The Linear-Quadratic Optimal Control Problem for Parabolic Systems with Boundary Control Through a Dirichlet Condition†

The Linear-Quadratic Optimal Control Problem for Parabolic Systems with Boundary Control Through a Dirichlet Condition†

Copyrighl © IFAC 3rd Svmposium Control of Distributed Pa ramrt('r Systcms Tou louse. France . 1982 THE LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEM FOR P...

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Copyrighl © IFAC 3rd Svmposium Control of Distributed Pa ramrt('r Systcms Tou louse. France . 1982

THE LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEM FOR PARABOLIC SYSTEMS WITH BOUNDARY CONTROL THROUGH A DIRICHLET CONDITIONt M. C. Delfour* and M. Sorine** *Centre d e recherch e de math ematiqu es apphqu ees, UniveTSite d e M ontreal, C.P.6 128, Succ. A , MOl/tr ea l. Qu ebec. H3C 3J7 Canada **INRIA . DomainI' d e Volu ceau , B. P. I05, 78 153. L e Chesnay, Franc e

Abstract. Th e obj ec t of thi s paper i s th e study of th e l inear quadratic optimal con trol pr ob l em associated with th e boundary co ntrol of a parabolic sys t em through a Uirich l e t condi ti on and distributed observations. The sys t em model is ob t ained by a transposi ti on. It is shown tha t by an appropriate chang e of variables this prob l em is equiva lent t o th e one s tudied by J. L.LIO ~S (1968). Riccati differential equa tion s are deri ved. Keywo rd s. 1.

Optimal co ntro l; parabolic systems;

Dirichl et boundary control .

I NTRODUC TIO~

LASIECKA and TRIGGIAN I ( 1981)). It is f e lt that our approach is simpler and more trans paren t. In particular it c l early estab l ishes th a t " modulo a change of variable" this prob lem is the same as the one s tudi ed by J.L.LIONS (1968, Chap t er 3). Cl a im s th a t J . L. LIONS ' ( 1968) techniques do not app ly a r e th ere fore unfounded. De t ai l ed proofs of the theorems will be found in a forthcoming paper .

The object of thi s paper is the s tud y of th e lin ear quadra tic (LQ) optimal con trol sys t em associa t ed with th e boundary control of a pa r abo lic sys t em through a Dirichlet condi tion and di s tribut ed observations. The system model is obtained by t he tr ansposi ti on of an adjoint i somo rphi sm (cf. L I O~S and ~~GENES (1968 a) , J.L.LIONS (1968, p.210 )) . By an appropriate change of variables the problem is s hown t o be equiva l ent to the standard problem s tudied by J.L.LIONS (1968, Chapter 3). In par ticul ar our prob l em is equiva l ent t o the LQ prob lem fo r parabolic sys tems with contro l through a Neumann con dition and boundary obse rv a ti on (cf .N. SOR I NE ( 198 1b, 1977) fo r a detailed ana l ysis). In the time-varying case limita ti ons a r e encountered in th e der i vation of a Ricca ti differential equa ti on as predicted by J.L. LI O\ S (1968, Chapter 3) . However under addi ti onal hypoth eses on th e obser va t ion opera t or it can be done. In th e time-in varia nt case M. SORINE ( 198 I a,b) has ext ended th e results of J.L.LIONS ( 196 8 , Chapter 3) for th e asso ciated Riccati differential eq ua ti ons . So hi s r esults app l y to our problem and a com plet e characterization of th e decoup l ing opera t or is obtained. For th e case of a final observation yeT) in L2 (Q), th e appropria t e space of boundary contro l ha s bee n introduced by J.L.LIO\S ( 1979, 1980) and a study of LQ optimal contro l pr oblem ca n be done a l so along the se line s .

No t a ti on . Given two Hilbert spaces X and Y, £(X,Y) denot es the Banach spa ce of all continuous linear maps from X to Y. Given an interval I c. R, CO(I;£s(X,Y)) will be th e space of a ll Q:I ~ £(X,Y ) such that for a ll x i n X the map t ~ Q(t)x, I ~ Y is bounded a nd cont inuo us; Cl(I ;£s (X,Y) ) will be the subspa ce of CO{ I;£ s(X ,Y) ) s uc h th a t th e map

t

~

d

(ft[Q(t )x ]:I

~

Y belongs

t o CO(I;Y). In a simi lar wa y we defin e LOO (I;£ s (X, Y)) an d C+ (X) the cone of ope rators n such t hat s P. E £s(X ', X), 'tJ'4! E X' 'tJ1jr E x ',< ncp ,Ijr >X = < cp ,nIjr ) X'
<: ,)X

betwe~n

2.

STATHIE:\T OF THE PROB LH I

Consider th e following parabolic system

Severa l authors have studied th e problem considered in th is paper (A. r.BALAKR I SH\A\ ( 19 78), I.LASIECK.-\ (1978, 1980a,b), a nd

A( t );.. yi

This research was suppo r ted a t th e

L

+

=u

~~

f

on L, y(O)

in Q,

= yO

i n Q,

(1)

where ~ is a bounded open COO domain of Rn hith boundary r, T > 0 i s a r ea l number ,

Uni v ersit~ de Montr~al by ~SERC Grant A- 8730 and a FCAC Gran t from the Mini s t~r e

Q = ] 0, T [ x )c ,

de 1 ' Education du Quebec. 87

L = ]O,T[xr

(2)

M. C. Delfour and M. Sorine

88

_~n

A(t)


a ( ( )a
a--

that A*(y'YT) = w

f~(A* (t)v -~~ ,y)dt+V

(3)

and the coefficients of A verify the following condit ions: aa aa .. O --..21.. E L (Q), :la> ao' at' a ij , at

°

W

such that

=f~<'f,v)Ddt+
~

y

Vs ERn;

°

(4 )

Associate with the solution y of (2.1) the cost function J(u,y) =
°

+f~{(Q(t)Y+2q,y)+lul~}dt,

°

Given yO we want to study the following problem Inf{J(u,yo) : u E L2(0,T;L2(f))}.

(6)

OPERATIONAL DIFFERENTIAL EQUATION FOR THE STATE Y

following J.L.LIONS (1968) and LIONS-MAGENES (1968a,b) we proceed by transposition of an appropriate adjoint isomorphism . Define 4>

=

I = O}

2 1 (Q): v Z {v EH'

(A*(t)V-~~,V(T)):4>

n a at A*(t)t=-z. '-1 -a-Ca .. (t,x)-a-)+ao(t,xH (9) 1,JXj lJ Xi W

Lerr~a 1. Assume that Q is bounded and C and that the a's verify properties (4). Then t he maps (8) is an isomorphism.

o

= H2(Q)n H~(Q),

W(O,T;D,H)={YEL2(0,T;D)I~~

2 EL (O,T;H)}

(10)

and notice that the space 4 coincides with the space W(O,T;D,H). We also know (cf. J.L.LIONS (1962)) that 2121. [D , Hlt = [H (Q)nHO(Q) ,L (Q)lt=Ho(Q)=\. (11)

°

v'" Wry) :4

-+

R,

w(v) = foTDdt+<'y ,v(O»v' 2

f E L (0,T;D')' yO E V'.

(12)

(15)

Identify 11' and H and define the canonical injections iv i * V -+ H = H' -+V V' (16) 2 and the boundary operators B(t):L (f) -+ D' and

B*(t):D B* (t)w ~B

=

-+

L2(f)

aw I aVA*(t) f'

(t)u,w/ D

(u,

aw I) aVA*(t) f L2(f)

(17)

Both operators are strongly measurable and bounded on [O,T] by hypotheses on the a's and Q Using (15) to (17) the original equation (1) becomes A*(t)*y +~~ = B(t)u+iOf in L2 (0,T;D'), y(O) = i~Yo

ln

V',

(18)

So it naturally extends to initial conditions yO in V', rig~t-hand sides f in L2(0,T;D') and controls u in L2(Z). The operator A*(t)* E £(H,D') is an extension of the operator A(t) E £(D,H) defined in (3). In th e sequel we shall write A(t) instead of A*(t)* and keep in mind that A(t) E £(D,H)n£(H,D')n£(\",V'). 0 ~o tation .

SOLUTIO:-; OF THE OPTUtAL COi'iTROL PROBLHt

Embed our original problem (1) into the following larger family: d)'

A(t)y+Cit = B(t)u+f, y(O)=y Since the functional w belongs to 4 ', there exists a unique pair (y'YT) E L2 (Q)XV' such

~toreover

where y(O) and yet) are to be interpreted as values of the function in C(O,T;V') which is almost everywhere equal to y in W(O,T;H,D'). ((A*(t)*:H-+D' is the topological transpose of A*(t):D-+H). 0

4.

Introduce the functional

(14 )

T av fO(A*(t)v-at ,y)dt+Ddt+
Proof.Cf. C.BARDOS (1971), L.TARTAR (1972).0 This is the isomorphism we shall transpose. Define

2 L (O,T;D')XV'

for all y in W(O,T;H,D') and v in W(O,T;D,H)

(7)

L2(Q)XH~(Q) , (8)

-+

-+

. . dz ., { zEL 2 (O,T,H). dt EL 2 (O,T,D )}.

and consider the continuous linear map v-tAv =

Vy = ((A*(t))*y+~~,y(O)):

is an isomorphism (W(O,T;H,D ') =

(5)

1 2 2 where V = H (Q), H = L (Q), U = L (f), u E L2(Z), l E V, q E L2(O,T;H), LEC+(V), Q EL w (O,T; C+s(H)).

3.

-+

W(O,T;H,D')

a.e. in Q.

(13)

Theorem 2. Under the hypotheses of Lemma 1, the lin ear map

~~,j=l a ij (t,x)SiSj ~ ao(t,x)

or equivalently

°E

2 2 f E L (O,T;D'), u E L (O,T;U).

V',

(19)

The Linear-Quadratic Optimal Control Problem Associate ~ith yo and u the cost function (2.5) . Theorem 3. (i) Fer each yO in V', there exists a unique minimizing control u* in L2(0,T;U) ~hich is completely characterized by the follo~ing optimality system dy = B(t)u*+f, y(O)=y , A(t)y~

°

y E W(O,T;II,D')

(20)

u* = -B* (t)p(t)

(21)

A*(t)P-~ = Q(t)y+q, p(T)=Ly(T)+t, P E W(O,T;D,f!)

(22)

(ii) Moreover the optimal control u* belongs to HL!(L) = L2(0,T;H!(f)rH!(0,T;L2(f)). If, in addition, yo E 11 and f E L2(0,T;V'), the optimal trajectory belongs to W(O,T;V,V') c l-ll,l(Q) = L2(O,T;Hl(Q))nll!(0,T;L2(Q)). 0

89

decoupling operator Pet) E £(H,H) . We know now that the compactness of the injection of V into H is not required. In the timevarying case no such equation is currently available for H's in L~(O,T;£s(U,V')); in the time-invariant case (A,S and Q constant), M.SORINE (1981a,b) has been able to make sense of and obtain a Riccati differential equation. We shall see in the next section how these new results apply to our problem. For the Dirichlet problem, B(t) '£(U,H). if we introduce the additional hypotheses Ho~ever

Q E

L~(O,T;£s(V'

and make N

y(t)=A

t~e

-1

,V)), L E £ (D' ,D)

(26)

new change of variables N

(t)y(t), p(t)=A* (t)p(t),

(27)

our new problem is covered in J.L.LIONS (1968, Chapter 3). DECOUPLING OF THE OPTIMALITY SYSTHI

5.

By invariant embedding of (19)-(5) with respect to the initial time we have: Theorem 4. There exists an operator valued function P(.) E CO([O,T];C+(V)) and a function r(.):[O,Tj -+ V such hat: pet) = P(t)y(t)+r(t).

°

A*!(t)p(t),

~ t ~ T. (24) This is possible since under hypotheses (4) the map A (resp. A*) is an isomorphism from L2(0,T;V) onto L2(0,T;V'), from L2(0,T;D(A)) (resp. LL(O,T;D(A*))) onto L2(0,T;II) and from L2(0,T;H) onto L2(0,T;D(A*)') (resp. L2(0,T;D(A)')).

Theorem 5. (i) With the change of variables (24) the optimality system (20)-(22) becomes d~ =A -! f-BB*p, -- - -y(O)=A -! y AY+

dt

-

dn

A*~-~

°

Y E W(O,T;D,H)

(23)

0

This is as far as we can go for Q E L~(O,T; £s(H,H)). In order to obtain a better understar.ding of the difficulties we make the following change in the variables in the optimality system (4.2)-(4.4) yet) = A-!(t)y(t), pet)

Theorem 6. Let f L2 (0,T;V')' yO E H, q E L2(0,T;V), t E 0 and Land Q verify (26). (i) Under the change of variables (27) the optimality system (20)-(22) becomes ~N dv -1 NN N 1 Ay+dt = A f-BB*p, y(O)=A- y ,

° yEW(O,T;V,V')

! -_ ! =Qy+A* q, p(T)=Ly(T)+A* t,

A*P-~ = QY+A*q,

p(T)

P E W(O,T;V,V')

(28)

-1

N

N

where A = A+A ~' is V-H coercive, BN A-IB E £(U,H), Q = A*QA E £(V,V'), L = A*LA E £(H,H) and (A-I), denotes the timederivative of A-I. (ii)NThere exist an operator valued function P(.) E CO([O,T]; C+(H)) and aNfunction r(.): [O,T] -+ H such tfiat pet) = P(t)y(t)+r(t). P is the unique solution of the Riccati differential equation in the sense that for each ~ in W(O,T;D(A),H), (~)(t) = P(t)~(t) is the unique solution in W(O,T;V,V') of the equation

-~t(~)+P~A*~+PA~+PBB*~ (P~)

(T) =

L~

=

~,

(T) .

(29)

Similarly r is the unique solution in W(O,T;V,V') of the equation N N dr N 1 A*r-~PBB*r = P(A- f)+A*q, r(T)=A*t. NNN

(25)

(30)

P E W(O,T;V,V'),

(iii) The operators P and P and the functions rand r are related as follo~s

where X=A+A-!(A!)~ is V-H coercive l ~ A-h E L~(O,T;£s(U,V')), Q = A*!QAl E

P (t) =A _1 (t) *P(t ) A-I (t), r (t) =A -1 (t) r (t) (31)

L~iO,T;£s(V,V'))' I = A*!LA! E C+(H) aod (A2)' denotes the time derivative of At (which makes sense, BARDOS (1971)). 0 In the form (5.3) we are back to the problem dealt with by J.L.LIONS (1968, Chapter 3). Under the additional hypothesis that ~ E L~(O,T;£ (U,H)), he derived a Riccati differential equation for the associated

~

So

E £ (D(A*)' ,D(A*)) and r(t) D(A*) °P(.) tpet) T, such that pet) = P(t)y(t)+r(t) E CO([O,T]:C+(D(A*))). Pis ~

~

~Ioreover

the unique sOluti5n of the Riccati differential equation in the sense that for each ~ in W(O,T;H,D(A*) '), (~) (t)=P(t) ~ (t) is the unique solution in W(O,T;D(A d /2),V) of the equation -

~t(~)+P~A*~+PA~+PBB*~

(~) (T)

=

L~ (T) .

~ (32)

N. C. Delfour and N. Sorine

90

~

In particular for all h in H, the map t p(t)h is the unique solution in W(O,T; D(A* 3 2),V) of the following equation

_~(t)h+A*(t)P(t)h+P(t)A(t)h+P(t)BB*P(t)h

tions of (39) or (42) is unique is completely characterized. This material is not included here since it requires additional definitions and notation.

dt

=Q(t)h,

O
(33)

P(T)=L.

The function r is the unique solution in W(O,T;D(A*3 2),V) of A*r- ~PBB*r = Pf+q, r(T)=t.

0

(34)

THE TIME-INVARIANT CASE

6.

Assume that the coefficients of A are constant in (3) and that Q(t) = Q (constant) in £(H,H). The change of variables (24) reduces to

-yet) = A-! y(t), -pet) = (A*) ! p(t),

(35)

where A (resp. A*) are isomorphllm~in £(V,V'), £(D(A),H) (resp.£(D(A*l,H)) and £(H,D(A*)') (resp. £(H,D(A)')). Theorem 7. Assume that f=O, t=O, q=O and yO E V'. (i) The optimality system (25) reduces to Ay+£f = -BB*p, yea) A-!yO,

y

E

\'I(O,T;V,V')

A*P-~ = Qy, p(T)

(36)

Ly(T),

P E W(O,T;V,V'). (ii) There exists P E CO([O,T];C+(H)) such that s pet) = P(t)7(t), t :'" T. 0 (37)

° :'"

As we said in section 5, the derivation of the Riccati differential equation for the optimalit y system (36) requires a special technique developped by M.SORINE (1981a,b) . We summarize the main results in the next theorem. Theorem 8. (M.SORINE (1981b)) (i) P is the solution in CO([O,T];C+(H) nCl([O,T[;£ (H,H)n£ (V,V))(38) s

s

s

of the following equation in [O,T[ -

~:

+PA+A*P+PBS*P=Q in £(V,V'), P(T)=L.(39)

(ii) P and P are related as pet) = (A*)-!P(t)A-!

( 40)

and P is the solution in the class P ( . ) E CO([O,T;C+(V)) s nCl([O,T[;£ (D',H) n£ (H,D)) s s

(41 )

of the following equation in [O,T[ dP _ ~PA+A*P+PBB*P=Q in £(H,H), P(T)=L.D (42) Remark . In M.SORINE (1981b) the class of operator-valued functions in which the solu-

REFERENCES Balakrishnan, A.V . (1978) . Boundary control of parabolic equations : L-Q-R theory . In Theory of nonlinear operators (Proc. Fifth Intern.Summer School, Central Inst. Math.Mech . Acad.Sci . GDR, Berlin 1977) Abh.Akad.Wiss.DDR, Abt.Math.Naturwiss Tech., 6, Akademie-Verlag, Berlin. pp.1l-23. Balakrishnan, A. V. (1981). On a class of Riccati equations in a Hilbert space. App1.~lath.Optim .• 2, 159-174. Bardos, C. (1971) . A regularity theorem for parabolic equations. Journal of Functional Analysis, 2. Lasiecka, I. (1978). Boundary control of parabolic systems: regularity of optimal solutions. Appl.Math.Optim., 4, 301-327. Lasiecka, I. (1980a). State constraint control problems for parabolic systems: regularity of optimal solutions. ~. Math.Optim., ~, 1-29 . Lasiecka, I. (1980b). Unified theory for abstract parabolic boundary problems a semigroup approach. Appl.Math.Optim., 6, 287-334. Lasiecka, I. and Triggiani, R. (1981). Dirichlet boundary control problem for parabolic equations with quadratic cost: analyticity and Riccati's feedback synthesis. Communication at 10th IFIP Conference on System Modelling and Optimization, New York, September 1971. Lions, J. L. (1968). Contr6le optimal des systemes gouvern~s par des ~quations aux d~riv~es partielles. Dunod, Paris . Lions, J.L. (1979). Nouveaux espaces fonctionnels en theorie du controle des systemes distribues. C.R.Acad.Sc.Paris, Ser.A, 289, 315-319. Lion~L.-cT962 ) . Espaces d'interpolation et domaines de puissances fractionnaires d'operateurs. J.Math.Soc.Japan, 14. Lions, J.L. (1980). Function spaces arid optimal control of distributed systems, Universidade Federal de Rio de Janeiro. Lions, J.L. and ~lagenes, E. (1968a). Problemes aux limites non homogenes et ~ cations. Vol.l, Dunod, Paris. Lions, J.L . and Magenes, E. (1968b). Problemes aux limites non homogenes et ~ cations. Vol.2, Dunod, Paris. Sorine, M. (198la) . Un result at d'existence et d'unicite pour l'equation de Riccati stationnaire. Rapport INRIA no. SS . Sorine, M. (198lb). Sur le semigroupe non lineaire associe a l'equation de Riccati. Rapport du CR~~ no. 1055, Universite de ~Iontreal .

Sorine, ~1. (1977). Sur les equations de Chandrasekhar. C.R.Acad.Sc.Paris t.285, Ser.A, p. 911. Tartar, L. (1972). Interpolation non-lineaire et regularite. Journal of Functional Analysis, ~, 469-489.