Copyrighl © IFAC 3rd Svmposium Control of Distributed Pa ramrt('r Systcms Tou louse. France . 1982
THE LINEARQUADRATIC 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 timevarying 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 timein 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)
1·
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 LIONSMAGENES (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 aCa .. (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~thand 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)vat ,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 vtAv =
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 LinearQuadratic 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 lll,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 timeinvariant 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 ! fBB*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 fBB*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 VH coercive, BN AIB E £(U,H), Q = A*QA E £(V,V'), L = A*LA E £(H,H) and (AI), denotes the timederivative of AI. (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 VH coercive l ~ Ah E L~(O,T;£s(U,V')), Q = A*!QAl E
P (t) =A _1 (t) *P(t ) AI (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 TIMEINVARIANT 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 operatorvalued functions in which the solu
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Sorine, ~1. (1977). Sur les equations de Chandrasekhar. C.R.Acad.Sc.Paris t.285, Ser.A, p. 911. Tartar, L. (1972). Interpolation nonlineaire et regularite. Journal of Functional Analysis, ~, 469489.