2w-p~wt w(O,-)

w12 Regularity with

these,

Abstract operators

results

= Wo: 8w

~ ~IX

in

wt(O,-)

in n,

Now,

= wI

in

~ o

for p r o b l e m

we associate

settlno.

= ~(x-x0)u(t)

with

(6.54) (6.54)

however,

we

(0, T]x~ = Q,

the same

introduce

(6.54b)

(O,T]xF

are given

(6.54a)

~ ~,

in [T.4].

cost

(6.54c)

Consistently

functional

the p o s i t i v e

(6.26).

self-adjoint

67

c~h ~r

~h = ~2h, re(A) = {h e H4(O):

hlF =

~ h = -~h,

hlr = ~ ( r

~(~)

= (h,

H2(n):

where the equality with ~(A~) [Gr].

Thus,

problem

(6.64)

= o} = H ~ ( n )

in (6.26)

which fits the abstract model

I: I -

(6.ss)

;

norms)

(e.s6)

Is standard

version

= &(x-xO)u

(1.1), Bu =

-pY~

= ~(A~),

(equivalent

admits the abstract

wtt+Aw+pBwt

A =

= o)

(1.2) w i t h

I°I

;

(6.57)

R = I

6 ( x-x 0 )u

on the spaces Y = ~(A½)xL2(n),

U = R 1.

62 h = a2h, ~(62) = {h e H4(n):

hit = ~ l r

From

(6.56),

= 6hlr

we have

= a~~ h

r

= 0}

c

~(A)

(6.58) By Green's second theorem,

we have

(~2f, f) = (d2f, f) =

o~

f dr- ~f ~

dr'+ (dfl2dF

F

= f(af)2dN,

(6.59)

n

f e ~(~2);

(Af, f) = (A2f, f) =

~

f dF- Af ~

r = |(af)2d/3, r

(6.60)

dF+

F

(6.61)

(Af)2d~ n

f e ~(A);

(6.62)

n where the boundary terms on the right hand side of (6.59) and still v a n i s h if f is only in ~(B), Thus,

by extension,

or ~(AM),

respectively,

(6.61)

see

(6.56).

we get

(~2f, f) = (Af, f) = I(af)2d~,

f ~ ~(~)

= ~(A~),

(6.63)

n and thus a f o u t l o r l particular a~a!ytic

establish

the

results

of [C-T.1-2]

apply:

that the operator A in (6.57)

semiqrouD e At o__~nY.

These

in

uenera~es

an s,c,,

68

AssumDtlon

(-A)-~B e Z(U,Y).

11.3]:

that a s s u m p t i o n argument below

Again,

(1.3J holds true for T = I.

iT is I m m e d i a t e to see In fact,

as ~n the

(6.30), we find that

(-A)-~Bu =

=

-

e Y.

6 ( x - x °)

(6.641

0

In effect, a s s u m p t i o n {1.3) holds ~ u ~ for p r o b l e m (6.54) for all w i t h ~n < ~ < I, n ~ 3, e x a c t l y as in the case of p r o b l e m (6.1). To see thls~

w i t h A and B as in (6.55),

=

/~

1° :l -A

=

(6.56), we d e n o t e for c o n v e n i e n c e

I ° }

;

A.~ =

A

-

0

I

-~

_~

;

-~*

,

;

~, =

_~

w h e r e the a d j o i n t s are wlth respect to Y.

;

(6.6s)

Since 3 in (6.56)

is

s e l f - a d j o l n t on L2(~), we have

:D(A.~) = :D(A3. ) = ~)(A~) As a c o n s e q u e n c e of

(part of)

= ~)(A}4).

(6.66)

(6.58), we have in our p r e s e n t case

[C-T.4] ~((-A~) Y+~) ¢ ~ 1 ( - 4 ) Y) c ~ ( ( _ ~ ) q - a ) , and ~+z i.e.,

< I.

Then,

to o b t a l n

Bu ~ [~9((-4)7]',

(-A~)-~B e ~(U,Y),

0 < ~ < 1,

~n < q < 1, as desired,

d u a l i t y with respect to Y,

it s u f f i c e s via the

right h a n d side of (6.67) to have Bu ~ [~((-AM)7-&)] " s h o w n to be

true

in Example 6.1,

(6.67)

n p r e c i s e l y for ~ < ~-z

But this was < I, in the

a r g u m e n t b e l o w (6.31). Alternatively,

we may write,

using E x a m p l e 6.1,

(-A~)-~B = ( - ~ ) - ~ ( - A ~ ) 7 - & ( - A ~ ) - ( q - Z ) B

since

(-AB)-~(-A~) q-~

(6.66) and the

closed

is bounded,

* 7-&

for (-A~)

graph theorem.

that

~ X(U,Y), * -q

(-AB)

(6.66)

Is b o u n d e d by

69

Assumption (6.54)

(H.I)=(1.5).

since

~t w a s

Th~s

already

The Finite

Cost

Condltlon

These

hold

true,

also

Conc~us~@n: Conclusion: that

(only)

(1.9)

since

e At

is u n i f o r m l y

to p r o b l e m

T < ~.

Theorem

2.1 a p p l i e s

to p r o b l e m

GL

is c l o s e d

6.6.

((losable);

condition

(The

case a = I ~n the w a v e

see Remark

[C-T.2]

(5.10).

and R =

I.

(6.54). (6.54)

2.1,

with

any

in p a r t i c u l a r

G

the

(2.25).

[C-T.I-2])

equation

with

On s o m e a strong

wtt-dw-pdw t = 6(x-xO)u(t)

~n

w(O,-)

in

(6.69)

= Wo;

wt(O,-)

c a n be w r i t t e n

precisely

abstractly

smooth degree

bounded of d a m p i n g

as p r o b l e m

We begin

(6.35),

= Q;

(6.69a)

fl;

(6.69b)

(O,T]×F

= ~,

(6.69c)

as

(6.70)

except = H2(~}

that

now A

is

n H~(~).

(6.71)

by selecting

Y = ~(A~)×L2(n) cost

(O,T]xfl

= 6(.-xO)u(t),

A = -d, ~(A)

1.

= wI

in

wtt+Aw+pAwt

The

is a s.c.

> O:

Problem

Case

e At

Condlt~on

stable

5.1 a p p l i e s

{ i.e.,

for p r o b l e m

that

a n d The D e t e c t a b i l l t v

Theorem

c R n we consider and p

true

(6.63)

T = ~.

sufficient

Example

holds

below

semigroup on Y, w h i l e ~ w a s s h o w n a b o v e to be < I for n ~ 3.

analytic,

such

assumption

observed

functional

fo~ T = ~

= H~(n)xL2(~);

U = R I,

(6.72)

is t h e n

2 )+luCt)12}dt. 3(u,w) = f {llw(t)ll2. +llwt(t)llL2(n o H~(~) Assumption (6.40),

(1.3):

since

(-A)-IB

A a n d B are

~ ~(U,Y). the s a m e

The

case

expressions

~ = I follows as

in

c6.73)

as

(6.39) : w e

in

require

70

A-16

~ ~(A~),

only

for d ~ m ~ = n = I.

have

(~A}-~Bu

Assummtlon

~.e.,

To r e m o v e Case

2.

We

~ L2(~), Indeed

~ Y provided

~H,1] = At e on Y

semigroup

A-~

$ ~ H-I(~).

~ = ~ works

A-~6

e L2(~),

Now A generates

[C-T.2]

and ~ = M

limitation

now

the

as

~n

~.e.,

~1.5).

the

choose

i.e.,

this

(6.41),

a contraction

n = I of C a s e spaces

is true

(6.42):

analytic

above.

1, w e n o w

and cost

consider

functional

for

(6.74)

U = RI:

Y = L2(Q~)xL2(~);

we

for n = 1.

< 1 was shown

following

And

¢o

J(u,w

15dt

=

0

Assumption that

the

(1.3): case

(-A)-~B

q = 1 requires

• (A) c H2(n),

this

which

is t r u e

in t u r n

n < 4.

Indeed,

e < ¼,

This

Indeed,

H2(n)

show

that:

that

Consequently,

as in

c H2-2e(~),

(#)

that

(5.40),

e ~(Ae), this

we

for a n y

[H2(O)] ", n 2 > ~ or u • R 1,

~ < ~ with

x+py

[C-T.4]

with a

• ~(Ae)}. = 1 to A

on ~ ( ~ ) x L 2 ( ~ )

see

that

or A - ( 1 - e ) ~

latter

6 6

provided

e L2(~):

than

we see

Since

e ~((-A)e),

true

of

rather

(6.40),

provided

i.e.,

(-A)-IBu

holds

the p r o o f

recalling

we see

using

~ ~2(C).

c C(~),

(1.3)

(6.74)

true if a n d o n l y if A - 1 6

~ ( A l-e)

~-16

m((-A) e) = {x,y

by adapting

on L 2 ( ~ ) x L 2 ( ~ )

[C-T.4]. holds

that

condition

we h a v e

As before,

is t h e n s a t i s f i e d

provided

we s h a l l

result.follows

defined

now

requirement

n ~ 3, so that

n ~ 3.

• ~(U,Y}.

as

conditlon

e L2(~ ) .

condition

holds

in

(#)

Since

true

in

c a s e 6 e [ H 2 - 2 ~ ( ~ ) ] ". i.e., p r o v i d e d H 2 - 2 8 ( ~ } c C(~), i.e., n 2 - 2 e > ~. For e < ~ w e h a v e 4 - 4 e > 3 ~ n, as d e s i r e d .

provided

AssumDtlon

analytic

semigroup IIR(A,A)II was

shown

{H.1}

=

e At o n Y

{1.5]. (not

The operator

contraction

A generates

now)

~ C/IA I for Re A > 0 ~n the n o r m above

for n ~ 3.

since,

a s.c.,

as one

of Z(Y)

sees

[C-T.2].

readily, That

~ < 1

71

(1.9).

F i n i t e cost C o n d i t i o n

w i t h e i t h e r choice of Y,

Concluslon:

T = ~.

and to the cost Conclusion:

T h e o r e m 5.1 applies to the cost

(6.75)

T < ~.

condition

7.

for n 5 3 for any G that makes GL c l o s e d

(2.25}.

E x a m p l e s of partial d l f f e r ~ n t l a l e a u a t l o n p r o b l e m s s a t l s f v l n u {H.2) we i l l u s t r a t e

the a p p l i c a b i l l t y

w i t h i n the class of d y n a m i c s subject to h y p o t h e s i s examples

include:

of T h e o r e m 5.2

(H.2) = (1.6).

the w a v e e q u a t i o n w i t h D l r l c h l e t control

Our

(Section

the E u l e r - B e r n o u l l i e q u a t i o n w i t h D i r i c h l e t / N e u m a n n controls;

w i t h c o n t r o l s as d i s p l a c e m e n t / b e n d l n g moment Schr~d~nger equation h y p e r b o l i c systems

w i t h D i r l c h l e t control

(Section Y.5);

control as a 'bending moment' 7.1.

for n = I

see R e m a r k 2.1 in p a r t i c u l a r the (only) s u f f i c i e n t

In this section,

7.1);

(6.73)

for n ~ 3.

T h e o r e m 2.1 applles w l t h Y as in (6.72) for n = 1

and to Y as in (6.74) (closable);

The s e m l g r o u p e At is u n i f o r m l y stable

(6.72) or (6.74).

flnal/y,

(Section 7.2); (Section 7.4);

and

the first-order

Klrchoff p l a t e w i t h b o u n d a r y

(Section 7.6).

C l a s s (H.2): S e c o n d order h v D e r b o l l o e u u a t l o n s w l t h D l r l c h l e t boundary ~ontrol We c o n s i d e r the f o l l o w i n g problem: Wtt = a W w(O,.)

in (O,T]×Q = Q,

= w O, wt(O,- } = w I in ~,

wl~ ~ U

r e p l a c e -~ by any s e c o n d order,

r e g u l a r i t y theory

{Wo,Wl} E L2{~)xH-I(~)

J(u,w)

(7.1c)

(In (7.1a) we m a y

e l l l p t ~ c o p e r a t o r w i t h time

s y m m e t r i c c o e f f i c i e n t s of its p r l n c i p a l part,

changes in the a n a l y s i s below.) (optimal)

(7.1b)

in (O,T]xF m Z,

w h e r e we take the b o u n d a r y control u ~ L2(~).

independent,

(7.1a)

w i t h minimal

Consistently with established

[Lio.l],

[L-T.2],

[LLT], we take

and the cost functional

2 r )}at. {llw(t)llL2(n )+iiwt(t)ll2H_s(~).+lu(t)iL2( 2

= 0

17.21

71

(1.9).

F i n i t e cost C o n d i t i o n

w i t h e i t h e r choice of Y,

Concluslon:

T = ~.

and to the cost Conclusion:

T h e o r e m 5.1 applies to the cost

(6.75)

T < ~.

condition

7.

for n 5 3 for any G that makes GL c l o s e d

(2.25}.

E x a m p l e s of partial d l f f e r ~ n t l a l e a u a t l o n p r o b l e m s s a t l s f v l n u {H.2) we i l l u s t r a t e

the a p p l i c a b i l l t y

w i t h i n the class of d y n a m i c s subject to h y p o t h e s i s examples

include:

of T h e o r e m 5.2

(H.2) = (1.6).

the w a v e e q u a t i o n w i t h D l r l c h l e t control

Our

(Section

the E u l e r - B e r n o u l l i e q u a t i o n w i t h D i r i c h l e t / N e u m a n n controls;

w i t h c o n t r o l s as d i s p l a c e m e n t / b e n d l n g moment Schr~d~nger equation h y p e r b o l i c systems

w i t h D i r l c h l e t control

(Section Y.5);

control as a 'bending moment' 7.1.

for n = I

see R e m a r k 2.1 in p a r t i c u l a r the (only) s u f f i c i e n t

In this section,

7.1);

(6.73)

for n ~ 3.

T h e o r e m 2.1 applles w l t h Y as in (6.72) for n = 1

and to Y as in (6.74) (closable);

The s e m l g r o u p e At is u n i f o r m l y stable

(6.72) or (6.74).

flnal/y,

(Section 7.2); (Section 7.4);

and

the first-order

Klrchoff p l a t e w i t h b o u n d a r y

(Section 7.6).

C l a s s (H.2): S e c o n d order h v D e r b o l l o e u u a t l o n s w l t h D l r l c h l e t boundary ~ontrol We c o n s i d e r the f o l l o w i n g problem: Wtt = a W w(O,.)

in (O,T]×Q = Q,

= w O, wt(O,- } = w I in ~,

wl~ ~ U

r e p l a c e -~ by any s e c o n d order,

r e g u l a r i t y theory

{Wo,Wl} E L2{~)xH-I(~)

J(u,w)

(7.1c)

(In (7.1a) we m a y

e l l l p t ~ c o p e r a t o r w i t h time

s y m m e t r i c c o e f f i c i e n t s of its p r l n c i p a l part,

changes in the a n a l y s i s below.) (optimal)

(7.1b)

in (O,T]xF m Z,

w h e r e we take the b o u n d a r y control u ~ L2(~).

independent,

(7.1a)

w i t h minimal

Consistently with established

[Lio.l],

[L-T.2],

[LLT], we take

and the cost functional

2 r )}at. {llw(t)llL2(n )+iiwt(t)ll2H_s(~).+lu(t)iL2( 2

= 0

17.21

72 Abstract

settlna

[T.2],

[L-T.I],

into the abstract model ad~olnt

[L-T.2].

To put problem

(7.1),

(7.2)

(1.1),

(1.2), we introduce the positive self1 ) and define the operator Ah = -~h, ~(A) = H2(Q) 0 HO(Q

operators A =

I° II ;

-A

where D is the Dirlchlet Eq.

Bu

I°I

=

; R = I.

0

17.31

ADu

map encountered

before

in Example

6.3,

16.47),

Dv = y,=, {~y = 0 i n n ,

Ylr = v},

(7.4)

U = L2(V).

(7.5)

and the spaces Z = Y = L2(Q]xH-I(~); The Diriohlet map satisfies Assumptlon

(1.3):

the regularity property

(-A)-~B ~ ~(U,¥).

From

16.48).

(7.3) with u ( L 2 ( r ) ,

we

obtain (-A)-IBu =

1° II°I l:uL =

-I

a [ortiori,

by the r e g u l a r i t y

0

~ Y,

17.6)

ADu

(6.48) of D, and a s s u m p t i o n

11.3) holds

true with q = 1. ASSUmption

(H.2) = 11.6).

B

= D z 2 = - ~-j

z2

Moreover,

(7.3) we calculate

since D

- ~-~

.

(7.7)

we h a v e B'e A*tj

where ~(t)

= ~(t,~O,~l)

~(°")o: with

From

z2zlI

solves

~o" ~t ( ° ' ' )

= ua ~ ,

[Zl,Z2]

e Y,

the c o r r e s p o n d i n g

: ~I

homogeneous

(7.8) problem

in (O,T]×~ ~ Q,

(7.9a)

in ~,

(7.9b)

in (O,T]xF m Z,

17.9c)

73

~0 = -A-Iz 2 e $(A ~) = H~(n); Thus, hy

(7.8),

(H.2) = (1.6}

(7.10),

an equivalent

~1 = Zl e L2(Q)"

formulation

(7.10)

of a s s u m p t i o n

is the inequality

(7.111 Z for the trace of the s o l u t i o n to problem

(7.9).

It should be noted

that inequality

(7.11) does NOT follow from a prioa'i

regularity ~(t)

~ C([O,T];H~(~})

(7.10).

Inequality

(7.11)

It was first e s t a b l i s h e d was first proved,

is an independent

in [L-T.I],

interior

~ L2(~)xH-l(n).

a p u r e l y operator

{w, wt} ~ C([O,T]~

independently see also

technique,

Finite Cost C o n d i t i o n (i)

that indeed

(1.9)

(exponential)

it was proved by

Inequality

and that,

(7.11) was proved

by a multipller

technique;

treatment.

Sufficient

conditions

which would imply

is satisfied are: uniform s t a b i l i z a t i o n

of p r o b l e m

(7.1) on the

space Y = L2(Q)×H-I(Q ) by means of an L 2 ( O , ~ ; L 2 ( F ) ) (ii)

it

techniques,

(7.11) holds true,

L2(~)xH-l(n)).

(1.9}.

operator

via a d u a l i t y argument,

for a comprehensive

that the F.C.C.

In these r e f e r e n c e s

(7.1) with u ~ L2(~),

and directly also in [Lio.1],

[LLT]

result.

result holds true:

for problem

Next,

interior

(7.9),

trace regularity

[L-T.2]:

regularity

{w, wt} E L2(O,T;L2(~)×H-I(~}}

in fact,

to problem

by means of p s e u d o - d i f f e r e n t i a l

that the following

{Wo,Wl}

of the s o l u t i o n

(optimal}

exact c o n t r o l l a b i l i t y from the origin)

of problem

(7.1)

over a finite interval

space Y = L2(~)xH-I(Q),

feedback u;

(to or, equivalently, [O,T],

on the state

within the class of L2(0, T;L2(F))-

controls u. A solution consequently,

controllability additional

to the u n i f o r m s t a b i l i z a t i o n

via a known result of D. Russell problem

geometrical

s t r i c t l y convex n). geometrical

(Iii was first o b t a i n e d

(1973)

exact controllability,

on ~, except

all of F, was e s t a b l i s h e d

in [Lio.2]

(i), and of the exact

in [L-T.12],

condition on ~ (which includes

Later,

conditions

problem

the

under s o m e

class of

this time without

for smooth r, if u is applied by relying on a lower bound

to

74 inequal~ty,

(7.11)

inequality

is s u f f i c i e n t l y large

with t h e

r e v e r s e d i n e q u a l i t y sign,

dZ > C T I l { ~ O , ~ l } l l

.

Z This the

latter

inequality in

the

form.

(7.12)

indeed,

in

of

A direct

this

work

approach

of

the

to

to

solve

(7.12) [t.-T.;t2],

is

the

in

operator

[H]--by

[Lio.1],

uniform

essentially

albeit

using

[L-L-T]

for

stabilization contained

Jn a less

exac~ controllability

input-solution

show the key inequallty,

obtained

had been used in

[L-T.12]

such Inequality

estimates

surjectivity

was explicitly

methods that

(7.11)--and

problem;

(7.12]

(~)xL2(~)

Inequality

same m u l t i p l i e r

if T

(twice the diameter of n)

also

transparent

based on the

and multiplier

methods

to

in the case w h e r e u acts o n l y on a p o r t i o n of

the b o u n d a r y ~0 is g i v e n in [T.3].

Later,

g e o m e t r i c optics m e t h o d s - -

fJrst i n t r o d u c e d in [Lit. ] for exact c o n t r o l l a b i l i t y q u e s t i o n s - p r o v i d e d sharp sufficient c o n d i t i o n s for i n e q u a l i t y true, w i t h ~ r e p l a c e d by a s u b p o r t l o n ~O c Z

(7.12)

[B-L-R].

to hold

The u n i f o r m

s t a b i l i z a t i o n p r o b l e m now holds w i t h no geometrical conditions in [L-T.25],

if the feedback

acts on all of F.

of the Finite Cost C o n d i t i o n v e r s i o n thereof,

(1.9)

In any case,

for p r o b l e m

(7.I},

the v a l i d l t y

or a more general

is firmly established.

We note, however,

that if the control u in (7.1c)

is sought

w i t h i n the class of f l n i t e l y m a n y a c t u a t o r s

u(t,x)

with

3 < ~,

gj

e L2(l')

arbitrary

3 = ~ J=l

but

g,,(x),u~(t) d J fixed,

and /dje

L2(O,T), then

c o n t r o l l a b i l i t y on any [O,T] w i t h i n the class of g j-controls p o s s i b l e for problem dim n = 1 IT.8]. section

exact

is not

(7.1} on the r e q u i r e d s p a c e Y in (7.5), unless

Thls comment applies to all other cases in this

IT.8] and will not be repeated.

Detectab~lltv Condition

However,

(5.17~-{5,~9).

This h o l d s true s i n c e R = I.

we find instructive to glve a n o n - t r l v l a l example for

the w a v e d y n a m i c s

(7.1) w i t h p e n a l i z a t i o n in L2(~)xH-l(n)

i n v o l v i n g an

75

observation cost

operator

function

{7.2)

R which

is not p o s i t i v e

we c o n s i d e r

3(u,w) = o{llw(t)ll~ 2 (~)+llmwt(t)ll2g-z(Q)" where

m(x)

support

is a s m o o t h

on a p r o p e r

up w t o n l y

on ~0'

f ~ H-I(~), order

K = diag[O,-I], now y = p.d.e,

Define

so that

to s a t i s f y

non-negative

subdomain

problem

the

I

w]x ~

geometric

optics

Conclusion: cases

~

(R2f)(x)

picks

= m(x)f(x),

operator.

(5.17)-(5.1g),

we

corresponding

wtt=

compact

to

-Aw-R2wt;

In

take (5.18)

i.e.,

Is

the

= wI

in O;

o

Z.

in

requires

that

stable

is the c a s e

the y - p r o b l e m ,

in the

equivalently

topology

of

If all r a y s

if a n d o n l y

of

[B-L-R].

Theorem

5.2 a p p l i e s

We h a v e

already

to p r o b l e m

(7.1)

in the

two

described.

:

T < ~.

for the w a v e Theorem

3.2

equation

7.2.

there.

smoothing

Class

1.

problem

to be a p p l i c a b l e

on R as d e s c r i b e d additional

Case

with

functional

in Q;

meet the set n x [ O , T ]

T = ~.

of the

damping'

the {w, wt}-problem, be u n i f o r m l y this

o n ~,

the n e w

R 1 = 0,

problem

equation

= w o, w t ( 0 , . )

The Detectability C o n d i t i o n

And

defined

= ~w-mw t

w(0,.)

L2(~)xH-I(~).

)}at '

Condition

feedback

'viscous Wtt

2 *lu(t)lL2(r

(self-ad~oint) m u l t i p l i c a t i o n

or the a b s t r a c t

with

Instead

so that

R = diag[R1,R2],

R 2 Is a

so that

function

n o of Q,

the D e t e c t a b i l i t y

(A+KR)y,

definite.

now

noted

(7.1), for

(7.1)

Finally,

Theorem

where

requires

Theorem

on R as d e s c r i b e d

that

(7.2),

some

holds

minimal

3.3 a p p l i e s

in R e m a r k

3.1

R = Identity,

for

true while

smoothing (7.1)

with

3.4.

~H.2): Euler-Bernoulll equatlons with boundary contro~

We c o n s i d e r

on a n y s m o o t h

bounded

~ c Rn:

76 I wtt+Z~2w = 0 w(0,.) w]:~

-

= w 0, wt(O,- ) = w I 0

theory,

control

(7.13a)

in D,

(7.13b)

in

[ ~wWlz s U with boundary

in (O,T]xf] = Q,

(O,T]xr

=

~',

(7.13c)

in Z,

u e L2(~ ).

the cost functional

Consistently

to be minimized

(7.13d) with optimal

regularity

is

+llu(t)l122 (r.)dt. 0

(7.14)

In)

To put problem (7.13), (7.14) into the abstract Abstract se~tinq. model (1.1), (1.2), we introduce the positive self-adjoint operator = {h e H4(~):

Ah = ~2h, ~(A) and define

hlF

=

~-~Irah

o}.

=

(7.15)

the operators A =

I°I

;

-A

where G 2 is the appropriate

y = G2 v ~

[°I

Bu =

;

0

R = I

(7.16)

AG2u

Green map:

(~2y

= o i n n;

vlr

= O, ~ v { r

= v),

(7.17)

and the spaces Y ~ L2(QlxH-2(~ 1,

AssumDtlon

{1.3):

(-A)-~B ~ ~(U,Y).

L2(? ) ~ L2(~ ). we readily obtain

(-AI-IBu

=

I: -

and assumption Assumption Appendix

(1.3)

(7.181

Since G 2 is certainly

bounded

(7.16) with u e L2(?):

0

=

e Y,

AG2u

holds true for problem

(H.2) = (1.6).

C],

from

U ~ L2(r 1.

One can show that

(7.13) with ~ = I. [L-T.9],

[FLT,

(7.19)

7T

s*eA*tlYZl

= a@(t)

ly l

where { ( t )

= ~(t,{O,{1)

{

+tt.~+

y

e

o

@o' @t ( ° ' ' )

=

+lz ~ ~ t z

= 91

~ o

(7.20)

y,

solves the corresponding =

#(o,-)

,

Ir

homogeneous problem

in (0, T]xO = Q,

(7.21a)

in n,

(7.21b)

in (0, T]xF m ~,

(7.21c)

with

tO = - A - l y 2 Thus,

by

(H.2)

=

(7.20), (1.6)

(7.22)

e ~(A V=) = H02(O): ~1 = Yl e L2{O).

(7.22),

an e q u i v a l e n t

formulation

of a s s u m p t i o n

is the i n e q u a l i t y

fl~12~

~

(7.23)

CTII{+o'+I}II2 ~(n)×L2(n )

for the trace of the s o l u t i o n

to p r o b l e m

wave e q u a t i o n

it s h o u l d

be n o t e d

that

(optimal)

~nterior

regularity

of s e c t i o n

does N O T f o l l o w ~(t)

a priori

from

It is an i n d e p e n d e n t

regularity

s m o o t h ~.

for p r o b l e m

(7.13).

Finite Cost

CODdi~ion

(7.13}

(I.9).

holds

true, within

geometrical

on ~

without

conditions Uniform

geometrical

Detectabilltv

As in the case of the

(H.2)

problem.

for any T > 0, the class

[Lio.2],

conditions

holds

inequality

(7.21),

indeed = (1.6}

conslderatlons

equation

stabilization

Conditions

assumption

The same

wave

space Y = L2(Q)×H-2(n)

all of F.

result w h i c h

Thus,

the case of the p r e c e d i n g of p r o b l e m

(7.21).

of the solution to the p r o b l e m

~ C([O,T];H~(~))

for a n y general

7.1,

[Lio.2],

holds

true

controllability

u, with no

if the c o n t r o l

can also he established,

acts

on

likewise

[O-T].

{5.171-{5.19}.

This holds

true s i n c e R = I.

Conclusion:

T = ®.

Theorem

5.2 a p p l i e s

to p r o b l e m

(7.13),

~oncluslon:

T < w:

Theorem

3.1 a p p l i e s

to

(7.14)

whil e Theorem 3.3 r e q u i r e s

true

on the state

of L 2 ( ~ ) - c o n t r o l s

[L-T.28]

(7.22).

a p p l y n o w as in

Exact

in fact,

(7.23)

a stronger

(7.13),

smoothing

assumption

(7.14). where on R.

R = I;

78 Case 2.

We now consider on any smooth ~ ¢ Rn: wtt+d2w = O w(O,.) = w 0, wt(O,.)

= wI

w[~ = u

in Q,

(7.24a)

in Q,

(~.24b)

in Z,

(7.24c)

(7.24d) Consistently w i t h optimal regularity

with boundary sontro] u e L2(~ ). theory [Lio.2],

[L-T.11],

the cost

5

H-*(n

functional

to be minimized

is

(7.2s)

where

V"

~s

the

dual

space

of

V defined

by

V = {h e Ha(n): hlr = ~ahV I r =

Abstract settlnu. To put problem (7.24), (7.25) model (1.1), (1.2), we introduce the operators A = I ~ -

I I; 0

Bu = ] 0 1 ; AGlU

with A the operator in (7.15) and

G1

(7.26)

0}

into the abstract

R = ~

the appropriate G r e e n

y = GIV c=~ {d2y = 0 in n; Y[r = v, ~YlF = 0},

(7.27)

map: (7.28)

and the spaces Y = H-1ln)xV"

= [~(A~)]'x[~(A~)] "

(7.29)

where with equivalent norms • (A¼) = H~(~):

Assumption

{1.3):

(-A)-TB

e

Z(U,Y).

~(A ~1 = V.

17.S01

It is plainly satisfied with,

= I, as one sees by proceeding as in (7.19).

say,

79

AsSUmDtlon

(H.2) = (1.~).

One can show that

[L-T.9],

[F-L-T,

Appendix C], w i t h y = [yl,Y2 ],

• A'tlyl I ly 2

B e

y e Y,

(7.31)

where @(t) = @ ( t , @ O , @ l ) solves the c o r r e s p o n d i n g h o m o g e n e o u s p r o b l e m (7.21),

this time h o w e v e r w i t h initial data, ~O = A - ~ Y 2 e ~(A~)

Thus, by (7.31) and

= V; ~1 = -A-~Yl E ~(A ~) = H~(~).

(7.32),

(7.32)

an e q u l v a l e n t f o r m u l a t i o n of a s s u m p t i o n

(H.2) = (1.6) is the inequality, dZ

< c 11(~ , ~ . } 1 1 2 . , " T 0 a VxH~ (Q)

for the trace of the s o l u t i o n to p r o b l e m prior cases,

inequality

(7.32).

Again,

as in

(optimal)

of the s o l u t i o n to p r o b l e m

It is an independent r e g u l a r i t y result w h i c h holds

indeed true [Lio.2], assumption

(7.32).

(7.33) does NOT follow from a prJor~

interior r e g u l a r i t y @(t) m C([O,T];V) (7.21),

(7.21),

(7.33)

[L-T.II],

for any general smooth n.

(H.2) = (1.6) holds true for p r o b l e m

F1nlte Cost C o n d i t i o n

(1.9).

Thus

(7.24).

The same c o n s i d e r a t i o n s apply now as in

the case of the w a v e e q u a t i o n in S e c t i o n 7.1 and of the E u l e r - B e r n o u l l i problem

(7.13).

Exact c o n t r o l l a b i l i t y of p r o b l e m

(7.24) on the state

space H-I(Q)xV" holds true for any T > 0 w i t h i n the class of L 2 ( Z ) - c o n t r o l s u, at least under some g e o m e t r i c a l eliminate g e o m e t r i c a l second control [L-T.11].

conditions,

in the B.C.

c o n d i t i o n s on ~.

one may add, however,

(7.24d).

under the same c o n d i t i o n s as exact c o n t r o l l a b i l i t y IT.?]: i.e., u n d e r s o m e g e o m e t r i c a l (7.24c)

a suitable

This and more is p r o v e d in

U n i f o r m s t a b i l i z a t i o n results on H'I(Q)xV"

feedback in the B.C.

To

can also be g i v e n

results

[B-T],

c o n d i t l o n s on ~ if o n l y one

is used; or else w i t h no g e o m e t r i c a l

conditions on n if two feedbacks are used In the B.C.

(7.24c) and

(7.24d) respectively.

In any case,

holds true for p r o b l e m

(7.24) under some g e o m e t r i c a l c o n d i t i o n s on ~,

the Finite cost C o n d i t i o n

or else w i t h no g e o m e t r i c a l c o n d i t i o n s on ~, control In (7.24d) and In the cost

(7.25).

(1.9)

if one adds a s e c o n d

80

Detectabilltv C o n d i t i o n s

f5.17}-(5.!9 ~.

This holds true since R = I,

Conclusion:

T = ~.

T h e o r e m 5.2 a p p l i e s also to p r o b l e m

~

T < ~.

Theorem 3.1 applies to (7.24) w i t h R = I while

:

(7.24).

T h e o r e m 3 . 3 require additional r e g u l a r i t y on R (Remark 3 . 2 ) . Case

3.

We now c o n s i d e r

any s m o o t h

on

[

Q

R n,

c

wtt+62w = 0 w(O,.)

= w 0, wt(0,.)

in Q;

(7.34a)

= w I in Q;

(7.34b)

in Z;

(7.34c)

in Z,

(7.34d)

wIx = 0 wlx = u

w i t h b o u n d a r y control u ~ L2(~). t h e o r y [L-T.14],

[Lio.2],

C o n s i s t e n t l y w i t h optimal r e g u l a r i t y

we take the f o l l o w i n g cost functional

¢O

2 +llwt (t)lI2 H-"1 (n) +]u(t) IL2(F o

)}dr

(7.35)

(Q)

1 with initial data {w0,wl} e H0(~)xH-l(f] ).

A b s t r a c t settlnu. model

(1.1),

To put p r o b l e m

(7.35) into the abstract

(7.34),

(1.2), we introduce the o p e r a t o r s A h = ~2h, ~(A)

= {h ~ H4(n):

hIF =

AhtF = 0};

(7.36a)

w h e r e S 4 Is the a p p r o p r i a t e G r e e n map

7 = S4v ~ and the spaces

(a2y

1 Y = HO(Q)xH-I(Q)

AssumDtlon say,

(1.3):

= 0 in n;

Y[r

= ~(A~)x[~(A~)]';

(-A)-TB E Z(U,Y).

= v},

(7.37)

U = L2(F ).

(7.38)

= o, a y [ r

It is p l a l n l y s a t l s f ~ e d with,

q = I, as one sees by p r o c e e d i n g as in (7.19),

G 4 G ~(L2(F),L2(~)).

since

81 AssumDtlon

(H.2) = ¢I.~I.

B

Y2

One can show that

= G4A Y2 ;

where ~(t) = ~(t,@o,@1)

[L-T.14],

(7.3g)

{Y2

B

solves the c o r r e s p o n d i n g

#it+a2# = 0 @(0,.)

= tO , ~t(0,.)

#{2 = ~ { X

[L-T.15],

= ~I

~ 0

homogeneous

problem,

in Q;

(7.40a)

in O;

(7.40b)

in 2,

(7.40c)

with tO = A-ly 2 E ~(A~); Thus,

by

(7.39),

(7.41),

(7.41)

~I = Yl e ~(A¼).

an equivalent

formulation

of a s s u m p t i o n

(H.2) = (1.6) is the inequality 2

for the trace of the solution of problem cases,

it should be noted that inequality

a priori

(optimal)

equivalent norms

indeed true assumption

(7.421

is an independent

[L-T.14],

where the

[Lio.2]

h{r = ahlr = 0}.

regularity

for a n y general

{ji,9).

for the Euler-Bernoulli (7.38)

Orlglnally, equation

by using however

(7.43)

result w h i c h holds smooth ~.

(H.2) = (1.6) holds true for p r o b l e m

required space

does NOT follow from

[Gr.1]

Finite Cos~ Condition [O,T]

(7.421

As in preceding

interior r e g u l a r i t y @(t) E C([O,T];~(A~)),

~(A ~) = V = {h E H3(O): Inequality

(7.401.

Thus,

(7.40).

exact c o n t r o l / a b i l l t y

(7.34a)

on any

was shown on the

two controls:

w{~ = u I G

H OI(O,T:L2(U))

and Awl~ = u 2 ~ L2(~ ) [L-T .15; .Thm. . 1 2].

is equivalent

to the inequality

[Llo.2]

This

(7.44)

Z

(A~ )x~ (A¼)

82

for p r o b l e m

(7.40),

(7.41), see

[L-T.15,

a~t u~D-- G

o b s e r v e d in [Leb.l] that the term ~absorbed.'

Lemma 3.2].

It was later

L2(~ ) in (7.44) can be

{This is a non-trlvlal improvement,

not a lower order term w i t h respect to ~

a# t

since ~

in L2(~),

usual compactness/uniqueness a r g u m e n t does not apply). improvement has the important equivalent

c o n t r o l l a b l e on [O,T] on the space

Indeed,

and hence the This

is indeed e x a c t l y

(7.38) as desired.

Thus,

[Las.7].

for p r o b l e m

Exact controllability has also been e x t e n d e d to the

Detectabilitv Conditlon

(5.17]-(5.19}.

[H.I].

This holds true since R = I.

Conclusi@n: T = =.

Theorem 5.2 applies to p r o b l e m

Conclu~lon:

Same as i n

T < ~.

(7.34),

(7.35).

Case 2.

We now consider on any smooth n • R n,

{

Wtt+d2W = 0 w(0,.}

in Q;

(7.45a)

= Wo; wt(O,. } = w I in Q;

(7.45b}

w[E = u

w[x

= 0

w i t h b o u n d a r y control u e L2(~ ). theory

(7.34).

the idea ~n [Leb.1] is useful also in the proof w h i c h

t w o - d i m e n s l o n a l plate model w i t h physical moment

Case 4.

the

(i.9) does h o l d true for p r o b l e m

e s t a b l i s h e s the c o r r e s p o n d i n g u n i f o r m s t a b i l i z a t i o n result (7.34)

is

f o r m u l a t i o n that p r o b l e m

(7.34) w i t h just one control u E L2(~ ) in (7.34d}

r e q u i r e d Finite Cost C o n d i t i o n

in L2(~)

[L-T.15],

[Lio.2],

in E ;

(7.4sc)

i n E,

(7.45d}

C o n s i s t e n t l y w i t h optimal regularity

the cost functional

to be m i n i m i z e d for

{Wo.Wl} e H-1(n)xV" is:

+llw t (t)II~,+llu( t ) 11~2 ( r } }at, o

w h e r e V" is the dual of V in (7.43), ~(A ¼ ) = H~(n),

hence

(7.46)

(fl)

and w h e r e we note that

(with e q u i v a l e n t norms}

Y = H-I{O)xV"

= [~(A~)]'x[:D(A~}] "

(7,47)

83

with A the operator defined in (7.36a). variable ~ = -A~w,

By m a k i n g the change of

w solution of problem

variable ~ satisfies precisely problem

(7.45),

(7.34),

(7.47) is mapped into the space in (7.38). problem

(7.45),

satisfies

(7.34),

(regularity, (7.46),

(7.46)

for w is converted

(7.35).

Hence,

the new

Thus,

with U = L2(F),

into a p r o b l e m

for ~ w h i c h

all the required p r o p e r t i e s

exact controllability,

are equivalent

(7.35).

Thus,

we find that

and that the space in

uniform stabilization)

to the corresponding

properties

Theorem 5.2 applies also to prcblem

for (7.45),

for

(7.34),

(7.45),

(7.46).

For sake of completeness we note now that A is the same as in (7.36b),

while B is now

0 1; BU = IAG3U

y = G3v ¢=~ { Ay 2 = 0 in n;

B* IYl t

* -~

Y2

where #(t)

= G3A

ylr

= V: dylr = 0};

• A'tlYll

Y2;

B e

Iy21 =

is the solution of the same problem

8 (~(t) 8u

(7.48)

(7.49)

"

(7.40) with

(7,5o) i.e., 7.3.

the same regularity of the initial data as in (7.41). qlass (H.2): contro,~,

Sch~dlnaer

In n c R n we conslder

theory proved

Abstract settlnq. abstract model

(7.51a)

y(0,.)

in n;

(7.51b)

in Z,

(7.sic}

= u

= Y0

C o n s i s t e n t l y wlth the optimal we take YO E H-I(o)

regularity

and the cost functional

2 }dr. = f0{llY(t)il2H-~(Q)-+]Iu(t)i]L2(F)

[L-T.25]

(1.1),

equation

in Q;

in [L-T.25],

J(u,y)

the Schr~dinger

boundarv

Yt = -i~y

YIZ with control u e L2(~ ) .

equation wlth Dlrlchlet

To put problem

(7.51),

(7.52)

(7.52)

into the

(1.2)p we take the following operators

where D is the same Dirichlet map introduced

in Eq.

(3.47):

and spaces

84

A = i.4,

A. = -/',;

B =-IAD; w h e r e ~(t) = ~(t,~O)

~(A)

R = I;

= H2(~)

= I, by

(7.53),

~ssumptlon

(7.55a)

#(O,-) = 40

in n;

(7.55b)

in X;

(7.55c)

IZ = 0

U = L21F).

(-A)-~B ~ ~(U,Y).

According

for a s s u m p t i o n

(6.48)

(H.2) = (1.6)

(7.57)

of (7.55).

is the inequality

As in p r e c e d i n g

cases,

does NOT follow from a-priori

regularity #(t) E C([O,T];H~(n])

is an independent

an equivalent

HO(~)

for the trace of the solution

(7.57}

say with

for D.

to (7.54),

Z

interior

(7.56)

This is plainly true,

(7.54) via p r o p e r t y

note that inequality

problem:

~n Q;

(H.2) = (1.6).

formulation

(7.54)

~t = ia~

Y = H-I(~);

ti.3):

(7.s3)

B*e A ty = -i ~ ;

solves the followlng homogeneous

i Assumptlon

n HI(Ci);

regularity

optimal

with #0 ~ H~(n).

result,

we

Inequality

which is e s t a b l i s h e d

in

[L-T.25]. F i n i t e Cost Condition stabilization

of problem

L2(O,T;L2(C))-contrcls space H-I(~) geometrical of F.

See

~1.9).

Both exact c o n t r o l l a b i l i t y

(7.51)

conditions

on n,

[L-T.25]

u n i f o r m stabilization.

Thus,

(1.9) holds true for p r o b l e m Detectabilltv

on any

as well as u n i f o r m s t a b i l i z a t i o n

with L2(O,~;LI(F))

[Leb.1],

on the space H-I(~)

Conditions

and u n i f o r m [O,T],

with

on the same

feedback controls hold true without

if the control action is e x e r c i s e d for exact c o n t r o l l a b ~ l i t y a fortiorl (7.51),

and

on all

[L-T.25]

for

the Finite Cost C o n d i t i o n

(7.52).

t5.1T)-IS.19).

This holds true since R = I.

85

CoD~luslon:

T = =.

Theorem 5.2 applies

to problem

(7.51),

(7.52).

Conclusion:

T < ~.

Theorem 3.1 applles

to problem

(7.51),

(7.52),

where R = I. 7.4.

Theorem 3.3 requires additional

Class

tH.21: First-order

Consider

hvDerbollc

smoothing on R.

systems

the following not necessarily symmetric

first order h y p e r b o l i c

or d i s s i p a t i v e

system in the unknown y((1,~2 ..... ~n ) e R m

n in (0, T]xf2;

(7.58a)

Y[t=0 = Y0 e [L2(~)] m

in ~:

(7.58b)

M(u)y(t,u)

in (0, T)xF,

(7.58c)

"

where Aj,

j=0

~

= U(t,a)

respect.

functions under

e L2(O,T;[L2(F)]k )

M, are smooth mxm,

the assumptions

F being non-characteristlc,

and

of

(after a s i m i l a r i t y M : [I,S]

Without

kxm,

matrix valued

(a) strict hyperbollclty,

and of

(b)

(c) rank M(a)

for the number of negative elgenvalues outward unit normal.

respect,

= k ~ m; here k stands n of A N =j=l ~ A~N~,j 2 N = IN 1 ..... N m]

loss of generality,

we may assume that

transformation)

I: kxk identity; S: kx(m-k)

A N : dlag[AN, AN],__

+ A; < O, A N > O,

(?.59) where AN is a kxk m a t r i x having the same negative and A + N is a (m-k)x(m-k) AN .

With

eigenvalues

m a t r i x having the same poslt~ve

(7.58) we associate

of AN;

elgenvalues

of

the cost

T J(u,Y)

= [ I"Ry(t)"2 ]m+,U(t), } dt 0 ~ [L2(n) [L2(F)]k

(7 60) "

for T < ~ or T = ~, with R E ~([L2(~)]m).

Abstract

settlnq.

abstract

form

[C-L],

(1.I},

[D-L-S]

To put problem

(7.58)

in the

(1.2), we choose

Z = Y = [L2(~)]m;

U = [L2(F)]k;

(7.61)

86 A = first order differential operator F with homogeneous b o u n d a r y conditions,

(7.62)

where

n Fy = Z Aj(¢)aty;

B = ADI:

(7.63)

A-IB = DZ;

J=0 where

(up to a translation) Dlg=

f ¢=, {Ff = O in f~;

DI: c o n t i n u o u s

(1.$):

(7.65)

[L2(F)]k ~ [L2(f))]m.

It is well known that A generates Assumption

(7.64)

Mf = g in F);

a s.c. s e m l g r o u p e At on Y.

(-A)-TB e ~(U;Y).

This follows with ~ = 1 from

(7.63)-(7.64). AssumDtlon available [C-L].

(H.2) = (1.6). from [Kr.1],

The required regularity p r o p e r t y

[Rau.1],

and is put in a semigroup

These formulas w~ll be n e e d e d

treatments

of Section

10.4 in P a r t

in the numerical

If.

We have

[H.2] is

framework

in

approximation

[C-L],

t (Lu)(t)

(7.66)

= AIexp[A(t-T)]DlU(r)dr; O

B*x = ANx-Ir

,

x = [x-,x+],

It is readily verlfled

dim x- = k;

that the Y-adjoint

, n ,~ )c3jy A y = -Y- A (~"

n - Y~ a

J=l

~(A

*

*

T

The r e g u l a r i t y

results of [Kr.l]

characteristic

systems yield

-i

[Rau.l]

(7.67)

T

(7.6s)

T

yields

-

for s t r i c t l y hyperbolic

~ CTtlXll [L2l~l]m

the desired estimate

(7.69)

(7.70)

S A N , Im_k].

lleA tXllL2(O,T;[L2(~)]m)

w h i c h combined with

T

J=0 jAj(¢)y + A (~);

= [-(AN)

(7.67)

of A is

A * h e [L2(~ ) ] m and M * h = 0};

) = {h ~ [L2(~)]m;

M

A

dim x + = m-k.

non-

(7.71)

(H.2) as in

87

HB'eA txll ~ CTHXll ]m L2(0, T;[L2(F)] k) [L2(n)

F i n i t e Cost C o n d i t i o n Cost C o n d i t i o n

(~,%).

A s u f f i c i e n t c o n d i t i o n for the F i n i t e

(1.9) to hold is the f o l l o w i n g

controllability'

property:

(7.72)

'exact null

there exists T > 0 such that for any

YO G [L2(n)]m , there exists u E L2(0, T;[L2(r)] k) such that the c o r r e s p o n d i n g s o l u t i o n of p r o b l e m

(7.58) s a t i s f i e s Y(T) = 0.

p r o p e r t y has been proved in the o n e - d i m e n s l o n a l finite open interval

[Ru.2,

Detectabilltv Condltlon Conclusion:

T = ~.

Thm.

case w h e r e then ~ is a

3.2].

(5.17}-(5.19):

holds true with,

T h e o r e m 3.1 a p p l i e s to p r o b l e m

Riccati o p e r a t o r P(t)

that R * R A ~ • Z(Y)

from (3.6).

two cases, and yields,

(3.25).

•

(7.60)

for

(Section 3.4)

That this P(t) solves the DRE (3.21)

r e q u i r e s one a d d i t i o n a l minimal assumption; [C-L].

Flnally,

requires a d d i t i o n a l smoothing; satisfy assumption

(8.58)

case.

(7.58),

any R ~Z([L2(n)] m) and yields a " v l s o o s ~ t y solution"

for all x,z • ~(A)

R = I.

P r o g r e s s on null c o n t r o l l a b i l i t y is

needed to apply T h e o r e m 5.2 also in the m u l t i d i m e n s i o n a l

T < ~.

say,

T h e o r e m 5.2 on the ARE applies to p r o b l e m

with R = I and n one dimensional.

Conclusion:

This

e.g.,

u n i q u e n e s s as in T h e o r e m 3.3

e.g.,

R ' R A E ~(Y)

[D-L-T~

p. 34] to

Below, we shall v e r i f y d i r e c t l y

(3.25) in

w h e r e then the e x i s t e n c e and u n i q u e n e s s T h e o r e m 3.3 a p p l i e s in particular,

B*P(tlx = AN[P(t)x]-IF:

that continuous

[L2(F)]k ~ c ( [ o , T ] ; [ L 2 ( n ) ] m ) .

(7.73) Case I.

Take R to be a hounded,

l-rank o p e r a t o r on Y: Rx = (x,a)b,

a,b • Y, so that R x = (x,b)a and R Rx = (x,o)c,

c = llb]~a. Then,

for

u 6 U, R * R e A t B u = (u,B*e A tc)c and T

~

,

[[R*ReAtBullydt ~ [lOll

llullU ~

llB*eA tCl]L2IO, T;U )

0 const T HclJ HuH U ,

(7.74)

88

and

(3.25) holds true.

Thus Theorem 3.3 applies.

Note that we have

taken any a,b e y, which would not s a t i s f y the sufficient R * R A • ~(Y) Qase 2.

condltlon

in general.

We now take

R'R: continuous [L2(n)] m ~ [H~+e(~)] m. The d i f f e r e n t i a b i l i t y

theorem

e A t : continuous Instead of proving

(7.75)

in [Rau.1] gives

W~+~(~)]m ~ C([0, T];[H~+a(Q)]m) [-0

((3.25):

(7.76)

R*ReAtB e ~(U;LI(0, T;Y)) , we shall

prove the dual statement, see (7.61),

equivalently

T

,

g ~ "JB*e A tR*Rg(t}dt:

continuous

L(0,T;Y)

~ U.

(7.77)

0 Indeed,

for such g we use

(7.67) with AN invertible and standard

trace

theory to obtain T [l~B*e A 0

tR*Rg(t)dt~U .< c T

(by ( 7 . 2 6 ) )

cT

(by (7.75))

sup S[eA tR*Rg(t)ll 0~t~T [H~+e(Q)] m

sup fiR Rg(t)II ._~+~ 0~t~T [n o (Q)

~ c T supHg(t)H

]m

]m = CTIIgHL (O,T;Y) [52(~)

as desired, 7.5.

and

(7.77)

is proved.

Class (H.2}: Kirchoff plate w i t h b o u n d a r y ~oment In n c R n, we conslder wtt-P~Wtt+~2w

the Kirchoff

in the bending

plate in (O,T]x~ ~ Q,

(7.78a)

in ~,

(7.78b)

wl~ e 0

in (O,T]xV,

(7.78c)

Aw[z = u

in ~,

(7.78d)

w(0,.)

= 0

control

= w o, wt(0,. ) = w I

89

with p take

> O a constant,

in L2(~).

[L-T.16].

and w i t h

Opt~mai

Consistently

Just

one b o u n d a r y

regularity

theory

with

results,

these

control

of p r o b l e m we

u which

(7.78)

take

the

we

is g i v e n

following

in

cost

functional oo

J(u,w)

fllw(t)ll2~

=

0

with

initial

Abstract model

settinq.

To put p r o b l e m

(1.2),

we ~ntroduce

A h = ~2h,

~(A)

as

in

(6.27)

A =

and

G 2 is the s a m e

satisfies D as

in

Eq.

of E x a m p l e

;

(6.48).

;

map

defined

in t e r m s

of

define

Y = [HZ(~) n HO(OI]xH

I1.3):

u ~ L 2(F).

we plainly

AssumDtlon

(1.3)

{H.21

the abstract operators

= O}

0 H~(~}

the

(

operators

;

in

(6.45)

R = I,

in E x a m p l e

the D i r l c h l e t

(7.80)

map

D:

6.3,

which

G 2 = -A-~D,

with

the s p a c e s

1~1 = ~ ( A ~ l x ~ ( A ~ ) ;

(-A)-~B e Z(U,Y).

By ( 7 . 8 0 )

and

u = L2IF).

(7.81)

(7.811

with

have

=

and assumption

(7.79)

AG2u

We a l s o

Assumption

= dhlF

and define

A =

into

self-adjolnt

hlr

= H2(n)

6.1,

Bu =

Green

(6.47)

(7.79)

the p o s i t i v e

~ ( A ~)

0

-

(7.78),

= {h ~ H4(O};

A ~ h = -~h,

the s a m e

)dr

"

{Wo, Wl} e [H2(~) n H ~ ( n ) I x H ~ ( ~ ) .

data

(1.1),

+llw~(t)ll (~)+llu(t)llL2(F 2

H'(n)

=

holds

= il.6).

true

One

0

for p r o b l e m

can show

that

=

(7.78).

[L-T.16]

~ Y"

(7.82)

90

B*e A*t Zl I = ~ 1 z2 ou

(Y.83}

IZ "

where @(t) = @(t,@O,@l) s o l v e s the c o r r e s p o n d i n g

homogeneous problem (7.84a)

#(o,.) = #o~ ~t (°'') = #z

(7.84b)

+lz

(7.B4c)

~ ~¢Iz

-

o

with

#0 = ( i + p A ~ ) - i z 2

• ~(A");

(7.SSa)

~1 ~ - ( I + p A ½ ) - I A ½ z l e ~(A½)" Thus,

by (7.83),

(7.85), an equivalent

(7.85b)

f o r m u l a t i o n of a s s u m p t i o n

(H.2) = (1,6) is the i n e q u a l i t y

X This i n e q u a l i t y holds indeed true, as r e c e n t l y shown in [L-T.16] m u l t i p l i e r methods. problem

Thus,

assumption

by

(H.2) = (1.6) holds true for

(7.78) for general smooth ~.

F i n i t e Cost C o n d i t i o n

fl.9).

The s a m e

as in S e c t i o n 7.1

considerations

for the w a v e e q u a t i o n and S e c t i o n 7,2 for the E u l e r - B e r n o u l l i e q u a t i o n apply.

It was r e c e n t l y proved that p r o b l e m

(7.78)

c o n t r o l l a b l e for s u f f i c i e n t l y large T > O on the s t a t e

is e x a c t l y

space

Y = [H2(~) n H~(O)]xH~(O) w i t h i n the class of L 2 ( Z ) - c o n t r o l s u, w i t h no geometrical

c o n d i t i o n s on ~

(except r smooth), the F.C.C.

if u is a p p l i e d to all

of r [L-T.16].

As a c o n s e q u e n c e ,

(1.9) holds true,

(Problem (7.78)

Is also u n i f o r m l y s t a b i l i z a b l e under some g e o m e t r i c a l

c o n d i t i o n s on n, e.g., strict c o n v e x i t y [L-T.16].) Detectabilltv Condition

(5.17)-(5.19).

This holds true s i n c e R = I.

Concluslon:

T = ~.

T h e o r e m 5,2 applies to p r o b l e m s

Conclusion:

T < ~.

T h e o r e m 3.1 applies to p r o b l e m

(7.78), (7.78),

(7.79). (7.79),

w h e r e R = I, while Theorem 3.3 requires a d d l t l o n a l s m o o t h i n g on R.

91

7.6.

C l a s s (H.2); as a bendlnu We return

A two-dlmenslonal moment

to the E u l e r - B e r n o u l l i

that n o w ~ c R 2 is a s m o o t h u acts

as a

plate

(physical)

model

equation

two-dlmenslonal

bending

moment.

w~th

of S e c t i o n

domain

More

boundary 7.2,

except

the c o n t r o l

and now

precisely,

control

we consider

the

problem wtt+~2w

I

w(0,.)

w[z

in

= O = w 0,

wt(0,.)

in

control

0 ~ ~ < I (physically

u E L2(Z } . 0 < ~

(7.87a) (T.87b)

(O,T]xF

= ~;

(7.87c)

in ~,

[~w+(I-H)BIW] ~ = u

with b o u n d a r y

= Q;

in n;

= wI

0

=

(O,T]×~

In

(7.87d)

< ~) w h i l e

the

(7.87d)

the c o n s t a n t

boundary

p

is

operator

B 1 takes

the f o r m

B1w =

8 w = - k ~-g 8w k ~-g

82w

(?.8s)

8T2 k(x)

being

(7.87c).

the

curvature,

Consistently

we t a k e

the

following

as

with cost

the tangential d e r i v a t i v e optimal

regularity

initial

Abstract (1.1), Ah

data

settlnq.

(1.2), -= A2h,

and d e f i n e

we

(n)

{Wo,Wl}

To put p r o b l e m

the o p e r a t o r s

[: :] ;

G

~s

the a p p r o p r i a t e

y = G v ¢== { d 2 y

= 0 in n;

Into

(7.B9)

the a b s t r a c t

self-adjoint

hIF = 0, ~ h + ( l - p ) B l h l r

BU =

-

where

+lu(t)tZ2(F:)}dt

(7.87)-(7.89)

the p o s i t i v e ,

= {U - H4(C]):

A =

by

[L-T.14],

~ Hl(f])xH-l(~).

introduce

~(A)

a s in

functional

o with

vanishes

theory

[0] ;

model

operator = 0},

R = I,

(7.90)

17.911

Gu

Green yIF

map defined = 0;

by

[dy+(1-p)BlY] F = v},

(7.92)

92

and the spaces

y = H~(n)H-I(~)

Assumptlon

(1.3).

L2(F} ~ L2(~),

(-A)-~B E L(U,Y).

we readily obtain from (7.91) w i t h u E L2(F) : 0

0

G

= -I

Assumption problem

(7.93)

Since G is certainly bounded

(-A)-IBu =

and A s s u m p t i o n

U = L2(I').

= ~(A~)x[~(A~)]';

(1.3) holds true for problem

(.H.2) = (1.6}.

~ Y,

{7.94)

G {7.87) with ~ = 1.

One can show [Hor.1]

that as in (7.39)

for

(7.34) we have

B

where ~(t)

" A't[Yl] e Y2

= ~(t.~PO.~Pl)

nonhomogeneous

boundary

= _ a d,(t),z ~-u solves

the

*

y = [yl,y2]

corresponding

(7.95)

problem with

conditions

i, t t . ~ , = (~-.ID(k ~ ( ~ , I ) ,(o

e Y,

I

= *0

°

in Q=

(7.96a)

~(~I 1

'(~t (0"" } = @I E ~(A~DlJ

i n CI;

@ = 0

on Z;

(7.96b) (7.96c) (7.96d)

With

+I = A;1A~Yz e :%(A~D) = H - l ( { ] ) ,

(7.97)

where AD :Is the p o s i t i v e s e l f - a d j o i n t operator defined by ADh -- -~h,

}CAD) = H2(C~)nH01(C~),

and D is the Dirichlet map as defined (7.96),

in ( 7 , 4 ) .

an equivalent formulation of A s s u m p t i o n

inequa I it7

Thus,

(?.98) b y (7.95),

(H.2} = (1.6)

Is the

93 2

dr"

at

<

CTII{<~O,~I}II 2

for the trace of the s o l u t i o n to p r o b l e m

it

s h o u l d be noted that I n e q u a l i t y

(7.99)

(7.96).

As in p r e v i o u s cases,

(7.99) does NOT follow from a priori

(optimal) I n t e r i o r r e g u l a r i t y of the s o l u t i o n to the p r o b l e m

(7.96),

(7.97).

It is an i n d e p e n d e n t r e g u l a r i t y result w h i c h holds true

[Hot.l],

for any general s m o o t h ~.

true for p r o b l e m

Thus A s s u m p t i o n

(H.2) = (1.6) holds

(7.87).

Exact controllabllity on any

Flnlte Cost Condltlon. Euler-Bernoulll plate

[0, T] for the within

(7.87) on the state space Y = H 0I(Q)xH-I(~)

the class of L 2 ( ~ ) - c o n t r o l s u is equivalent to the I n e q u a l l t y

'

,

which indeed holds true w i t h no g e o m e t r i c a l c o n d i t i o n s on ~, control u acts,

as assumed,

on all of F [Hot.l]

if the

(in this reference,

an

e x t e n s i o n of this result w h i c h allows the control functions to act o n l y on a p o r t i o n of the b o u n d a r y may also be found:

course, g e o m e t r i c a l c o n d i t i o n s are needed). Euler-Bernoulll problem absorb the term below

{7.44)).

However,

(7.34)

in this case,

of

As in the case of the

(Case 3), a novel d i f f i c u l t y is to

~t ~ L21~) by the t e r m ~ - ~

(&~) • L2(~ ) (see c o m m e n t s

This is done in [Hot.l] u s i n g the idea of [Leb.1].

a d d i t i o n a l d i f f i c u l t i e s arise because of the i n c l u s i o n of the

boundary o p e r a t o r B 1 In (7.87) w h i c h gives rise to the n o n - h o m o g e n e o u s rlght-hand side in e q u a t i o n

(7.96).

To bound the a d d i t i o n a l

which arise from the r l g h t - h a n d side of {7.96),

a s p e c i f i c S c h r O d i n g e r e q u a t i o n must be established.

Exact

controllability on [0, T] w i t h just one control u • L2(~) shown to hold on the space Finite Cost C o n d i t i o n

(7.93) as desired.

terms

a r e g u l a r i t y result for

Thus,

can then be

the r e q u i r e d

(1.g) does hold true for problem

(7.87).

Both

the idea in [Leb.1] and the r e g u l a r i t y e s t i m a t e for the S c h r ~ d l n g e r e q u a t i o n are also u s e d in the proof of t h e s t a b i l i z a t i o n result for p r o b l e m [Las.7] absent.

for the case p = 1,

corresponding uniform

(7.87) o b t a i n e d in [Hor.2],

following

i.e., w h e n the b o u n d a r y operator B1 is

94

7.7.

Class

{H.2):

Wave e u u a t l o n w l t h interior DOin~ control

We consider the f o l l o w i n g interior point control p r o b l e m for the wave e q u a t i o n wtt = A W + 6(x)u(t)

w(O,.) = WO; wt(O,.)

wlz --w h e r e 6(x)

-- w 1

0

in (O,T]xO = Q;

(7.101a)

in ~:

(7.101b)

in (O,T]xr = Z,

(7.101c)

is the Dirac mass +I at the point 0 (origin),

an interior point of the open bounded d o m a i n ~ = R n,

assumed to be

n = 1,2,3.

The

control u is a s s u m e d in L2(O,T). ~ o n - s m o o t h ~ n g o b s e r v a t i o n R. r e g u l a r i t y theory,

Consistently with established

[L2(~)xH-I(~) {Wo'Wl } ~ Y = YlxY2 def~ IH~O(~)×[H~o(Q)]'|

JL~l~(n)×L2(n) and,

initially,

(optimal)

d e s c r i b e d b e l o w in (7.111), we take n = 3;

(7.102a)

n = 2;

(7.102b)

=

(7.1o2¢)

n

1,

the cost functional T

i2)dt.

(7.103}

0 where we recall

[T.10] that

(7.1o4) for the p o s i t i v e s e l f - a d j o i n t o p e r a t o r A h = -dh, Z(A) = H2(O)nH~(Q).

A b s t r a c t setting. model

(1.1),

To put p r o b l e m

(7.101)-(7.103}

A =

; -

B =

;

R = I;

{1.3).

(-A)-~B e ~(U,Y).

(7.105) w i t h u e RI:

(7.105)

0

Z = Y (as d e f i n e d in (7.102));

Assumption

into the a b s t r a c t

(1.2), we take

U = ~1.

(7.1o6)

As in (7.6), we compute via

95

(7.1o7)

w h i c h we the

shall

show

second-order

to b e l o n g

to Y.

differential

To this

operator

end,

A we h a v e

we

recall

$(A u/2)

that

for

c H~(n),

hence, 6 ~ [Ha(n)] " c

after

recalling

n = 3;

Sobolev

a = l+&

z = A-a~26 E A-Is

[~(Aa/2)] "

embedding,

for n = 2;

L 2 ( Q ) we h a v e

by

[4-~ +~/2 z ~ ~(A~ 4-~

+z/2 z : ~ ( A ~

A-~ +el2 z Thus,

by

holds

true with

(1.12);

with

L2(~),

u = ~+e for

£ > O,

for n = 1.

(7.108)

Thus with

(7.10g),

-1 z

= Aa/2

(7.107),

Assumption

where

a = ~+&

A-a~26 e

or

c L2(~),

-z/2}

c $ ( A ~)

= H~0(~),

2(A~ - e l 2 ) ¢ ~(A ~)

(7.110),

and

(7.102),

H~(n),

we see

that

n = 3;

(7.110a)

n

2;

(7.1lOb)

n

1.

(7.110c)

Assumption

(1.3)

~ = 1.

(H.2)

i.e.,

-~/2)

=

(1.6).

This

case,

in our

is e q u i v a l e n t

to the

statement

to its d u a l that

version

for p r o b l e m

(7.101)

w i t h w O = w I = 0 w e have: L: u ~ { w ( t ) , w t ( t ) } = y(t):

w i t h Y in proofs Meyer;

(7.102).

have

been

(7.111)

see

[Lio, 2, p.

b y L. N i r e n b e r g )

and

IT.10];

variable)

that o n e w o u l d

Conclusion: analysis,

that

the r e g u l a r i t y

measured obtain

T < ~ we o b t a i n

Theorem

R). 3.1

optimal pair holds true for problem

true:

for n = 2,

space

On

~ C([O,T];Y)

is

Lions;

1, see "½+e"

order,

(7.111)

for n = 3 s e v e r a l

(by J. L.

only property

(non-smoothing that

27]

(7.111)

in S o b o l e v

by u s i n g

L2(O,T)

is i n d e e d

given,

We n o t e space

Property

continuous

[T.IO].

sharper

than

by Y.

(in the

the r e g u l a r i t y

(7.108}.

the b a s i s

of

the

foregolng

on the polntwlse synthesis of the (7.101)-(7.103).

96

S m o o t h l n u o b s e r v a t i o n R.

We now turn to the applicability of Theorem

3.3 on the e x i s t e n c e and u n i q u e n e s s of the Riocatl o p e r a t o r u n d e r s m o o t h i n g a c t i o n of the o b s e r v a t i o n o p e r a t o r R e Z(Y,Z)

(possibly)

final state o b s e r v a t i o n on G e ~(Y;W)

y(t)

in the cost

and

(1.2), w i t h

= [w(t),wt(t)].

AssumDtlon

~A.I) = ~3.25) on R.

If C(t) t

is the c o s i n e o p e r a t o r on

L2(Q) g e n e r a t e d by A and S(t) = [ C(?)dr is the c o r r e s p o n d i n g slne 0 operator:

c o n t i n u o u s L2(~ ) ~ C([0, T];~(A~)),

with u E ~i

via

(7.105),

[-As(t)

(7.109),

a-1

c(t)

w h e r e z = A-a126 e L2(Q)

6u

Aa/2C(t)zu

c(t)6u

and the above p r o p e r t y of S(t), we o b t a i n from

eAtBu E

[

(?.1z2)

+z/2)],),

(7.112),

n = 3;

(7.113a)

C([O,T];[~(A&/2)],x[~(A~ +&/2)],),

n = 2;

(7.113b)

C([0, T ] ~ ( A ¼

n = 1.

(7.113c)

- ~ / 2 ) x [ ~ ( A ¼ +z/2)],),

(3.113) we see that a-fortlorJ A s s u m p t i o n

(A.1) = (3.25)

s a t i s f i e d p r o v i d e d that the o b s e r v a t i o n o p e r a t o r R G ~(Y,Z), (7.102),

•

Recalling the v a l u e s of a in

by (7.I08).

C([O,T];[~(A~ +z/2)],x[~(A~

Thus, by

we compute p r e l ~ m i n a r i l y

is

Y as in

satisfies

s

R R: c o n t l n u o u s Cy(A¼ + ~ I 2 ) ] , × C ~ ( A ~ +z12)],

I

[ ~ ( A a l 2 ) ] , x [ ~ ( A M +z12)],

[~(A¼ -~/2)x[2~(A~ +~/2)].

[

L2(~)×[~(A~)]"

w h i c h r e q u i r e s that R * R be smoothing:

~sSUmDtion

(A.21 = (3.26~ o~ G.

r e q u i r e in order to s a t i s f y o b s e r v a t l o n G ~ Z(Y,W)

makes

n = 3;

(7.114a)

(A¼)x[~(A¼)] "

n = 2;

(7.114b)

(A~)×L2(~)

n = I,

(7.114c)

e.g.,

R = A-I~8 -E

If g e L I ( O , T ; U ) arbitrary,

(A.2) = (3.26),

we need to

that the final state

g7

T

T

.

(7.z15)

f ( B ' e A tG*X, g(t)Iudt = ~(x, s e A t B g { t ) ) y d t 0

well defined,

0

for x ~ Y.

(7.113) w h e r e C([O,T])

But eAtBg(t) has the r e g u l a r i t y e x p r e s s e d by

there ~s r e p l a c e d by LI(O,T) now.

Thus,

G must

t

have the same by (7.114);

(smoothing)

e.g.,

p r o p e r t i e s as the o p e r a t o r R R, as d e s c r i b e d

G = A -~ -~

Conclusion:

T < ~

the d y n a m i c s

(7.101) w i t h interior point control on the s p a c e s

described

above,

(smoothing R and G).

p r o v i d e d R and G are

T h e o r e m 3.3 is a p p l l c a b l e to

(smoothing)

sense that R R and G s a t i s f y the lifting p r o p e r t y

operators,

in the

(7.114).

R e m a r k 7.2.

A similar analysis holds true for p r o b l e m (7.101a-b) w i t h a] w~ = O: the D i r i c h l e t B.C. (7.1Olc) r e p l a c e d by the N e u m a n n B.C. ~ _ corresponding

regularity

results are g i v e n in [T.10] and they c o i n c i d e

In terms of spaces based on domains of fractional p o w e r s w i t h those of (7.Z01).

Case T = ~. problem

Exact c o n t r o l l a b i l i t y

(Y.I01) w i t h

(hence u n i f o r m s t a b i l ~ z a t i o n )

(finitely many)

i n t e r i o r point control(s)

L2(O,T) on the space Y of r e g u l a r i t y in (7.102) Thus,

in

is not p o s s i b l e

IT.8].

the c o r r e s p o n d i n g p r o b l e m w i t h cost g i v e n by (7.103} w i t h T =

cannot a d m i t a Riccat~

7°8°

for

Class

fH.2|:

theory.

K i r c h h o f f e u u a t l o n w i t h interior D o l n t control

We c o n s i d e r the following interior

point control p r o b l e m for the

Kirchhoff equation w t t - P A W t t + ~ 2 w = ~(x)u(t)

in (0, T]xn = 0;

(7.116a)

w(o,-)

in Q;

(7.116b)

in (0,T]xr = ~;

(7.116c)

in ~,

(7.116d)

wlZ wlz

= w 0, wt(O,. ) = w I

~ 0 ~ o

w h e r e as in (7.28a), p > 0 is a constant, + 1 e x e r c i s e d at the origin,

open b o u n d e d d o m a i n ~ c R n, n = 1, 2, 3. a s s u m e d in L2(O,T).

and w h e r e 5 is the Dirac mass

a s s u m e d to be an interior p o i n t of the A g a i n the control u is

98

~on-smoothina theory

R.

observat~o~

[T.11].

described

Consistently

below

with

in (7.1271,

def I"~(A~)×~(A~)'

{w0,wl}

and,

e

y = ylxY2

the cost

~nltially,

~

(optimall

regularity

we take (7.117a)

n = 3;

I~(A~)x~(A~)"

n = 2:

(7.117b)

Im(Am1×m(A~1, L

n = I,

(7.117c)

functional T

wl

(7.1i8)

=

0 In (7.117)

we h a v e

as in S e c t i o n

7.5,

that A is the p o s i t i v e or in (6.27),

A h = ~2h; Then

~(A}

self-adjoint

operator

defined

by

= {h e H4(n):

hIF = d h l F

= 0}.

(7.119)

1 = H0(~]I;

(7.Z201

[Gr.1] = {h e H3(f)): hlr = A h l r

• (A ~1

A~4h = -Ah;

~%bstract settlna. model

(1.1),

dynamics

~(A ~) = H 2 ( ~ ) O H o ( ~ ) .

To put p r o b l e m

(1.2),

(7.116)

= 0}; ~(A ¼)

(7.1161-(7.1181

we first o b s e r v e

can be r e w r l t t e n

via

(I+pA)~)wtt

into

(7.120),

abstractly

(7.1211

the a b s t r a c t

(7.121),

that

the

as

(7.122)

+ Aw = 8u,

and t h e n we take as in {7.80), A =

;

;

B =

-A

(as d e f l n e d

Z =Y

Assumption

A =

(I+pA~)-IA:

R =

I.

(7.123)

( Z+pA ~ )

11~3l.

(-AJ-~B

(-AI-IBu

in (7.117));

~e Z ( U , Y ) .

With

U = R 1.

u E ~ 1 we c o m p u t e

=

= -I

[I +pA ½ )-16u

,

(7.1241

99 which we shall

show

the f o u r t h - o r d e r

to b e l o n g

operator

A

to Y.

in

To t h i s

(7.119)

6 E [Ha(~)]" c [~(Aal4)]" where ~

takes

z = A-a/46

A-16

Thus

e L2(~),

= Aa/4

by

holds

on the v a l u e s

(7.124),

AssumptiQn (I.12);

by

we recall

in

that

c Ha(G),

~-a146 E L2(G), (7.109).

Thus

for hence

(7.125)

with

(7.109),

~-7: +e/¢z e m(A ~ -e/4) a ~(A~),

n = 3;

(7.126a)

A -~ +e/4z e ~(A ~ -e/4) c ~(A~),

n = 2;

(7,126b)

A -~ +e/4z e ~(A ~ -~/4) c ~(A~),

n = 1.

(7.126c)

I

(7.126),

and

(7.117),

we see

that

assumption

(1.3)

ff = 1.

(H,2}

i.e.,

we have

-1 z =

true with

or

described

end,

we have ~(Aa/4}

=

(1,6|.

in our

This

case,

is e q u i v a l e n t

to the s t a t e m e n t

to its d u a l that

version

for p r o b l e m

(7.115)

w i t h w 0 = w I = 0 we h a v e L: u ~ { w ( t ) , w t ( t ) } = y(t):

w i t h Y as has b e e n

in

preceding (in the

regularlty

smoqth~a

(1.2),

holds

that true

under

R e Z(Y,Z)

with y(t)

(A,1)

that

by u s i n g

and

R).

final

only

"~+~" than

(3.25)

o n R.

We now

smoothing

The

sharper

the

foregoing

syntheses

turn

of

the

and uniqueness

action

observation

to the

of

of the

the observation

G e Z(Y,W)

in the c o s t

= [w(t),wt(t)].

=

of the

(7.125).

of the

the p o l n t w i s e

on the e x i s t e n c e

state

is

order,

property

(7.127)

(7.101)

(7.116)-(7.118).

R a n d G.

(possibly)

(7.127)

space

(7.127)

property

of p r o b l e m

O n the b a s i s

3.1 on

for p r o b l e m

~ C([O,T);Y)

regularity

in S o b o l e v

Theorem

3.3

of the

the r e g u l a r i t y

obtain

operators

L2(O,T)

in the c a s e

measured

of T h e o r e m

operator

As

(non-smoothlng

observation

Assumptio~

note

one would

T < ~

applicability

operator

we

we obtain

palr

The validity [T.11].

variable)

that

qoncluslon: analysis,

Rlccatl

in

section,

space

optlmul

(7.117).

provided

continuous

operator

[L-T.16;

App.

C],

i00

- A = _ ( Z + p A ~ )-i A = -" A ~ + Z

p

a bounded

perturbation

of

-A~/p,

I

(I+P4~) -i

p2

p2

generates

a s.c.

cosine

(7.i28)

o p e r a t o r ~(t)

t o n L2(~)

w l t h ~%(t) = I Z ( r ) d r :

(7.122),

(7.123),

continuous

L2(/~) ~ C([0, T ] ; ~ ( A ~ ) ) .

From

0 we then o b t a i n w l t h u E

eAtBu

[-A~ (t) ! =

where

z = A-~/46

(7.109)

Thus,

(I-I

(I+pA½)-lA¼~(t)zu

i A~/4

( i + p A ~ ) - i~ ( t ) zu

(7.125).

a n d the a b o v e p r o p e r t y

by

(7.130)

satisfied (7.117).

{

Recalling

,

{7.129)

the v a l u e s

of ~(t) w e o b t a i n f r o m

of u in

(7.129),

C([O,T];~(A~s-£/4)x~(A~-~/4)},

n = S;

(7.130a)

C([O,T];$D(A~-&/4)x~9(A}~-~/4)),

n = 2;

(7.130b)

C([O,T];~(AY~-E/4)x~9(AY~-z/4)),

n = 1.

(7.130c)

we see that a - f o r t J o r i

provided

(I+pA)~ )-16

IA I 4

~ L2((]) by

eAtBu ~

~(tl]

that

assumption

the o b s e r v a t i o n

operator

(A.1)

= (3.25)

R E ~(Y,Z),

is

Y as in

satisfies

R R: c o n t i n u o u s ~ (A~-e/4 )x~ (A~ - e / 4 )

(A~-~/4)x~(A ~-~/41

. .

(A~-~ 14 )x~ (A~-z 14 ]

which requires Assumption (7.115)),

that R R be s m o o t h i n g ;

(A,2)

= (3.26)

we o b t a i n

the o p e r a t o r

on G.

n = 3;

(7.131a)

(A~)x~(A~),

n = 2;

(7.131b)

(A")x~(A~),

n = 1;

(7.131c)

[i e.g.,

R

= A

1/16

As in the p r e c e d i n g

that G m u s t h a v e

R R, as d e s c r i b e d

(A~)x~(A~),

by

section

the same s m o o t h i n g

(7.131);

e.g.,

(see

properties

G = A -~-£

on

101

C_ooncluslon:

T < ~

the d y n a m i c s

(7.116}

identified

(smoothing with

above provided

(7.127),

Theorem

3.3 is a p p l i c a b l e

control

on the s p a c e s

R and G are s m o o t h i n g

If the B.C. dwl~

then the c o r r e s p o n d i n g prop e r t y

point

the l i f t i n g p r o p e r t y

that R R and G s a t i s f y

Remark 7-3-

R and G).

interior

~ 0 in

Kirchhoff

(7.117),

has

aw~ ZI by~-

is r e p l a c e d

~ O,

the same r e g u l a r i t y

in terms of the s p a c e s

of the new o p e r a t o r A w h i c h

incorporates

in [T.11]

for the n e w t e c h n i c a l

to w h i c h we refer

in the sense

(?.131).

(7.116c)

problem

operators,

to

of f r a c t i o n a l

the n e w B.C.

This

issues

powers

is p r o v e d

that appear

in

this case. Case T = ~.

As in the p r e v i o u s

(hence u n i f o r m

stabilization)

interior

controls

point

regularity Remark

in (7.117)

7.4.

Similar

Euler-Bernoulli Schr~dinger regularity

in L2(O,T)

equations

theory

exact (7.116)

controllability with

is not p o s s i b l e

(finitely

many)

on the space Y of

IT.8]. analyses

equations

subsection,

for p r o b l e m

with

with

w o r k also

interior point control,

interior

is p r o v i d e d

considerations/conclusions

in the f o l l o w i n g

point

in IT.10]

of S e c t i o n s

controls.

and

two cases:

and

Their

[T.12].

All

7.7 and 7.8

hold

(i)

(il) sharp

the

true mutatis

mutandls.

8.

E~amDle

of a partial

differential

equation

problem satlsfvlna

(H.2R) The p r e s e n t

section

4.1 and 4.2 by m e a n s for h y p e r b o l i c

mixed problems

example w h i c h m o t i v a t e d All a s s u m p t i o n s this case.

serves

type

as we shall

[L-T.20],

[M.I] will be s h o w n 8.1.

Boundary problems

type,

type.

see,

[L-T.23],

control/boundarv of N e u m a n n tVDe.

r, w e c o n s i d e r

is,

of the class

this

while

to be i n s u f f i c i e n t

of both T h e o r e m observation

This

be s h o w n task will

t h e o r y of s e c o n d - o r d e r

With D a n open b o u n d e d smooth boundary

of N e u m a n n

4.1 and 4.2 will

the recent s h a r p regularity of N e u m a n n

control/boundary

the i n t r o d u c t i o n

of T h e o r e m

However,

as an i l l u s t r a t i o n

of a b o u n d a r y

and

earlier

problem

in fact,

(H.2R)

a key

= (1.8).

to be s a t i s f i e d

crltlca]2y

hyperbolic theory

rely on

equations

[L-M] and

inadequate.

o b @ e ~ v a t i o n for h y p e r b o l i c m i x e d A p p l i c a t i o n of T h e o r e m s 4.1 and 4.2

domain

in R n, n > 2, w i t h s u f f i c i e n t l y

the f o l l o w i n g m i x e d

problem

in

of N e u m a n n

101

C_ooncluslon:

T < ~

the d y n a m i c s

(7.116}

identified

(smoothing with

above provided

(7.127),

Theorem

3.3 is a p p l i c a b l e

control

on the s p a c e s

R and G are s m o o t h i n g

If the B.C. dwl~

then the c o r r e s p o n d i n g prop e r t y

point

the l i f t i n g p r o p e r t y

that R R and G s a t i s f y

Remark 7-3-

R and G).

interior

~ 0 in

Kirchhoff

(7.117),

has

aw~ ZI by~-

is r e p l a c e d

~ O,

the same r e g u l a r i t y

in terms of the s p a c e s

of the new o p e r a t o r A w h i c h

incorporates

in [T.11]

for the n e w t e c h n i c a l

to w h i c h we refer

in the sense

(?.131).

(7.116c)

problem

operators,

to

of f r a c t i o n a l

the n e w B.C.

This

issues

powers

is p r o v e d

that appear

in

this case. Case T = ~.

As in the p r e v i o u s

(hence u n i f o r m

stabilization)

interior

controls

point

regularity Remark

in (7.117)

7.4.

Similar

Euler-Bernoulli Schr~dinger regularity

in L2(O,T)

equations

theory

exact (7.116)

controllability with

is not p o s s i b l e

(finitely

many)

on the space Y of

IT.8]. analyses

equations

subsection,

for p r o b l e m

with

with

w o r k also

interior point control,

interior

is p r o v i d e d

considerations/conclusions

in the f o l l o w i n g

point

in IT.10]

of S e c t i o n s

controls.

and

two cases:

and

Their

[T.12].

All

7.7 and 7.8

hold

(i)

(il) sharp

the

true mutatis

mutandls.

8.

E~amDle

of a partial

differential

equation

problem satlsfvlna

(H.2R) The p r e s e n t

section

4.1 and 4.2 by m e a n s for h y p e r b o l i c

mixed problems

example w h i c h m o t i v a t e d All a s s u m p t i o n s this case.

serves

type

as we shall

[L-T.20],

[M.I] will be s h o w n 8.1.

Boundary problems

type,

type.

see,

[L-T.23],

control/boundarv of N e u m a n n tVDe.

r, w e c o n s i d e r

is,

of the class

this

while

to be i n s u f f i c i e n t

of both T h e o r e m observation

This

be s h o w n task will

t h e o r y of s e c o n d - o r d e r

With D a n open b o u n d e d smooth boundary

of N e u m a n n

4.1 and 4.2 will

the recent s h a r p regularity of N e u m a n n

control/boundary

the i n t r o d u c t i o n

of T h e o r e m

However,

as an i l l u s t r a t i o n

of a b o u n d a r y

and

earlier

problem

in fact,

(H.2R)

a key

= (1.8).

to be s a t i s f i e d

crltlca]2y

hyperbolic theory

rely on

equations

[L-M] and

inadequate.

o b @ e ~ v a t i o n for h y p e r b o l i c m i x e d A p p l i c a t i o n of T h e o r e m s 4.1 and 4.2

domain

in R n, n > 2, w i t h s u f f i c i e n t l y

the f o l l o w i n g m i x e d

problem

in

of N e u m a n n

102

Wtt-Aw+w

w(o,.)

(8.1a)

= O

= w0; wt(0,.)

wI

=

= %~

(As n o t e d

in R e m a r k

regular.)

8.1 below,

The optimal

preassigned,

control

(8.1b)

in

n;

in

Z =

(O,T]xr.

(8.zc)

the c a s e d i m ~ = 1 is m u c h problem

is now:

with

more

O < T <

mlnlmlze T

{ l l w ( t } l r f l ~ 2 l r l + l l u t l L 22( r

J(u,w) =

) }at

(8.2)

0 over

all u

We s h a l l

show

the p r o b l e m Abstract as

L2(O,T;L2(F))

e

that

settlna.

follows.

observation

in S e c t i o n

The abstract

The a b s t r a c t space

B of m o d e l

Ix

A =

-Ah =

;

(&-l);

v = Ng,=~ N: H s ( r ) L2(~)

Finally,

R: Y ~ ~(R)

~emark With

8.1.

When

reference

solution

w solution

for

Y and

(8.1)

due

to u.

~s a s p e c i a l i z a t i o n

(i.I),

for

of

of

(1.2).

the m i x e d

U of m o d e l

problem

(1.1),

and

(8.1)

is

the

Z are

A and

continuously.

4,1

setting

spaces

Y = HI(Q)xL2(~); The operators

with

optimal control p r o b l e m

this

considered

= L2(~),

(1.1)

I°I

Bu = ~(A)

((~-z)v

ANu

;

(8.3)

I°i

(8.4)

8 h F = O}; ~-QI

(8 5)

A-IBu

= {h e H2(~):

8v

= o i n n; ~ ' ~ l r

=

;

= g};

(8.e)

~ Hs+~(~) ~ H~(n)

(8.?a)

c H~-2P(n)

~ Z = L2(F):

(8.1),

= ~ ( A ~-p)

operator

R Y2

d i m Q = i, the s i t u a t i o n

yield

Z = L2(F ) .

are

the o b s e r v a t i o n

to p r o b l e m

formulas)

U = L2(F);

elementary

= YI[F

(8.7b1

R is:

= NAy

drastically methods

1.

(8.81

simplifies.

(including

explicit

103

L: U e L2{Z ) ~ {w, wt} ~ C { [ O , T ] ; H 1 ( n ) x L 2 ( n ) ) ,

(8.9)

RL: u E L2(~ ) ~ w[Z ~ H I ( o , T : R 2 ) , while t h e s e results a r e d e f i n i t e l y dim ~ = 1, the s e t t i n g yielding existence

of S e c t i o n

and uniqueness,

R e m a r k 8.2.

(Sharp r e g u l a r i t y

to [L-T.20],

[L-T.23]

only a few r e s u l t s

false 3.3,

for d i m D ~ 2 [L-T.21].

in particular T h e o r e m

is a p p l i c a b l e .

t h e o r y of p r o b l e m

for f u r t h e r

results

a n d proofs,

where ~,$ are constants

~ w12

u = ~ = ~ for a sphere;

= ~-&; $ = ~ for a g e n e r a l t h e o r y as in [L-M.1],

assumptions d o i n g so,

we shall

verifications briefly

[M.1],

(H.2R),

p o i n t out

in R e m a r k

greater

(h.O)

Instead,

through

the s e v e r a l

to [L-T.IO].

Verification by (8.4),

of

that all

(h.5)

where

sharp

where ~,~

required

are v e r i f i e d . these

regularity

> ~.

In

theory

Instead,

the

For d e t a i l s

we

I

(1,3):

(-A)-~B.

This

(H.2R)

= (1.8).

Using

is c e r t a l n l y

satisfied

with ~ = 1

(8,7).

Verification R*z = [Nz,0],

of

R* the Y - a d J o l n t

B [zl,z 2] = Z21F = N A z 2. S(t)

regularlty

I

earlier t h e o r y w i t h a = @ = M w o u l d be i n s u f f i c i e n t . refer

bounded domain

in e a r l l e r

places

o n the r e c e n t

8.2 above,

than M:

for a p a r a l l e l p l p e d ;

We s h a l l n o w v e r i f y

and

{8.11) (8,12)

of the s m o o t h

~ = ~ = ~-~

domain.

r e l y crltLcaIIy

recalled

we q u o t e here

HPlX),

e

~ = ~ and @ = ~.

of a s s u m p t i o n s . (1.3),

W h i l e we r e f e r

w 0 = w I = O,

strictly

< a ~ ~ w h i c h m a y d e p e n d on the g e o m e t r y

Verification

I

(8.1))

e C([O,T];H~(~)×H~-I{n)),

u = 0; {w0,w~} ~ Xi(n)xn2(n)

n; e x a m p l e s :

For

3.3

w h e n d i m n ~ 2:

L: u e L2(X ) ~ {w, wt}

continuously,

(8.10)

the s i n e o p e r a t o r

(8.8)

of R. s e e

Then we obtain

associated

w i t h A:

for R, we c o m p u t e

(8.3).

that

Moreover,

for z e Z = L2(F)

and w i t h

104

B*e A tR*z = AS(t)NzJ~

see e.g.,

[L-T.20],

[L-T.23],

e H = - ~ - 2 p + ( ~ - u ) ( ~ - 2 p ) (~)

(8.13)

c L2121,

(8.14)

[L-T.24], where in g o i n g from

(8.13) to

(8.14) we use c r u c i a l l y that $ >_ ~ > ~4 (since p in (8.7) is arbitrarlly small)

as in the sharp theory of R e m a r k 8.2, w h i l e ~ = ~ = ~ as in the

more classical

theory

V e r l f i c a t l o n of

would

fail to yield

{h.O) = {4.1|.

V e r ~ f i c a t l o n of (h.ll = (4.2).

(8.14).

Immediate:

R in (8.8)

From (8.8),

is b o u n d e d Y ~ U.

(8.4), we c o m p u t e

u ~ U = L2(F)

(8.1s)

R e A t s u = S(t)ANuI~ e L2(Z),

and

we

are

in the same s i t u a t i o n of (8.13),

V e r ~ f l c a t l o n of see

(4.22),

(H.21 = (4,3).

(8.14).

It is immediate:

for x = [Xl,X 2] ~ Y,

and by (4.2Y), ReAtx = [C(t)Xl+S(t)x2]12 ~ C([0, TI;H~(UI),

(8.16)

w i t h C(t) and S(t) cosine and sine operators a s s o c i a t e d w i t h A, w h e r e now classical to p r o d u c e

interior theory plus s t a n d a r d trace theory are sufficient

(8.16),

Thus,

which a f o r t i o r i

so far, T h e o r e m

Y = HI(~)xL2(~).

with

well,

verifies

(h.2).

4.1 is a p p l l c a b l e

to p r o b l e m

(8.1),

(8.2)

We shall now see that T h e o r e m 4.2 a p p l i e s as

with ~[t,T]

V e r i f i c a t l o n of

= H~(Xt)

= L2(t'T;H~(F))

{h.3) = (4.15).

c l a s s i c a l r e g u l a r i t y theory

N H~(t,T;L2(F)).

(8.17)

With ~[0, T] g i v e n by (8.17), more

[M.1] is s u f f i c i e n t to y i e l d

(4.15)

(while

s h a r p r e g u l a r i t y theory [L-T.23] yields an even s t r o n g e r result).

V e r i f i c a t i o n of and

~h.4) = f4.16).

By (I.I0),

for L O, R

above

(8.16), we shall s h o w that w i t h x = [Xl,X 2] ~ Y, we h a v e

(8.13),

105 *

{LoR

*

[Re

A'X

]}It)

(8.18)

= # ( t ) l ~ " e H~=12),

where

{

@tZ

= &@-@

#(T,.)

= ~t(T,-)

= 0

~l~'~l2 = *1~

T(t)

= @(t;@O,~l)

Plainly

from

This u s e d

(8.20)

a more

[L-T.23]

would (h.4)

~e~iflcatlon case

with

in p r o b l e m

invoking

(8.17),

= ReAtx

we

in ~;

(8.19b)

~;

(8.19c)

Xl

gives

result

a stronger

and @12

t h e n ~]2 E H½{2) [M.I]

(while

result).

e H½(2).

as d e s i r e d ,

sharp

Thus

(8.20)

x2 = tl"

= ~0'

x E Y, we h a v e ~ E HI(Q)

classical

just

regularity

(8.18)

by

theory

is p r o v e d

and,

by

is v e r i f i e d .

of

(h.5)

=

t # 0 is s i m i l a r .

u ~ H~{2)

(8.19a)

in

= O(t)x1+S(t)x2;

(8.19)

yield

in Q;

(4.17). By

We s h a l l

(1.9),

(1.10),

verify

(h.5)

we s h a l l

for

show

t = O.

that

The

for

have

{L;R*R~0u}(t)

= ~(t)12

e H2~-½12),

(8.21)

where ctt

=

/".,:;-<~

(T, • ) = ;t(T, • ) = 0

a~ I~ = wlx and w{t)

= w(t;O,O)

the m o r e

classical

is the s o l u t i o n result

of

[M.1]

of

in Q;

(8.22a)

in •;

(8.22b)

in 2,

(8.220)

(8.1)

for w O = w I = O.

on p r o b l e m

(8.1)

with

By using

w 0 = w I = 0 we

have:

continuously, stronger

while

the s h a r p

[L-T.20],

[L-T.23]

yields

the

result

u e ~(2) continuously,

where

may s t i l l

the

(8.22c);

theory

use

however,

~ w]~ e H2~-~CZ)

a > ~ is d e f i n e d

in R e m a r k

conservative

regularity

it is at t h e

level

(8.241 8.2.

(8.23)

of a n a l y z i n g

At

for w12 the

this

stage

which

resultlng

we

enters

106 of ~I~ that we crlticatly use the counterpart

regularity (8.24)

(~.e.,

of estimate

the sharp theory):

w[~ e H~(~7 ~ ¢1~ ~ H2a-~(~), and (8.21) follows.

But 2a-~ > ~, see Remark 8.2; thus

the injection H2a-~(~} ~ H~(~} Putting together

(8.21),

(8.25),

and (h.5) = (4.17)

not follow instead, Conclusion. (H.2R),

~

H~lZl

(8.27)

Conculsion

(8.277 would

We have verified all the required assumptions (h.5) for problem

(8.1),

(8.2).

to this problem.

(1.3),

Thus,

both

We obtain the

specialization:

Theorem 8.1. observation (i)

~(z)

is verified.

4.1 and 4.2 are applicable

following

(8.26)

using the earlier theory a = ~.

(h.O) through

Theorems

is compact.

and (8.28), we obtain

L~R*RL0: c o m p a c t as desired,

(8.25)

Wi~h reference problem

(8.1),

to the optimal boundary control/boundary

(8.2), we have:

there exists a unique solution of the DRE, v x,z q HI(~)xL2(~), d

(P(t)x,z)

-

I = (EIIF'ZIIF)L2(F)-(P(t)X'AZ) H_(Q)xL2(n ) H_(~)xL2(n ) I

(P(t)Ax'Z)Hl(~)xL2(n) +([P(t)x]21F,[P(t)z]21F)L2(F

1

w i t h P(T) = O, where we write P(t)x = {[P(t)x]1,[P(t)x]2} the two components (ii)

Uniqueness

(8.287

for

in Hl(n)xL2(~).

is within the class of the following properties: ~w

_> O,

0 ( t < T (* ~n HI(QTxL2(~));

(8.29}

(ii I )

P(t)

( i i 2)

P(t) E ~(HI(fl)×L2(Q);C([0, T];HI(D)×L2(Q));

(8.30)

(~i 3)

tf[P(t)x][FI[C([O,T];L2(F))

(a.31)

=

P

(t)

~ CTllXl! 1 H (N)xL2(N)

107

(lli)

The p o l n t w l s e

feedback r e p r e s e n t a t l o n

of the u n i q u e optimal pair

for the problem starting at t = 0 is:

lwt(t;Wo, Wl) (iv)

The optimal

cost is

lw°( • ;Wo,Wl) I

(V)

The optimal

lu°(.,t;X) lH~

sup

ly°(.,t;x) l

0~t~T

where x = {w0, wl},

Remark 8.3. problem

(8.33)

the regularity

~ CTIXlH 1

(~t)

(Q)xL2(Q)

~

C([t,T];Hl(n)xL2(n))

properties:

(8.34)

;

CT]X I HI(Q)xL2(Q)

(8.35)

y0 = /w0,w0%

We can also use the setting of Section

(8.1),

problem.

3

pair satisfies

sup

O~t~T

2

(8.2);

However,

in particular,

3.3 to treat

we can apply Theorem 3.3 to this

in order %o do so, we must take now 1-a

y

= Ha(~)xHU.-l{~)

instead of the smoother

= 2)(A¢/2)x[2)(A 2 ) ] ,

(8.36)

space Y = HI(~])xL2(Q) as in (8.3).

To t h i s

end, all we need is the following: Ve~ificatlon of a s s u m m t l o n ~n (8.36}.

(A.Z) = (3.~5) w i t h U = z = L2(F) a~d Y as

Since ~ > ~ in the sharp theory

trace theory that the operator R (D1rlchlet R ~ Z(Y;Z). (8.15)

Moreover,

This,

•

( R e m a r k 8.2),

trace)

we h a v e by

in (8.8) satisfies

ReAtBu ~ L2{~ ) = L2(O,T;Z ) w i t h u ~ U = L2(F) by

combined wlth

R =

e L(Z;Y),

shows

(3.25) as desired.

II.

ApproxlmatiQntheory

9.

Num~rlcal a m p r o x l m a t l o n s of the s o l u t i o n to the_abstract Differential and Alaebralc Riccatl Euuatlons The main goal of thls section

numerical

algorithm

is twofold:

for the computation

(1) to formulate a

of the s o l u t i o n

to the

•

107

(lli)

The p o l n t w l s e

feedback r e p r e s e n t a t l o n

of the u n i q u e optimal pair

for the problem starting at t = 0 is:

lwt(t;Wo, Wl) (iv)

The optimal

cost is

lw°( • ;Wo,Wl) I

(V)

The optimal

lu°(.,t;X) lH~

sup

ly°(.,t;x) l

0~t~T

where x = {w0, wl},

Remark 8.3. problem

(8.33)

the regularity

~ CTIXlH 1

(~t)

(Q)xL2(Q)

~

C([t,T];Hl(n)xL2(n))

properties:

(8.34)

;

CT]X I HI(Q)xL2(Q)

(8.35)

y0 = /w0,w0%

We can also use the setting of Section

(8.1),

problem.

3

pair satisfies

sup

O~t~T

2

(8.2);

However,

in particular,

3.3 to treat

we can apply Theorem 3.3 to this

in order %o do so, we must take now 1-a

y

= Ha(~)xHU.-l{~)

instead of the smoother

= 2)(A¢/2)x[2)(A 2 ) ] ,

(8.36)

space Y = HI(~])xL2(Q) as in (8.3).

To t h i s

end, all we need is the following: Ve~ificatlon of a s s u m m t l o n ~n (8.36}.

(A.Z) = (3.~5) w i t h U = z = L2(F) a~d Y as

Since ~ > ~ in the sharp theory

trace theory that the operator R (D1rlchlet R ~ Z(Y;Z). (8.15)

Moreover,

This,

•

( R e m a r k 8.2),

trace)

we h a v e by

in (8.8) satisfies

ReAtBu ~ L2{~ ) = L2(O,T;Z ) w i t h u ~ U = L2(F) by

combined wlth

R =

e L(Z;Y),

shows

(3.25) as desired.

II.

ApproxlmatiQntheory

9.

Num~rlcal a m p r o x l m a t l o n s of the s o l u t i o n to the_abstract Differential and Alaebralc Riccatl Euuatlons The main goal of thls section

numerical

algorithm

is twofold:

for the computation

(1) to formulate a

of the s o l u t i o n

to the

•

108 D i f f e r e n t i a l and A l g e b r a i c Riccati E q u a t i o n s (5.1);

(DRE)

(3.25) and

(ARE)

(ii} to present the relevant c o n v e r g e n c e results. To b e g i n with, we i n t r o d u c e a family of a p p r o x i m a t i n g subspaces

V h c y 0 ~(B

), where h, 0 < h ~ h 0 < ~,

d i s c r e t i z a t i o n w h i c h tends to zero.

is a p a r a m e t e r of

Let ~ h be the o r t h o g o n a l

p r o j e c t i o n of Y onto V h, with the usual a p p r o x i m a t i n g p r o p e r t y ll~hY-yIIy ~ O,

y e Y.

(9.1)

Let Ah: V h ~ V h and Bh: U ~ V h be a p p r o x i m a t i o n s of A, r e s p e c t i v e l y B, w h i c h s a t i s f y the usual,

natural requirements:

(1)

~ h A - 1 - A ; I ~ h ~ O, s t r o n g l y in Y;

(9.2a)

(il)

llA-l(Bh-B)ully ~ O, u ~ U.

(9.2b)

We consider

the following a p p r o x i m a t i o n of the DRE

(3.25) and ARE

(0.I): (Ph(t)Xh,Yh)y+(AhPh(t)Xh, Yh)y+(Ph(t)AhXh, Yh)y+(RXh, RYh)Z = (BhPh[t)Xh,BhPh(t)Yh)u ;

(Ph(T)Xh, Yh}y = (G Gxh, Yh),

DRE h (9.3)

v xh, Y h ~ Vh;

(AhPhXh,Yh)y+(PhAhXh, Yh)y+(RXh, RYh)z = (BhPhXh, BhPhYh) U • xh, Y h e V h. Our m a i n goal is to prove that,

under natural a s s u m p t i o n s w h i c h are the

d i s c r e t e c o u n t e r p a r t of the h y p o t h e s i s of the c o n t i n u o u s case, we have

(ARE) h (9.4)

(H.I) = (1.5) or

(among other things):

(H.2) = (i.6)

in the case of

the D i f f e r e n t i a l Riccatl E q u a t i o n P h ( t ) ~ h X ~ P(t)x,

s t r o n g l y in C([O,T];Y),

x e Y;

(9.5)

B h P h ( t ) ~ h X ~ B P(t)x,

s t r o n g l y in C([0, T];U},

x E Y;

(9.6)

and in the case of the A l g e b r a i c Riccati Equation, (i)

Ph ~ p"

s t r o n g l y in Y;

(g.7)

109

B h P h ~ B P, in a technical sense to be made precise;

(9.8)

(ill) A l t h o u g h there are a n u m b e r of p a p e r s in the l i t e r a t u r e w h i c h deal w i t h the problem of a p p r o x i m a t i n g RE, most of these w o r k s [B-K.1],

[K-S.1],

bounded.

[I-T.1],

treat the case w h e r e the input o p e r a t o r B is

W h e n instead B is g e n u i n e l y unbounded,

difficulties arise.

[G.1],

an a r r a y of new

Some of them are the same w h i c h are a l r e a d y

encountered in the continuous case treatment; some others are new, are i n t r i n s i c a l l y c o n n e c t e d w i t h the a p p r o x i m a t i n g schemes.

and

We llst a

few.

(a)

Open loop approximation.

Consider

the input ~ s o l u t i o n

operator t

(9.1o)

(Lu)(t) = f e A ( t - T ) B u ( r ) d r . 0 Under either h y p o t h e s i s the o p e r a t o r L is continuous:

(H.I) = (1.5), or else h y p o t h e s i s L2(0,T;U) ~ L2(O,T;Y)

C([O,T];Y)

in the case of a s s u m p t i o n

assumption

(H.1) w i t h Y < ~, or with ~ = M, w h e n A has a Riesz basis on

Y).

(H.2),

(H.2),

(indeed

and also in ~he case of

In the c o r r e s p o n d i n g approximation theory,

the q u e s t i o n arises

whether the d i s c r e t e map t (LhU)(t)

= f eAh(t-rlBhu(rldr

(9.11)

0 which is continuous:

L2(0, T;U) ~ L2(0, T;Vh),

operator L2(O,T;U) ~ L2(0, T;Y), h.

(For instance,

is a l s o c o n t i n u o u s as an

u n i f o r m l y w i t h respect to the p a r a m e t e r

one may take B h = ~ h B, w h e r e we note that ~ h B is

well d e f i n e d s i n c e V h ~ ~ ( B

) by assumption:

(~hBU, V h ) y = (BU, Vh) Y = (u,B vh)U). In the case where B is bounded,

this s t a b i l i t y r e q u i r e m e n t

the a p p r o x i m a t i o n of A is consistent, follows via Trotter-Kato theorem.

i.e.,

Instead,

subject

(9.12) is true if

to (9.2),

as it

in the case w h e r e B is

110

unbounded,

special c a r e

a p p r o x i m a t i o n scheme, (b)

must be exercised to select a s u i t a b l e

which guarantees the above s t a b i l i t y requirement.

~ p o r o x l m a t l o n of oaln ooerators B Ph(t),

here is that even if (9.5) clear that

(9.6)

(resp.

(resp.

(9.7)) holds true,

(9.8)) will also follow.

(9.6),

Thus,

B is g e n u i n e l y unbounded,

w h i c h are not present

such as it arises in b o u n d a r y control and

In the B - b o u n d e d case.

offers new challenges

In order to cope w l t h

we n e e d - - a s in the c o n t i n u o u s c a s e - - d i s t i n g u i s h

b e t w e e n d y n a m i c s w h i c h satisfy a s s u m p t i o n which satisfy assumption

(H.I) = (1.5) and dynamics

(H.2) = (1.6).

9.1.

A p D r o x l m a t l o n fo~ ~he fH.1)-class

9.1.1

ApDroximatlon assumptlon~

A D D r o x i m a t l o n of A.

as to obtain

in the case w h e r e the operator

for partial differential equations,

these difficulties,

special c a r e

(9.8) for the gain o p e r a t o r s .

a theory of a p p r o x i m a t i o n s

point control

The problem

it is far from

Thus,

must be given in s e l e c t i n g the a p p r o x i m a t i n g schemes, convergence

B Ph"

Let Ah: V h ~ V h be an a p p r o x i m a t i o n of A w h i c h

s a t i s f i e s the f o l l o w i n g requirements: (A.1)

(uniform analytlcity) IAheAh t IX(y)

~ ~C e (w+z)t •

t > 0

(9.13)

d i s c r e t e a n a l o g of (H.1), where the constant C is u n i f o r m w i t h respect to h; (A.2)

I~hA-I-A;IEhlZ(y)

~ % p p r o ~ m a t l o n of B.

< Oh s

for some s > O.

(9.14)

We shall assume that the o p e r a t o r B: U ~

and Bh: U ~ V h s a t i s f y the following " a p p r o x i m a t i o n "

[~(A )]"

properties,

where

and s w e r e d e f i n e d above (A.3)

('inverse a p p r o x i m a t i o n property')

* JIB*XhJ~u+NBhXh~IU _< C h-~ sllXhllH , (A.4)

f

%

HBS(]~h-I)X~Iu <_ C hS'l-V'Hx~l

.

,

V x h ~ Vh;

(9.15)

x G ~(A

(9.16)

);

111 * lib * X-Sh~hXll U

{A.5)

(If, in particular,

~ C h s(l-~

)llxll

,

x e ~(A* );

(9.17)

~(A*)

we take B h = ~ h B, then (A.5)

is c o n t a i n e d

In

(A.4).) HB ]~hxHU .< C]I(A ) xll H .

(A,6)

Remark 9.1.

Notice that

(A.2) throughout

approximation properties. the original

(finite elements,

spectral approximations).

(A.4) are the standard

Moreover,

they are satisfied

finite differences,

to our knowledge,

it is satisfied

are, however,

a number of significant

underlying s e m i g r o u p s Co~$e~ences From follow

e

Aht

(A.2) and

physical examples

form Is not coercive,

are uniformly analytic

of a D D r o x i m a t l n o

(see [Las.1,

(A.1},

assumDtlons

the following

(e.g.,

of There

damped

while the

(see Section

i0).

on A and B

"rough" data estimates

Appendix]).

leAht~h-~heAt. ~ ( y )

(1)

For instance,

form a s s o c i a t e d with A h (see Lemma 4.2 in (Las.l]).

elastic systems,) where the billnear

~ ChS8 e(~+~)tte ,

(9.19)

where 0 ~ e ~ 1 and e > 0 can be a r b i t r a r i l y small: (li)

I~hR(~,A)-R(l,Ah)~hlZ(y)

containing some ~/2

(lii)

~ C h s, s > 0

c° "A";, where Z a p p [A) = closed triangular in A e Z appl

uniformly

the axis

sector

[-~,a] and delineated by the two rays a+p ±I@ for

.

(~(A),Y) uniformly

(9.i0)

< ~ < 2~; a = w+z;

leAht~h-~heA*tl

in t > 0 on compact subintervals.

is

for most of the schemes and

condition for (A.1) to hold is the u n i f o r m c o e r c l t l v l t y

the bllinear

9.1.2

(A.I)

in each case.

examples which arise from analytlc semlgroup problems. sufficient

by

mixed methods,

The property of u n i f o r m a n a l y t l c l t y

not a standard a s s u m p t i o n and needs to be v e r i f i e d However,

(9.18)

They are consistent with the r e g u l a r i t y of

operators A and B.

typical schemes

x e ~((A*)~).

~ C hs

('9.21)

112 R e m a r k 9.2.

We c o n s i d e r the special case of c o e r c i v e b i l i n e a r forms,

and s h o w that in this case a s s u m p t i o n W i t h Y the g i v e n Hilbert space,

(A.1) is a u t o m a t i c a l l y satisfied.

let W be a n o t h e r H i l b e r t s p a c e for

w h i c h the i d e n t i t y W c-~ y is continuous.

A s s u m e that the o p e r a t o r A

s a t i s f i e s further the following conditions: (i)

c o n t i n u i t y of s e s q u i l l n e a r form on W: there exists a constant K s u c h that

l(Ax, y)yl ~ KIIxUwllyll w (li)

(G~rding inequality)

v x,y e w;

there exist p o s i t i v e c o n s t a n t s cl,c 2 such

%hat 2 Re(-Ax, x)y ~ cl~[xHW

-

c211xll~,

v x ~ W,

so that -A+c2I is W - e l l i p t i c or coercive. Then, as is well known,

e.g.

[Sh., p. 99],

the o p e r a t o r A

a c t u a l l y g e n e r a t e s an a n a l y t i c s e m i g r o u p on Y. W i t h V h the a p p r o x i m a t i n g subspaces i n t r o d u c e d before,

define

Ah: V h ~ V h by (AhXh, Yh)y = (AXh, Yh) Y. Then, conditions

xh, Y h ~ V.

A h s a t i s f i e s a u t o m a t i c a l l y the c o n t i n u i t y and G a r d i n g

(1) and

(ii) above w i t h the same c o n s t a n t s K, cl,

i n d e p e n d e n t l y of h > 0.

Therefore,

the very

same

c2,

a r g u m e n t w h i c h proves

a n a l y t l c i t y of e At in the continuous case, verbatim)

once a p p l i e d ( e s s e n t i a l l y Aht yields that e s a t i s f i e s the uniform

to the d i s c r e t e case,

analyticlty condition

(A.1), w i t h constant C i n d e p e n d e n t of h.

9.1.3. A p p r o x i m a t i o n of d y n a m i c s and of contPol D r o b ~ e m s ~ Riccatl eauatlon

•

Related

We n o w i n t r o d u c e an a p p r o x i m a t i o n of the control p r o b l e m and of the c o r r e s p o n d i n g RE. Control oroblem.

G i v e n the a p p r o x i m a t i n g d y n a m i c s Yh(t) c V h such that

yh(t)

= AhYh(t)+BhU(t);

y(O)

= ~hy 0

(9.22)

minimize T J(U'Yh(U))

~

~[IRYh(t)12+lu~t)l~ ~dtz 0

.

(9.231

113

where

T < ~ in the

optimal

solution

~ccatl

Eauat~on.

Riccati

Equation

case

to

of

the DRE

(9.22),

(9.23)

and T = @@ in c a s e (which w e s h a l l

The approximation

of the ARE.

see

later

The

to exist)

the D i f f e r e n t l a l / A l g e b r a l c

of

is d e f i n e d by e q u a t i o n

(DREh)

=

(9.3),

(ARE) h =

{9.4)

wlth B h .

9.1.4.

Main

Differential

Theorem

Ricca~i

9.1.

(DRE)

G G • Z(Y;~(A*)). and

(A.1)

=

Then solution

of

E~uation.

Assume

The m a i n

hypothesis

In a d d i t i o n ,

(9.13) there

Ph(t)

following

of a o D r o x i m a t l n u s c h e m e s

results

through exists

(A.6)

=

(H.I)

(9.3)

=

is

(1.5)

and moreover

the a p p r o x i m a t i n g (9.18)

h 0 > 0 such

the D R E h =

properties

let

result

for ali

exists,

(9.1)

true.

hold

that

that

properties

0 < h < h 0'

is unique,

and

the

satisfies

the

as h~O: cT • Y 0 < p ~ I ;

l[Ph(t)~h-P(t)]X[c([O,T];y)

I[B P ( t ) - B u o uOl

h-

If,

o

~ O,

Ph(t)~h]xlC([O,T];U)

~

x ~ Y;

O,

o

V h c ~(A~)

and

the

(9.25)

x e Y; I

C¢[O,T);U)+IYh -y IC([O.T];y)+I

in a d d i t i o n ,

following

(9.24)

norm

(9.26)

~

o.

(9

equivalence

"

holds

{A Xht ~ {AhXh{,

(9.28)

then IA*~[Ph(t)~h-P(t)]Xlc([O,T];y)

Alqebralc in t h e

Riccatl

case

conditions satisfied

Equatlon.

of the ARE, which

it

guarantee

In o r d e r

to o b t a i n

is n e c e s s a r y that

b y the a p p r o x i m a t i n g

I

(9.29)

approximation

results

to i m p o s e

the F i n i t e problem

X ~ Y.

~ O,

Cost

(notice

some

approximation

Condition that

this

(i.9) does

is

not

114 a u t o m a t i c a l l y follow from the fact that the F.C.C. c o n t i n u o u s problem). sufflclent,

holds true for the

B e l o w we impose c o n d i t i o n s w h i c h are o n l y

but w h i c h are s a t i s f i e d by all a n a l y t i c e x a m p l e s to be

c o n s i d e r e d in Section I0. For the present (H.1) = (1.5),

'analytic'

class,

the Finite Cost C o n d i t i o n will be g u a r a n t e e d by the

followlng Stabillzabllity Condition (S.C.)

subject to a s s u m p t i o n

[B F • Z(Y,U) ,^(A+BF)t } j=

(S.C.)

such that the s.c. a n a l y t i c s e m l g r o u p

[as g u a r a n t e e d by (1.3))

is e x p o n e n t i a l l y

(9.30)

]stable on Y:

/

-~F t

L

e(A+BF)tI[z(Y) ~ MF e

for s o m e ~F > O.

Our main results are f o r m u l a t e d in the theorems below.

T h e o r e m 9.2.

(ARE)

[L-T.2],

[L-T.19] Assume:

I. The continuous h y p o t h e s i s Stabilization Condition and,

(9.30},

(H.I) = (1.5),

the a b o v e

the D e t e c t a b i l i t y C o n d i t i o n

(5.10},

in a d d i t i o n either R > 0

ii)

(9.31)

ii)

or

A-1KR: Y ~ Y compact; *^*-i

either B A

{((i) ii)

: Y ~ U compact

(9.32) or F: Y ~ U compact.

II. The a p p r o x i m a t i o n p r o p e r t i e s (A.6} = (9.18).

Then there

s o l u t i o n Ph to t h e e q u a t i o n following convergence

(9.1),

(A.1) = (9.13) t h r o u g h

exists h 0 > 0 s u c h that for all h < ho, (ARE) h = (9.4) exists,

the

is u n i q u e and the

p r o p e r t i e s hold:

le-Ah'ptjz(y

) .< C e - ~ p t P

> 0,

(9.33)

w h e r e Ah, p - Ah-]]hBB'Ph;

^ "1-p IAh PhlZ(Y)

~ 1~*½-Pp A½-P -'" h h lZ(y)

IIPhT[h-PIIz(y)

.< C h ~0 ~ 0

~ C, as h i 0 ,

for any 0 < p < 1;

v ~'0 < s ( 1 - ~ ) ;

(9.34)

(9.35)

115

llShPh~h-B PIIL(Y;U) ~ 0 for all. ¢0 < s(l-~),

~'~pt

sUp e t>_O

as hgO, x e Y,

0

<_ C h

llUh(%,~hX)-uO(t,x}H~(y;U)

I l y ~ ( . ,~hx)-yO( • , N x ) I I ~ ( Y ; L 2

for all ~o < s(l-7)

(9.36)

as hiO;

( o,~;Y))

~0

•,, o;

(9.387

(9.39}

~ c h aO ~ O;

and for all z > O, as h$O,

s u p t & eWP tllyh{, O.t,llhx)_y " " O , t ,x)llz(y) <_ C h ~ O~ t_>O

O;

(9.407

for all SO < s(1-~),

13(u:(. on nhxT,V:(.,~hXT)_j(uO(, o. x), o(.,x)l Moreover,

if in addition,

.

~.

c

h

~0

for some 0 < 8 < 1, V h c ~(~e)

ll(~*)eXhlly _< Cell(A~)exhlly , or

-.

(9.41)

O.

and

(~*)e(~h-1)e ~ Z(Vh, Y),

(9.42)

then ^*

e

}I(A ) (Ph~h-P)XJ}y ~ 0

(9.43)

as h~O, x ~ Y, 0 ~ ~ < i;

II(A )e (Ph]~h-P)A xlly-, 0

as h$O,

Assumption

is c e r t a i n l y

the case when A is coercive and A h is a s t a n d a r d Galerkln

Remark 9.4.

of A: l.e.,

Theorem 9.3.

(ARE)

Thls

[]

(or, more generally,

and A2: Y D ~((-A171-z)

take e = ~ in (9.44).

true wlth ~ = ~.

(AhXh, Yh) y = (AXh, Yh) y.

If A is self-adjoint

with A 1 self-adjoint

holds true:

typlcally holds

(9.44)

11

Remark 9.S.

approximation

(9.42)

x ~ Y, 0 _< ~ < ~.

If A = AI+A 2,

~ Y is bounded),

one can

M

(i) The following u n i f o r m e x p o n e n t i a l

stabllity

116

II-~ ( Y ) " "< ~; e

under the same assumptions (ii)

(9.45)

,

of Theorem 9.2.

Moreover,

sup t~O

e

m

~(y)

Theorem 9.2 provides

the basic convergence

results

for the optimal solutions

of the approximating

the c o r r e s p o n d i n g

operators, and gain operators,

quantities

Riccatl

of the original problem

(I.i),

It states

once acted upon by the discrete

g i v e n by Uh(t,~hX)

= -B PhYh(t,x)

yields

(with rates)

(9.22),

(9.23),

to the same

(1.2).

The advantage of Theorem 9.2 is this:

original system,

problem

(9.46)

that the

feedback control

(uniformly)

law

exponentially

stable solutions. R e m a r k 9.5.

Instead of the original

introduce an equivalent C21Xhly. discrete adjolnt

inner product

In some situations,

inner product operators

R e m a r k 9,6.

inner product (Xh, Yh)Yh,

where CllXhl Y ~ IXhlYh

it is more convenient

( , )Yh as to simplify

for the discrete problem.

The literature

(Xh, Yh) Y , one can

on a p p r o x i m a t i n g

to work with a

the computations

for the

• schemes of optimal

control

problems and related Riccatl equations g e n e r a l l y assumes (i) convergence properties

of the

'open loop'

solutions,

i.e.,

of the maps u ~ y of the continuous problem; (ii)

"uniform s t a b i l i z a b i l i t y / d e t e c t a b i l i t y "

approximating In contrast,

hypotheses

for-the

problems. our basic assumptions

are:

(a) stabilizabillty/detectability hypotheses the continuous

(S.C.)/(D.C.)

of

system;

(b) a "uniform analyticlty"

hypothesis

(A.1) on the

approximations. Starting properties

from

(a) and

(b), we then derive both the c o n v e r g e n c e

of the open loop and the uniform s t a b i l i z a b i l i t y /

117

d e t e c t a b i l i t y hypotheses--(1) assumptions

and

(il) a b o v e - - w h l c h are taken as

in other treatments.

Thus,

the theory p r e s e n t e d here is

"optimal," in the sense that it assumes o n l y what is s t r i c t l y needed. Indeed,

it can be shown that a s s u m p t i o n s

only sufficient,

but also necessary,

(A.I),

{S.C.)/(D.C.)

are not

for the m a i n t h e o r e m s presented

here. These c o n s i d e r a t i o n s are an important aspect of the entire theory since,

in the case where B is an u n b o u n d e d operator,

the

requirement of c o n v e r g e n c e L h ~ L of the open loop s o l u t i o n s is a v e r y strong a s s u m p t i o n as remarked before.

Generally,

bounded,

it may well h a p p e n that the

and the scheme is consistent,

scheme is not even stable;

i.e.,

even w h e n L is

L h may not be u n i f o r m l y b o u n d e d in h.

The p r o p e r t i e s of the c o m p o s i t i o n eAts may not be r e t a i n e d in the Aht approximation e B h. Special care must be e x e r c i s e d in a p p r o x i m a t i n g B. T h e o r e m 9.2 p r o v i d e s rate of c o n v e r g e n c e ~(h s(1-Y)) a p p r o x i m a t i n g problem.

This rate is, in general,

for the

non-optimal,

as it

does not reflect the r e g u l a r i t y properties of the o r i g i n a l c o n t i n u o u s problem.

More precisely,

the regularity p r o p e r t i e s of the Riccatl

operator

(given by (5.2)},

together with the approximation

property

(A.2) s u g g e s t s that the optimal rate of c o n v e r g e n c e of Rlccatl Operators r e c o n s t r u c t i n g this r e g u l a r i t y s h o u l d be l(Ph-Pllz(y)

Similarly,

because of estimate

(Theorem 5 . 2 ) ,

(9.19),

one would expect

s e m i g r o u p Would r e t a i n

eAP t

{

that

convergence

and because

exp(Ap)t

the approximating

properties

eAh'Ph t

-

(9.47)

= ~(hS(1-~)).

[z(YI

similar

C h s8

~ ~

is

analytic

feedback to

(9.19),

i.e.,

~t

e

(9.4a)

where

Ah, Ph

If the o p e r a t o r B is b o u n d e d above rates of c o n v e r g e n c e Even more,

(i.e., B e ~(U,Y)

(9.47),

if A-~B ~ Z(U,Y),

Ah-BhB;P h .

(9.49)

and ~ = 0),

then the

(9.48) are g i v e n by T h e o r e m 9.2.

Theorem 9.2 p r o v i d e s the c o n v e r g e n c e rates

118

equal

to ~ ( h S ( 1 - q ' / %tl-E),

following

question

at c o n v e r g e n c e , A-qB

(9.47}

e ~[U;Y),

Below

~ > 0

we shall

provided,

instance,

A h,

of c o n v e r g e n c e

operator

'nonoptimal'

a positive

B h.

While

are valid

if ~ > O.

to o b t a i n

in the u n b o u n d e d

(particularly,

in the answer

care

for a n y

case,

interesting

is g i v e n

consistent

subject

to

will

Thus,

the o p t i m a l

to the a b o v e

i.e.,

case

results

7 > M)?

question,

with

require,

the

(i.e.,

(9.47),

(9.48))

hypotheses

imposed

on the approximations of

of

A h,

Bh

optimal

(for rates

in general, the u n b o u n d e d

B.

Finally, optimal

rates

it s h o u l d require

the n e c e s s i t y optimal

of

'rough

asserts,

preserve

The

in the

the s i n g u l a r

crucial

role

roughly

speaking,

that

we shall

and

additional

A h are

the

following.

(A.7)

Let U

c U c U be r0 _ r1

(1)

I[A;IBh-A-IB]Uly

two H i l b e r t

IBh_~-IB]ul

abstract

result

perturbations with

results.

for the o p e r a t o r s

spaces

hOlUlu

Ch

such

that

;

rOluIu

• r0

where

0 ~ r 0 ~ s, a n d p

Let Y

be a n o t h e r

> O.

Hilbert

space

r1 y

D ~9(A 1-&) n ~ ( ~ i - £ ) rI

and

the

by the

r1

cii) JEI

of

of

of d i s c r e t i z a t i o n .

our m a i n

~ C

bounded

the

because

{at the origin) is p l a y e d

stabilizability'

approximatlon p r o p e r t i e s

The

is so,

a perturbation

relatively

'uniform

of the p a r a m e t e r formulate

end

with

of B - u n b o u n d e d ,

This

behavior

to this

together

analytlcity'

case

analysis.

estimates

independent

Below,

that

delicate

data'

'uniform

estimates

be n o t e d

a more

'tracing'

solutions.

so-called

(A.8)

the

the r a t e

approximations

(A.1)-(A.S),

the

rates when

to the s e l e c t i o n

the c o n v e r g e n c e

B h = B or B h = ~ h B)

additional

which

(9.48)

very s p e c i a l

however,

~ ( h S ( l - ~ ) / t 1-&)

are

is it p o s s i b l e

and

provide

approximations

which

arises:

such

that

for s o m e

a > O,

B h and

119 (i)

^ -1+~ A B • Z(Url;Y);

(1i)

*"*-1 *^*-1 I[BhAh -B A ]YlUrl

A-IB e ~(Ur0;Yrl), rl c

h

lylz

,

r1

where 0 ~ r I ~ s. (A.9)

B •^A ~-2+6 e ~(Y;Ur6);

(i)

B*A*-I+eA -l+& e Z(Y;U).

(ii) There exists n > 1 such that [B*~*-I~-IB.n J

Theorem g.4. [Las.6] assume hypotheses C independent

¢ Z(U, Uro).

In addition to the hypotheses of T h e o r e m g . 2 ,

(A.7)-{A,9).

Then with

• > 0 arbitrarily

small,

of h and t,

(i)

IP-Phl~(y)

(ii}

IBhPh-B P[~(y) ~ C h-~(s+~)[hS+hr0+hrl].

~ c[hs(1-~)+hr0+hrl];

Theorem g.5. [Las.6] Assume t h e same hypotheses as above. exists w0 > 0 such that for any ~ > 0, t > 0, (i)

-~0 t 0 }(y 0 -yh)(t)Iy ~ Cetl_z IXly[h s( I-~) +hr0*hrl];

(il)

[ (u -u h) (t) IU <

(lii)

Ie

0

0

-

Then, there

-Wot Ce IXly[hS(1-~)+hr0+hrl]: t ~ -E

Apt eAPh t

Apt

Iz(y)+le

(A-BB * Ph)t

IzcY)

-e

C e-~Oth -'f(s+~) [hS+h r 0+h r 1]; (iv)

and

IeAP t - e APht l~(y) <- ~Ce -w ot

Corollary 9.6.

[hS( 1-¢ )+hrO+hrl]"

Let x • $9(A), then

l(u°-u°)(t)lu _< c e

-~ t

.

o [hSl1-~)+h

r0

+h

r1

]IXly~

120

0

O

< C e-~0t[hS(1-e)+hr0+hrl]IxIy.

{(y -Yh)(t)ly

Corollary "boost-strap"

'goodness' i.e.,

9.5 gives

Indeed,

follows Url

=

from

U,

takes

with

Corollary

of T h e o r e m s

Assume

(A.9)

the o p t i m a l

In fact,

operator

B,

the additional

show

in S e c t i o n

boundary For

instance,

that

controls,

"Nitsche

satisfying

the usual

all

Galerkin

complies

wlth

provided

for the s t r o n g l y

assumptions

or p o i n t

controls

(A.7) in

with (with

(A.8)

w e take

B is u n b o u n d e d

(A.7),

(A.9)

hold

and true

9 . 4 a n d 9.5,

provides

of

(A.6)-(A.9). plate

10).

where

in order

to

A h and B h comply

r O = r I = s.

the heat can

equation

indeed

problems

an example

approximation

•

in t h e c a s e of u n b o u n d e d

the approximations

examples

T h e n "the s t a t e m e n t s

r 0 = r I = s(!-7).

the requirements

(see S e c t i o n

satisfied

if B ~ ~ ( U ; Y ) .

Let B h ~ ~ h B.

(A.6)-(A.9)

damped

the

we have

such approximations

scheme"

trivially

o n the

B is b o u n d e d ,

Similarly,

the o p e r a t o r

in the c a s e of p a r a b o l i c

algorithm case,

holds

of T h e o r e m s

that

I0 i n c o n c r e t e

controls

(A.2).

true with

hypotheses

provided

r I depend

to t a k e U r o = U a n d

of c o n v e r g e n c e

it is n e c e s s a r y

with

are automatically

(A.I)-(A.6).

rates

of r 0 a n d

then assumptions

In v i e w of t h e r e s u l t s obtain

the u s u a l

of c o n v e r g e n c e

If the o p e r a t o r

case where

9.4 a n d 9.5 h o l d

9.5 a n d

[L-T°19].

t a k e B h = ~ h B o r B h = B, a n d

the h y p o t h e s i s

B h = ~ h B o r B h = B,

9.7.

rates

B h.

~t is e n o u g h

general

[L-T.2],

the v a l u e s

(A.Y)-(A.9)

r O = r I = s(1-~).

from Theorem or

the optimal

a n d r I = s.

In t h e m o r e one

[F.I]

then one can

assumptions

r 0 = r I = s.

= Y,

easily

in

of the a p p r o x i m a t i o n s

additional

Yrl

as

O n the o t h e r hand,

B • ~(U;Y),

r 0 = s)

follows

argument

Theorem r 0 = r I = s.

9.6

_

with

with

be constructed. Dirichlet

boundary

of a n a p p r o x i m a t i n g

(A.6)-(A.9).

of t h e

W e shall

elliptic

Similar equations

In the N e u m a n n operator

examples with

can be

either

boundary

121 The a p p r o x i m a t i o n

Theorems

9.2 and 9.3 In the general

is proved in [L-T.19].

Here the techniques

ideas of the continuous

problem

approximating properties properties

[Las.2].

and it follows

[D-I],

together with

[L.I] and convergence The proof of Theorem

from the arguments

given in

Theorems 9.4 and 9.6 are proved ~n [Las.6].

9.2.

Approximation

for ~he

9.2.1

ADDroximatinu

assumDtlons

ApDroximat~o~

of A.

(B.1)

(H,2l-class

We assume as h ~ O: A;l~h-~hA-1 ^*-1

~ O

^*-1

Ah

gh-gh A

[ eAh t [ Z ( Y )

(B.2)

ADDroxlmatlon

of B.

T

(B.3)

IF.I],

of analytic semigroups

for the open loop problem,

9.1 on the DRE is simpler, [L-T.19].

[L-T.8],

case ~ < 1

rely on a c o m b i n a t i o n of

in Y;

strongly in Y.

~ 0

~ C

e~t

(9.50)

t > O.

(9.51)

We assume as h ~ O:

s

* Aht 2 JlBh e ~hXludt

2 < CTIXJy

(discrete analog of

(H,2)).

(9.52)

0 With reference (B.4)

to L and L h defined by (9.10) ~ 0

I(Lh-L)Ulc([O,T};y)

and

(9.11), we assume

for u E L2(O,T;U);

(9.53a)

e

l(Lh~h-L

Sufficient conditions assumptions

[Las.3]:

)f152(O,T;U)

~ 0

for assumptions (B.I)-(B.3)

for f ~ LI(O,T;Y).

(B.4)

(9.53b)

to hold are the following

together with

(s.4 S) (i)

^-1 IA ( B h - B ) U l y

(ii)

^-I ^-1 ,I(A h - A " ) B h U ]Y, -~ O;

~ 0;

u ~ U; u ~ U;

(9.54) (9.55)

122

(iii)

[(Bh~h-B

(iv)

IBh(Ah

.

9.2.2.

)A

X l U ~ 0;

^*--I

^*-i

gh-gh A

A o D r o x i m a t l o n of d v n a m i c e Ricoati Euuation

Control

(9.56)

x ~ Y;

}Xlo ~ O;

X ~

a n d of c o n t r o l

problem.

G i v e n the approximating d y n a m l c s

problem.

Yh(t)

(9.57)

Y.

Yh(t)

Related

e V h s u c h that

(9.58)

= A h Y h ( t ) + B h u ( t ), Yh(0) = ~hY(0)

minimize T

j(U,Yh(U) ) = ~[IRYh(t) Iz+lU(t)2 l~]a t

(9.59)

0 w i t h T < ~ for t h e D R E a n d T = ~ for the ARE. Riccatl

Equation.

The a p p r o x i m a t i n g

( D R E ) h = (9.3) a n d

9.2.3.

Riccati

Theorem

(DRE)

[Las.5]

for the c o n t i n u o u s

(B.I)

= (9.50)

through

I. A s s u m e h y p o t h e s i s

problem

(B.4)

= (9.53).

Then,

l y ~ ( - , ~ h X ) - y O ( . , X ) I c ( [ 0 , T];y) {j (uh, 0 Yh)-j(u0,y0)|I 0

(iii)

In a d d i t i o n ,

assume

T h e n as h~0:

= (1.6) a n d hypotheses

as h~0:

~ 0,

x ~ Y; x ~ Y;

~ 0.

(9.60) (9.51) (9.62)

that as hi0,

T * I[B~[eAht~h-eA't]R~Rg(t)dtlu0

(H.2)

a n d the a p p r o x i m a t i o n

I P h ( . ) ~ h X - P ( . ) X l c ( [ 0 , T ] ; y ) ~ 0,

II.

in

Euuations

(5.0)

(ii)

are g i v e n

results

Differential

(1)

Equations

(ARE) h = (9.4).

Aooroxlmatlnu

9.8.

Riccatl

~ 0,

g E C([O,T];Y).

(9.63)

123

x

I B * [ P h ( - ) ~ h - P ( - ) ] X l C ( [ 0 , T ] ; U ) ~ O,

Remark 9.7.

Note

that in Part I of Theorem 9.8,

e

Y.

•

in o r d e r

(9.64)

to obtain

convergence to the optimal s o l u t i o n s and to the Rlccatl o p e r a t o r in (9.60)-(9.62), R e ~(Y,Z)

no s m o o t h i n g a s s u m p t i o n on R is imposed;

as in (5.0).

However,

in Part II,

c o n v e r g e n c e of the gain o p e r a t o r s B Ph(t),

in order

in g e n e r a l

fails with R E ~(Y;Z)

C o r o l l a r y 5.4.

t h a t R R has

It can be e a s i l y

is a g r o u p - - t h a t c o n c l u s i o n only.

(9.64)

See comments b e l o w

M

T h e o r e m 9.8 is proved in [Las.5]: assumptions

to o b t a i n

it Is e s s e n t i a l

a r e g u l a r i z i n g effect as p o s t u l a t e d by (9.63), shown--for example w h e n exp(At)

here s i m p l y

(B.I)-(B.2)

Imply h y p o t h e s i s

It is e n o u g h to n o t i c e that (3.7) of T h e o r e m 3.1 in

[Las.5]. ~ l g e b r a l c Riccatl Eauatlon.

In the more d e l i c a t e a p p r o x i m a t i o n case of

ARE, we need a d i s c r e t e c o u n t e r p a r t of the Finite Cost C o n d i t i o n which w o u l d then g u a r a n t e e ARE h = (9.4).

Also,

II, on the DRE, In this case,

(1.9),

s o l v a b i l i t y of the finite d i m e n s i o n a l

in contrast w i t h the results of T h e o r e m 9.8, P a r t

no s m o o t h i n g a s s u m p t i o n on the o b s e r v a t i o n R is needed.

one obtains c o n v e r g e n c e of the g a i n operators,

as u n b o u n d e d o p e r a t o r s

(9.75) below.

This,

again,

but o n l y

is in llne w i t h the

continuous theory. Theorem 9.9.

[Las.3].

Assume

I. the c o n t i n u o u s h y p o t h e s e s (D.C.)

=

(H.2),

(F.C.C.) = (1.9) and

(5.17)-(~.19).

II. The a p p r o x i m a t i o n p r o p e r t i e s

(B.I)-(B.4)

and,

In addition:

(F.C.C.) h (uniform Finite Cost Condition): B a > 0; V Y0 ~ Y" ~ u h G L 2 ( 0 , ~ ; U ) such that J(U, Yh(U)) ( ~ly01 ~. (D.C.)h (uniform D e t e c t a b i l i t y Condltion):

(9.65)

There exist Kh: Z ~ V h such

that

IX~Xhl z S C[l~Xhlu+lXhly],

(9,66~

124 and AKht

Ie where

AKh = Ah-KhR. I.

-wit

I~(y)

~ c e

(9.67)

,

Then:

(convergence

of Hiccati

operators)

IPh, hX-PXly ~ O, Ap,ht

Ie

(9.68)

x e y, -~0 t

Xnl Y ~ 0 e

(9.69)

IXhly,

8

where Ap, h = Ah-BhBhP h.

(convergence of optimal solutions)

II.

0

0

0

0

0

0

~ O;

(9.70)

Iyh- Y IL2(O,®;y) -. O;

(9.71)

lUh-U I L 2 ( O , . ; U )

lyh- Y Ic(O,o.;y)

(9.72)

-* O.

Hence ht

le AP, le Ill.

(iS)

~ O;

(9.73)

Ap h t Apt " ~hx-e Xlc(o,~;y ) ~ O.

(9.74)

(convergence

(1)

Ap~

~hx-e

of "gain"

IBhPh, hXlu For each x E ~(AF)

XJL2(O,~;y)

operators)

]B PXlu,

x e ~(A).

there exists a sequence

(9.75)

x h E Yh such that

x h ~ x in Y and m

IBhPhXh-B Pxl The proof of the above theorem provides operators the model.

theorem

We notice

O.

m

(9.78)

is given in [Las.3].

us with the convergence

and the gain feedbacks

~

theory

with minimal

that the convergence

This

for the Riccati

assumptions

imposed

of the gain operator

on

holds

on a dense set in Y, and not on the whole space Y.

This

with the continuous

B P is only densely

defined.

theory,

where

the gain operator

Is consistent

125

Remark 9.8. analytic

Note

that

semigroup)

nume r i c a l l y continuous

in both

the above

the p r o p e r t i e s case.

and u n b o u n d e d

operators

B, we o b t a i n s t r o n g

on the full space Y, w h i l e operators

regularity

9.2.4.

(i)

"optimally"

Note

(1.5),

analytlcity arbitrary

9.1--we

(9.67).

these p r o p e r t i e s (A.1)

semigroup,

=

these p r o p e r t i e s

in general

They m a y fail,

[L-M],

fact:

[M.1],

(ii)

All

•

need

rather

(delay) These

"uniform

to be e s t a b l i s h e d

sensitive

Negative

equations,

condltlons

assumptions

problem.

requirements of c o n s i s t e n c y

Indeed,

are

to the

A s h o u l d be scheme.

(B.1)-(B.4)

(B.I)

if an examples

are r e l a t e d

operator

They are c o n s i s t e n t

scheme. and scheme

wlth spline

by the c h o s e n a p p ~ o x l m a t l n g

ones.

Conditlon"

in the case of an

of the a p p r o x i m a t i o n

of the o r i g i n a l

the r e m a i n i n g

minimal

imposed on the c o n t i n u o u s usual

problem).

to a s s u m p t i o n

from the

is selected.

[P].

The s p e c t r u m

approximated

in fact,

the

the a n a l y t i c

in the case of B bounded,

scheme

in the case of r e t a r d e d

"faithfully"

and,

even

approximating

approximations following

are

with

case subject

Instead,

conditions

even

hold

the " u n i f o r m F i n i t e Cost

can be d e d u c e d (9.13).

Indeed,

known,

contrast

In the a n a l y t i c

choice

inappropriate

with

" un i f o r m D e t e c t a b i l i t y

for a s p e c i f i c

dependent.

as

the

our a p p r o x i m a t i o n

of the c o n t i n u o u s

to a s s u m e

and the

independently these

case,

and gain operators

is in a g r e e m e n t (i.e.,

9.9--In

need

(9.65)

condition"

s.c.

converge

In the a n a l y t i c

the p r o p e r t i e s

in T h e o r e m

(F.C.C.) h =

(D.C.) h = (9.66),

again, theory

CO semlgroups

on the a s s u m p t i o n s that

of S e c t i o n

Condition"

[H.I) =

This,

in the

of the Riccatl

convergence

operators

and

as they r e c o n s t r u c t

the g a i n o p e r a t o r s

set.

of b o t h Riccati

s p a c e Y.

Discussion

situation

on some d e n s e

of the c o n t i n u o u s

reconstructs

are optimal,

C O semlgroup

that ~n the case of general

operators

on the e n t i r e

abstract

the s o l u t i o n w h i c h are present

unbounded

uniform c o n v e r g e n c e

(general

results

of

This means

cases

are v e r y n a t u r a l

with

the h y p o t h e s e s

and

(B.2)

of the a p p r o x i m a t i o n

are the

of the o r i g i n a l

semigroup and its adjolnt. Hypothesis assumptions

(B.3)

grouped

that A - I B be b o u n d e d

Bh = ~hB).

is a d i s c r e t e

in (B.4s) (and,

counterpart

a r e in llne w l t h

in fact,

of

(H.2),

while

the c o n t i n u o u s

they are s a t i s f i e d

the

property

if one takes

126

9.2.5.

Literature Most of the l i t e r a t u r e dealing w i t h a p p r o x i m a t i o n schemes for

Riccatl E q u a t i o n s for a r b i t r a r y C 0 - s e m i g r o u p treats the case of the input o p e r a t o r

B bounded,

see e.g.,

[G],

[I.l],

[KS].

In the

B - u n b o u n d e d case and w i t h a r b i t r a r y C o - s e m i g r o u p s , we are aware, a d d i t i o n to [Las.3],

[Las.5] of only one paper

[I-T] w h e r e the

a p p r o x i m a t i o n s of ARE are d i s c u s s e d subject to the c o n d i t i o n the a d d i t i o n a l r e q u i r e m e n t [P-S] of Part I.

Since,

in

(H.2) and

that the o b s e r v a t i o n R is s m o o t h i n g like in

as a l r e a d y discussed,

the f r a m e w o r k of [P-S]

is not a p p l i c a b l e to all the e x a m p l e s of S e c t i o n s 9.1,

9.2,

9.3,

9.4

the treatment of If-T] cannot be a p p l i e d either.

10.

E x a m p l e s of, n u m e r $ c a l a p p r o x i m a t i o n ~q~u~be classes and {H,2)

(H.1)

Except for the case of f i r s t - o r d e r h y p e r b o l i c systems,

in this

s e c t i o n we shall c o n c e n t r a t e o n l y on the more d e m a n d i n g a p p r o x i m a t i o n case for the ARE, w h e r e m o r e c o n d i t i o n s n e e d to be satisfied. i l l u s t r a t e the a p p l i c a b i l i t y (class

We shall

of the a p p r o x i m a t i o n T h e o r e m 9.3-9.6

(H.I)) and of the a p p r o x i m a t i o n T h e o r e m 9 . 9

(class

few e x a m p l e s taken from the c o n t i n u o u s S e c t i o n s 6, Y.

(H.2))

in a

For a full

treatment of the case of the heat e q u a t i o n w i t h D l r l c h l e t b o u n d a r y control,

10.1.

we refer to [L-T.I].

Class

(H.1}:

We r e t u r n

Heat e q u a t i o n w i t h D i r i q h l ~ t b o u n d a r v control

to the c o n t i n u o u s p r o b l e m of s e c t i o n 6.1,

a p p l y the a p p r o x i m a t i n g theory

C h o l c e of V h.

[L-T.1],

[L-T.19],

to w h i c h we

[Las.6].

We shall select as the a p p r o x i m a t i n g space V h c H~(Q)

be a space of s p l l n e s

(linear,

quadratic,

to

etc.) w h i c h comply w i t h the

usual a p p r o x i m a t i o n properties:

II~hY-Yll H e (n)

C hS-~IIyll

s ~ 2;

s-~ ~ O;

0 ~ ~ ~ I;

(10.1)

H s (n)"

inverse a p p r o x i m a t i o n p r o p e r t i e s HyhlIH~(~ ) ~ C h-UllyhlIL2(n),

[B]: 0 ~ a ~ I,

(10.2i)

126

9.2.5.

Literature Most of the l i t e r a t u r e dealing w i t h a p p r o x i m a t i o n schemes for

Riccatl E q u a t i o n s for a r b i t r a r y C 0 - s e m i g r o u p treats the case of the input o p e r a t o r

B bounded,

see e.g.,

[G],

[I.l],

[KS].

In the

B - u n b o u n d e d case and w i t h a r b i t r a r y C o - s e m i g r o u p s , we are aware, a d d i t i o n to [Las.3],

[Las.5] of only one paper

[I-T] w h e r e the

a p p r o x i m a t i o n s of ARE are d i s c u s s e d subject to the c o n d i t i o n the a d d i t i o n a l r e q u i r e m e n t [P-S] of Part I.

Since,

in

(H.2) and

that the o b s e r v a t i o n R is s m o o t h i n g like in

as a l r e a d y discussed,

the f r a m e w o r k of [P-S]

is not a p p l i c a b l e to all the e x a m p l e s of S e c t i o n s 9.1,

9.2,

9.3,

9.4

the treatment of If-T] cannot be a p p l i e d either.

10.

E x a m p l e s of, n u m e r $ c a l a p p r o x i m a t i o n ~q~u~be classes and {H,2)

(H.1)

Except for the case of f i r s t - o r d e r h y p e r b o l i c systems,

in this

s e c t i o n we shall c o n c e n t r a t e o n l y on the more d e m a n d i n g a p p r o x i m a t i o n case for the ARE, w h e r e m o r e c o n d i t i o n s n e e d to be satisfied. i l l u s t r a t e the a p p l i c a b i l i t y (class

We shall

of the a p p r o x i m a t i o n T h e o r e m 9.3-9.6

(H.I)) and of the a p p r o x i m a t i o n T h e o r e m 9 . 9

(class

few e x a m p l e s taken from the c o n t i n u o u s S e c t i o n s 6, Y.

(H.2))

in a

For a full

treatment of the case of the heat e q u a t i o n w i t h D l r l c h l e t b o u n d a r y control,

10.1.

we refer to [L-T.I].

Class

(H.1}:

We r e t u r n

Heat e q u a t i o n w i t h D i r i q h l ~ t b o u n d a r v control

to the c o n t i n u o u s p r o b l e m of s e c t i o n 6.1,

a p p l y the a p p r o x i m a t i n g theory

C h o l c e of V h.

[L-T.1],

[L-T.19],

to w h i c h we

[Las.6].

We shall select as the a p p r o x i m a t i n g space V h c H~(Q)

be a space of s p l l n e s

(linear,

quadratic,

to

etc.) w h i c h comply w i t h the

usual a p p r o x i m a t i o n properties:

II~hY-Yll H e (n)

C hS-~IIyll

s ~ 2;

s-~ ~ O;

0 ~ ~ ~ I;

(10.1)

H s (n)"

inverse a p p r o x i m a t i o n p r o p e r t i e s HyhlIH~(~ ) ~ C h-UllyhlIL2(n),

[B]: 0 ~ a ~ I,

(10.2i)

127 l[~u (Y-~hY)II

~ C hS-~,,y[,HS(~ ),

~ < s ~ 2,

(10.2ii)

L2(F)

L2(F) where ~h is the orthogonal projection of L2(Q) onto V h.

Choice of A h.

We define Ah: V h ~ V h as usual, where the inner products

are in L2: (AhXh, Yh) n = (AXh, Yh) ~ -fVXh.VYhd~+c2(xh, Yh) Q ,

Choice of B h.

xh, Y h e V h.

(i0.3)

With reference to (6.5), we define Bh: U ~ V h by (10.4a)

B h = -QhAD1 • D I as in (6.6), (8.7), and we notice that (L2-inner products) • 8Yh. (BhU,Yh) ~ = -(ADlU, Yh) Q = -(u, DiAYh) F = (u, u~-~--}F.

(10.4b)

Hence 8Y h

•

(zo.5)

BhY h = ~y~--.

~pproxlmatlna control problem.

Th~s is given by the O.D.E. problem:

i(~h,~h)n+!~yh.t~hd n - c 2 ItyhI~Q2 = ( u , ~ u #h)F"

~h ~ Vh;

(10.8) [(Yh(0),#h) n = (yCOl,~hl n The optimal feedback control for the approximating flnlte-dlmensional problem is 8

u~(t;O,y O) = - ~5- PhY~(t;O, Y0), where Ph satisfies the following discrete Algebraic Riccati Equation

128

-~VPhXh'~TYhd~-fVXh'VPhYhdQ+(Xh'Yh)~ n n

Verification (9.32).

of ~ s s u m p t l o n s

These are plainly

= ~+z

in o u r case.

bounded}, thus B (A)

Assumntlon satisfied

Because

(A.1} = (9.13) see

U = L2(F)

.

of the c o m p a c t n e s s

in t u r n that A - I B

of A -1

U ~ Y, and

[B-S]

for the s e l f - a d j o i n t

(A.2)

(A.3)

That

this

approximations case and

[Las.l]

is

of e l l i p t i c for the

case.

The s t a n d a r d

elliptic

approximation

< C h2

h o l d s w i t h s = 2.

= (9.15).

By

(10.5)

a nd

(10.21il),

(A.3)

is s a t i s f i e d

(conservatively)

we o b t a i n w i t h

(A.4) = (9.16).

By

(10.5)

implies =

bssumDtlon coincides

(A.4)

2(l-~-e)

and

<

(A.5) = (9.17). with

(i0.2ii)

(A.5).

applied

w i t h s = 2,

(io.9)

~ C h~llxllH2(~)

in v i e w of the fact that O(A)

= ½-2a

(lo.8)

for s = 2, ~ = ~+~.

IIB'(~hX-X)IIL2(F ) = ll~u (~hX-X)IIL2(F)

s(l-~)

is

a n d Y = L2(~),

Assumption

which

with

(since ~ is

is c o m p a c t

IIB*Yhll u = IIB~Yhll u = lieu yhlIL2(F ) ~ C h-~llYhllL2(n ) . Thus

(I0.7)

( 9 . 3 1 ) and

AssumDtlons

(uniform analytlcity).

(A.2) = (9.~4).

so that

Assumption

V Xh, Yh e V°n

9.2.

II~hA-I-A;I~hlIE(L2(n))

[B-A],

•

since R = I and A-~B ~ ~(U,Y)

from r e s u l t s o n G a l e r k i n

non-self-adjoint

Assumption

PhXh • ~u PhYh)Y

is c o m p a c t Y ~ U, as d e s i r e d .

follows

operators, general

of T h e o r e m satisfied

this t h e n i m p l l e s

= (~

c H2(~)

and

½.

Since

in our c a s e Bh~ *h =

B*~h

, (A.4)

129

AssumDtlon {A.6| -- (9.18|.

From (lO.2iii) applied wlth s -- ~+z and

from the trace theorem, we obtain {{B']]hX{'L2(r) = {{~u ]]hX}{L2(r) -< "~u([[h-r)x}}L2(F)+{{~u X{'L2(r ,

c he {{x{}H~+a

+C{{X{IH~+~

(n) (A.6) follows now from ~(A *~+~)

c

.

(lO.10)

Co)

H~+2~(~).

THUS, we have verified all the assump~lons of Theorems 9.2 and 9.3 in the c a s e of the heat equation problem with Dirlchlet boundary control as in (10.1). Then, appllcatlon of Theorem 9.2 yields the following convergence results: ~0

(i)

nPh~h-P~{~(L2(n)) ~ C h

(1i)

{{~ [BhHh-P]{{Z(L2(O);L2(F)) -. 0

(iii)

II o Oi{ Yh -y Z(L2(~);L2(O,~;L2(~))

+ sup

t~o

Moreover,

,

E 0 < ~;

(I0.II)

as h~,O;

Pttz.. 0.t,_y0 ZO llYh{ ; (t)lIz(L2(~)) ~ C h ,

(10.12)

~0

½"

<

(10.13)

if we use the feedback law g i v e n by

^

a

uh{t) = - ~-~ Ph y(t), and we insert it Into the original dynamics Yt = (&+C2)Y

then the corresponding system is exponentially stable in L2(~) uniformly in the parameter h. Remark 10.1.

The rate of convergence G(h ~-e) guaranteed by

(10o11)-(I0.13) is not optimal, In view of the regularity P e Z(L2(~};H2-~(~)) of the Rlccati operator (see (5.2)), one Would expect that the optimal rate of convergence should be of the order of ~(h2(1-£)).

Indeed, we shall show that this is possible,

but for

130

different,

a p p r o p r i a t e a p p r o x i m a t i o n s of A h and B h.

order to obtain the optimal rates of c o n v e r g e n c e

M o r e precisely,

(G(h2(1-£))),

in

care

must be e x e r c i s e d in s e l e c t i n g the a p p r o x i m a t i o n of the P o i s s o n o p e r a t o r A -1. variational

Since the D l r i c h l e t p r o b l e m does not admit a natural

formulation,

extra a t t e n t i o n must be paid to the

a p p r o x i m a t i o n of the b o u n d a r y conditions.

Thus,

optimal rate

(~(h2(1-~))),

approximates

'well' the b o u n d a r y conditions.

in o r d e r to o b t a i n the

we n e e d to i n t r o d u c e an a p p r o x i m a t i o n w h i c h For this p u r p o s e we shall

use the e l l i p t i c a p p r o x i m a t i o n of the P o l s s o n o p e r a t o r due to N i t s c h e

IN. I~. W i t h V h d e f i n e d by (10.1), d e f i n e d as

(10.2) w i t h s > ~,

let Ah: V h ~ V h be

(see [N.1])

(AhXh'Yh) ~ a(xh'Yh)

~ a(xh'Yh)

- (~u Xh'Yh)F

_ (Xh, ~-~ 8 yh)F + ~ h-I (xh, Yh)F+c2(xh, Yh)~ in the L2-norms,

(10.14)

where ~ > 0 is s u f f i c i e n t l y large and c 2 as in (6.1a).

The a p p r o x i m a t i n g f i n i t e - d l m e n s l o n a l Riccati o p e r a t o r Ph: Vh ~ Vh s a t i s f i e s the following A p p r o x i m a t i n g A l g e b r a i c Riccatl Equation: (ARE h)

-(AhPhXh, Yh)-(PhAhXh, Yh)+(xh, Yh ) = ((~u - p h - l ) P h X h ' ( ~ u - p h - 1 ) P h Y h ) F "

(10.15)

We shall now v e r i f y the a s s u m p t i o n s of T h e o r e m s 9.4 and 9.5 on optimal

rates.

HvDotheses

(A.1) = (9.13) - (A.2) = f9,14)

(with s = 2) are well k n o w n

for the N i t s c h e ' s a p p r o x i m a t i o n A h d e f i n e d in (I0.14)

Hypotheses have

|A.31 = {9.15) - {A.5) = (9.17).

(see [B-S-T-W]).

In the D i r i c h l e ¢ case we

(see [ C h - L ] ) B~Xh = ~-Q 8 x h + p h ixhl F .

Thus,

hypothesis

property

(10.16)

(A.3) is the result of the inverse a p p r o x i m a t i o n

(IO.2iii).

xIF = O, we o b t a i n

"

8

Since B x = ~-Q x and for x E ~(A),

we have

131

I(s*-s~)Xlu

= I ~ ~ + I 'h-lXlr - ~ "h" - ~'h-l~h"l',~Ir~

×l~2(r)

~u(gh-I)xlL2(r)+ph-lll~h-I) (by the approximation

property

(10.2~i) and (10.

C h-~h21XlH2lQ)' as desired for (A.5) to hold.

(A.6)

=

[9.18}.

(i0.iy)

xi~(A)

As for (A.4), we have as desired

(~-~.)=Iu ~ C h~ IxlH2(n).

tB*(Z-~h)=IU = I ~

HvDothesls

C h2(1-~) I

1))

It involves only B

(10,18)

(not A h, Bh) and was

verified before in (10.10). HYPothesis

(A.7).

Let z h = AhlBhu and z = A-1Bu.

We have

a(zh, Xh)+~(Zh, Xh) = (u, ~u Xh)r + Bh-1(U'Xh)r

"

(lo.19)

and A(¢,8)z+~z = O, Since

(10.20}

z~r = u.

(10.19) defines an elliptic approximation of z, t h e

convergence

results of IN,I] apply to yield

}Z-ZhlL2(r SO (A.7)(ii)

) ( C h21ZiH2(n)

,

(lo.21)

holds with r 0 = 2 and Ur0 = H~(r) c L2(r ).

As for (A.Y)(J),

we shall prove that

"--1 s ~-1 I [ A h h-A B ] } Z ( L 2 ( r ) ; L 2 ( ~ ) so ( A . 7 ) ( i )

c h21ulH ~ ( r )

is satisfied

To assert

with

)

= ¢(h~),

(lO22)

p = ~ and Urz = L 2 ( r ) .

(I0.22), we use a duality argument,

.^*-1 l[S *^*-1 A -BhA h ]XIu

= I~

(~L~-~)~t%(r)

^-1-A^-1 )xlL2(?) + ph-ll(Ah

< C

h-~h21xlL2(n)

132

Hvuothesls r I -- 2.

fA,8).

Part

Here we have Url

U

L2(F) and Hrl

(i) of (A.8) is t r i v i a l l y satisfied.

H~(~) ~ ~(AY~);

As for part

(ii),

we c o m p u t e *^*-1

I[BhAh

*^*-1

*

-B A

*-1

]%'1¥ -< IBh[Ah

^*-1

h-~hrlA-lyl r

(by (A.3)-(A.5))

H if r. ln

(10.1)

^*-1

+

(~)

*^*-1

-B A

]%,lL2(r)

h-~hr IZ-z%,I H r ( ~ )

3~ c

HVDOthesls

*

]%'ln2(r)+l[BhIlh A

-[[h A

(A.9) .

c h21YtH~

hr-~lylHr_2tn)

(10.23)

(n)

We have

(10.24) From e l l i p t i c

theory [L-M],

~-IAD G ~(H~(r); Ha+~ (n)),

(lo.25)

= >_ 0,

(lO.26)

~-I~-IAD • ~(H~(F);H~+~(n)). Thus,

applying

(10.26) w i t h ~ = )~ and trace t h e o r y yields

B * ~ * - I ~ - I B • ;Z(H)~(r);HF'(r)), w h i c h proves part

(i) of

(A.9)

(we recall

(lo.27)

that U r0

For part

= H~(r)).

(ii) we use

B*^*-2*~ A = ~ a' E ~ - 2 + z " and s i n c e

.%-2+e e Z(L2(n); Trace T h e o r e m implies part Finally,

part

(ii) of

H4-2E

(O)).

(lo.28)

(A.9).

(ill) is s a t i s f i e d w i t h n = 2.

a p p l i e d w i t h ~ = 0 and trace theory,

Indeed,

B'~.*-I~.-1B E ;Z(L2(F); H I ( F ) ) . R e p e a t e d a p p l i c a t i o n of

(10.26)

(10.26),

g~ves

this time w i t h a = 1 g i v e s

(10.29)

133 B*~*-1~-1 B ~ Z ( H I ( F ) ;

H2(F)).

(lo.3o)

Combining (IO.29) with (i0.30) gives the desired result of (A.9)(lli). Thus, we have verified all assumptions of Theorems 9.4 and g,5. Application of these theorems to the heat equation (3.1) with Dirlchlet control yields the following results. Theorem iO.1, I.

Assume that ~ < s ~ 2 in (I0.I).

Then

The unique solution Ph: Vh ~ Vh to (AREh) = (10.15)

satisfies the follow~ng estimate:

]Ph-P[~{L2(Q)) II.

~ C h2(1-e).

There exists ~0 > 0 and C > 0 such that 0 t

lyh()IL2(Q)

-Uot ~

C

e

lYoln2(n )

,

where y~(t) satisfies in the L2-norms

(y~(O),x h) = (Y0,Xh). III.

C h2(l-a) [Yh(t)-yO(t) IL2(f]) <.

-~0 t

tl_a

e

lYoIL2(~ )

IV. lYh(t)_yO(t)iL2(Q)

•

[Yh(t)-yO(t)[L2(Q)

< C h 2(l-&)

-WO t

t~-- e

~ C h ~-a

e-~O t

ly0[B2(n);

lYo]L2(n ) ,

where Yh(t) is the solution to the orlginal heat equatlon (6.I) wlth 8

feedback boundary conditions u(t) = - ~

Remark 10.2.

The rates of convergence

PhYhlt)

in (6.1c).

•

provided by Theorem i0.I are

optimal in the sense that they reconstruct the optimal regularity of the original solutions, The presence of factor ~ 1 in Part III is

134

consistent with semigroups. R @ m a r k 10.3. equation

'rough data estimates'

a v a i l a b l e for p a r a b o l i c

• To e m p h a s i z e a contrast,

(6.1a-b) w i t h N e u m a n control

Y = L2(~ ) (rather

than cost

we shall n o w c o n s i d e r the heat (6.12) and cost

a p p r o x i m a t i o n treatment is g i v e n in [Las.6]. can take the usual G a l e r k i n a p p r o x i m a t i o n s B h = ~h B.

The detailed

Now since ~ < ~, one

for A h and s i n g l y define

This choices produce optimal rate G(h 2(1-z) ) of convergence

w i t h any spline a p p r o x i m a t i o n of order s ~ 2.

10.2.

(6.22) where then

(6.13)) w h e r e then Y = HI(~).

|

Class (H.1): Th~ s~rKcturallv d a m p e d plate p r o b l e m in E x a m p l e 3.1 We return to Example 6.1, model

(6.25)

in S e c t i o n 6, w i t h

n = dim n ~ 3. ~ b o l c e of V h. H~(~)

We shall select the a p p r o x i m a t i n g space V h c H2(~) n

to be a s p a c e of spllnes

(e.g.,

cubic splines),

w h i c h comply with

the usual a p p r o x i m a t i o n p r o p e r t i e s [QhZ-Z[Htln)

~ C hS-t[z[

HS(fl)"

z e Hs(fl)

n H~(n),

0 <_ ~ _< 2; ~ _< s < r;

[ZhlHa(n)

< C

h-SlZhIHa_S(n),

0 _< a <

w h e r e Qh is the o r t h o g o n a l p r o j e c t i o n of L2(Q)

2,

(zo.3i) (lO.32)

onto V h and w h e r e r is

the o r d e r of approximation.

C h o i c e of A h-

We let A h = Q h A Q h : V h ~ Vh:

i.e., (Xh~h,~h)O = (a~h,&Th)O = ( A ~ h , ~ ? h ) O

IA~hlL2(n) = l~hln2(nl; where

(10.34)

is a c o n s e q u e n c e of

,

~h,~ h : Vh,

(10.33)

la~h] ~ ]~hlH2(n )" ~h " Vh' (10.34) (10.33).

From elliptic estimates

135

[(A-1-AhIQh)Z[H2(fl )

h2[z152{fl);

c

(Io.3s)

h21zlH2(n), z e re(A½).

c

Choice of Ah and B h.

To begin with, we let Yh s VhlXVh2"

where Vhl

consists o~ the elements o~ v h e q u i p p e d w i t h ~orm I~hlv~ = l ~ h l % ~ n ~ and Vh2 conslsts of the elements of V h equipped with the L2(~)-norm. Next, we define

We shall write x h = [Xhl,Xh2 ] • Yh"

(10.36)

Ah: Yh ~ Yh: Ah = -A h

Bh: L2IF) ~ Yh: BhU =

-A

I°1

(~hU, Vh}~ = Vh(xO)u.

(i0.37)

~h u

F~nally,

we let ~h: y ~ Yh be defined as

Qh s

Computation Q ~ d ~ o l n t $

A h and B h.

To compute the adJolnts of A h and

B h, we use the inner products generated by the topology on Vhl and Vh2. We find, as in the continuous case, A h --

IA°h

-A

; BhX h

= xh2

(x 0 )

as it follows from (AhXh' Yh)Yh = (Xh'AhYh)Yh and {BhU, Xh)Yh and (U, BhXh)u,

respectlvely.

Approxlmatlnq approximating

control problem. With the above notation, dynamics (9.22) is now

the

136 '(yh,@h)+(AhYh,@h)+(A~yh,@h) (AhYh,~h)

= (Ayn, A~h),

(Yh(O),~h) where all inner products The optimal is given by u~(t)

= @h2(xO)u;

all ~h ~ Vh;

= (y0,~h);

(10.38)

{Yh(0),#h ) = (yl,~h),

are in L2(~ ) .

feedback

control

~ -[Ph2~h(t)][xO],

for the flnlte-dimensional

problem

where

"PhlYhl+Ph2Yh2

m PhlYh;

Ph3Yhl+Ph4Yh2

~ Ph2Yh ,

PhYh ~

and Ph satisfies

(I0.39)

the following

algebraic

equation

with L2(~)-inner

products

-(AhXh2,PhlY h) + (AhXhl-A~Xh2,Ph2y h)

(AhPhlXh, Yh2)

-

+ (Ph2Xh,AhYhl-A~Yh2) + (AhXhl,Yhl) + (Xh2,Yh2) = (Ph2Xh)(xO)(Ph2Yh)(XO).

Ver~flcatlon Theorems

of assumptlons

(A.I) = (9.13) (9.32).

(A.6) = (9.18),

follows

E L2(Q),

{A.1} = {9.13}.

of the continuous

from

In order

to apply

the approximating

assumptions

as well as assumptions

the last two are plainly

(9.32)

A-(~-&)6

Assumption [C-T.2]

through

Indeed,

R = I, while essence,

9.2.

of Theorems

9.2 and 9.3, we need to verify

(10.40)

(6.30)

wh~le A -~ ~s

This follows

satisfied:

(9.31)

and the argument compact

(9.31),

since

below ~t (in

on L2(n).

by applying

t h e arguments of

case to the finite-dlmensional

operator

given

by (10.36}. Assumption holds

(A.2} = (9.14).

By (10.35),

we have that

(A.2) with s = 2

true:

I CA;1-A-

)XhIy °

) C h2[IXhllH2(Q)+[Xh2[L2(Q)

] = C h2[IxhIy].

137 The same result holds for the adjoint A , in vlew of its definition. AssumDt~o~

(A.3J = (9.15).

approximation

property

By Sobo]ev embedding and the inverse

(10.32), we have for any • > O,

• = JB Xh[ U lXh2(xO)l

~ C h -n/2-~

~ ClXh2[Hn/Z+~(n) C

IXh21L2(n)

h-n/2-¢[Xh[ Y ,

n n and (A.3) follows since ~s = (~ +~)2 > ~.

Assumptlon

(A.4) = (9.16}.

Is*(ghX-X) lu

By (10.31} we compute

= [(QhX2)(x°)-x2(x°)lR1

~ CIQhX2-X21Hn/2+~(n )

C h 2 - n / 2 -~tIx211H2(Q ) ~ C h 2 - n / 2 Since 2(1-I)

-~llxll~(A,) "

= 2(I- ~n -~) < 2- ~n -~, and (A.4) is satlsfied.

A~qmptlon

(A.5) = ~9.17).

It coincides with

Assumptlon

(A.6} = (9.18).

we compute

(A.4).

llB"T[hX[lU = llXh2(x0}llL2(n) <- CllXh21]Hn/2+z (~) <. Cl]Xhil (A, ~ ) as in [C-T.4], ~(A~T) c H 4~ (O)xH2~ (~) and 21 = 2( ~ +~) = ~n +2~ > ~n +s.

THUS, we have verified all the assumptions 9.3.

Thus,

convergence

(ii)

or

Theorem 9.2 applies to our problem and ylelds the following results: i]Ph~h-Pll ~ (H2 (~)xL2 (~)) _< C h

{i)

of Theorem 9.2 and

II

%0

,

l[(Ph~h-P) Ix=x 0.Z(H2(nlxL2(n } ;RI)

equivalently

4-n ~o < -'~--;

--.o

as h~O;

138

IB "Ph~h-B where Ph is computed

(ili)

sup e t_>O

~pt

P "II ,'~(H2 (~)×L2 (n) ;R)

from

-~ 0

(10.40).

~[u~(t}-uO(t)~l

_< C h

~0

~ (H2 (n)xL2(n) ;R)

e

~0

_< C h

Application result:

as h~O,

4-n , eO < ---~--.

of Theorem 9.3 to our problem yields

Let uh(t)

the following

be a feedback law given by

u~(t) = -[Ph2Y(t)][x O] which we insert

into the original dynamics wtt+~2w_p~wt

Then the c o r r e s p o n d i n g exponentially approximates numerical

stable

* W}r = ~Wlr = 0. = 8(x-x 0 )Uh;

in the topology of H2(~)xL2(~) feedback dynamics.

(in h) and u n i f o r m l y

This means

that the

a l g o r i t h m provides a feedback control w h i c h yields uniform results

We conclude

for the original

system.

this section by pointing

out that the other examples

6 dealing with s t r u c t u r a l l y damped plate problems

dealt wlth by a simllar a p p r o x i m a t i n g 10.3.

to obtain

feedback system is u n i f o r m l y

the original

(in h} s t a b i l i t y of Section

(6.25)

Class

(H.2):

we verify the a p p l i c a b i l i t y

treated in Section 7.1,

Let V h ~ H~(~)

Let Qh be the L2-orthogonal that V h enjoys

can be

[L-T.19].

of the

Theorem 9.g to the wave equation with L 2 ( Z ) - D i r i c h l e t

b o u n d a r y control, Choice of V h.

see

The wave equation with Dirichlet b o u n d a r y control

In this subsection, approximating

scheme,

Part I, Eq.

be an a p p r o x i m a t i n g

p r o j e c t i o n of L2(~)

the following a p p r o x i m a t i n g

(7.1).

subspace

onto V h.

properties:

of L2(n ). We assume

139

(I0.41)

(iii) IZh IH (Q)

(iv)

I:~

C h-SlZh IH _s(o ).

(qhz-z)lL2(O)

! C

I,

o:

h~l~l H~(n) .

It is well known that the above approximation propertles are satisfied for, say, spline approximations

(of order r, r ~ i) defined

on a uniform mesh. Also, modal (eigenfunction), or spectral (polynomial) approximations are typical examples of schemes which satisfy the requirements in (10.41). Choice of A h.

We let

A h = QhAQh: V h ~ Vh: i.e.,

(Ah~h,$h) = I ~ h . V ~ h d ~ . n

From the definition of A h, it follows immediately that

]A~Zh]L2(n ) = IA~ZhlL2{n), and A

is an isomorphism H 01(~) ~ L2(Q ) N V h (with a norm uniform in h).

Since A~ is self-adjoint on L 2 ( n ) , ~somorphism

L2(D) - H-I(D) n V h.

l(~hlqh-~-l~zl

C hoic@ of A h and B h.

We have that A~ is also an Moreover,

elliptic estimates give

s ( ~ ~ C h2-Slz~n2(fl) "

Ho

0 f s ~ I.

We introduce the space Yh = VhlXVh2' where Vhl

(resp. Vh2) is the space V h equipped with norm L2(n)

IvhlVhl

(i0,¢2)

]Vh}L2(O);

]vhlVh2

(resp. Ah~-norm}:

(0}.

By (I0.42), Y. is topologically equivalent to Y. n A h and Bh: U ~ Yh are defined by

The operators

140

Ah =

I°11oi

I°

Ah

~h u

=

QhADU

.

i,o..)

well defined since (~hU,@h)L2(n) Finally,

= (ADU,#h)L2(n)

°#hI

(10,44)

= [u, v;r~-jn2(~).

we let ~h: Yh ~ Yh be as before, below (10.37): a diagonal

matrix with Qh on the main diagonal.

Computation of ad~olntS A h an_.__~dB h. the topologies of Vhl, Vh2.

These are computed with respect to

As in the continuous case, one readily

obtains * a A~I Xh2. A h* = -A h and BhX h =~-~

ApDroximatlnu control DrQblem. With the above notation, version of the state equation is equivalent to

(10.45)

the discrete

(10.46) t(Yh(O),@h)n = (VO,~h) n, (;lh(O),@h)~l = (yl,@h)fl, for all @h ~ Vh' in the L2-inner products of ~ and r, where

h Yh(t) =/;Iyt~ ~ =

(x) = Yhl; Yh2 (t) = Yh (t)'

The optimal feedback control for the finite-dimensional

problem

(9.22) is given by

u~(t)

= _ ~-ff8 A ~ l ~ h 2 l Y h ( t ) )

(10.47a)

(I0.47b)

and Ph satisfies the Riccatl equation

-(PhlXh, Yh2)~ + (Ph2Xh, Yhl)~ - (Xh2,EhlYh) n

141

+ (Xh1,Ph2Yh) n + (Xhl,Yhl) n + (A;lXh2,Yh2)~] ~-i~

for

all

Thus,

xh =

iXh2j;

Yh =

lYh2j =

y

r

•

Vhl×Vh2 = Yh'

if one can apply %he theory p r e s e n t e d

once i n s e r t e d into the d y n a m i c s

in T h e o r e m 9.9,

(10.46) y i e l d s a

f e e d b a c k s e m i g r o u p w h i c h is u n i f o r m l y e x p o n e n t i a l l y stable, c o n v e r g e n t to the

original, inflnite-dlmenslonal

and

through

(D.C.) h = (9.66)-(9.67),

(F.C.C.) in

(B.I) = (9.50)

Part

= (1.9), and

i.e.,

(F.C.C.)h = (9.65),

since the o t h e r a s s u m p t i o n s

(H.2) = (1.5),

have a l r e a d y b e e n v e r i f i e d

I, Section 7.I.

Assumptions

(9.1),

(B.I),

approximating properties (10.42),

(B.4) = (9.53),

(D.C.) = (5.17)-(5.10)

and it is

solution.

It remains to v e r i f y the a s s u m p t i o n s of T h e o r e m 9.9; conditions

then

Riccati operator Ph

the f e e d b a c k law (10.47) c o m p u t e d w i t h the aid of (from (IO.48)),

(10.48)

one can show

(B.2),

(B.4),

(10.41),

([Las.3])

(D.C.)h.

By u s i n g the

together w i t h the e l l i p t i c e s t i m a t e s

that h y p o t h e s e s

(B.I),

(B.2),

(B.4) are

indeed s a t i s f i e d for our d y n a m i c s of the w a v e e q u a t i o n w i t h Dirlchlet control

(7.1).

Indeed,

assumption

a u t o m a t i c a l l y satisfied, Assumption

(D.C.)h = (9.66)-(9.67)

is

since R = I in our case.

[9.1). IghX-Xly ~

IQhXl-XIIL2(Q)+IA-½[QhX2-X2]IL2(Q ),

and since the inverse a p p r o x i m a t i o n holds,

IQhX-Xl (Aa

)

~ O,

we have

x e ~(A~),

I~} ~ ~.

w h i c h implies the d e s i r e d conclusion. Aht Assumotlons

tB.1) = (9.50).

x h = ~lhX; x = (x1,x2).

(B.2) = (9.51~.

Then zh(t } satisfies:

Let Zh(t)

= e

Xh;

142 &h, 1(t) = -Zh, 2(t ) (~h, 1(t),@h)n+(VZh, 1(t),V@h)n = O, or equivalently, (Zh, l(t),#h)+(AhZhl(t),#h) n = O. Take #h ~ AhlZh, l(t)"

~ [d~ h

-~.

We have

Zn, l

(t)l

2 2 L2(D)+IZh, 1(t)lL2(~)] = O,

which gives •

2

+

2

2

IZh, 2(tllH-l(~ ) IZh, l(tlIL2(~)

= [~h, 1(t}IH-1(n )+IZh . 1(t)IL2(n) lqhX21H-l(~)

and proves (B.2) = (9.51).

÷I qhXl[L2(n)'

As for {B.I) = (9.50), we have

]A~inhx-A-1~Iy ~ JA~Inh~-~hA-Ix[y+l(~h-I)A-ixIy. Convergence to zero of the second form on the right hand side of the above expresslon follows from the convergence properties of ~h o It remains to estimate that iAh1~hX-~hA-IXly = l(khiQh-A-1)x21L2(n)

= l(AhlQh-A-1)QhX21L2(n)+IA-1(QhX2-x2)IL2(n)

C h21QhX21L2(~)+IQhX2-X21H_l(n) C hlQhX21Hl(n)+IQhX2-X21H_l(n)

~ 0,

where we have used (I0.42), the inverse approximation property, and convergence properties of Qh on H-I(~). Slnce A

= -A h, %he proof of (B.I) for A h ~s the same.

143

Assumption Part

(!)

(B.4) =

=

(9.53).

{9.54).

IA-I(Bh-B)Uly = IA-I(Qh-I)ADUlL2(~).

Denote T h : A-I(Qh-I)AD.

Then the adJolnt T h of Th: L2{F) ~ L2(n) is

given by ThZ = and

(Qh-I)A-Iz

IThZlL2(F) < C h%!~t-lzlH2(Ol

which proves that T h ~ O, as desired.

Part ( I i I =

(9.55).

I(A~I-A-I~BhuIy= 1(~-~-1~hUlL2in) Let T h m (Ahl-~-l)~h:

L2(F ) . L2(n }. ~

Then --

--

ThZ = ~- Qh(AhlQh -A 1)z and IThZlL2(F)

: C h-½-¢IQhlA;1Qh-A-llZlHl(n)

C h-½-ellAhlQh-A-1)zl

which,

via

duality,

The proofs (via

transposition)

that

~n p a r t s

(i)

proves

part

from the and

(ii),

Discussion on hypotheses

-

(9.56)

proofs we h a v e

(B.3)

,

[~),

part ( i f i l

of

1,~, ~ C h - ½ - Z h t Z l L 2 ( n )

=

of

(resp. part

proved

(9.52)

~nd

(t)

(iv)

= (9.57))

(resp.

uniform

(Notice

convergence.)

(F.C.C.} h =

delicate is the issue of validity of hypotheses

(tt)).

follow

(9.65).

More

(B.3) and (F.C.C.) h in

t h e c a s e of the wave equation problem with Dirichlet control (7.1) under study. Although both conditions are v e r y natural--or they are

144

d i s c r e t e c o u n t e r p a r t of p r o p e r t i e s w h i c h hold true in the c o n t i n u o u s c a s e - - t h e i r v a l i d i t y in a p p r o x i m a t i n g p r o b l e m

(7.1) may well d e p e n d on

the s p e c i f i c s e l e c t i o n of the numerical scheme adopted. pathology, say,

should n o t - - r e a l l y - - b e surprising.

or

in the case,

of d e l a y d i f f e r e n t i a l e q u a t i o n s w h i c h are u n i f o r m l y s t a b i l i z a b l e

to b e g i n with, Condition

(7.1),

it is k n o w n that the v a l i d i t y of the u n i f o r m F~n~te Cost

(F.C.C.)h depends on the p a r t i c u l a r n u m e r i c a l scheme selected

(see [L-M], conditions

form,

Indeed,

This fact,

[M.I],

[P]}.

(B.3) and

Before we analyze further the v a l i d l t y of

(F.C.C.) h in our present case of the wave equation

it w~ll be e x p e d i e n t to re-wrlte them in an explicit, as they a p p l y to our case.

Let Sh(t)

•

It can be s h o w n [Las.3]

equivalent

that

V h be a s e m i - d i s c r e t e s o l u t i o n of the

f o l l o w i n g ODE p r o b l e m (~h(t), #h)n +

~h(t).V~hdr

(10.49)

= O, V #h ~ Vh"

n Then:

(i)

Condition

(B.3) = (9.52)

is e q u i v a l e n t to the f o l l o w i n g

inequality:

I ~-j a *hlL2(Z~) < CT[l+h(°)lL2 ~Q) , +llV%h (0) I'~2(n)] -

(ii)

Condition

(F.C.C.) h = (9.65)

is satisfied,

(i0.50)

provided

(I0.5i)

R e m a r k 10.4.

N o t i c e that both c o n d i t i o n

(10.50),

(10.51) a r e

satisfied

for the c o n t i n u o u s s o l u t i o n s of the h o m o g e n e o u s w a v e equation. fact,

in this case,

they are e q u i v a l e n t

regularity assumption

to, respectively,

(H.2) = (1.5) and the exact c o n t r o l l a b i l i t y of

the w a v e e q u a t i o n w ~ t h b o u n d a r y control as in (7.1c), [Lio.1-2],

[H,1],

In

the

IT.3].

Moreover,

[L-T.1-3],

the a p p r o x i m a t i n g s u b s p a c e V h

c o n s i s t s of p o l y n o m i a l s d e f i n e d on Q, of d e g r e e N(h) ~ ~, as hiO, if T is s u c h a polynomial,

~ E Vh,

p o l y n o m i a l of d e g r e e N and hence

then p l a i n l y

then

(X-Xo)-VT is also a

(X-Xo)-VT G V h as w e l l .

Thus the same

145

m u l t i p l i e r t e c h n i q u e w h i c h yields the c o n t i n u o u s c o u n t e r p a r t of (10.50),

~

(10.51) w o u l d give

.

(10.50) and

(10.51) as well.

Thus we have

A s s u m e that the a p p r o x i m a t i n g s p a c e V h c o n s i s t s of

p o l y n o m i a l s ~ of degree N(h) ~ ~ as h~0 d e f i n e d on n. inequalitles

(10.50) and

(10.51) hold true.

of T h e o r e m g.g a p p l y in this case.

To q u e s t i o n w h e t h e r these e s t i m a t e s (see our e a r l l e r remark),

a full and general answer.

(10.50),

(10.51) are always

(10.41)

is far from being

and we are not in a p o s i t i o n to give

It is clearly a technical

issue w h i c h

depends very m u c h on the specific a l g o r i t h m employed. natural

(in fact,

necessary)

continuous counterpart

for our problem,

Below,

(satisfied,

The general case,

open question. [G-L-L],

their v a l i d i t y for a

an open p r o b l e m of n u m e r i c a l

we shall p r o v i d e an a f f i r m a t i v e answer,

additional hypothesis tions).

Although very

and a l t h o u g h they have a

for the original system,

s p e c i f i c n u m e r i c a l scheme is, in general, analysis.

the results

•

s a t i s f i e d for schemes w h i c h c o m p l y only w i t h obvious

Then

Consequently,

for instance,

but u n d e r an

by modal a p p r o x i m a -

say of spllne approximations,

is still an

(Here, we report some p r o m i s i n g numerical c o m p u t a t i o n s

[D-G-K-W] w h i c h confirm numerically the v a l i d i t y of these

c o n d i t i o n s In the case of finite d i f f e r e n c e s and m i x e d finite elements.)

Lemma 10.4.

[Las.3] Assume that the f o l l o w i n g c o m m u t a t i v i t y p r o p e r t y

holds true Qh A = AQ h. Then,

any a p p r o x l m a t l o n scheme w h i c h c o m p l i e s w i t h

satisfies

the i n e q u a l i t i e s

R e m a r k 10..5.

for instance,

10.4.

Class

(10.41) and

(I0.50) and (10.51) as well.

As noted above,

given,

(10.52) (10.42)

•

examples of such an a p p r o x i m a t i o n are

by modal approximations.

(H.2~: F i r s t , o r d e r h y p e r b o l i c Systems

In this section,

following [Las.5], we shall v e r i f y the

a p p l i c a b i l i t y of the a p p r o x i m a t i n g T h e o r e m 9.8 on the D i f f e r e n t i a l Riccatl Equation, S e c t i o n 7.4.

T < ~ for the f i r s t - o r d e r h y p e r b o l i c p r o b l e m of

146 Qholue of V h. Let ~h c Hl(n) let Qh be the c o r r e s p o n d i n g

be an approximating subspace L2(n)-orthogonal

that ~h enjoys the following a p p r o x l m a t l o n

i~IL2(r~ IQhZ-ZlK~(nl ~ C Typical

IzlHr(n) ,

0

~ r

Vh = [~h ]m;

C h o i c e of A h.

~

2;

in (7.63),

the above identity

in the notation of Section 7.4,

A .

The subspaces i.e.,

include spaces of Vh

(10.54)

is equivalent

Since

w Yh, Vh ~ V h , (7.64),

(7.58c).

~ Yh, Vh ~ V h

see

approximation

(Y.52),

(10.55)

k

This is an important

the b o u n d a r y conditions

By

to

V h are not required to s a t i s f y

V h # Z(A).

• (A ) are not compatible, conformal.

(10.54)

The a p p r o x i m a t i o n

,

(7.62),

U = [L2(F)]

framework requires a s i m u l t a n e o u s adjoint

I.

(m-tlmes).

(AhYh, Vh) Q = (F(x,a)Yh, Vh)n+(MYh, A;v;) U ,

conditions;

<

Let Ah: V h ~ V h be d e f i n e d by

(7.67),

10.6.

~ ~

p r o ~ e c t i o n ~h are then

~h = [Qh' "'''Qh ]

where F, A, D I, M are defined

Remark

(10.53)

subspaces

(AhYh, Vh)y = (F(x,a)Yh, Vh)y+(ADIMYh, Vh)y

(7.63),

0

defined on a u n i f o r m grid.

of Y = [L2(~)]m and its c o r r e s p o n d i n g

We assume

z h e ~h;

examples of the above a p p r o x i m a t i n g

spl~ne a p p r o x i m a t i o n s

projection.

properties

C h-~IZhlL2(Q ) ,

h r-~

of L2(~) and

feature,

the boundary since our

of both A and the

a s s o c i a t e d w i t h ~(A)

(Y.69),

and

the spaces V h cannot be

•

Choice of B. We take B h = ~ h B = ~h(AD1),

i.e.,

by (7.67},

(BHU, Vh) ~ = (u, DIA Vh) F = (U, ANVh} F

(I0.56)

147 t

@

C o m m u t a t i o n s of a d J o i n t s A h and B h.

One r e a d i l y obtains

(see (5.14) Jn

[Las.5]) *

=

+

+

T

-

-

+

(AhYh,Vh) Y = (F (x,a)Yh, V h ) y - ( A N Y h + M Agyh, Vh) U , where F obtains

~s the formal adjoint of F, see

(7.68).

(10.57)

A l s o from (10.56) one

(see (7.67))

BhV h

=

A p p r o x i m a t l n u control problems.

B

Vh

=

ANVhl r

(i0.58)

.

With the a b o v e notation,

v e r s i o n of the state e q u a t i o n is e q u i v a l e n t

the d i s c r e t e

to

( Y h ( t ) , ~ h ) y = ( F ( x , 8 ) y h ( t ) , ~ h ) y + ( N Y h - U , AN~h)U •

f

(lo.59)

(Yh(O),~h)y

(yo,~h)y ,

v ~h e Vh"

The optimal f e e d b a c k control for the finite d i m e n s i o n a l p r o b l e m is g i v e n by

= -AitP hct w h e r e Ph(t)

y°ct) 1 I , •

(i060

s a t i s f i e s the DRE h s

(Ph(t)Yh, Vh)y = -(R Ry h , v h ) Y - ( P h ( t ) A h y h , v h ) Y - ( A h p h ( t ) y h , v h ) Y + (AN[Ph(t)y hI ,AN[Ph(t)Yh]-) u , Ph(T) Thus,

= 0,

V Yh, Vh 6 V h .

in order to apply T h e o r e m 9.8 on the DRE

the a s s u m p t i o n s

(B.II = (9.50)

through

(T < *),

(B.4) = (9.53).

fact, we c/aim that these a s s u m p t i o n s are satisfied, following additional hypothesis

(10.61)

we must v e r i f y As a matter of

however,

under the

(which may in fact be relaxed):

the m a t r i c e s Aj(~) are s y m m e t r i c and b l o c k d i a g o n a l

A~

(10.62)

0

A, = 10~ Aj+ 1;

, = o,1,2,.--n

This a s s u m p t i o n is a u t o m a t i c a l l y s a t i s f i e d if dim ~ = I. u n d e r an a s s u m p t i o n weaker than (10.62}

In fact,

it is s h o w n in [Las.5]

that

148

both A h and A h ~ a t l s f y the f o l l o w ~ n g c o e r c ~ v l t y e s t i m a t e

for some

> O, w h i c h we w r i t e only for Ah:

-(Ahyh,~yh)y

~ ~lYhl~-ClYhl~

(10.6a)

.

Here R is a c e r t a i n invertible m a t r i x (symmetrizer).

These e s t i m a t e s

for A h and A h are key e l e m e n t s in p r o v i n g that a s s u m p t i o n s (B.I) = (9.50) and T h e o r e m 5.1]).

(B.2) = (9,51) are then s a t i s f i e d

'In effect

(10.62) may be dispensed,

(see [Las.5; as long as (10.63)

can be o b t a i n e d w i t h an a p p r o p r i a t e R, and s i m i l a r l y for A

.

It is

well k n o w n that in the continuous case such e s t i m a t e s always hold true [Kr.l] by u s i n g a p s e u d o - d l f f e r e n t i a l hand,

s y m m e t r i z e r R.

in order to prove the s t a b i l i t y e s t i m a t e

On the other

(B.2) = (9.51),

one

needs that Ry h c V h w h i c h of course is not true if R is a pseudodifferential

operator.

Similarly,

inequality

(5.23)

in [Las.5] asserts

that leAh t

YhlFIL2(O,T;[L2(r)]k Inequality

(10.64) c o m b i n e d w i t h

assumption

(B.3) = (9.52).

cT[Yhl

)

(10.58) yields the v a l i d i t y of

Finally,

the v a l i d i t y of

u n d e r the present a s s u m p t i o n is e s t a b l i s h e d

Conclusion.

All r e q u i r e d a s s u m p t i o n s

the a d d i t i o n a l h y p o t h e s i s

(10.64)

[L2(n)]m

(I0.52).

(B.1)-(B.4)

Thus,

(B.4) = (9.53ab)

in [Las.5].

are s a t i s f i e d under

T h e o r e m 9.8,

Part I applies

and yields in p a r t i c u l a r IPh(t)~hX-P(t)xl

~ O,

x G [L2(~)]m,

(10.65)

C([O,T];[L2(Q)]m) 0

0

+lu~-u°l

lYh-Y IC([O,T];[L2(~I]m)

w h e r e P(t}

, n

~ o,

(lo.o6)

L2(O,T;[L2(FI]k)

is the v i s c o s i t y s o l u t i o n of the DRE

(3.21) g i v e n by (3.6).

In order to obtain a b o n a f i d e s o l u t i o n to the DRE and to claim convergence

of the gain operators,

we need to make a d d i t i o n a l

a s s u m p t i o n s on the o b s e r v a t i o n o p e r a t o r

R.

Here

o u r s e l v e s to note that the s u f f i c i e n t c o n d i t i o n

we s h a l l

confine

(9.63) of Part II of

T h e o r e m 9.8 is indeed s a t i s f i e d in each of the cases I and 2 c o n s i d e r e d

149

at the end of S e c t i o n Y and w h e r e R*Rx =

(x,c}c: T

7.4,

w h e r e R is a finite

R obeys h y p o t h e s i s

(T.75).

rank b o u n d e d

We shall

verify

operator

(9.63)

let g E C([0, T];Y)

•

T

*

IfB'ce AhtH h - e A't ]

= 0

2

c(g(t),C)ydt]u

0 T

A*t

~

T

{flB*Ce h Hh_eA t]cl~dt ) f{(g(t),C)ul2dt. 0 0 Convergence [Las.5:

to zero of the

Theorem

5.2].

the g a i n o p e r a t o r s conclusion

holds

In either

11.

first term in

Thus T h e o r e m

holds

case,

Ph(t)

two-part

limitatlons--the

and P(t)

but proof

are b o n a f i d e

significant

(9.54)

rank.

in on

The same

is omitted.

solutions

of DRE.

and A l g e b r a i c

Riccatl

of their n u m e r i c a l

operators

B, subject

regularity =

(1.6}

(first

Theoretical

(H.I)

class.

where

A generates

unbounded

Equations

{H.2).

years--by

control,

and,

see

few p.d.e,

class

as well

(1.1)

as the numerical computations.

with unbounded

(but not m u t u a l l y

labeled

(H.2R)

comprises

an s.c.

analytic

in the technical

p.d.e,

problems

conditions,

and

The r e g u l a r i t y

purely

hyperbolic

form)

operator

for actual

dynamics

=

(H.1)

(1.8)

=

obvious

the

(1.5)

(second

exclusive) and

form).

aspects

This

as" A,

w h i c h were

concerning

to a b s t r a c t

approximation

to two d i f f e r e n t

hypotheses,

to c o l l e c t - - w i t h l n

results

of solutions

We have c o n s i d e r e d a rather general

Qlass

is p r o v e d

II c o n c l u s i o n

(7.75),

paper a t t e m p t s

most

(and u n i q u e n e s s )

Differential analysis

11.1.

9.8 Part

as hlO

(10.67)

Conclusions

existence

(H.2)

(10.67)

true in our case w i t h R finite

true for R s a t i s f y i n g

The present space

on

with

palrs

moreover,

requirement

(H.2)

hold

true

for the S c h r 6 d i n g e r

of physical

4.1.

importance

in (1.1),

shown

in recent

for a v a r i e t y with

certain

equation

However, where

as

(l.S}.

has been

as of plate p r o b l e m s

the e x a m p l e s of Sections

problems

of o p e r a t o r s

and B is "almost

sense of a s s u m p t i o n

techniques--to

as well

{A,B}

semigroup

of boundary

with Dirichlet

there

are still

a gap still

exists,

a at

149

at the end of S e c t i o n Y and w h e r e R*Rx =

(x,c}c: T

7.4,

w h e r e R is a finite

R obeys h y p o t h e s i s

(T.75).

rank b o u n d e d

We shall

verify

operator

(9.63)

let g E C([0, T];Y)

•

T

*

IfB'ce AhtH h - e A't ]

= 0

2

c(g(t),C)ydt]u

0 T

A*t

~

T

{flB*Ce h Hh_eA t]cl~dt ) f{(g(t),C)ul2dt. 0 0 Convergence [Las.5:

to zero of the

Theorem

5.2].

the g a i n o p e r a t o r s conclusion

holds

In either

11.

first term in

Thus T h e o r e m

holds

case,

Ph(t)

two-part

limitatlons--the

and P(t)

but proof

are b o n a f i d e

significant

(9.54)

rank.

in on

The same

is omitted.

solutions

of DRE.

and A l g e b r a i c

Riccatl

of their n u m e r i c a l

operators

B, subject

regularity =

(1.6}

(first

Theoretical

(H.I)

class.

where

A generates

unbounded

Equations

{H.2).

years--by

control,

and,

see

few p.d.e,

class

as well

(1.1)

as the numerical computations.

with unbounded

(but not m u t u a l l y

labeled

(H.2R)

comprises

an s.c.

analytic

in the technical

p.d.e,

problems

conditions,

and

The r e g u l a r i t y

purely

hyperbolic

form)

operator

for actual

dynamics

=

(H.1)

(1.8)

=

obvious

the

(1.5)

(second

exclusive) and

form).

aspects

This

as" A,

w h i c h were

concerning

to a b s t r a c t

approximation

to two d i f f e r e n t

hypotheses,

to c o l l e c t - - w i t h l n

results

of solutions

We have c o n s i d e r e d a rather general

Qlass

is p r o v e d

II c o n c l u s i o n

(7.75),

paper a t t e m p t s

most

(and u n i q u e n e s s )

Differential analysis

11.1.

9.8 Part

as hlO

(10.67)

Conclusions

existence

(H.2)

(10.67)

true in our case w i t h R finite

true for R s a t i s f y i n g

The present space

on

with

palrs

moreover,

requirement

(H.2)

hold

true

for the S c h r 6 d i n g e r

of physical

4.1.

importance

in (1.1),

shown

in recent

for a v a r i e t y with

certain

equation

However, where

as

(l.S}.

has been

as of plate p r o b l e m s

the e x a m p l e s of Sections

problems

of o p e r a t o r s

and B is "almost

sense of a s s u m p t i o n

techniques--to

as well

{A,B}

semigroup

of boundary

with Dirichlet

there

are still

a gap still

exists,

a at

150

present,

between

space w h e r e Finite

the space w h e r e

exact

Cost C o n d i t i o n

mentioned

(H.2)

is s a t i s f i e d

controllability/uniform

have been ascertained.

(1,9))

In the case of ~ a v @ e u u a t i o n s equations)

Dirlchlet

control

this s p a c e

of s e c t i o n

that a s s u m p t i o n

regularity

These

control

the full

4.1),

(H.2)

holds

stabilization

8.2.

The s o l u t i o n

a space definitely H1(n)xL2(n)

larger

of finite

[L-T.20-23],

true:

holds

This

d i m Q > 2.

Thus,

true.

However,

results

stabilization--needed presently spaces

available

of sharp

of exact

to v e r i f y

regularity

assumption

(H.2)

relations exact

where

to v e r i f y here

conditions, equations

forces

"energy"

Cost

control.

instead

order

the s i t u a t i o n

is,

and b e n d i n g

moments

space H2(~)xL2(n),

stabilization results

uniform

are a v a i l a b l e

(1.9)--are not

in the

in s e c t i o n

7.2 in

regularity,

where

results Thus,

are also

for these

is not

available

fourth-order

as for w a v e

equations

order

the c o u n t e r p a r t

boundary of w a v e

case of waves,

we

true for p l a t e s with, in r e l a t i o n

exact

results

to the

ILL].

that a fully c o n s i s t e n t

discussed

numerical

say,

controllability/uniform

Numerlca~ a s p e c t s As to the n u m e r i c a l

that

conditions,

rather

instead

if

boundary

as c o n t r o l s

where

on the

of optimal

As in the latter (H.2)

in

is true.

is as c o m p l e t e

instead,

that a s s u m p t i o n

depend

for p l a t e s w i t h h i g h e r

control.

in

the s p a c e

results

Cost C o n d i t i o n

we have seen

Condition.

However,

in time

with L2(~)-controls,

spaces

the s i t u a t i o n

with Neumann

have at p r e s e n t

shown

for lower

identified

the F i n i t e

(in space),

with Dirlchlet

11.2.

true

(H.2)

Droblems,

in

to H I ( Q ) x L 2 ( Q ) ,

controllability/

controllability/unlform s t a b i l i z a t i o n

problems

shear

holds

to e x p l i c i t l y

in r e l a t i o n

the F i n i t e

only on Hl(~)xL2(n)

In the case of p l a t e s

than)

regularity

which moreover

is not true

of

However,

topology

see the s h a r p

to

as m e n t i o n e d

than

in Remark 8.2,

5.2

exact

lives p o i n t w i s e

(H.2}

to the

is the space

likewise

(with w e a k e r

order

It is in r e l a t i o n

to L 2 ( ~ ) - c o n t r o l s

"energy,"

summarized

of Q.

second

theory of T h e o r e m

for dim n ~ 2, the situation is not so satisfactory,

geometry

the

cases a r e

(as o p p o s e d

in this case of dim Q = 1, w h e r e

controllability/unlform

Remark

(smoother)

(hence

(or m o r e generally,

with Neumann boundary

in dim ~ = I on the space Hl(n)xL2(n).

optimal

the

below.

hyperbolic

applies

and

stabilization

in this paper,

we have

theory can be d e v e l o p e d

for

151

d y n a m i c a l m o d e l s w h i c h s a t i s f y either a s s u m p t i o n (H.2)

=

Class

(H.IJ.

(H.I) = (1.5), or else

(1.6).

('analytic'

In the case of models w h i c h comply w ~ t h a s s u m p t i o n class),

optimal c o n v e r g e n c e

the numerical results

theory is optimal,

(which are in llne w i t h the p r o p e r t i e s of

the original c o n t i n u o u s solutions) the a p p r o x i m a t i n g subspaces.

under m i n i ~ l

assumptions

Class (H.2},

(H.2).

the n u m e r i c a l

etc.

theory is less satisfactory. 'optimal,'

assumptions.

However,

The b a s i c

theory of

in the sense that optimal c o n v e r g e n c e

c o n c l u s i o n s are o b t a i n e d u n d e r only

'natural'

approximating

the q u e s t i o n remains as to w h e t h e r or not a few

'natural' a p p r o x i m a t i n g a s s u m p t i o n s are a c t u a l l y s a t i s f i e d by

various approximating subspaces delicate

modal or

In the c a s e of d y n a m i c s w h i c h comply w i t h a s s u m p t i o n

T h e o r e m 9.2 is, yes,

of these

imposed on

These are just basic c o n v e r g e n c e

p r o p e r t i e s w h i c h are s a t i s f i e d by all s p l l n e approximations, s p e c t r a l approxlmatlons,

(H.1)

as it p r o v i d e s

issue,

and our L e m m a s

(algorithms).

This is,

in fact, a

10.3 and 10.4 p r o v i d e Just one answer.

The issue is p u r e l y technical: w h e t h e r a g i v e n a p p r o x i m a t i n g s u b s p a c e w i l l g u a r a n t e e the d e s i r e d e x p r e s s e d by i n e q u a l i t i e s these lemmas,

(and

'natural') s t a b i l i t y r e s u l t s as

(10.50),

(10.51).

In full g e n e r a l i t y b e y o n d

this is p r e s e n t l y an open question.

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I. Laslecka, Galerkln approximations of abstract parabolic boundary value problems with rough boundary data; L P theory, Math-.~Ccmp.~, Vol. 47, No. 175 (1988), 55-75..

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I. Lasiecka, Approximations of the solutions of infinite dimensional Algebraic Riccatl Equations with unbounded input operators, Numerical Funct. Anal. and ODtlmiz.

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I. Lasiecka and A. Manltlus, Differentlablllty and convergence of approximating semlgroups for retarded functional differential equations, SIAM J. Numerical Anal. 25 (1988), 883-907.

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I. Lasiecka and R. Trlgglanl, An L2-Theory for the Quadratic Optimal Cost Problem of Hyperbollc Equations with Control in the Dirichlet B.C., Workshop on Control Theory for Distributed Parameter Systems and Applications, University of Graz, Austria (July 1982); Lecture Notes LNICS, Vol. 54, Sprlnger-Verlag (1982), 138-153.

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I. Lasiecka and R. Triggianl, Riccati equations for hyperbolic partial differential equations with L2(0, T;L2(F))-Dirlchlet boundary terms, S_IAM J. Control and ODtimiz.. VOI. 24 (1988), 884--924.

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I. Lasiecka and R. Triggiani, The regulator problem for parabolic equations with D i r i c h l e t boundary control; Part I: Riccati's feedback synthesis and regularity of optimal solutions, ~DDI. Ma~b. ~ d o m ~ Z , , Vol. 16 (1987), 147-168.

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I. Lasiecka and R. Triggiani, The regulator problem for parabolic equations with Dirlchlet boundary control; Part II: Galerkin approximation, ApDI. Matb, and Optlmiz., Vol. 16 (1987), 198-216.

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I. Lasiecka and R. Triggiani, Infinite horizon quadratic cost problems for boundary control problems, proc@edings 2Oth CDC Conference. Los Angeles (December 1987), 1005-1010.

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I. Laslecka and R. Trlggianl, Differential Riccatl Equations with unbounded coefficients: Appllaatlons to boundary control/boundary observation hyperbolic problems, J. of Nonlinear Analysis. to appear.

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I. Lasiecka and R . Triggiani, Exact controllability of the Euler-Bernoulli equation with controls in Dirlohlet and Neumann boundary conditions: A n o n - c o n s e r v a t i v e case, SIAM J. Control and Optimiz., 27 (1989), 330-373.

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I. Lasiecka and R. Triggiani, Uniform exponential energy decay of w a v e equations in a bounded region with L2(O,~;L2(F)-feedback control in the Dirlchlet boundary conditions, J. Diff. Eons, 66 (1987), 340-390.

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I. Laslecka and R. Trlgglanl, Regularity theory for a class of nonhomogeneous Euler-Bernoulll equations: A cosine operator approach, Bollett. Undone Mathem. Itallana UMI (7), 3-B(1989), 199-228.

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I. Lasiecka and R. Tr~ggianl, Exact controllability of the Euler-Bernoulll equation with boundary controls for displacement and moment, J, Math. A n a l . & App,., 146 (1990), 1-33.

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I. Laslecka and R. Trlgglanl, Regularity, exact controllability and uniform stabilization of Kirchoff plates via only the bending moment, 1989, J, Diff. Eans., to appear.

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I. Lasiecka and R. Trlggiani, Stabilization and structural assignment of Dirlchlet boundary feedback parabolic equations, SIAM J. Control and ODtimlz., vol. 21 (1983), 766-803.

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I. Laslecka and R. Trlggiani, Stabilization of Neumann boundary feedback parabolic equations: the case of trace in the feedback loop, ADDI. Mathe. and ODtlmlz.. vol. 10 (i983), 307-350.

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I. Laslecka and R. Triggianl, Numerical approximation for abstract systems modelled by analytic semigroups, and applications, Mathematics of Computation, to appear.

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I. Lasiecka, and R. Trlgglanl, Sharp regularity results for mixed second order hyperbolic equatlons of Neumann type: The L2-boundary case, Ann. d~ Matem. Pura e ADDI., tO appear.

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I. Laslecka and R. Trlggianl, Trace regularity of the solutions of the wave equation with homogeneous Neumann boundary conditions and compactly supported data, J, Math. Anal. and A p ~ . , vol. 141 (1989), 49-71.

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I. Laslecka and R. Triggianl, Riccatl Differential Equations with unbounded coefficients and non-smooth terminal condltlon--The case of analytic semlgroups, SIAM q. Math. A~al., to appear.

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I. Laslecka and R. Trlgglanl, Regularity theory of hyperbolic equations wlth non-homogeneous Neumann boundary cond~tlons. Part II: General boundary data, J. Diff. Eons., to appear.

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I. Laslecka and R. Triggianl, Recent advances in regularity of second order hyperbolic mixed problems, and applications, to appear in book series, Dynamics ReDorted, Wiley-Teubner.

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I. L a s i e o k a and R. Trigglani, Optimal regularity, exact c o n t r o l l a b i l i t y and u n i f o r m s t a b i l i z a t i o n of Schz~Ldinger e q u a t i o n s w i t h D i r i o h l e t control, piff. and Integral Eqns., to appear.

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I. L a s i e c k a and R. Triggiani, Exact c o n t r o l l a b i l i t y and u n i f o r m s t a b i l i z a t i o n of E u l e r - B e r n o u l l i e q u a t i o n s w i t h b o u n d a r y control o n l y on dwl~, B o l l e t t i n o Unione Matem. Italiano.

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