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

Springer-Vedag Bedin Heidelberg NewYork

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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
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 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 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 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 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

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, ~

~ 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


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
~, 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 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

O, as h$O,

s u p t & eWP tllyh{, O.t,llhx)_y " " O , t ,x)llz(y) 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
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


_ 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]) 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)