Proceedlngr ol25th Confemnce on Declalon and Control Athens, O m * December 1986 TA= 8:OO

EXACT CONTROLLABILITY OF DISTRIBUTED SYSTEMS AN INTRODUCTION

Jacques-Louis LIONS

COLLEGE DE FRANCE and C.N.E.S. (CENTRE NATIONAL D'ETUDES SPATIALES)

1. - INTRODUCTION. FORMAL EXPOSITION OF THE METHOD HUM. The b o w d u t y conditions are

We consider a ne.#^ system of evokhtion in a domain fi of Rn (n=l,2,3 in the applications). (3) B.y J = v j , I 5 j 5 m , on r x ]O,T C .

One can act on the system on the boundary of or at some given subregions of R , by the so-called COW?W~ &nCtion4 or contrioL u h b h .

The boundary operators B are such that the pro- j

blem ( 1 ) ( 2 ) ( 3 ) is well set - in suitable function

The problem of exact controllability is the follo- spaces.

wing : is it p o b b i b l e to 6-ind control variables that drive the system t o U g i v e n A M e in a given (finite) time T ? And how to cofi5&~~& such controls, in case

We refer to J.L. Lions and E. Magenes [I] , vol. 1

and 2 , for sufficient (or necessary and sufficient) conditions on the B.'s such that ( 1 ) ( 2 ) ( 3 ) is indeed

J

they do exist ? well set

Since the system is supposed to be fieah, it is In (3) the v.'s are functions to be chosen :

t he contrrol v c v h b l a . Given T , we want to find - id J

eauivalent to ask whether it is possible to drive the system to rest. p o b d i b h - control variables v such that, if y(v)

denotes the solution of (1)(2)(3) (the state of the pro- blem) , then

j The importance of this type question is obvious.

It is closely related (in a sense, it is equivalent) to the problem of the b,tab&zatian of systems : is it possible to find feedbacks (i.e. formulaes relating the ( 4 ) y(T ; v) = y'(T ; v) = 0.

control variables to the state of the system) such that the system is ARabLe - more precisely such that the sys-

tem goes to zero e x p o n e n t i a e e y daSt as the time t +. OD. Remark 1.

The operator A can be a b y h t m of partial diffe- More precisely, let us assume that the state y rential operators, 0

of the system is given by the solution of the evolution equation Remark 2 .

( 1 ) y" + Ay = 0 in R x l0,TC A typical example is the following. We suppose that 2

where y" = , and where A is an elliptic operator of

order 2m at which is supposed to be b Y n u n e t t L i C . The (5) A 5 - A , A = Laplacien,

state is given by the solution of ( I ) , subject to s o that ( 1 ) becomes the W v e eyllation .inLtid con&ons and to boundatry conditionb.

The initial data are given : (6) y" - Ay = 0 ,

( 2 ) Y(0) = yo , Y'(0) = y 1 subject to ( 2 ) . If we take B = Identity, then

j

where yo and yl are functions in R (in appropria- ") = " On x 'o*T' *

te function spaces) and which span all the spaces.

731 CH2344-0/86/0000-0731 $1.00 @ 1986 IEEE

Is it possible to find v such that the system is driven to rest at time T ?

Clearly this question has to be made more precise by specifying the QunCtion dpace6 where yo , y are given and where v has to be found. But one conditionis obvious : due to finite speed of wave propagation, T has to be given h g e enough. 0

1

Remark 3.

The example given is the above Remark 2 is an hy- petrbofic system. But the problem - and the general method we are going to present - is n o t confined to hyperbolic systems.

As a typical example, we can consider

(8) y" + A y = 0 2

subject to (2) and to 2 Boundary conditions involving the control variables. U

Remark 4 .

In fact one can add conbZ4dnL5 on the v ' s . F O h

h t a n c e one can take in (7 ) v = 0 on r* X lO,TC, r, c r. In other words :

j

v on To Xl0,TC

( 9 ) y =

o on ( r \ rJ x l0,~C.

Given T is it possible for every yo, y 1 to find v such that the system is driven to rest at time T ? The problem depends now on T and on Y o . Remark 5 .

Another type of constraints is : dome of the v 's j

are zero. An example of such a situation will be presented

in Section 2 below, for

of (8) subject to

lY'0 on

the system given by the solution

r X lO,TC,

ro XJO,TC T w o ) x IO,TC.

In (IO) one has only a control on one boundary condition and one can act only on p& of the boundary.

0

Let us present now .&I a do& Uny the gWnehae method for solving the above questions.

Let us introduce the boundary operators C

Bj , in 1 S j 5 m , which are comp&mentahy of the following sense : if rp and $ are smooth functions in 51 , then

(11) IQ((Arp)$ - rpA$)dx = (c.9 B $-B.9C.$)dT.

i '

' 1 ~ 1 j J J

Let us start from the equation

9'' + A@= O ,

(12) p(0) = rpo , V'(0) = 9 , 1

B.P = 0 , 1 S j 5 m on r x 3 0 , T C , J

Given 9' , ' P I - in Space6 t o be chosen meh on (12) admits a unique solution,

We consider then the backward problem - which is also well set - :

$" + A$ = 0 , ( I 3 ) $(T) = $'(T) = D ,

Bj$ = - Cj p on r x l O , T [ .

Given {vO , 9 1 , (13) admits a uni'quc soluti.on, and 1

we define an operator A acting on {@' , rg } by 1

( 1 4 ) A{VO ,'D 1 I$'(o) , -$(o)} . 1

Of course this operator A depends on T. kdwne now that we can dind @ m t i o n bpaCe6 whehe

Then, given yo,yl , there exists { V 0 , q } such A i~ inveht ibLe.

I

that

(15) A { q o , r p } = {y', -yo}. 1

Then 2f we d e 6 . h

v j = - cj9 9

(16) rp = solution of (12) with @0,91 given by (15)

then y(v) = $ and therefore ( 4 ) holds true and we have found - & a cOt16LU~otive manneh - controls v 's which drive the system to rest.

j

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Let us now observe the following property : (i) when does (20) define a norm ?

(17) <A{P0,rppl} , {V0,p1}> = f (Cjq) 2 dT dt. (ii) what kind of information can we have on the J=' rxro,TJ function space F ? D

Remark 7 . [To obtain (l7), multiply (13) by rp and use Green's formula. In the left hand side of (17), the scalar pro- If

We arrive now to the main point ; let us assume A m de&i.nined a nom (for To and r suitable), then the same method applies. It gives exact controllability by acting on@ on To and by using only dume boundary conditions (corresponding to j E J ) . 0

that we have the following UniquMc4A ThLOam :

'e" + A . p = 0 in R x lO,T[ ,

(18) B . V = 0 on T X IO,T[: , 1 5 j S m , J Remark 8 .

c . p = 0 on r x lO,T[ , I 5 j S m 3 The method indicated above consists in defining

implies a Hilbert space F corresponding to a uniqueness theo- rem. We shall call this the method of the HiLbetLt Apace

' p = o

Then

abdO&ed wLth UniQUenedA, in short Hilbert-Uniqueness Method (HUM ... ) . 0

Remark 9.

Let us give an example of a different nature. Let b be given in R. Let the state be given by

d e d h t ~ a nom on the couples IFo , d l . Let 1 1 / I F be this norm, and Let F be t h e completion 06 Amooth & ~ ~ c t i o r z d (subject to appropriate Boundary conditions) ; subject to F .d a H a b a t Apace.

(25) y" - Ay = v 6(x-b) , v = v(t) = control variable,

(26) y = 0 on Txl0,Ti:

Then, one can verify that and to (2).

(21) A E%F ; F ' ) , A*= A

and by (20) one has

(22) A .d an .damohpkidm &om F onto F ' .

Again for every yo , yl , in suitable spaces, is it possible to find v such that (4) holds true ?

Applying HUM, we are led to the following question Let 'f be the solution of

Therefore if yo , yl are given such that I ! # I ' - = 0 ,

then the above program is in order and the exact control- lability problem is solved. 0 Let us consider the expression

Remark 6.

The key questions are now :

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If b is chosen in a suitable manner, assuming all eigenvalues of -A for Dirichlet boundary condit- ion to be d h t p h , then, for T large enough, ( 2 7 ) does define a norm. Hence a (new) Hilbert space F and an exact controllability result. 0

It is quite clear that this method can be applied in a very large number of situations. A short announce- ment has been given in J.L. Lions [ l ] . Our goal here is simply to give some details on the method in a speci- fic example. A more comprehensive presentation is given in the J. Von Neumann lecture given by the Author in July 1986, J.L. Lions r 2 1 . A (hopefully) comprehensive presentation will be given in two volumes in preparation J.L. Lions C31 .

Of course the above method is not the only one at hand for attacking problems of exact controllability.

A direct method, based (essentially) on the proper- ties of the solutions of the Cauchy problem ( R = Rn) , has been introduced by D. Russel [ I1 and used by several authors. We mention in particular the recent work of W. Littman Cll .

D. Russel [ I1 observed that one could use the experimental decrease of solutions of stabilized pro - blems to solve the exact controllability question. This leads to the question : can we d.ind d e e d b a d g iv ing exponential decheue 06 t h e A O L U ~ ~ O ~ S ? Several interes- ting contributions to this question have been given,in particular by G. Chen [ I ] , J. Lagnese [ I ] , I. Lasiecka and R. Triggiani [ I 1 .

A d y d t e m a t i c method is the following. One consi-

ders the state y(v) given by the solutions of (1 ) (2 )

(3) and one introduces the cod2 d u c t i o n (v={v,, . . . ,vm})

where 1 1 I ( is appropriately chosen. If the system is exactly controllable, thereexiss

v's such that J(v) <a. Indeed we can find v's such that the v.'s and y(v) are with compact support in T.

J One considers then the problem of optimal control

( 2 9 ) inf J(v) . V

Let u be the solution of ( 2 9 ) . One can write the optimality system (0.S) giving u , y(u) =y. One can show that y is exponentiaUy decheabhg. One can next show that there is a d e c o u m g of the O.S., leading to a feedback formula. Therefore :

I thehe A a Feedback 6 0 h m d . a g i v i n g exponwt la t ( 3 0 ) n.tabdXzation any Lime thehe A a Uniquenudb t k o -

(and one can apply HUM). 0

Of course (30) does not solve everything .,, It remains to obtain "explicit formula" - which can be technically extremely complicated, This part of the "general program" will be presented in J.L. Lions C31, Volume 2. D

Remark 10.

There is another part to the general program how does HUM method behave under p m b m % n A ?

We can consider three types of perturbations

(i) A b l g U & L t pentwrbatioa ;

for instance,

let y, be the state of the system

(31) yg + EA y, - AyE = 0 , 2

y, subject to (2) and (IO).

One can then show (cf . J.L. Lions c 4 3 ) that this

system is wzidohmey exactly controllable, i.e, there exists To &depmdent o d E such that, for any T >To, the system (31) (2) (10) is exactly controllatile.

Let uE be t h e control driving the system to rest using HUM. Then, as E+O,

( 3 2 ) - 6 uE -+ uo in L (X), C = YX (0,T)

where u is the HUM control driving

2

y" - Ay = 0

to rest, by acting on C , i.e.

y = uo on C.

[We start with yo E L (n), y E H - l ( n ) , All this can be 2 1

done by acting only on a (suitable) pWd of r . Cf. J.L. Lions C41 for technical details], 0

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(ii) Homogenization theohy

One can consider the problems of Exact Control- lability for systems governed by

( 3 3 ) y", + AEyE = o

where the AE's correspond to operators with Rapid& vcVujing coe66icients. How does exact controllability behave when replacing AE by its "homogenized wehsion"?

(iii) Similar question for special domains R such as pehdollated domains, or vehy tkin domains, or pluri- dimensional domains (domains which have several compo- nents of different dimensions).

E

These questions will be studied in subsequent

papers. 0

Remark 11.

The method HUM is conbhct iwe . One solves (15) by conjugate gradient techniques, and one solves next (12) and one obtains

v = - C.Y. j J

We refer to R. Glowinski and C.L. Li [ I1 and to the presentation made in J.L. Lions C51, based on the above work. 0

Remark 12.

In a sense HUM method gives "the best" way to drive the given system to rest. Among all v's driving the system to rest at time T , the one given by HUM minimizes 1 vi drdt. 0

%(O,T)

Remark 13 .

One of the two basic questions which remains is (cf. Remark 6 , (ii)) : what information do we have on F ?

If one considers the wave equation

p'-&=o, ' f (o )=Yo , p'(o)='p , ' f=O on rx(0,T) 1

is, doh T enough, a norm e q u i v d e n t to the norm of H:(R) XL (Q) ( ' ) . More precisely, one has (for ewehy T > O )

2

where c does not depend on T (cf. J.L. Lions C61 ) ;

and there is a To> 0 such that for T > To one has

This last inequality is due to L.F. Ho [ I 3 ( 2 ) . In other words, in that case, for T>To, one

has, up to an equivalence of noms

MORE ON H U M IN A SPECIAL EXAMPLE.

We now consider the system (8)(10) of Section 1 :

p " + A y = O 2

y = 0 on Tx(0,T) ,

Let us now give precisions on ro cr. Let xo be chosen arbitrarily in Rn ; we set

and we define

where in ( 4 ) V(X) = unit normal vector to T , direc- ted towards the exterior of R.

( * ) H;(W ={Y/ y , a axl ,..., h c L 2 ( ~ ) , axn LP= o on r}.

('1 L.F. Ho [ I ] shows an inequality of type ( 3 6 ) using only pcvLt of T (cf. also Section 2)

then

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We shall take in (2) Remark 2.

We consider the initial states

where yo is given in L2(n) and y1 is given in H-2(n) ( ').

T h a e e d 1 2 To m g e enough (depending on xo1 duch that 60h

( 7 ) T > To

and for every y0<L2(Q), Y'EH-~(R) , t h a t e x A ~ 2 v E L2(r(x0) X (0,T)) duch that id y(v) d e n o t a t h e doLlLtion 0 6 ( 1 ) (2) ( 6 ) , then

In other words the system is exactly control- lable by acting on one boundary condition, on paht of r.

A few remarks are now in order.

Remark I.

The best possible To in ( 7 ) is not known. One can show (cf. J.L. Lions C21, [31, Vo1.l) that

where T(xo) is given by

(IO) T(xo) = + ln-2/ . 0 2po

Here R(xo) is the smallest number such that 0 is contained in the ball centered at xo and with radius R(xo) ; X and po are the best constants such that

0

We use HUM in the following way, Let cp be the solution of

The second inequality in (13) is valid with

r(xo) replaced by 7 and for T arbitrary (of course c will depend on T). Cf. J .L . Lions C6l. The first ine- quality is obtained by multiplying by % &!! and after integration by parts. Cf. J.L. Lions C2Ic31 . It is an extension of the method of L . F . Ho [ I 1 .

a x ,

With the general terminology of Section 1, we define

and we obtain

(15) F = Ho(n) XL (a) . 2 2

The operator A E ~ F ; F ' ) is constructed as follows :

we solve (12) ; then J, is the unique solution of

(') H-2(R) denotes the dual space of H:(Q) =

={(PI Day EL (n) , Y la1 SZ, y=O and *= 0 on TI. 2 av

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(16)

and

$ " + A + = O , 2

$(T) = $'(T) = 0

= 0 on Tx(O,T) ,

so that A is an isomorphism of F onto F'. One

solves then

and one takes

among all controls in L (r(xo)x(O,T)) which drive the system to rest at time T. 0

2

Remark 3 .

Actually inequality (13) gives a (new) &4UeflUA theo4em : if y satisfies

yt l + A C P = o , (P<L~(O,T ; ~ ~ ( 0 ) ) , 2 2

9 = *= 0 on TX(O,T) ,

AV= 0 on r(x0)x(O,T) av

and if T > T(x0)

then LQ= 0

Remark 4 .

In the above example, we have been able t o

chahactehize F. In other cases one can get pcvLtiae information on F,

but sufficient to conclude that "reasonable" initial

data can be driven to 0 . We refer to results with

I. Lasiecka and R. Triggiani, and to J.L. Lions C2l C3l C41. cl

Remark 5 .

For controllability and stabilization problems for other models of plates equations, cf. J. Lagnese and J.L. Lions r11. 3

BIBLIOGRAPHY

Chen, G. [ I ] Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain. J.M.P.A. ( 9 ) 58 (1979) ,p . 249-274.

Glowinski, R. and Li, C.L. To appear.

Ho, L.F. C11 C.R.A.S., Paris, 1986.

Lagnese, J. Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Diff. Equations. 50 ( 2 ) , ( 1 9 8 1 ) , p. 163-182.

Lagnese, J. and Lions, J,L. . To appear. Lasiecka, I. and Triggiani, R. c 1 9 Various papers to

appear. See in particular : Uniform exponential energy decay in a bounded region with L (O,~;L~(~))- feedback control in the Dirichlet boundary conditions. J. Diff. Equations. To appear.

2

Lions, J.L. c11 ContrElabilitE exacte des systsmes dis- tribuss. C.R.A.S. 3 0 2 , Paris, ( 1 9 8 6 ) , p. 471-475.

c21 Exact controllability, Stabilization and Perturbations. J. Von Neumann Lecture, SIAM, July 1986. S U N Review, to appear.

C41 Exact controllability and Singular Per- turbations. An Example. Dedicated to P . LAX. Berkeley, 1986. To appear.

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Lions, J.L. I51 Numerical Approach to Exact Controlla- bility. Moscow Conference on "Modern Problems in Numerical Analysis", Sept. 1986, to appear.

C6l ContrGle optimal des syst6mes distri- bu6s singuliers. Dunod, Paris, 1983.

Lions, J.L. and Magenes, E. r l l Probllmes aux limites non homoglnes et applications. Dunod, Paris, 1968, vol. 1 and 2.

Littman, W. [11 24th C.D.C. Conference, Fort Landerdale, 1985.

Russel, D.L.CI1 Controllability and Stabilizability Theory for Linear Partial Differential Equations. Recent Progress and Open Questions. S U M Review 20 (1978), p. 639-739.

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