C.P.6128, Suet.A, MontrEal (QuEbec). Canada, H3C 3J7. I Introduction. This paper deals with the linear quadratic optimal control problem over an infinite time.
Linear
Quadratic
Control
Problem
without
Stabilizability
Giuseppe Da Prato (t) Scuola Normale Superiore Piazza dei Cavalieri 7, 56126 Pisa,Italy Michel Delfour(z) Centre de recherches matt~matiques et D6partement de MathEmadque et de Statistique Universit6 de MontrEal C.P.6128, Suet.A, MontrEal (QuEbec) Canada, H3C 3J7 I Introduction. This paper deals with the linear quadratic optimal control problem over an infinite time horizon for infinite dimensional systems in Hilbert spaces with bounded control and observation operators. In [1] the authors have recently constructed examples where the system is not stabilizable and yet the algebraic Riceafi equation has a positive self-adjoint unbounded solution. This phenomenon is intimately related to the fact that stabilizability only occurs for a dense subset of initial conditions. The object of this paper is to fill up this gap in the theory. Under no stabilizability hypothesis we a priori define the set 2; of initial states which can be stabilized and show that it can be given a natural Hilbert space structure. When Z is dense in the space of initial conditions, we construct the minimum positive self adjoint unbounded solution to the algebraic Riccati equation. A new technique is introduced to directly obtain the semigroup associated with the closed loop system and the properties of the feedback operator. If the usual detectability hypothesis is added we recover the fact that the closed loop system is exponentially stable. Examples are also included to illustrate the theoretical considerations. Extensions to systems with unbounded control and observation operators are possible and will be reported in a forthcoming paper. We felt that it was more instructive to first illustrate the phenomenon and the main features of the theory for the bounded case.
2 Problem formulation.
Let H ( state space), U ( control space ) and Y ( observation space ) be three Hilbcrt spaces. Let A: D(A)CH-~H be the infinitesimal generator of a strongly continuous semigroup etA. Let BeL(U;H) and C~L(H:Y) be the control and observation operators. Consider the system
(1) Work partially supported by the Italian National Project M,P.I. 40% "Equazioni di Evoluzionc e Applicazioni Fisico-Matcmaticho" (2) 'ntis rcscaxeh has been suppoxted in part by Canada Natural Sciences and "Engineering Research Coundl Grant A8730 while the author was a Killa.m fellow from the Canada Council.
127 x'(s) = Ax(s) + Bu(s) (2.1)
x(O)
,
s>0
= h
and thc associated cost function
(2.2)
J(u,h) = o~ ICx(s)12+lu(s)lz} ds
Dcnotc by V the value function V(h) = inf{ J(u,h) ; ue L2(0,oo;U))
(2.3) with domain (2.4)
dom
V = {hell
; V(h)
0 x(0) = h
(3.6)
and the observation x(s) ( the observation operator C is the identity). Consider the cost function
(3.7)
J(u,h) = J{ Ix(s)12+lu(s)l 2 } ds
It is well known that if the pair (A,B) is stabilizable, then there exists a bounded symmetric normegative linear operator P** on H which is the minimum normegative solution of the algebraic Riccati equation (3.8)
P**A+A*P. -P= BB*P, + I = 0
However, it is easy to check that the unbounded operator (3.9)
P . e k = (k+l)ek, kEN
is the only solution to (3.8). This means that only initial conditions h in the domain D(P~/2) of 1:)1/2
[
tx
(3.10)
:;
[0,oo] is convex, strict and lower semi-continuous, with domain (4.4)
Z = {hEH ; ¢(h) < oo}
130 L e m m a 4.1. The following statements hoM : (i) For all h and k in Z, (P(.)h,k) is bounded. (ii) Z is a vector subspace of H. (iii) For all h and k in Y., the foUowing limit exists
(4.5)
v(h,k) = lim (P(t)h,k) t-.¢.o*
Moreover qt is a bilinear form on Y_,xZand
(4.6)
~(h,h) = ~b(h),
X/hE H
Proof. (i) For all h and k in r. and t_>0, we have
(4.7) l(P(Oh,k)l2 _ h. Moreover there exists Z20 such that
(4.1O) and
Ihnl2 + O(hn) --*X
131
(4.11)
0(hn) ~ ~.-Ihl2.
By lower semi-continuity of 0, we have (4.12)
X-lhl2 = lnim_m.~(h . n) > (~(h)
and, by definition of Z, h belongs to Z. Finally, for each ¢>0. there exists a positive integer N(£) such that Ihn-hml~ = Ihn-hml2+ 0(hn-hm) < e , Vm,n->N(e) As n goes to inf'mity we get Ih-hml2+ ~(h-hm) < £ , Vm2.N(£) by continuity of the nonm in H and lower semi-continuity of ~. This shows that hn'-'> h in Z and completes the proof.# We have constructed the space Y. of initial conditions for which the expression (P(t)h,h) has a limit. In general its closure in H will not be dense and it will be natural to decompose H as a direct sum (4.13)
H = Z(BZ ±
where E is the closure of Z in H and Z ± is the orthogonal complement to X. In the sequel we identify the elements of the dual H' of H with those of H. We shall denote by Z' the dual of Z and by t0-0 and tn>T A
ut~ --->~ in L2(0,T;U)-weak, ^xt, --->~ in LZ(0,T;H)-weak
But for tn>T T (P(tn)h,h) -> J{ICxt (s)12+lut,(s)12}ds
and by weak lower semicontinuity T
~b(h)>_ J{IC~i(sll2+l~(s)12lds As T goes toinfinity (5.18)
dd(h)> J~lC~(s)i2+i~(s)12}ds = J(~,h)
Combining (5.18) and (5.13) it follows that there exists fl = fl (.,h)~ L2(0,,,o;U) such that J(~,h) __.0(h) < J(u,h) , Vue L2(0,oo;U) It follows V(h) ~ J(~,h) ~ ~b(h) < V(h). This establishes (5.2). As for the uniqueness of 0, assume that Ol and {t2 are two optimal controls in L2(0,oo;U). Then
136 J({~l'h) = J(~2,h)= V(h). So for 61 ~ 62
J((~l + 62)/2'h) = l[J({~l'h)+J((62'h)] -J(({~l "~2 )/2'h) = = V(h)-J((61 - ~2)/2,h) < V(h)- 41-II~1 - ~2 112< V(h) which contradicts the optimality of ClI and 6 2 . •-
A
(u) Let u t be defined by (5.13), then ^ 2 IlutllL~o,**;u ) < (P(t)h,h) < ihl2
(5.19)
Moreover, since the optimal control is unique, in the step (i) we have proved that tlim~t= 6,
in L2(0,*o;U) weak, for any he Z
We now prove that Aut---> uA in L2(0,*o;U)-strong. By optimality of the pair (xt,ut) on [0,t] jt(ut,h) = Inf{jt(v,h) ; vE L2(0,o*;U) } whcre t
jt(v,h) = J{ICx(s;v)12+lv(s)12}ds We want to prove that
U~nJ'(ueh) = J(~,h) By definition of the minimizing element u t on [0,t] t
Jt(ut,h ) a .]t(~ (.,h),h) = Oj'{IC~.(s,~(.,h))12+l~(s,h)12} ds and nccc~sarily litmsupjt(ut,h) < ~lC~(s,6(.,h))12+l~(s,h)l 2} ds = J(6 (.,h),h) We have shown in Section (i) that 0t---> D,
in L2(0,**;U) weak
137
and we can show by the same technique that {C~,} is bounded in L2(0,**;Y) and that weak subsequenees {C~t.} converging to some y in L2(0,**;Y) can be extracted : C~t.---> y, in L2(0,o*;Y) weak By continuity of the state x(.;u) with respect to the control u on a t'mite time interval [0,T], T>0, the map u ~ x(.;u) : L2(0,T;U)--> L2(0,T;I-I) is weakly continuous and finally u --)
Cx(.;u) : L2(0,T;U)--~ Lz(0.T;Y)
is also weakly continuous. This implies that for all T>0, y = C~(~,h) in L2(0,T;Y) and hence in L2(0,*~;Y). As a result 0(") ~, in L2(O,,~;U) weak and C,~t--¢ C~, in L2(O,~;Y) weak. But the functional (v,y) •---> ~ly(s)12+lv(s)lZ}ds) : LI(0,~;U)xLZ(0,,,,,;Y) --~ R
6"
is lower weakly continuous and necessarily limt~** inf ~ IC~tl2 + I~tl2]ds > ~lC~l 2 + I~12]ds o o that is
~im"mfJt(ut,h) _>J((~,h) Finally J({~,h) < ltim£mfjt(ut,h) < lirat~.supjt(u,,h), < J(~,h) and this proves that ~m2'(u,,h) = JC~,h) .
The strong continuity will now be obtained by the following simple computation ,C~- C~P + fistula 2 = ,c.~,, 2 + Jlfltti2 + ilc~.tt 2 + ,~II 2 -2(c~,,c9.) -2(Q,,0) =
= jt(ut,h) + J(~,h) -2(C~t,Ct ) -2(~r{~). As t goes to ~ jt(ut,h) --->J(n,h) and by weak convergence (c~,,c~) -> (c~,c~) = ,cRit2 .
138
(0t,0) ---> (0.0) -II0112 . So we conclude that tlirn {liCit - C~II2 + II~at-ull2} = 2J(~,h) - 2[11C~112+ I1~112] = 0 and
~t--~ O, in L2(0,,,~;U)-strong and C~t---+ C~¢, in L2(0,=,;Y)-strong. By (5.18) and by the Uniform Boundnedess Theorem it follows that the mapping X -~ L2(O,~;U), h -~ 0 (.,h) is linear and continuous. (iii) We first remark that, by Bellman's Optimality Principle we have ~ (t,h) ~ Z for all h~ Z and (5.20)
~(t+s,h) = ~(t;~(s,h)) , Vt20, Vs~0
(5.21)
V(~ (t,h)) = ~lC~(s,h)12+lfi(s,h)12}ds t
Thus Sx(t) is a linear operator in Z for all ~0. We prove now that St(t) is bounded in Z. By (5.17) we have I
(5.22)
~ (t,h) = etAh + Je(t'S)hB~a(s,h)ds
It follows that for any T>0 there exists Cr>0 such that
(5.23)
IS (t,h)q 0
,[ ~(t ',h)-h = f{ {Ci(t+s'h)'Ct'(s'hL " t C~,'(s,h)12+1 ;0+s'h)'0(s'h) - t
} ds
o
As t goes to zero the Fkrst two terms go to zero and necessarily oo
l~ r[ O(t+s)-Q(s) ,O(s)12ds t--*0 d
t
-
= 0
which implies v~ = ~' and ~E H l (0,oo;O), X/hE D(A;c ). By (5.22) it follows that hE D(A) and (5.6) foUows. (v) We have shown in (ii) that the map h.--~ 0(.,h) : Z ~ L2(0,oo;U), is linear and continuous. In particular h.--> ~'(.,h) : = ~(., Axh): D(Az)---) LX(0,oo;U) is also continuous. Hence h---->~(.,h) : D(Ax)..-4 H1(0,=,;U) is linear and continuous when D(A~;) is endowed with the graph norm topology : IIhll~(Ad = llhl~ + llAxhll 2 . In particular, t~(oo) = O, ~ E C([O,oo];U) and thc map h--+ (~(0,h): D(Ar)--> U islinearand continuous. W e denote itby K. Equivalendy K is a cloud linearunbounded operator from Z to U with domain
D(K) = [ hEX: Khe U I D D(Ax) In view of this and identity (5.6) k/hE n(Ar.), Axh = Ah + BO(0,h) = [A+BK]h
140
Conversely, if he D(A)nD(K), Azh = Ah +BKh ~ h~ D(Az) anfd D(Ax) = D(A)c~D(K). (vi) To relate K and the limit of P(t), we go back to formula (4.2) with heZ, u = Q(-,h) and x =
~(.,h) : t
(5.24)
t
Jt~(s,h)+B'P(t-s)~(s,h)Fds =
z +
JOC~'(S,h)~'+ta(S,h))=]dS
As t goes to infinity we obtain t
lira fl~(s,h)+B*P(t-s)~(s,h)12ds = 0 t---->**
Setting P(r) = 0 for r~_0, then (5.25)
~
fl~(s,h)+B*P(t-s)~(s,h)12ds = 0
since
~im f&~,h))2~ = 0. I--).**t"
Now repeat thc same estimate with Azh instcad o f h and ~(.,Azb) = ~'(.,h) , ~(.,Azh ) = ~'(-,h). Then by the same argument (5.26)
~
>'(s,h)+B*P(t-s)~'(s,h)12ds
=0
Introduce the notation and use (5.25) and (5.26) : A
*
A
vt(s ) = u(s,h)+B P(t-s)x(s,h),
vt---)0 in Lz(0,~;U)
wt(s) = 0'(s,h)+B*P(t-s)~'(s,h),
wt---)0 in L2(0?,,;U)
For h in D(Az) , differentiate (5.24) with respect to t t
~t ~(0h,h> + t0(0,h)+B P(t)~,2+2J( But for t'>-.t
= iC~(t,h)12+l~(t,h)l 2
141 - ->0 ~ d > 0 and notice that t
Hence o
-
0, ~ ~ (7.6)
- ~ t , ~ ) = bzlxl(t,x) + bz2xz(t,x) + j2_==lgj(t)Vj(t,x), t>0, ~ xl(O,x ) = hl(X ), ~e f~ x2(O,x) = h2(x), ~,e xl(t, ~) = O, x2(t, ~) = O, t>O, ~ a ~
where we assume that cx, bij : R + R
are given real numbers, with ~ > O, and ¢1,",¢j;
~x""VJ ~ C(~) are linearly independent functions. Choose H= L2(~'2)xL2(~'2), U = RJxR j ; setting
(7.7)
Ix,]
x= x2 ,h=
[h,] h2
u= r (rl,'',f,)] Bu= 4>i(t''): L(gr..,gj) j ,
LJ =i Vj(t,') ]
and (7.8)
b=[
bit b i 2 ] b21 b22
we can write system (7.6) in the abstract form (2.1). The spectrum o(A) of A consists in two sequences of semi-simple eigenvalues {~.±(k)}k~N and the accumulation point (7.10)
~ , = -b22
The eigcnvalues L±(k) are defined by (7.11)
2L±(k) = ½ {-tXgk+ Tr(b)+[(-CqXk+ Tr(b))2+4(CqXkb2z- det(b)] }
where are the eigenvalues of the Laplacian with Dirichlet boundary conditions. Now it is easy to check hypothesis 7.1, so that we can apply Proposition 7.1.#
147 References
[1] G.Da Prato and M.C.Dclfour, Stabilization and unbounded solutions of thc Riccati equation, Proc. 27 th IEEE Confcrcncc on Dccision and Control, pp.352-357, IEEE Publications, N.Y., 1988 [2] J.Evans, The Stability of Ncrvc Impulses, I: Linear Approximations, Indiana Univ. Math. Journal, 21 ,pp. 877-885, 1972.
