The Linear Quadratic Optimal Control Problem Over an Infinite Time Horizon for a Class of Distributed Parameter Systems

THE LINEAR QUADRATIC OPTIMAL CONTROL PROBLEM OVER A N INFINITE TIME HORIZON FOR A CLASS OF DISTRIBUTED PARAMETER SYSTEMS M.C. Delfour Centre de Recherches Mathimatiques, Universiti de Montrkal, Montrial, Quebec, H3C 317, Canada

ABSTRACT This paper surveys currently available theory for linear infinite dimensional systems the evolution of which can be described by semigroups of operators of class Co. For systems of linear ordinary differential equations the infinite-time quadratic cost problem is well-studied (cf .R.BROCKETT [l] , R.E.KALE.lAN [l] , [Z] , J.C.WILLEE1S [l] , W.M.WONW! [I]). This problem has also been studied for certain classes of infinite-dimensional systems (cf. J.L.LIONS [l], LUKES-RUSSELL [I], R.DATK0 [3], [S] , [7]). This paper makes use of the work of R.DATK0 [2] on Stability Theory in Hilbert spaces and J.ZABCZYK [l] on the concept of detectability in Hilbert spaces. In doing this, we insist on an approach which clarifies the system-theoretic relationship between controllability, stabilizability, stability and existence of a solution of an associated equation of Riccati type. This theory covers certain classes of distributed controls; it also covers hereditary systems which can be looked at as distributed parameter systems with boundary control. At this time, it does not seem possible to systematically deal with mundary control problems. However some results are now available (cf. H.O.FATTORIN1 [I], [Z], [3], [4], H.L.KOH [l], J.L.LIONS [I]) 1. INTRODUCTION For systems described by ordinary differential equations the infinite-time quadratic cost problem is well-studied (cf. R.BROCKETT [I], R.E.KALMAN [1],[2], J.C.WILLEMS [I], W.M.WONHAM This problem has been studied for certain classes of infinite-dimensional systems. J.L. [l]). LIONS [l] has studied this problem for abstract evolution equations of parabolic type and given a complete solution to the problem. LUKES-RUSSELL [I] have studied this problem for abstract evolution equations of the type dt

=

Ax

+

Bu,

x(0)

=

xo Cv(A),

where A is gn unbounded spectral operator (cf. DUNFORD-SCHWARTZ [I]) and B is also an unbounded operator satisfying certain conditions. LUKES-RUSSELL [l] also allow unbounded operators in the cost function. Using an approach originally due to R.E.KALMAN [2] they obtain an operational differential equation of Riccati type to characterize the time-varying feedback gain in the finite time case. They also show that under an appropriate stabilizability hypothesis the solution to the infinite-time quadratic cost problem can be obtained in feedback form, where the "feedback gain" is characterized by the solution of an operator equation of quadratic type. The same problem has also been studied by R.DATK0 [S]. The objective of this paper is to survey available results for a special class of infinite dimensional control systems, the evolution of which is characterized by a semi-group of operators of class Co. The approach we use here is different from that of LUKES-RUSSELL [l] as well as R.DATK0 [3] and constitutes a synthesis of the work of J.L.LIONS [l] and DELFOURMcCALLA-MITTER [I]. Complete results and detailed argument are to be found in a forthcoming monograph (cf. BENSOUSSAN-DELFOUR-MITTER [I]). Our results also make use of the work of R. DATKO [2] on Stability Theory in Hilbert Spaces and J.ZABCZYK [l] on the concept of detectability in Hilbert spaces. In doing this, we insist on an approach which clarifies the systemtheoretic relationship between controllability, stabilizability, stability and existence of a

* This research was supported by the National Research Council of Canada Grant A-8730 and by a FCAC Grant of the Ministry of Education of Quebec. 57

M. C. Delfour

solution of an associated operator equation of Riccati type. This theory covers certain classes of distributed controls; it also covers hereditary systems which can be viewed as distributed parameter systems with boundary control. As this time, it does not seem possible to systematically deal with boundary control problems. However, in a different framework, results are now available (cf. H.L.KOH [l]). Notation

IX

I ly

and and inner product ( , )X Let X and Y be two real Hilbert spaces with norms / and ( , )y. The space of all continuous linear maps T:X -c Y endowed with the natural norm

will be denoted L(X,Y). When X = Y we rator of T in L(X,Y) is an element of Y' are the topological dual of X and Y. adjoint operator T is positive, T ? 0,

shall use the notation L(X). The transposed opeL(Yt ,XI) which will be denoted T*, where XI and T in L(X) is s e l f - a d j o i n t if T*= T; a selfif for all x in X (x,Tx) ? 0.

2. PRELIMINARIES AND PROBLEM FORMULATION Let X,U and Y be real Hilbert spaces. Let B be an element of L(U,X) and let A be an We assume that A is the infinitesimal unbounded closed operator on X with domain V(A). generator of a strongly continuous semi-group {S(t) :t ? 0) of class C0' We denote by V(A) the domain of A endowed with the graph norm

We write i:V -+ X the continuous dense injection of V into X. We also introduce the topological duals V' and XI of V and X, respectively. We identify elements of X and XI and denote by i* the adjoint map of i: i* XI-+ V'.

i V+X We now consider the system

or more generally x(t)

=

S(0)xo

+ j

t S(t-s)Bv(s) 0

ds.

Equation (2.4) can be looked at as a "weak solution" of equation (2.3) and any solution of (2.3) will be of the form (2.4). We associate with the control function v and the trajectory x the cost function

where H and N belong to L(X,Y) c z 0 such that

and L(U),

respectively. Ploreover there exists a constant

2 Vv, ( N V , ~ )2~ clvl,,. Given x0 , the - optimal control problem consists in minimizing the cost function (2.5) over 1 all v in Lloc(O,-;U) 2 Inf{J(v,xo) : v € Lloc(O,m;U)).

In order to make sense of the problem (2.7), it is necessary to introduce concepts of stability to characterize the asymptotic behaviour of solutions of system (2.3) as the time t goes to infinity. Consider the uncontrolled system i = Ax in [O,-[, x(0) = x0

The Linear Quadratic Optimal Control Problem with observation 2

(i) A is said to be L -stable with respect to H if

Definition 3.1.

-

i w t ) l 2 dt -= -, 0 2 2 (ii) A is said to be L -stable if it is L -stable with respect to the identity I with Y = X. (iii) The pair (A,B) is said to be stabilizable with respect to H if

i

axo,

Vxo, 3v

€

2 L (0,-;U)

00

2 such that j I~x(t)l~ dt

c

0

-

(iv) The pair (A,B) is said to be stabilizable if it is stabilizable with respect to the identity I with Y = X. o Theorem 3.2. The following statements are equivalent: 2 (i) A is L -stable with respect to H; lii) There exists an element B of L(X) such that

such that (iii) There exists a positive self-adjoint element D of L(x) * * * * A D i + i D A + i H H = 0 in L(V,Vt).

(3.6)

Proof: Cf. R.DATK0 [ 2 ] , DELFOUR-McCALLA-MITTER [I], BENSOUSSAN-DELFOUR-MITTER [l]).o Corollary 1. If D in L(X) is a positive self-adjoint solution of ( 3 . 6 ) , then D 2 B. Moreover for all x in X, the map t -+ (A(t)x,DA(t)x) is a monotonically decreasing function of t and for all x and y in X lim (A(t)x,DA(t)y) t+ Corollary 2.

=

(x,Dy) - (x,By).

Any of the statements in Theorem 3.2 implies that Vx

=

0

and

Vx

-

X, lim BS(t)x 3. tThe above results are specialized in the following theorem on L2-stability. €

V, lim HS(t)x t-

q

€

Theorem 3.3. The following statements are equivalent. (i) A is L2 -stable. (ii) There exists an element B of L(X) such that

(iii) There exists a positive self-adjoint element D of L(X)

such that

(iv) (v)

is strictly negative. The type o of the semi-group A(t) 0 There exist p c 0 and M s 1 such that for all x in X and t

(vi)

lim S (t) t-

=

0 in L ( X ) .

2

0

o

4. THE OPTIMAL CONTROL PROBLEM Ih! [O,-[

In order to make sense of problem (2.7) we must check that for each xo there exists at 2 leas: one v in Lloc(O,-;U) such that the cost J(v,xo) be finite. This is precisely the stabilizability of the pair (A,B) with respect to H.

PI. C . Delfour Theorem 4 . 1 .

Let

control function

u

-

ments o f back law

L

L

1oc

(

O

(A,B)

b e 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 H, t h e n t h e r e e x i s t s a unique

"

in U

L ; ~ ~ ( O , - ; U ) which minimizes Moreover t h i s c o n t r o l f u n c t i o n

where n i s a p o s i t i v e s e l f - a d j o i n t element o f L ( x ) . solution o f t h e operator R i c c a t i equation A

*

ni

J(v,xo)

*

*

*

+ i HA + i [H H-nRTI]i

=

0,

u

f o r a given

xo

over a l l e l e -

can be s y n t h e s i z e d v i a t h e f e e d -

o f X i s t h e minimal

The t r a n s f o x m a t i o n

i n L(V,V1),

R

BN

=

-1 * B .

There i s a c o n n e c t i o n between t h e e x i s t e n c e o f s o l u t i o n s t o t h e R i c c a t i e q u a t i o n ( 4 . 2 ) and t h e s t a b i l i z a b i l i t y o f t h e p a i r (A,B) w i t h r e s p e c t t o H. Theorem 4.2. The p a i r (A,B) 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 H i f and o n l y i f t h e r e e x i s t s a p o s i t i v e s e l f - a d j o i n t s o l u t i o n n t o t h e R i c c a t i e q u a t i o n ( 4 . 2 ) such t h a t

where

x

i s t h e s o l u t i o n o f t h e closed Zoop system

In g e n e r a l !I w i l l n o t b e t h e unique p o s i t i v e s e l f - a d j o i n t s o l u t i o n o f ( 4 . 2 ) and t h e closed2 loop syFtem (4.4) w i l l n o t b e L - s t a b l e . It

N e v e r t h e l e s s a s u f f i c i e n t c o n d i t i o n i s a v a i l a b l e i n o r d e r t o answer t h o s e two q u e s t i o n s . makes u s e o f t h e c l a s s i c a l concept o f d e t e c t a b i l i t y a s i n t r o d u c e d by W.M.WONHAM [ I ] . D e f i n i t i o n 4.3.

The p a i r

i s s a i d t o be detectable i f t h e p a i r

(A,H)

(A',H*)

is s t a b i l i z -

a b l e (A' i s t h e i n f i n i t e s i m a l g e n e r a t o r o f t h e a d j o i n t semi-group { ~ ( t ) * ]o f { S ( t ) ) . Remark. Notice t h a t we d i s t i n g u i s h between A~ and t h e o p e r a t o r A*:H' + V ' d u a l o f A:V -+ H. We have j u s t s e e n t h a t when t h e p a i r (A,B) 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 H, it i s poss i b l e t o c o n s t r u c t a c o n s t a n t feedback K = - N - ~ B *i ~n L(X,U) i n o r d e r t o s t a b i l i z e t h e p a i r (A,B) w i t h r e s p e c t t o H. Hence D e f i n i t i o n 3 . 1 ( i i i ) is completely e q u i v a l e n t t o D e f i n i t i o n 4.4. ( i ) The p a i r (A,B) i s s a i d t o b e stabiZizabZe by feedback w i t h respect t o H if t h e r e e x i s t s K i n L(X,U) such t h a t

axe,

OD

00

I

:1

I H X ( ~ )dt + 0

I

2 I K X ( l~U ) d t <

-.

(4.51

0

( i i ) The p a i r (A,B) i s s a i d t o b e stabiZizabZe by feedback i f t h e r e e x i s t s such t h a t m 2 vx0, f J H X ( ~d It <~ - ~. 0 The g e n e r a l i z a t i o n o f t h e r e s u l t s o f W.M.WONHAM s i o n a l c a s e i s due t o J . ZABCZYK [ I ] . Theorem -4 . 5 .

K

in

L(X,U) (4 61

[ l ] and V . K U ~ E R A [ l ] t o t h e i n f i n i t e dimen-

Let t h e p a i r (A,B) be 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

H

and l e t t h e p a i r (A,H)

2

be d e t e c t a b l e . Then t h e c l o s e d l o o p system (4.4) i s L - s t a b l e and i 7 i s t h e unique p o s i t i v e s e l f - a d j o i n t solution o f equation (4.2). 5 . RELATIONSHIPS BETWEEN CONTROLLABILITY AND STABILIZABILITY Assume f o r a moment t h a t X = R", U = R ~ , Y = R' and t h a t t h e o p e r a t o r s A and B a r e m a t r i c e s o f dimension nxn and nxm, r e s p e c t i v e l y . This i s t h e s o - c a l l e d f i n i t e dimensional c a s e , where t h e concept o f c o n t r o l l a b i l i t y i s d e f i n e d and c h a r a c t e r i z e d a s f o l l o w s : Definition 5.1.

The p a i r (A,B) is s a i d t o b e controZZable i f Vxo € X , 3 T ,

where

x(t;xo,v)

2 0 , 3v t L (0,T;U)

such t h a t

is t h e solution of t h e d i f f e r e n t i a l eauation

x(T;xo,v)

=

0,

The Linear Quadratic Optimal Control Problem

Theorem 5.2. The following conditions are equivalent: (i) (A,B) controllable; (ii) Given any spectrum a of a real nxn matrix, there exists an mxn matrix K such that the spectrum, a(A+BK), of A+BK is exactly a; An - 1 B]=n. (iii)Rank[A,AB ,..., This theorem now says that when the pair (A,B) is controllable it is necessarily stabilizable by feedback. The converse is obviously not true. When X,U and Y are infinite dimensional spaces, Definition 5.1 can be retained, but conditions (ii) and (iii) are difficult to generalize. However the following straightforward result remains true. Theorem 5.3.

The pair (A,B) is stabilizable if the pair (A,B) is controllable.

For the infinite dimensional problem, the concept of controllability (to the origin) as introduced in Definition 5.1 differs from the concepts of exact or approximate reachability as studied by H.O.FATTORIN1 [l] to 131, D.L.RUSSELL [l] to [6], R.TRIGGIAN1 [I] to [6] and M.SLEE1ROD [l] to [4]. The last two authors have done an extensive study of the relationship between the two concepts of reachability and stabilizability. In the same spirit, E.W.KAMEN [Z] has developed an algebraic approach to this problem for hereditary differential systems. Given the system (2.3), we define for each t > 0 the reachable set K(t) at time t from 0, t 2 K(t) = {j S(t-s)Bv(s) ds:v € L (O,t;U)), K- = U K(t). (5.3) 0 t>O

Xo =

Definition 5.4. The pair (A,B) has the approximate (resp.exact) K_

=

X

(resp. Km

=

X).

reachability property if (5.4)

o

When (A,B) has the exact reachability property, it is stabilizable. However it is a very strong property as can be seen from the following theorem. Theorem 5.5. If B:U X is compact and if X is infinite dimensional, then (A,B) never has the exact reachability property (cf. R.TRIGGIAN1 [4],[5]). -+

When we go to the weaker concept of approximate reachability, we need to add extra hypotheses on the spectrum a(A) of A in order to insure stabilizability. Given 6

7

0 we define

Spectrum decomposition assumption. There exist 6 > 0 and a rectifiable simple closed curve (or more generally, a finite number of such curves) so as to enclose an open set containing aU(A) in its interior and as@) in its exterior. Under the above assumption the operator A may be decomposed according to the decompositivn X = X +X . there exists a projection operator P on XU = PX along Xs = (I-P)X such that U

S'

Theorem 5.6. We make the following hypotheses: (i) the spectrum #(A) spectrum decomposition assumption for some 6 > 0, (ii)

of A satisfies the

(jii) the space XU is finite dimensional. Then (A,B) is stabilizable if (A,B) has the approximate reachability property (cf.R.TRIGGIAN1

r411. n

M. C. Delfour

This theorem covers many interesting classes of infinite dimensional systems including hereditary differential systems in the Hilbertian product space framework (cf.bI.C.DELFOUR 131, However there are counterexamples to the implication that apDELFOUR-McCALLA-MITTER [l]). proximate reachability implies stabilizability. The following example can be found in R. TRIGGIANI [4]. Let X = E 2 , A be the left shift operator (bounded, but not compact) and let 1 1 ,... ) E 8'. It can be shown that the pair (A,B) has the approximate reachB = (1, 7,..., ability property, but it is not stabilizable and does not have the exact reachability property. This is due to the fact that o(A) is the unit disc and that the essential spectrum a (A) is the unit circumference. Since B:R -+ E 2 is compact, it is clear that no compact ess (A) which has a non-zero intersection with the right perturbation of A will affect a ess hand side of the complex plane. Example.

Recent work (cf. A.MANITIUS [l] and DELFOUR-MANITIUS [l]) has shown that approximate reachability is still too strong for the stabilizability of hereditary differential systems. A weaker concept of F-reachability has been introduced and complete results and examples have been given. This shows that we have not yet found the right minima2 co:~cept of reachability which would fit a22 infinite dimensional systems. General ideas and concepts begin to emerge, but a systematic case by case study would be extremely useful and would help understand what is really pertinent in this problem. 6. EXAMPLES

We shall give in this section a list of examples to which the general theory developed here may apply 6.1. Second order parabolic systems This type of problem is studied in full detail in the book of J.L.LIOKS [I]. The reader will find in this reference numerous examples. Notice that for such systems the operator A (and hence A*) is stable. As a result the conditions of stabilizability and detectability are automatically verified. 6.2. First order hyperbolic systems Such problems have been studied by N.BARDOS [I] and J.L.LIONS [I]. Under appropriate hypotheses we can make sense of such problems for distributed controls. 6.3. Boundary control In many distributed parameter systems, the control is exerted on the boundary and the previous framework is not completely appropriate for that situation. Boundary control problems have been studied by H.O.FATTORIN1 [1],[4], D.L.RUSSELL [I] to [S], GRAHAM-RUSSELL [I], and R.DELVER [I]. In some instances it is possible to reformulate the problem in such a way that the boundary control becomes a distributed control. This can be achieved by lifting the original problem to a big enough space that the control becomes distributed (cf. V.P.KHATSKEVICH [I], B.FRIEDMAN [I]) . Another approach has been suggested by A.V.BALAKRISNAN [2]; it results in a distributed conof the original trol problem with respect to the derivative 9 and the initial value v(0) boundary control v. Although our theory does not apply directly to these systems, similar methods can be developed (see A.V.BALAKRISNA3 [2]). An interesting example of boundary control is given by hereditary differential systems (cf. DELFOUR-MITTER [l]). When we consider the system in state form this system is controlled through a differential equation on the boundary. This problem cannot be dealt with in the present framework if we choose as state space the continuous function or a Sobolev space. 2 However in the product space M2 = !Rnx~(-a,O;iRn) this problem reduces to a distributed control problem and the above theory can be readily applied (cf. DELFOUR-McCALLA-MITTER [l]). Detailed result on exact and approximate reachability and the relationship with stabilizability are now available (cf. MANITIUS-TRIGGIANI [l] to [ 3 ] , A .MANITIUS [I], DELFOUR-MANITIUS [l]). Another problem which is not covered in the framework of this paper is the occurence of delays in the control function. Interesting approaches have been suggested by A.V.BALAKRISNAN [2], CURTAIN-PRITCHARD [3], A.ICHIKAWA [l] ,[2], D.C.WASHBURN [I].

The Linear Quadratic Optimal Control Problem

A.V.BALAKR1SNAf-J [I], Introduction to Optimization Theory in a Hilbert Space, Springer-Irerlag, Berlin, 1971. A.V.BALAKRISNAN [2], Identification and stochastic control of a class of distributed systems with boundary noise, in Control Theory, Numerical Methods and Computer Systems Modelling, A.Bensoussan and J.L.Lions,eds., 163-178, Springer-\'erlag, New York, 1975. N.BARDOS [l], Thgse de doctorat d'ktat, Paris 1969 A.BENSOUSSAN, M.C.DELFOUR and S.K.MITTER [I], Representation and control of infinite dimensional system (Reports ESL-P-602,603 and 604, Electronic Systems Laboratory, MIT, Cambridge,Mass.;June 1975). A.BENSOUSSAN, M.C.DELFOUR and S.K.MITTER [Z], The linear quadratic optimal control problem for infinite dimensional systems over an infinite horizon; survey and examples, Proc. 1976 IEEE Conference on Decision and Control, 746-751, Clearwater,Fla., December 1976. K.P.M.BHAT [l], Regulator theory for evolution systems, Proc. 1976 Allerton Conference on Circuit and System Theory, U. of Illinois at Urbana-Champaign. R.W.BROCKETT [l], Finite Dimensional Linear Systems, J.Wiley, New York, 1970. J.P.COMBOT [I], Doctoral dissertation, Purdue University, Lafayette, Indiana, 1974. R.F .CURTAIN and A.J .PRITCHARD [l] , The infinite-dimensional Riccati equation, J .Math.Anal. Appl. 47 (1974), 43-57. R.F.CURTAIN and A.J.PRITCHARD [2], The infinite-dimensionalRiccati equation for systems defined by evolution operators,SIAbl J.on Control and Optimization 14 (1976),951-983. R.F.CURTAIN and A.J.PRITCHARD [3], An abstract theory for unbounded control action for distributed parameter systems, Control Theory Centre Report 39, University of Warwick, Coventry. R.DATK0 [l], An extension of a theorem of A.M.Lyapunov to semi-groups of operators, J.Math. Anal.App1. 24 (1968), 290-295. R.DATK0 [ Z ] , Extending a theorem of A.M. Lyapunov to Hilbert space, J .Math.Anal.Appl. 32 (1970) 610-616. R.DATK0 [3], A linear control problem in an abstract Hilbert space, J.Differentia1 Equations 9 (1971), 346-359. R.DATK0 [4], Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J .Math.Anal. 3 (19721, 428-445. R.DATK0 [5], Unconstrained control problem with quadratic cost, SIAM J.Contro1 11 [1973), 32-52. R.DATK0 [6], Neutral autonomous functional equations with quadratic cost, SIAM J.Contro1 12 (1974), 70-82. R.DATK0 [7], Some linear nonautonomous control problems with quadratic cost, JDifferential Equations 21 (1976). R.DATK0 [8], The stabilization of linear functional differential equations, Internal Report (1976). R.DATK0 [9], Differential-difference equations in Hilbert spaces, Proc. 1976 Allerton Conference on Circuit and System Theory, U. of Illinois at Urbana-Champaign. M.C.DELFOUR [l], Theory of differential delay systems in the space M ~ stability ; and the Lyapunov equation, in Proceedings of the Symposium on Differential Delay and Functional Equations: Control and Stability, Ed. L.Markus, Control Theory Centre report 12 (1972), 12-15. M.C.DELFOUR [2], GSnCralisation des rSsultats de R.DATK0 sur les fonctions de Lyapunov quadratiques ddfinies sur un espace de Hilbert, CRM-Report 457, University of Montreal, January 1974.

64

M. C. Delfour

F1.C.DELFOUR [3], State theory of linear hereditary differential systems, to appear in J.Math Anal. and Appl . M.C.DELFOUR and A.E.lASITIUS [I], Control systems with de1ays:areas of applications and present status of the linear theory, Proc.Int.Symp. on New Trends in Systems Analysis, Dec. 1976, Springer-Verlag, to appear. Bl.C.DELFOUR, C.McCALLA and S.K.MITTER [I], Stability and the infinite-time quadratic costcontrol problem for linear hereditary differential systems,SIAhl J.Contro1 13 (1975),48-88 M.C.DELFOUR and S.K.PlITTER [I], Controllability, observability and optimal control of affine hereditary differential systems, SIMI J.Contro1 10 (1972), 298-328. M.C.DELFOUR and S.K.PIITTER [2], Controllability and observability for infinite-dimensional systems, SIN1 J.Contro1 10 (1972), 329-333. R.DELlrER [I], Boundary and interior control for partial differential equations, Can.J.Plath. 27 (1975), 200-217. N.DUNFORD and R.S.SCHWARTZ [I], Linear Operators I,II,III, Interscience, New York, 1967. H.O.FATTORIN1 [l], Some remarks on co~pletecontrollability, J.SIM1 Contrz~l4 (1966), 686-694. H.O.FATTORIN1 [2], On complete controllability of linear systems, J.Diff.Eq. 3 (1967),391-402. H.O.FATTORIN1 [3], Controllability of higher-order linear systems, blathematical Theory of Control, A.V.Balakrisnan and L.Neustadt,eds., Academic Press, New York, 1967, 301-311. H.O.FATTORIN1 [4], Boundary control systems, SIAM J.Contro1 6 (1968), 349-385. B.FRIEDMAN [I], Principles and Techniques of Applied Mathematics, \Vile!., N.Y., 1956. K.GLASHOFF and N.WECK [I], Boundary control of parabolic differential equations in arbitrary dimensions:supremum-norm problems, SIAM J. Control and Optimization 14 (1976),662-681. K.GRAHAM and D.L.RUSSELL [I], Boundary value control of the wave equation in a spherical region, SIAM J. Control 13 (1975), 174-196. E.HILLE and R.S.PHILLIPS [l], Functional Analysis and Semi-Groups, AMS, Providence,R.I.,1957. A.ICHIKAWA [I], Generation of a semigroup on some product space with applications to evolution equations with delay, Control Theory Centre report 52, University of Warwick, Coventry. A.ICHIK4WA [ Z ] , Optimal quadratic control and filtering for evolution equations with delay in control and observation, Control Theory Centre report 53, University of IVarwick, Coventry. R.E.KALMAN [I], On the general theory of control systems, Proc. 1st IFAC Congress, Bloscow, Butterworths, London 1960. R.E.KAUWN [2], in "Contributions to the Theory of Optimal Control", Bol.Soc.Mat.Mexicana 5 (1960), 102-119. E.W.KAMEN [I], Module structure of infinite-dimensional systems with applications to controllability, SIMI J. Control and Optimization 14 (1976), 389-408. E.N.KAMEN [2], State and input feedback in systems containing time delays, Proc. 1976 Allerton Conference on Circuit and System Theory, U. of Illinois at Urbana-Champaign. V.P.KHATSKEVICH [I], On the problem of the analytical design of regulators for distributedparameter systems under boundary-function control, Prikl.Mat.Meh. 35 (1971), 598-608 (Engl .transl. J .Appl.Plath.Mech. 35 (1971), 548-558 (1972)). H.L.KOH [I], Structure of Riccati equation solutions in optimal boundary control of hyperbolic equations with quadratic cost functionals, PIRC Technical Seminary Report 1642, Mathematical Research Center, Madison Wisc., June 1976. V.KUCERA [I], A contribution to matrix quadratic equations, IEEE Trans.Aut.Contro1 AC-17 (1972), 344-347.

The Linear Quadratic Optimal Control Problem V.JURJEVIC [I], Abstract control systems: controllability and observability, SIN1 J.Contro1 8 (1970), 424-439. J.HENRY [I], to appear. J.L.LIONS [I], Contr6le Optimal de Systsmes Gouvern6s par des Equations aux DGriv6es Partielles, Dunod, Paris,1968. (English translation, Springer-Verlag, Berlin, New York, 1971.) J.L.LIONS and E.MAGENES [l], Problsmes aux Limites ):on Homog2nes et Applications 1 et 2, Dunod, 1968; 3, Dunod, 1969, Paris. D.L.LUKES and D.L.RUSSELL [I], The quadratic criterion for distributed systems, SIN1 J. Control 7 (1969), 101-121. A.MANITIUS [I], Controllability, observability and stabilizability of retarded systems, Proc. 1976 IEEE Conference on Decision and Control, 752-758, December 1976. A.MANITIUS and R.TRIGGIAN1 [I], Function space controllability of linear retarded systems: a derivation from abstract operator conditions, Internal report CN1-605, Centre de Recherches Mathhatiques, Universit6 de Plontr6a1, Canada, March 1976. A.MANITIUS and R.TRIGGIAN1 [2], New results on functional controllability of time-delay systems, Proc. 1976 Conference on Information Sciences and Systems, The John Hopkins University, Baltimore, hlaryland (1976), 401-405. A.MANITIUS and R.TRIGGIAN1 [3], Function space controllability of retarded systems:a derivation from abstract operator conditions (announcement), Proceedings of International Conference on Dynamical Systems, Univ. of Florida, Gainesville,Florida, March 24-26 1976, Academic Press (to appear). A.PAZY [I], On the applicability of Lyapunov's theorem in Hilbert space, SIAM J. Math. Anal. 3 (1972), 291-294. A.J.PRITCHARD [I], Stability and control of distributed parameter systems, Proc.IEEE (1969), 1433-1438. A.J.PRITCHARD [2], The linear-quadratic problem for systems described by evolution equations, Control Theory Centre Report 10. D.L.RUSSELL [l], The quadratic criterion in boundary value control of linear symmetric hyperbolic systems, Control Theory Centre Report 7, 1972. D.L.RUSSELL [2], Quadratic performance criteria in boundary control of linear symmetric hyperbolic systems, SIAM J. Control 11 (1973), 475-509. D.L.RUSSELL [3], On the boundary-value controllability of linear symmetric hyperbolic systems, Mathematical theory of contrnl (Proc.Conf., Los Angeles, Calif.,1967), 312-321, Academic Press, New York, 1967. D.L.RUSSELL [4], Nonharmonic Fourier series in the control theory of distributed parameter systems, J.E.lath.Anal.App1. 18 (1967), 542-560. D.L.RUSSELL [5], A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Studies in Appl.Math. 52 (1973), 189-211. D.L.RUSSELL [6], Towards a general theory of control canonical structure, Proc. 1976 Allerton Conference on Circuit and System Theory, U. of Illinois at Urbana-Champaign. M.SLEMROD [l], The linear stabilization problem in Hilbert space, Journal of Functional Analysis 11 (1972), 334-345. M.SLEMROD [ 2 ] , An application of maximal dissipative sets in Control Theory, J.Math.Anal.App1. 46 (1974). 369-387. M.SLEMROD [3], A note on complete controllability and stabilizability for linear control systems in Hilbert space, SIMf J.Contro1 12 (1974), 500-508. M.SLEMROD [4], Asymptotic behavior of Co semigroups as determined by the spectrum of the generator, Rensselaer Polytechnic Institute internal report, January 1975.

M. C. Delfour

R.TRIGGIAN1 [l], Extensions of rank conditions for controllability and observability to Banach spaces and unbounded operators, SIAM J.Contx-01 and Optimization 14 (1976), 313-338. R.TRIGGIAN1 [Z], On the lack of exact controllability for mild solutions in Banach space, J. Math.Anal.App1. 50 (1975), 438-446. R.TRIGGIAN1 [3], Delayed control action controllable systems in Banach space, Control theory centre report 33, University of Harwick, England. R.TRIGGIAN1 [4], On the stabilizability problem in Banach space, J.Math.Anal.Appl.52 (1975), 383-403. R.TRIGGIAN1 151, Controllability and observability in Banach space with bounded operators, SIAM J.Contro1 13 (1975), 462-491. R.TRIGGIAN1 [6], Pathological asymptotic behavior of control systems in Banach space, J.Math. Anal.Appl.49 (1975), 411-429. R.TRIGGIAN1 171, On the relationship between first and second order controllable systems in Banach spaces, Proc.IFIP Working Conference on Distributed Parameter Systems, Rome (Italy), Springer-Verlag (to appear). R.B.VINTER and T.L.JOHNSON [l], Optimal control of non-symmetric hyperbolic systems in n variables on the half-space, MIT-report (unpublished). D.C.WASHBURN [l], A semigroup theoretic approach to the modelling of boundary input problems, in Proc. IFIP Working Conference on D.P.S.:modelling and identification, SpringerVerlag, 1976 (to appear). J.C.WILLEMS [I], Least squares stationary optimal control and the algebraic Riccati equation, IEEE Trans. Automatic Control AC-16 (1972), 621-634. A.WIRTH and A.J.PRITCHARD [I], Unbounded control and observation systems and their duality, Control Theory Centre report 44, University of Warwick, Coventry, June 1976. W.M.WONHAM [I], On a matrix Riccati equation of stochastic control, SIAM J. Control (1968), 681-697. W.M.WONHAM [Z], Linear multivariable control, a geometric approach, Springer-Verlag, New York, 1974. J.ZABCZYK [I], Remarks on the algebraic Riccati equation in Hilbert space, J.Appl.Math. and Optimization 2 (1976), 251-258. J.ZABCZYK 121, On semigroups corresponding to non-local boundary conditions with applications to system theory, Control theory centre report 49, University of Warwick, Coventry, Oct. 1976.