Domaine de Voluceau. Universite de ... AX + Bu, ~(0) = x0 E U(A) , where A is .... (2.7) InfiJ(v,x&. : v E Lloc(O,m;U)l. 2. 3. Asymptotic behaviour and L -stability,. 2.
TP2 = 5:30 THE LINEAR QUADRATIC OPTIMAL CONTROL PROBLEM FOR INFINITE DIMENSIONAL
SYSTEMS OVER AN INFINITE HORIZON; SURVEY AND EXAMPLES* A. Bensoussan
Abstract Survey of currently available theory for systems the evolution of which canbedescribed by semigroupsofoperatorsofclass Co. Connect i o n between t h e c o n c e p t s o f s t a b i l i z a b i l i t y and d e t e c t a b i l i t y and t h e problem of existence and uniqueness of solutions to the operator Riccati equation. Examples and open problems. 1.
S.K. Mitter
Delfour Centre Recherches de Mathematiques Department of Universite deMontr6al Montreal, Canada H3C 357
M.C.
IRIA-LABORIA Domaine deVoluceau Rocquencourt,France
Introduction.
For systems described by ordinary differential equationstheinfinite-timequadraticcostproblem i s well-studied(cf. R. BROCKETT El], R . E . K A L W [l] , [ 2 ] , J.C. WILLEMS [ l ] , W.M. WONHAM [ l ] ) .T h i s problem has been studied for certain classes of inJ . L . LIONS [ l ]h a s finite-dimensionalsystems. s t u d i e d t h i s problem for abstract evolution equations of parabolic type andgiven a completesolut i o nt ot h e problem. LUKES-RUSSELL [l] havestudied t h i s problem for abstract evolution equations of the type dx = AX + Bu, ~ ( 0 )= x0 E U ( A ) , (1) dt
-
where A i s an unbounded s p e c t r a l o p e r a t o r ( c f . DUNFORD-SCHWARTZ [ l ] ) and B i s a l s o an unbounded o p e r a t o rs a t i s f y i n gc e r t a i nc o n d i t i o n s . LUKESRUSSELL [ l ] a l s o a l l o w unbounded o p e r a t o r s i n t h e costfunction. Usinganapproach o r i g i n a l l y due t o R . E . KALMAN [2]theyobtainanoperationaldifferent i a l equation of Riccati type to charactqrize the time case. time-varyingfeedbackgaininthefinite They a l s o show t h a t under an a p p r o p r i a t e s t a b i l i z a bility hypothesis the solution to the infinite-time quadratic cost problem can be obtained in feedback is characterized by form, where t h e "feedbackgain" t h e s o l u t i o n o f an operator equation of quadratic R. type. The same problemhasalsobeenstudiedby DATKO [SI, Unfortunately R . DATKO [5] doesnot characterize the solution as a feedback c o n t r o l l e r a c t i n g on the "state" o f the system.
Electrical Engineering Massachusetts Institute of Technology Cambridge, Mass. 02139, U.S.A.
R . DATKO [3] and of LUKES-RUSSELL [l] aswellas c o n s t i t u t e s a s y n t h e s i s o f t h e work of J.L. LIONS [ 11 and DELFOUR-McCALLA-MITTER [ l ] , Complete r e s u l t s and d e t a i l l e d argument a r e t o befound i n a forthcoming monograph ( c f . BENSOUSSAN-DELFOURMITTER [ l ] ) . Our r e s u l t s a l s o make useofthe work of R. DATKO 121 on S t a b i l i t y Theory i n H i l b e r t spaces and J . ZABCZYK [ l ] on theconceptofdet e c t a b i l i t yi nH i l b e r ts p a c e s .I nd o i n gt h i s , we i n s i s t on anapproach which c l a r i f i e s t h e systemt h e o r e t i c r e l a t i o n s h i p between c o n t r o l l a b i l i t y , s t a b i l i z a b i l i t y , s t a b i l i t y and existence of a solution of an associated operator equation of Riccati type.
This theory covers certain classes of distributedcontrols; it alsocovershereditarysystems which canbelooked as distributed parameter system withboundarycontrol. A t t h i s time, i t does not seem possible to systematically deal with boundarycontrolproblems. However, i n a d i f f e r e n t framework, r e s u l t s are now a v a i l a b l e ( c f . D. L. RUSSELL [6]). Notation Let X and Y be two real Hilbert spaces with norms I I X and I l y and innerproduct ( , )x and ( , )y. The spaceofallcontinuouslinear maps T : X +- Y endowed with the natural norm
IT1 = sup{ITxly : / x i x
5
13
will bedenoted C(X,Y). When X=Y we s h a l lu s et h e T in n o t a t i o n c (X). The transposedoperatorof P(X,Y) i s an element of C. (Yl , X t ) which will be denoted T*, where X ' and Y ' arethetopological dualof X and Y. T i n C(X) i s s e l f - a d j o i n t i f T*=T;a s e l f - a d j o i n t o p e r a t o r T i s p o s i t i v e , T 2 0 , i f f o r a l l x i n X (x,Tx) 2 0.
2.
Preliminaries andproblemformulation.
i s to survey availThe o b j e c t i v e o f t h i s p a p e r a b l e r e s u l t s f o r a s p e c i a l class of i n f i n i t e dimensional control systems the evolution of which i s characterized by a semi-group of operators of class Co. The approach we usehere i s d i f f e r e n t from t h a t
Let X, U and Y berealHilbertspaces.Let B be an elementof L(U,X) and l e t A be an unbounded closedoperator on X with domain U(A). We assume that A is the infinitesimal generator of a strongly continuoussemi-groupCS(t) : t 2 0) of c l a s s C o . We denoteby U(A) t h e domain of A endowed with the graph norm
* This research was supported by NationalResearch
(2.1)
Councilof Canada Grant A-8730 and by a FCAC Grant of the Ministry of Education of Quebec.
We w r i t e i : V
2
llvi12 = IvI2 +-
+
2
IAv12.
X the continuous dense injection
of
V i n t o X. W e a l s oi n t r o d u c et h et o p o l o g i c a ld u a l s and X' of V and X, respectively. We i d e n t i f y elementsof X and X ' anddenote by i * t h e a d j o i n t map of i:
V I
v
(2 * 2)
->i
x
**
(iv) The p a i r (A,B) i s s a i dt ob es t a b i l i z a b l ei f it i s s t a b i l i z a b l e w i t h r e s p e c t t o t h e i d e n t i t y with Y=X. 0 Theorem 3.2. The followingstatementsareequivalent : ( i ) A i s L2 - s t a b l ew i t hr e s p e c tt o H;
z X' L> V'.
We now considerthesystem (2.3)
5(
x(0) = x. E V
= Ax + Bv i n [0,-[,
o r more generally
(ii)
t x ( t ) = S(0)xo
(2.4)
+
,/
Equation(2.4)canbelooked a t a s a "weak solution" ofequation(2.3) and any solutionof(2.3) will be of the form (2.4).
W e associate with the control function t h e t r a j e c t o r y x the cost function =
1 [IHx(t) l y2
+
(Nv,v)"
5
Asymptoticbehaviour
a j o i n ts o l u t i o no f (3.6), then D 2 B. Moreover f o r a l l x i n X , t h e map t 3 (A(t)x,DA(t)x) i s a monotonicallydecreasingfunctionof t and f o r a l l x and y i n X
2
c(vlu.
(3.7)
(3.8)
2
with observation y ( t ) = Hx(t).
(3.2)
Definition 3.1. ( i ) with respect to H i f
Any of thestatementsin
Theorem 3.2
V x E V , lim HS(t)x = 0 , and, t-
and L - s t a b i l i t y ,
k = Ax i n [O,-[, x(0) = x,,
lim (A(t)x,DA(t)y) = (x,Dy)-(x,By). t-
Corollary 2 . 'impliesthat
I n o r d e r t o make senseofproblem(2.7), it i s necessary to introduce concepts of s t a b i l i t y t o characterize the asymptotic behaviour of solutions as t h e time t goes t o i n f i n i t y . ofsystem(2.3) Consider the uncontrolled system (3.1)
.
+
Given x0 , theoptimalcontrolproblemconsists i n minimizing thecostfunction(2.5)over all v in L:OC(O,m;U) 2 : v E Lloc(O,m;U)l. (2.7) InfiJ(v,x&
3.
A*Di + i*DA + i*H*H = 0 i n I:(V,V').
R. DATKO 121, DELFOUR-McCALLA-MITTER Proof:Cf. T~~,BENSOUSSAN-DELFOUR-MITTER[ 11) 0 Corollar 1. I f D i n 1: (X) i s a p o s i t i v e s e l f -
(Nv(t),~(t))~ldt,
where H and N belong t o L ( X , Y ) and L(U), respect i v e l y . Moreover there e x i s t s a constant c > 0 such t h a t V v,
( i i i ) There exists a p o s i t i v es e l f - a d j o i n te l e ment D of I: (X) such t h a t (3.6)
0
(2.6)
t (3.5) V x,y E X, (Bx,y) = lim ,f(HA(t)x,HA(t)y)ydt; t- 0
v and
m
(2.5)J(v,xo)
There e x i s t s an element B o f L ( X ) such
that
S(t-s)Bv(s)ds.
0
V x E X, lim BS(t)x = 0. t*
0
The above r e s u l t s a r e s p e c i a l i z e d i n t h e following theorem on L 2 - s t a b i l i t y . Theorem 3.3. The followingstatementsareequiva1ent ( i ) A i s L2-stable; (ii) There e x i s t s an element B of t(X) such that t (3.9) V x,y E X, (Bx,y) = l i m ,/(S(t)x,S(t)y)dt; t+= 0
.
A i s s a i dt ob eL 2 - s t a b l e
( i i i ) Thereexists a p o s i t i v es e l f - a d j o i n t element D of L(X) such t h a t m
(3.3)
tl x,
0
IHx(t) I2dt
I
