Optimal control of stable weakly regular linear ... - Semantic Scholar

Math. Control Signals Systems (1997) 10:287-330 9 1997 Springer-Verlag London Limited

Mathematics of Control, Signals, and Systems

Optimal Control of Stable Weakly Regular Linear Systems* Martin Weiss# and George Weiss:~ Abstract. The paper extends quadratic optimal control theory to weakly regular linear systems, a rather broad class of infinite-dimensional systems with unbounded control and observation operators. We assume that the system is stable (in a sense to be defined) and that the associated Popov function is bounded from below. We study the properties of the optimally controlled system, of the optimal cost operator X, and the various Riccati equations which are satisfied by X. We introduce the concept of an optimal state feedback operator, which is an observation operator for the open-loop system, and which produces the optimal feedback system when its output is connected to the input of the system. We show that if the spectral factors of the Popov function are regular, then a (unique) optimal state feedback operator exists, and we give its formula in terms of X. Most of the formulas are quite reminiscent of the classical formulas from the finite-dimensional theory. However, an unexpected factor appears both in the formula of the optimal state feedback operator as well as in the main Riccati equation. We apply our theory to an extensive example. Key words. Regular linear system, Optimal cost operator, Riccati equation, Popov function, Spectral factor, Feedthrough operator.

1. Introduction

A well-posed linear system is called weakly regular if its transfer function has a weak limit at +o% along the real axis. Such a system may have unbounded control and observation operators, which may correspond, for example, to boundary control and boundary observation in a system described by partial differential equations. In this paper we investigate the quadratic optimal control problem for weakly regular linear systems. We consider the infinite horizon problem, that is, the time runs from zero to infinity, and we assume that the system to be controlled is

* Date received: April 22, 1996. Date revised: April 22, 1997. Part of the results reported here were obtained while the second author was visiting FUNDP Namur, under the Belgian Program on InterUniversity Poles of Attraction initiated by the Belgian state, Prime Minister's Office, Science Policy Programming. The scientific responsibility is assumed by the authors. t Faculty of Applied Physics, Techn. Univ. of Eindhoven, 5600 MB Eindhoven, The Netherlands. School of Engineering, University of Exeter, Exeter EX4 4QF, England. 287

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stable, in a sense which is made precise later. The cost to be minimized is the integral (with respect to time) of a bounded quadratic form in the input and the output of the system. Since the output might, in particular, be the state, this problem includes as a particular (and simpler) case the problem where the quadratic form is in the input and the state. A well-posed linear system is called regular if its transfer function has a strong limit at + m , along the real axis. Such systems have been studied extensively in the recent literature, see, for example, [W3], [W4], and the references therein. Weakly regular systems constitute a slight generalization of the class of regular linear systems, and we owe the reader an apology about why we chose to make this generalization. Indeed, the class of regular linear systems is quite general, and it has the important advantages of being closed under feedback and cascade connections, which is not true for weakly regular systems. If we could carry out our investigation within the class of regular systems, the generalization would not be justified. However, even if we start with a regular system, we have to introduce weakly regular systems later, or else make restrictive assumptions. This is because the dual of a regular system is not necessarily regular, while weak regularity is inherited by the dual. Since dual systems play an important role in the quadratic optimal control problem, it seems more natural to work entirely in the class of weakly regular systems. If the input and output spaces of the original system are finite-dimensional, then the difference between regularity and weak regularity disappears, for all the systems considered in this paper. There are two main approaches to solving linear quadratic optimal control problems on an infinite time horizon: (a) Solving an algebraic Riccati equation we obtain the optimal cost operator, and then the optimal feedback operator follows easily. For infinitedimensional systems, this approach has been studied extensively for bounded control and observation operators, as well as for the PritchardSalamon class, and the results are very similar to the finite-dimensional theory, see, for example, [CZ] and the references therein. For unbounded control operators the problem becomes much more technical, see [FLT], [LT1], or [LT2]. Some results in this direction (mainly for parabolic equations) can be found [BDDM]. (b) After finding a spectral factorization of the associated Popov function, the optimal feedback operator can be computed using various procedures, see [M] or [K]. Afterwards, the optimal cost operator is computed easily. For infinite-dimensional systems, this approach has been studied in a series of papers by Callier and Winkin, see [CWl], [CW2], and the references therein. They consider systems with bounded control and observation operators. In [CW2], to obtain the optimal feedback operator from the spectral factor, an operator diophantine equation has to be solved (see also [G1] and [G2]). When trying to generalize these approaches to infinite-dimensional systems with unbounded control and observation operators, we encounter additional difficulties: it is far from clear if the optimal feedback operator is admissible for the

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original semigroup, and even less clear if the closed-loop optimal system is weakly regular, or at least well-posed. It is also not easy to write a meaningful Riccati equation. We found that for weakly regular linear systems the first approach described above is very difficult, if not impossible. We propose following the spectralfactorization approach, assuming that a regular spectral factor of the Popov function is given. With this assumption, everything else works out nicely, almost as in the finite-dimensional case. However, in the formula linking the optimal cost operator with the optimal feedback operator, as well as in one term of the Riccati equation, a surprising new factor appears, a phenomenon discovered by Staffans [$3]. Staffans had started to work on similar problems before we started, and there were frequent contacts between him and us. He gave one of us early versions of [$3] and [$4] in the summer of 1995, and later we received from him [$6], [$7], [$8] and [$5], as well as other related material (all in several versions). There has been some influence in both directions (see also his comments on this in [$7]). In [$3] he considers systems whose impulse response is a matrix-valued finite measure. He assumes that a spectral factor is given which is also a finite measure. Then the optimal feedback can be computed from the spectral factor and the importance of the feedthrough operator of the spectral factor is revealed. The paper [$4] is the generalization of [$3] to unstable systems. The closest to our paper is his paper [$6], where the results from [$3] are extended to stable wellposed linear systems. His paper [$8] is the extension of this work to unstable wellposed linear systems, using his coprime factorization theory from [$7]. There is, unfortunately, a considerable overlap between the results in [$6] and our results. Nevertheless, our approaches are quite different: Staffans deals most of the time with well-posed linear systems (which are more general than our systems) and his analysis is entirely in the time domain, so that he is close to the spirit of Salamon [S1]. We assume that the reader has some familiarity with regular linear systems, especially with the basics presented in [W3] and with the feedback theory developed in [W4]. We often use concepts and results from these papers, without repeating them here. A few basic facts are recalled in Section 4, for easy reference. Our work originates in the Popov function approach to operator Riccati equations in the thesis of the first author [W6]. We want to emphasize that a good part of our results does not depend on the regular spectral factorization assumption. Thus, in Sections 7-11 we state and prove the results which we can obtain without having a regular spectral factor. In Section 11 we investigate the realization of spectral factors, and in Section 12 we show that if a regular spectral factor exists and its limit at + ~ is invertible, then the optimal feedback fits into the framework of the feedback theory from [W4]. 2. Formulation of the Problem

2.1. The Open-Loop System Now we describe in more precise terms the system to be controlled and we introduce notations which will be used throughout the paper. We assume that ~ is a

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weakly regular linear system with generating operators A, B, C, O. For the precise definition of a weakly regular system and its generating operators we refer to Section 4. Thus, Z is described by the equations =

Ax(t) + Bu(t),

(2.1)

y(t) = CAwx(t).

Here, A is the generator of a strongly continuous semigroup ~ on the Hilbert space o~, the state space. The space 5Yl is defined as ~(A) with the norm [Izll -- I1(/ I - A)zll, where fl ~ p(A), and Y'-I is the completion of 5f with respect to the norm [Izll_l = I1(/ I - A)-Iz[I. The choice of fl is not important, since different choices lead to equivalent norms on :gl and on s The Hilberty spaces q/ and q,' are the input and the output space of Z. We assume that o// is separable. B ~ s 5Y_1) is the control operator of Z, C~s qr is the observation operator of Z, and CAw is the weak A-extension of C, defined by CAwZ = weak lim (72(21 - A)-lz. (2.2) 2--++00

The domain of CAw consists of those z e s for which the above weak limit exists. In (2.1), x(t) is the state at time t > 0, u is the input function, and y is the output function. We assume that u~Lq2oc([0, oo),q/), which implies that y e L12oc([0,oo), ~). Equations (2.1) are satisfied for almost every t _> 0 (in particular, x(t) ~ D(CAw) for almost every t > 0). The extended output map of Z, q~: Y" --+ L2oc([0, oo), q,/), is given by

(q/xo)(t) = CAwTtXo,

for almost every

t > 0.

The extended input-output map of Z, ]F: L2oe([0, oo), a//) _.+/_~oe([0' oo), qr is

(lFu)(t) = CAw

ll't_~Bu(a) d~,

for almost every

t > 0.

With these notations, the initial state x(0) = x0 and the functions u and y from (2.1) are related by y = ~x0 + IFu. (2.3) We denote by G the transfer function of Z, G(s) = CAw(SI -- A ) - I B ,

which is a bounded and analytic s qC)-valued function defined on some right half-plane in C. Moreover, for any v e 0//, weak lim G(2)v = 0. 2---++oo

In the above weak limit, 2 is real. The existence of this limit means that g is weakly regular (see Section 4) and the fact that the right-hand side is zero means that the feedthrough operator of Z is zero. We recall the notation introduced so far: Z, oh,,~ , ~ , ~1, ~-1, A, B, C, CAw,T, ~P, IF, G.

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2.2. The Stability Assumptions

Our stability assumptions on the system Y~are the following: 9 Se(Sf, L2([0, oo), ~/)),

F 9 Le(L2([0, oo),

(2.4)

L2([0, oo),

(2.5)

These assumptions are satisfied, for example, if T is exponentially stable. However, they might also be satisfied by systems with an exponentially growing semigroup. It is easy to see that (2.4) is equivalent to the condition that C(sI - A)-lx0 9 H2(~r

(2.6)

for any x0 9 5f, where H 2 (Y/) is the Hardy space of qC-valued functions on the open right half-plane Co, which arise as Laplace transforms of functions in L2([0, oo), o2r In the terminology of Grabowski [G3] or Hansen and Weiss [HW], (2.4) means that C is an infinite-time admissible observation operator for T. Note that (2.6) can hold even if a part of a(A) is in Co: in this case, the function s ~ C(sl - A)-lx0 still has an analytic continuation to all of tE0. The assumption (2.5) is equivalent to the following condition on the transfer function: G 9 H~176 (s176 ~ ) ) , (2.7) where (for any Banach space ~ ) H ~~(L~) denotes the space of bounded analytic ~-valued functions on lE0. If E is stabilizable and detectable, then (2.7) is equivalent to the exponential stability of T. This follows by slightly adjusting the argument of Rebarber [R] to fit weakly regular systems. At this stage, the reader might find it useful if we summarize all the assumptions made on the system E: (a) E is a weakly regular linear system. (b) The feedthrough operator of Y. is zero. (c) E is stable, in the sense of (2.4) and (2.5). Our stability assumptions can be relaxed to stabilizability and detectability, as shown in [$8]. This makes the results considerably more complicated. It seems to be an open problem whether the main results extend to the case when the system only satisfies the finite cost condition. 2.3. The Optimality Criterion

Consider the quadratic cost function oo

Q

N*

910 0. Because of (2.7), this limit exists in the strong sense for almost every o9 ~ IR, see Theorem 4.5 (Fatou) in [RR]. We assume that ~ is strictly positive (i.e., positive and boundedly invertible) or, equivalently (see Section 7), that the Popov function is bounded from below: H(iog) > el,

for almost every

co ~ ]R,

where e > 0.

(2.12)

Since ~ determines the quadratic term in u in formula (2.9), condition (2.12) implies that J has a unique minimum (again, we refer to Section 7 for details).

3. Outline of the Results

In Section 4 we recall the definition of well-posed linear systems, of regular linear systems, and of other related concepts. We introduce weakly regular systems and describe their basic properties, using the weak A-extension. Section 5 is devoted to state trajectories and output functions of weakly regular systems on the time interval ( - ~ , 0]. In Section 6 we introduce the dual system and its anticausal version. Using the dual system, we give a representation of the operator •*, which is needed later. In Section 7 we show that under the assumptions of Section 2, for any initial


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state x0, the minimum of the cost J is attained for a unique input function U~ The optimal cost can be written as a quadratic form in xo, J(x0, U~

: (XXo, xo)x,

where X 9 5e(X) is the so-called optimal cost operator. We express u ~ and X by formulas which contain the inverse of the Toeplitz operator ~ . In Section 8 we introduce and analyze a certain interconnection of the system Z with its anticausal dual. The output of this interconnection is zero if and only if the input function is u~ In this case the state of the anticausal dual system is X applied to the state of E, at any moment of time. In Section 9 we show that the optimal state trajectories determine a strongly continuous semigroup T ~ on X. If A ~ denotes the generator of lr ~ then its domain ~ ( A ~ is difficult to characterize in terms of the initial data, at this high level of generality. The optimal input function u ~ is obtained from the state trajectory xOpt(t) _- at"'~ via a uniquely determined admissible observation operator F ~ which means that, for every x0 9 ~(A~ u~

= F~176

for all

t > 0.

If we denote by B~w the weak A-extension of B*, then, for every z 9 N(A~ RF~

= -- (n~kwX -[- UCAw)Z.

Thus, if R is invertible, then F ~ can be computed from X. Moreover, A~ = (A + BF~ holds for all z 9 ~(A~ In Section 10 we show that the optimal cost operator X satisfies several Riccatilike equations. If C is bounded, then these equations become simpler, and look like in the finite-dimensional theory. A drawback of these equations is that they hold on ~(A~ which is not a priori known. We would like to describe ~F~ as a closed-loop semigroup, in the framework of the feedback theory from [W4]. For this purpose, the above results are not satisfactory, since we do not know if F ~ is in some sense admissible for qF and we do not know if B is an admissible control operator for ~,opt. TO overcome these difficulties, we need the concept of spectral factorization. In Section 11 we consider spectral factors of the Popov function II. These are functions E 9 H ~ (&a(~ with E -1 9 H~176 (~q(q/)), which satisfy the identity I-I(ico) = E(ico)*E(ico),

for almost every

09 ~ IR.

If (2.12) holds, then spectral factors E exist, and we show that a spectral factor system corresponds to each E. This is a well-posed linear system having the same semigroup and the same control operator as 2, and its transfer function is E. We show by an example that E is not necessarily regular, even if q / = Y / = IE. However, we think that this is a rare occurrence for physically motivated systems, as are nonregular well-posed linear systems in general. Before outlining the contents of Section 12, we introduce a concept which is an important motivation for what follows. For any F: ~(A) ~ q/, we denote by FA

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the A-extension of F:

FAZ= lira F 2 ( M - A ) - l z ,

(3.1)

defined for those z s • for which the limit exists. 3.1. An operator F: ~(A) ~ q/is an optimal state feedback operator for the system X with the cost function J if the following three conditions hold: Definition

(1) A, B, F, and 0 are the generating operators of a regular linear system. (2) The feedback system s = Ax(t) + Bu(t), u(t) = Fax(t) + v(t) (with input v and output u) is well-posed. Equivalently, the ~(U)-valued transfer function I - F A ( S I - A)-IB has a uniformly bounded inverse on some right half-plane. (3) We have ~(A ~ c ~(FA) and the restriction of FA to ~ ( A ~ is F ~ Under the assumptions made so far, such an F does not necessarily exist. In Section 12 we find sufficient conditions for its existence. Assuming that such an F exists, let us see what are the immediate consequences of the properties in the deftnition, using the results listed earlier and the feedback theory from [W4]. It follows from (1) and (2) that the feedback system in (2) is regular. Now from (3) (and from the fact that A~ = (A + BF~ we conclude that A~ A + BFA with its natural (or maximal) domain. Hence, the semigroup corresponding to the feedback system X~ defined in condition (2) (shown in Fig. 1) is exactly 17~ If popt *A denotes the A-extension of F ~ (defined as in (3.1) but with F ~ and A ~ in place of F and A), then popt " a = FA (in particular, they have equal domains). Thus, F is the restriction of F~pt to ~(A), implying in particular that the optimal state feedback operator F is unique. We call X~ the optimal feedback system. Its generating operators are A ~ B, F ~ and I (in particular, B is an admissible control operator for qr~ The transfer function of X~ is

G~

=

(I - FA(SI -- A)-IB) -] = I + FA(SI -- A~

(3.2)

In Section 12 we assume that some (hence, every) spectral factor E is regular, i.e., the following limit exists for every v e r lim E(2)v = Dv. 24+00 The operator D e La(q/) is called the feedthrough operator of E. We assume further that D is invertible (if r finite-dimensional, then this follows from the invertibility of E). Thus, the generating operators of the spectral factor system are

Xo? +:~

F i g . 1.

A, B, F

The optimalfeedbacksystemZ~ If v

=

O, then u

=

u~


295

A,B, C ~, and D, for some C r ~ ~a(~Cl, if//). We define F = - D - 1 C r so that Z(S) = D ( [ - F A ( S [ -- A)-IB).

We show that F can be expressed in terms of X as follows:

Fz=-(D*D)-I(B~wX+NC)z,

for all

z~(A).

(3.3)

We prove that this F is the optimal state feedback operator for E and J. In particular, it follows with (3.2) that E-l(s) = (I + FA(Sl -- A~

= G~

Now it is possible to write the "true" Riccati equation on N(A):

A * X + XA + C* QC = (B~,wX

+ N C ) * ( D * D ) -1 (n~wX -[- NC).

(3.4)

Here, B*hwX + N C is a bounded operator from ~Y1 to ~//and all the terms in the Riccati equation have their range in ~Y~. In (3.3) and (3.4), where from finite-dimensional theory we are used to seeing R, we see D*D instead. This does not contradict the finite-dimensional theory, since if B is bounded, then D*D = R, implying that R is strictly positive. However, this is not the case in general, as can be seen from the following very simple example: take G(s) = e -~ (a delay line, v > 0) with the realization given in Section 2 of [W3]. The input and output space is C. Take R = 0, Q -= 1, and N = 0, then H(ir = 1, so that E(s) = 1, D = 1, and D*D ~ R. In Section 13 we solve the quadratic optimal control problem for a class of delay systems, showing that the problem can be reduced to the solution of a matrix discrete-time Riccati equation (which can be computed numerically).

4. Weakly Regular Systems We introduce some notation. Let ~ f be a Hilbert space. For z > 0, P~ is the projection of L2oc([0, 00), 3(e) onto L2([0, v), ~vf) (by truncation). St is the right shift by v on L12oc([0,~ ) , ~vf), or on/~oe((-0% 0], ~f), and S~* is the left shift by v on the same spaces. (If we restrict S~ and S* from L~o~ to L 2, then they are adjoint to each other.) For any u, v ~ L2oe([0, o0), ~ ) and any v > 0, the z-concatenation o f u and v, denoted u 0 v, is defined by u O v = P~u + S~v. Thus, (u ~ v)(t) = u(t) for t ~ [0, z), while (u ~ v)(t) = v(t - z) for t _> z. We recall the precise definition of a well-posed linear system, following [W3], slightly adjusted (without changing its meaning). In [W3], [W4], and other earlier papers, the term "abstract linear system" was used instead, but it seems that "well-posed" is much more to the point than "abstract."

296


Definition 4.1. Let ~//, Y', and Yr be Hilbert spaces, f2 = L2([0, o0),~) and F--L2([0, oe),~r A well-posed linear system on f~, &r, and F is a quadruple I2 = (T, ~, W, F), where: (i) T = (]Ft)t_>o is a strongly continuous semigroup of linear operators on ~r. (ii) q~ (~t)t> 0 is a family of bounded linear operators from f~ to ~ such that =

r

0 v) = T t ~ u + r 7?

for any u, v 9 f~ and any z, t ___0. (iii) W is a linear operator from ~r to L2oc([0, oc), ~/) such that, for any z > 0, W~ = P~qJ is in ~~ F) and, for any x0 9 Y', qJx0 = q~x00 WT~x0.

(4.1)

(iv) F is a liner operator from f~ to LlZo~([0,oe), ~r such that, for any r > 0, IF, = P~IF is in ~e(fL F) and, for any u, v 9 f2, lF(u O v) = lFu ~ (~@~u + lFv).

(4.2)

T

q/is the input space, YE is the state space, and ~/is the output space of 12. We recall some consequences of this definition, following [W2], [W3], and [W5]. Denote the generator of T by A and the spaces ~ and ~r_l are as in Section 2. It follows from assumptions (i) and (ii) above that there exists a unique B9s X-l), called the control operator of 12, such that, for all t > 0, 9 tu =

]Ft_crBu(cr) d~r.

(4.3)

The fact that @tu 9 Y( means that B is an admissible control operator for T. We see from (4.3) that qgtu depends only on Ptu, and so @t has a natural extension to 2 [0 , oe), q/). If x0 9 ~r is the initial state of Y~and u 9 L~oe([0, oo), q/) is its input Lioc( function, then the state trajectory of 12,x: [0, oe) --+ Y', is defined by x(t) -- Ttx0 + q),u,

(4.4)

for all t > 0. The function x is continuous and it satisfies the differential equation k(t) = Ax(t) + Bu(t),

(4.5)

in the strong sense, in Y'-I. The function x is the unique solution of (4.5) satisfying the initial condition x(0) ~ x0. If u has a Laplace transform it, then so does x and we have 2(s) = (sI - A ) - Bit(s), for all s with Re s sufficiently big. The operator W in Definition 4.1 is called the extended output map of 12. (In [W3] and in [W4] this operator was denoted tp~.) More generally, any operator which satisfies assumption (iii) in Definition 4.1 is called and extended output map for T. For every such 9 there exists a unique C 9 s qr called the observation operator of W, such that (Vexo)(t) = CT,xo,

(4.6)


297

for every x0 e Xt and every t > 0. This determines W, since X1 is dense in X. The function Yo = gTx0 has a Laplace transform Y0 and we have P0(s)= C(sI - A)-lxo, for all xo e X and for Re s sufficiently big. The A-extension of C is defined by CAz = lim C2(2I - A)-lz. (4.7) The domain ~(CA) consists of those z e X for which the above limit exists. If we replace C by CA in (4.6), then it holds for all x0 e X and almost every t > 0. If x0 e X1 and y = q~xo, then y e L2oe([0, ~ ) , ~/) and

~9= VAxo,

(4.8)

i.e., p(t) = CAqrtAxo, for almost every t > 0. The operator CAw, the weak A-extension of C, is defined as in (2.2). It is clear that CAw is an extension of CA, and CAw = CA if Y/is finite-dimensional. The operator IF in Definition 4.1 is called the extended input-output map of E. (In [W3] and in [W4], this operator was denoted IF~.) IF is shift-invariant, which means that S~IF = IFS~ holds for all z > 0. This implies that IF satisfies P~IF = P~IFP~,

for all

7 > 0,

(4.9)

a property known as causality. Using (4.9) we can extend IF to L~oe([0, ~ ) , q/). We can represent IF via the transfer function G of E, which is a bounded analytic 5~ Y/)-valued function on some right half-plane in ~. We do not distinguish between two transfer functions defined on different fight half-planes, if one is a restriction of the other. The connection between IF and G is as follows: If u e f~, then y = IFu has a Laplace transform ~ and, for Re s sufficiently big,

y(s) = G(s)ft(s).

(4.10)

If G is a bounded analytic ~e(q/, ~r function defined on some fight halfplane in 113,then a realization of G is a well-posed linear system Z whose transfer function is G. It was proved in [$2] that every G as above has realizations based on shift semigroups. We need this result only for G e H ~ (5r ~t)), but together with an additional property which follows from the construction in [$2]. Theorem 4.2. Every G e H~176 ~r has realizations E = (T, O, W, IF). For any such E, IF satisfies (2.5) (i.e., it is bounded). Moreover, E can be chosen such that tp satisfies (2.4) (i.e., it is bounded). Definition 4.3. The system E (or its transfer function G) is called weakly regular if the following weak limit exists in ~r for all v eog, weak lira G(),)v = Dv. .&--*+

(4.11)

In (4.11), 2 is real. Z (or G) is called regular if the limit in (4.11) exists in the norm topology of ~/. In either case, the operator D e &a(q/,~r defined by (4.11) is called the feedthrough operator of E (or of G).

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If ~r is finite-dimensional, then weak regularity equals regularity, of course. For the proofs of the following two theorems we refer to [W3] and [W5]. In fact, the proofs are in [W3] for regular linear systems and their extensions to weakly regular systems are straightforward.

Theorem 4.4. Weak regularity is equivalent to the fact that ( s I - A)-IBql c N( Caw),for some (hence, for any) s 9 p(A). Regularity is equivalent to the fact that (sI - A)-I Bql ~ N(CA). I f Z is weakly regular, then its transfer function is G(s) = CAw(SI -- A ) - I B q- D, and i f Z is regular, then Caw may be replaced by CA.

Theorem 4.5. Let Z be weakly regular. I f x is the state trajectory of Z corresponding to the initial state xo 9 3f and the input function u 9 L12oe([0,oo), ql) (as in (4.4)), then the output function of 2, y = Wxo + Fu, satisfies y(t) = CAwx(t) + Du(t),

(4.12)

for almost every t > O. I f t >_ 0 is such that both u and y are continuous from the right at t, then (using those right limits) (4.12) holds at t, in particular, x( t) 9 ~(Caw). I f Z is regular, then CAw may be replaced by CA. A,B, C,D are called the generating operators of Z, because Z is completely determined by them via (4.5) and (4.12).

5. System Behavior for Negative Time Until now we have considered the time to be positive. It is sometimes important to think of a well-posed linear system Z functioning on the time interval ( - o% 0]. We still use the notation from Definition 4.1 and from (4.5), (4.6), and (4.7). In addition, for 9 _> 0 we denote by P_~ y the truncation of the function y: ( - go, 0] qr to [-T, 0]. Definition 5.1. Let u 9 A function x: (-oo,0] ~ 3 f is a state trajectory of Y. vanishing at - c o , corresponding to the input function u if: (a) x satisfies (4.5), in the strong sense, in 3f-1, (b) limt--,_oo x(t) = O. Condition (a) above is equivalent to x(t) = qrHx(z) +

J'

qrt_,Bu(a) da,

(5.1)

.g

for all z, t 9 ( - oo, 0] with z _ t. It is easy to see (using the corresponding result for [0, oo)) that x is continuous (and hence bounded, by (b)). In general, state trajectories as in Definition 5.1 might not exist, or there might be infinitely many of them. The questions of existence and uniqueness of state


299

trajectories vanishing at - ~ are much more delicate than on [0, oo). Some aspects of these were discussed in [Wl]. Proposition 5.2 below gives sufficient conditions for existence and for uniqueness. The control operator B is called infinite-time admissible for ~? if the operators Ot from (4.3) are uniformly bounded (see [G3] or [HW] for details). In this case we can define the operator ~ ~ ~e(L2([0, oo), q/), ~r) by ~v = lim

T~ao

0

T~Bv(e) d~.

(5.2)

Indeed, it is easy to see that this limit exists in ~ . Note that we cannot write ~ in (5.2), since this Bochner integral might not exist. We have ~S~ = T ~ , for all ~ > 0 . For any function u defined on (-oo,0] or on [0, oo), we denote ~(t) = u ( - t ) , so that/~ is the reflection in time of u.

Proposition 5.2. I f B is infinite-time admissible for the semioroup ]r, then, for any u e L2((-oo, 0], ql), the function x : (-oo, 0] ~ W defined by x(t) = ~S*_t~ =

lira

T~-~

I'

T

~t_~Bu(~) d~

(5.3)

is a state trajectory of Z vanishing at -oo, corresponding to the input function u. I f T is uniformly bounded, then x defined above is unique with this property. Proof. It is easy to see that condition (a) from Definition 5.1 holds, because x satisfies (5.1). We have IIx(t)ll---I1~11" IIS*-,~ll, and clearly IlS*-,~ll ~ 0 as t ~ - o o , so that condition (b) holds as well. Let z be another state trajectory of Z vanishing at - ~ (for the same u), and denote e ( t ) = z ( t ) - x(t). Then e(t) = Tt_~e(z) holds for z < t < 0. If T is uniformly bounded, then from lim~_~_o~ e(z) = 0 we conclude that e(t) = 0 for all t < 0, so that x is unique. [] Proposition 5.3. Assume that Z is weakly regular and let u and x be as in Definition 5.1. Then for almost every t < 0 we have that x( t) e ~(CAw) and the function y defined by (4.12) is in/~2oc((-oo, 0], ~). I f t < 0 is such that both u and y are continuous from the right at t, then x( t) ~ ~(Caw). IrE is regular, then in the above statements CAw may be replaced by CA. Now assume that B is infinite-time admissible for qF and x is given by (5.3). Then, for every 9 > O, P_,y ~ L2([-z, 0], ~r depends continuously on u. We call y introduced above the output function of Z corresponding to u and x. Proof. Fix z > 0 and for t ~ [0, z] denote z(t) = x(t - ~), v(t) = u(t - "c). Then z is a segment of a state trajectory of Z, corresponding to the initial state z0 = x ( - r ) and the input function v, as in (4.4). By Theorem 4.5, z(t) e ~(CAw) for almost every t e [0, z]. Moreover, if we define the output function w via (4.12), i.e., w(t) : CAwZ(t) + Dr(t), then from w = P~[Wz(0) + Fv],

(5.4)

300


we see that w E L2([0, r], ~r Still by Theorem 4.5, if both v and w are continuous from the fight at t, then z(t)~ ~(CAw). Since w ( t ) = y ( t - ~), it follows that P_~y e La([-z, 0], ~). Since ~ was arbitrary, we have y e L~o~((-oo , 0], ~/). Now assume that B and x satisfy the additional assumptions. Then z(0) depends continuously on u, and since P~W and P ~ are bounded operators, it follows from (5.4) that w (and hence also P_~y) depends continuously on u. 9 We mention that if X is not assumed to be weakly regular, we can still define a unique output function y on (-o9,0], corresponding to u and x which are as in Definition 5.1, but this becomes slightly more complicated. Let ~ be a Hilbert space and y e L 2 ( ( - o o , 0 ] , ~ ) . We define the Laplace transform o f y on the left half-plane, i.e., for R e s < 0, as follows:

j

0

~(s) =

--

e-Sty(t) dt. 0, the adjoint of Wr = P r Y is

V~y =

IF~C*y(tr) dtr = edrw,

(6.4)

0 where w(a) = y ( T - tr), so that IIw[[ = Ilyf[. This shows that

I[~rll = 11~1[.

(6.5)

If (i) holds, then with (6.5) it follows that [l~ll --- IlVll, i.e., (ii) holds. Conversely, if (ii) holds, i.e., if II~drll _< M for all T > 0, then with (6.5) it follows that II~xoll -< M IIx011 for all x0 e Y" and all T > 0. This implies that Wx0 L2([0, ~z), Yr and IlVx011 -< M . [Ix011, so that (i) holds. The existence of the limit in (6.3) follows easily from (ii), and this limit is ~aw by the definition of ~d. It remains to prove the equality of the first two terms in


303

(6.3). It follows from (6.4) that this equality holds if w has bounded support. Since the set of such w is dense in L2([0, oo), ~'), (6.3) holds in general. 9 The following theorem and proposition give a description of the operator IF*.

Theorem 6.3. Let A, B, C, D be the generating operators of the weakly regular system Z. Suppose that the stability assumptions (2.4) and (2.5) hold for Y~. Let w 9 L2([0, oo), ~/) and define the function q: [0, oo) --. ~f by q(t) = ~P*Stw = lira

T-*ao t

]r*_tC*w(a ) da.

Then q(t) ~ N(B~w) fOr almost every t > O, and the function v = F*w satisfies v(t) = B~,wq(t ) + D*w(t),

for almost every

t > O.

(6.6)

Proof. Remember that we have denoted ~(t) = u(-t). We denote yd = ~ and xd = 0 (so that yd and x a are defined on ( - o e , 0]). By the definition of q and by (6.3) we have that, for all t < 0,

xa(t)

= v*S*,w = ~ds*_y.

Comparing this with (5.3) and using Proposition 5.2, we conclude that x d is a state trajectory of Z d vanishing at - 0 % corresponding to the input function yd. According to Proposition 5.3 we have xd(t) 9 N(B;tw) for almost every t < 0, and so we can define an output function ud via (6.1), for almost every t < 0. We want to apply Proposition 5.4 (with Z d in place of Z, yd in place of u, and u d in place of y). By Propositions 6.1 and 6.2, the assumptions of Proposition 5.4 are satisfied. Formula (5.5) becomes L2(i~.,og) = L2(i]Rq/) '

(6.7)

for every t / 9 L2((-oo, 0], q/), where Gd(ie)) = G(-io))*, by Proposition 6.1. If v is defined by (6.6), then clearly u d = ~, so that ~(io)) = ~d(--ico). It follows from (6.7) (by taking ~ = ~) that, for any p 9 L2([0, oe), og), @,/i>L2(iR,qz) = (G*~, ~>L2(iR,ql) = (W, G/~>L2(iN.,og) . Hence, by (4.10), = < w , F ~ > = < F * w , ~ > ,

so that indeed v = lF*w.

Proposition 6.4.

With the notation of Theorem 6.3,/f w 9 L 2 ([0, o0), ~), then also b 9 L2([0, oe), q/). Moreover, q(t) 9 ~(B~w ) and (6.6) hold for all t >_ O.

Proof. Under the assumptions, w belongs to the domain of the generator of the left shift semigroup (St)t_>0 , on the space L2([0, oo), ~J). Therefore, Stw is a continuously differentiable function of t, so that the same is true for q and for F*Stw = S~F*w = Sty. Hence, v is in the domain of the same generator (with

304


replaced by og), so that b ~ L2([0, oo),~ The fact that q is continuously differentiable allows us to define the continuous function r: [0, oo) -+ Y" by

r(t) = flq(t) + kl(t), where/~ s p(A*) is fixed. Then we can easily check (using (6.1)) that

q(t) = (ill - A*)-I It(t) + C*w(t)]. According to Theorem 4.4, q(t) ~ ~(B~w ) for all t > 0 and the function

h(t) : B~xwq(t) + D*w(t) = Gd(fl)w(t) + B~w(flI - A*)-lr(t) is continuous. Since, by (6.6), h(t) = v(t) for almost every t, and both functions are continuous, it follows that they are equal for all t > 0. 9 If y d is regular, then everywhere in this section B~w may be replaced by B~. Every well-posed linear system E = (~, ~, W, •) has an antieausal interpretation where, in the state trajectory x and in the input and output functions u and y, the time t is replaced by - t . More precisely, if u e L~oc((-o% 0], q/) and x0 ~ ~', then the anticausal state trajectory of E corresponding to the initial state x0 and the input function u is the function x: (-0% 0] --- ~r determined by

x ( - t ) = •txo + gPt~t. Similarly, the anticausal output function of E corresponding to x0 and u is the function y ~ L~oc((-oo, 0], Y/) determined by = ~Px0 + IF/~. Now suppose u ~ L2([0, oo), oR). An anticausal state trajectory of 2 vanishin9 at +o% corresponding to the input function u, is a function x: [0, oo) ~ ~r such that /~ and 2 are related as in Definition 5.1. For the existence and uniqueness of such trajectories we can appeal to Proposition 5.2. If Z is weakly regular, then the anticausal output function of ~ corresponding to u and x is defined by (4.12). The anticausal interpretation of a system is mostly applied to dual systems, which are then called anticausal dual systems. Some authors, such as Salamon [S1] or Staffans [$6], define the dual system to be anticausal, because it is the anticausal interpretation of the dual which appears in optimal control theory (see Section 8).

7. The Optimal Cost and the Optimal Input Function We use the notation and all the assumptions from Section 2, more specifically, the assumptions summarized at the end of Section 2.2 and the strict positivity of the Popov function, expressed in (2.12). In fact, the weak regularity of Z is not needed for the results of this section. They hold for well-posed linear systems. First we want to explain more clearly why (2.12) is equivalent to

~ > eI, which, in particular, implies the invertibility of ~.

(7.1)


305

Recall that, for any Hilbert space o~ and any z > 0, the operators of right and left shift by z on L2([0, 00), ~ ) are denoted S~ and S~, respectively. Let o~fl and ~vg2 be separable Hilbert spaces. A Toeplitz operator is a bounded operator Y from L2([0, o0),o~(1) to L2([0, m ) , ~ 2 ) which satisfies S*r = Y" for every z > 0, see, for example, [RR]. Every Toeplitz operator can be represented via a unique function ff e L ~ (iN, ~(Oefl, ~2)) (this means a weakly measurable essentially bounded function defined on the imaginary axis), called the symbol of 9-, in the following sense: If i denotes the Laplace transform of u, then, for any Ul E L2([0, ~), 3r and any u2 e L2([0, ~ ) , 3~v2),

l( ~" i l , i2)L2(iR,~2 ) 9

(~'Ul, U2) = ~

(7.2)

Conversely, every operator of the form (7.2) is a Toeplitz operator. For example, if 9- is pointwise application of an M e ~(W1,3r then ~b(io9)= M, for all o9 ~ IR. Suppose now that ~/gl = Jg2 = W. Using (7.2) it can be proved (by a short argument involving shifts on L2((-oo, ~3), ~ ) ) that Y-- is positive iff its symbol ~b is positive almost everywhere. Hence, 9- > e I i f f ~b(io9) >_ el for almost every o9 ~ lR. Thus, the equivalence of (2.12) and (7.1) follows from the following proposition. Recall that q/is assumed to be separable.

Proposition 7.1.

The operator ~ from (2.10) is a Toeplitz operator whose symbol is the Popov function II from (2.11).

Proof.

By the Paley-Wiener theorem [RR], for every ul, u2 ~ L2([0, oe), q/), (~Ul, U2) =- (RUl, u2) + (NFUl, u2) + (Ul, NIFu2) + (Q]Ful, FU2) 1

= ~ [(Ril, i2)L2(iR,q/) + (NGil, i2)L2(i~,q/)

+ (ill, NGi2)L2(iR,ql) + (QGil, Gi2)L2(iR,~)]

1

= ~ ( H . i l , i2)L2(ip..,q/) .

9

The following proposition gives us the "open-loop" solution of our optimal control problem, and the corresponding optimal cost operator X.

Proposition 7.2. For every xo ~ ~, min

u~L2([O,oo),q/)

J(xo, u) = (Xxo, x0)~r,

(7.3)

where X = X* ~ ~(:Y) is defined by X = W*QW - tP*(QIF + N*)g~-a(IF*Q + N)W. The minimum is attained for a unique input function U~

uOpt = _ ~ - 1 (IF*Q + N)~x0.

(7.4)

9iven by

(7.5)

306 Proof.

M. Weissand G. Weiss Using (2.9) we write the "completion of the square" formula

J(xo, u) = ([W* QW - W*(QIF + N*)~-I(IF*Q + N)W]Xo,Xo}~r + ( ~ [ u + ~-1 (IF*Q + N)Wxo], [u + .~-1 (IF,Q + N)Wxo]}L2([o,~o),~u)" Since ~ is strictly positive, the second term on the fight-hand side above is always positive, and it vanishes only if u = u~ with u ~ defined by (7.5). The minimal value for the cost function is the first term on the right-hand side, and this is exactly (Xxo, x0}~, with X from (7.4). 9 A problem with formulas (7.4) and (7.5) is that they contain ~-1, and this operator is not easy to compute. If a spectral factor E of II is known, then N-1 can be expressed in terms of E, see the comment after (11.5) in Section 11. Example 7.3. Let E be the delay line mentioned at the end of Section 3. We consider scalar signals, i.e., q / = ~ = C. With the realization from [W3], the state space of E is 5V = L2[-z, 0] and T is the left shift semigroup on Y'. B is the delta distribution at 0, while Cxo = x0(-z). Intuitively, the input signal enters the line at its right end, gets shifted to the left at unity speed, and is measured at the left end of the line. We have, for almost every t > 0,

(Vx0)(t) = f [x ~0

if if

t-z_O,

so that W*W = I (the identify on Y'). The transfer function of Y. is G(s) = e -~. For this system, consider the cost function (2.8) with Q, R real numbers and N = 0. Then the Popov function is I-I(ico) = R + Q and ~ is a multiple of the identity on L2[0, oo): ~ = (R + Q). I. The optimal control problem has a unique solution iffR + Q > 0 (which is possible also if Q < 0 or i f R < 0). Since IF* is the left shift by z on L2[0, oo), we have that IF*W = 0. Now from (7.4) we see that X = Q. I and from (7.5) we see that U ~ = 0. These expressions of X and u ~ could also have been obtained from even more elementary considerations, without using Proposition 7.2. For a more sophisticated example see Section 13. 8. The Interconnection of the System with Its Dual We continue to work with the assumptions and the notation of Section 2. We need the anticausal interpretation of the dual system ~d on the time interval [0, ~ ) , as introduced at the end of Section 6. If ya e L2([O, ~)~ ~t) is its input function, then we define the functions xa: [0, oo) -~ 5~ and ua ~ L2([O, ~ ) , q/) by x a ( t ) = ' e ' S * tY a ,

U

= IF*ya,

(8.1)

similarly as we have defined q and v in Theorem 6.3. Then it follows from this theorem, together with Proposition 5.2 and the definitions, that x a is an anticausal state trajectory of E d vanishing at +0% and ua is the corresponding anticausal


307

output function. In particular, these functions satisfy the equations -jca(t) = A*xa(t) + C*ya(t),

ua(t)

B*Awxa(t),

for almost every t > 0, and limt-.+oo xa(t) = 0. If we make the additional smoothness assumption that p a t L2([0, m), ~r then according to Proposition 6.4 we have that xa(t) ~ N(BTxw) for all t > 0,/t a ~ L2([0, o0), ~//), and ua(t) = B~xwxa(t),

for all

t > 0.

(8.2)

8.1. Let u and ya be arbitrary functions in L2([0, oG),a//) and in L2([0, ~ ) , ~), respectively. Suppose that x and y are the state trajectory and the output function of 12 corresponding to the input function u and some initial state. Suppose that x a and ua are the anticausal state trajectory and the output function of X a corresponding to the input function ya as in (8.1). Then, for every t >_ O, Proposition

-~

< y(o-), ya(o-) > do- -

do-.

t

(8.3)

t

Proof. If the operators B and C are bounded, then the proof of the identity (8.3) is very simple, from ( d / d t ) ( x , x a) = (2, x a) + (x, xa). However, in the general case, the argument is more involved. The system Z is time-invariant, implying that if xo, u, x(t), and y are related as in (4.4) and (2.3), then StY = ~Fx(t) + lFStu , (8.4) see the proof of Lemma 4.4 in [W3] (or use the functional equations from Definition 4.1 to prove this directly). Now from (8.1) and (8.4) we have = ,

9

a

9

9

a

= - 9

9 a

*

* * a

= - < S t u , F S t y > . From the shift-invariance of F we have F S t = StF*, so that, again using (8.1), = _ .

9

Now we interconnect I2 with the anticausal version of Z d via (8.5)

ya = Qy + N ' u ,

where u and y are the input and the output of Z. Thus, 12d receives its input ya from 12, but Z is not influenced by Z d, see Fig. 2. We denote by X ~ and by yOpt the state trajectory and the output function corresponding to the initial state x0 and the optimal input function U~ i.e., x~

= ]rtxo + (I)t u~ yOpt = qJX0 + ~:Tu~

for all

t > 0, (8.6)

308


U

Q

1

.c

Fig. 2. The interconnection of the system with its anticausal dual. This diagram visualizes formulas (7.5), (8.5), and (8.11).

Similarly, we denote by ya,Opt, Xa,opt, and Ua'~ the input function, the anticausal state trajectory, and the output function of E a defined as in (8.1), when the systems are connected as in (8.5) and the input function of E is optimal. Thus, ya,Opt ~- QyOpt _}_N.uOpt,

Lemma 8.2.

xa,Opt(t) = ' 1. ". . . .~tya'~ .

Ua,opt = •*ya,Opt .

(8.7)

Stu ~ is the optimal input for ~ with the initial State x~

Proof. This follows from the observation that the restriction of the optimal trajectory to [t, ~ ) is also optimal. In greater detail, the argument goes like this: From the directly verifiable formula

J(xo, u~ O V) = J(Xo, u ~ t

-~- J(x~

v) -- J(x~

S t u~

we see that J(xo, u ~ O V) attains its minimum (which is known to be J(xo, u~ for v = S t u~

t

Hence, J(x~

v) attains its minimum for the same v.

[]

The above lemma is also known as the "principle of optimality." Proposition 8.3.

With the notation from (8.6) and (8.7), we have xa,~

=

Xx~

for all t > O, where X is the optimal cost operator from (7.4). Proof.

Using the definition (8.7) of x a'~ and relation (8.4), we have Xa,opt(t) : ~ *

(QSty, opt _~_N,StuOpt)

= W*QtPx~

+ V*(QF + N*)St u~

(8.8)


309

From Lemma 8.2 and (7.5) we have (8.9)

S;uOpt ~_ _ ~ - 1 (]F*Q --]-N)Wx~

Substituting this into (8.8), we get xa,Opt(t) = tp, Q~xOpt( t ) _ tp, ( Q[7 + N , ) ~ - I ([7, 0 + N)q~xOpt( t). The right-hand side above is exactly Xx~ Lemma 8.4.

[]

With the notation from (8.6) and (8.7), Ru~ t = - u a , ~ t - Ny0pt.

Proof.

(8.10)

According to (8.1) and (8.5), we have, for every u ~ L2([0, oo), q/), Ua : F*(Qy + N ' u )

= ([7*Q + U)y - Uy + [7*U*u.

According to (2.3) and (7.5), this can be rewritten: u a + Uy = ([7*Q + U)q?xo + ([7*Q + U)lFu + [7*U*u = --~U ~

+ ( ~ -- R ) u

or, slightly rearranging, Ru + u a + Ny = ~ ( u - u~

(8.11)

(The above signal is denoted e in Fig. 2.) The right-hand side of (8.11) vanishes if U ~- U~ leading to (8.10). [] Proposition 8.5.

We define the space = {x ~ ~(CAw)lXx ~ ~(B~w)).

For every initial state xo ~ X and, for almost every t >_ O, x~

(8.12) ~ g and

R u ~ (t) = - ( B ~ w X + N C A w ) X ~ (t).

(8.13)

I f xo is such that/I ~ ~ L2([0, ~ ) , q/) and p ~ ~ L2([0, oo), ql), then for every t > 0 we have that x~ ~ g and (8.13) holds.

Proof. According to Theorem 4.5, if x is a state trajectory of X, then for almost every t > 0 we have that x(t) ~ ~(Chw) and y(t) = Cawx(t). In particular, this is true for yOpt and x ~ Similarly, if y a x a, and u a are as in (8.1), then (as explained after (8.1)) for almost every t > 0 we have that x~(t)~ ~(B~,w) and u~(t)= B~wXa(t). In particular, this is true for u a,~ and x a,~ Substituting these formulas into (8.10), we obtain that, for almost every t > 0, R u ~ (t) = -- [B~xwxa'~ (t) -q- N C A w X ~ (t)].

Now by Proposition 8.3, for almost every t > 0, X~ 6 ~ and (8.13) holds.

(8.14)

310


To prove the last part of the proposition, assume that/~opt e L2([0, oo), og) and j~opt ~ L2([0, oo),q/), in particular, u ~ and yOpt are continuous. According to Theorem 4.5 we have that, for every t > 0, x~ ~(CAw) and y~ CAwx~ From (8.7) we see that j~a,opt e L2([0, oo), ~). As explained at (8.2), /ta,~ oo),q/) and, for every t > 0 , xa,~ e~(B~w) and ua'~ B~wxa'~ Substituting these into (8.10), we get that, for every t > 0, (8.14) holds. Using Proposition 8.3 we get the desired conclusion. [] Remark 8.6. Propositions 8.1-8.3 and Lemma 8.4 did not use any regularity assumption, they hold for well-posed linear systems. If E and E d are regular, then in (8.12) and (8.13) we can replace CAw and B~w by CA and B~x. 9. The Semigroup "IF~ and the Operator F ~ We still work with the assumptions and the notation of Section 2. In addition, we use the notation B~w from (6.2), X from (7.4), and x ~ and yOpt from (8.6). Proposition 9.1. There exists a strongly continuous semioroup describes the optimal state trajectories of~: For every xo ~ Yf, x~ Proof.

: ]rtPtx0,

for all

t >_ O.

]F ~

on Yf which

(9.1)

For every t _> 0 we define the operator --t'lF~ e &a(sf) by T otp t .~0 . : 7~tXo -]-

(I)t u~ ,

(9.2)

is the optimal input corresponding to the initial state x0, given by (7.5). We note that (9.1) holds by the definition of x ~ in (8.6). It is clear that lr~ = I. We show that ]F ~ has the semigroup property. It is a ~0 consequence of the time-invariance of Y~that, for any initial state x0 and any input function u, if the state trajectory x(.) is defined by (4.4), then where u ~

X(T .qt_t) = TtX(T ) "q- f~tS~u.

This can be deduced from the functional equations in Definition 4.1, see formula (6.9) in [W4]. In particular, this is true with u ~ in place of u and x~ in place of x(.). However, according to Lemma 8.2, S~*u~ is the optimal input function corresponding to the initial state x~ Thus, by the definition of 1F~ we have x~

-q- t) : :

]rtx~

(T) -t- d~ - ~ , ~*,,opt oz-

~vtPtxopt (~.) :

nmoptnmopt.. II t It,c A,0.

Using (9.1) with z + t in place of t, we obtain the semigroup property for ,]~opt. Since, for any x0, the optimal trajectory X ~ is a state trajectory of Y,, it is continuous. Thus, "IF~ is strongly continuous. [] It is an important objective in optimal control theory to describe the optimal input function u ~ in state feedback form. Formula (7.5) is not adequate from this


311

point of view. The following proposition will bring us closer to this objective, but it will also leave substantial gaps.

Proposition 9.2.

We denote by A ~ the generator of T ~ from (9.1) and we regard its domain ~ ( A ~ as a Hilbert space with the norm II(~I-A~ where fl ~p(A~ There exists a unique bounded operator F~ ~ ( A ~ ~ q/such that, for every initial state xo ~ ~(A~

Ilzll~pt=

u~ Proof.

--- F~176

for all

t > O.

(9.3)

We define wopt: ~r __~ L2([0, ~ ) , ~ ) by ti/opt = _ ~ - 1 (]F*Q + N)~P,

(9.4)

so that U~ = ~p~ according to (7.5). We show that ~popt with ~r ~ satisfy the functional equation (4.1), i.e.,

~IJ~

=

~t/~

~ ~~lJoptqr~ ~t t--~o,

(9.5)

t

for all t > 0. This equality obviously holds on [0, t). For the interval [t, ~ ) it can be rewritten equivalently in the form S;~p~

_.~ uijopt'IFtPtx0"

Using (9.1) and the definition of wopt this reduces to (8.9). Thus, wopt is an extended output map for ~opt. By the representation theorem in [W2] (stated also in Section 4 at (4.6)), there exists a unique bounded operator F~ ~ ( A ~ ~ q/, the observation operator o f ~ ~ such that (9.3) holds for all x0 e ~(A~ 9

Proposition 9.3.

With the notation of the previous proposition, we have that ~(A~ t) c ~(CAw) and the restriction of CAw to ~ ( A ~ is bounded from ~ ( A ~ to ~. For every initial state xo ~ ~(A~ the optimal output function yOpt is

y~

--- CAwX~

for all

t > O.

(9.6)

Proof. We use the operator ~popt introduced in the proof of Proposition 9.2, so that u ~ = ~p~ We define another similar operator ~opt: ~ ~ L2([0, oo), ~/) by

@opt ~- ~ + F~I.topt I

According to (8.6), we have yOpt = ~.tOptx0" Using (8.4) and (9.5), it is not difficult to verify that ~,opt (like ~t~opt) is an extended output map for ~r~ Therefore, there exists a unique bounded C: N(A ~ ~ ~/ such that, for every x0 e ~(A~ ,e%,~opt.~0 holds for all t _> 0. In particular, it follows that yOpt is cony~ = wxtt tinuous. From Proposition 9.2 we see that u ~ is also continuous. From Theorem 4.5 it follows that x~ ~ ~(CAw) and (9.6) hold for all t _ 0. Taking t = 0, we conclude that C is the restriction of CAw to ~(A~ 9 Note that since, for any x0 ~ X, U~ and yopt are L 2 functions, F ~ are infinite-time admissible observation operators for ][opt.

and CAw

312


Theorem 9.4.

With the space ~ defined as in (8.12), we have that ~ ( A ~ and, for every xo ~ ~(A~

RF~ Moreover, if xo ~ ~(A~

= - ( B ~ w X + NCAw)Xo.

c (9.7)

then

A~

= (A + BF~

(9.8)

.

Proof. If x0 ~ ~(A~ then from Proposition 9.2 and from (4.8) (with A ~ in place of A and F ~ in place of C) we know that/t ~ ~ Z2([0, oo), qZ). By a similar argument, based on Proposition 9.3, we get that popt ~ L2([0, ~ ) , q/). Thus, by the last part of Proposition 8.5, in which we take t = 0, we obtain that x0 e g and Ru~ = - ( B * A w X + N C A w ) X o . Using (9.3) we obtain (9.7). To prove (9.8), we use the Laplace transform of U~ which, according to the theory explained after (4.6), is given by

s176

----F ~

- A~

(9.9)

for all complex s with Re s sufficiently big and for every x0 ~ 5f. Now apply the Laplace transformation to (9.2) and use (9.9) to obtain that (SI -- A~

=- ( s I - A)-lx0 + ( s I - A ) - l a ~ t ~ = ( s I - A)-Ix0 + (sI - A ) - I B F ~

- A~

0.

By taking x0 = ( s / - A~ where z0 s N(A~ and applying s / - A to all the terms, we get (sir - A)zo = ( s I - A~ + BF~ which is the same as (9.8).

9

Note that Propositions 9.1 and 9.2 and formula (9.8) hold without any regularity assumption. If E and E a are regular, then we can use CA and B~ in (9.6) and (9.7).

10. Some Riccati Equations In this section we show that X satisfies some Riccati-like equations. These equations involve ~(A~ which is not a priori known. The "true" Riccati equation, which holds on ~(A), is derived in Section 12. The notation is the same as in the previous section.

Proposition 10.1 The operator X from (7.4) satisfies the equation ( A g o , X x o ) + (z0, X A ~ xo ) = - ( C z 0 , (QCAw + N *F ~ ) xo ) ,

(10.1)

for every zo ~ ~ ( A ) and every xo ~ ~(A~

We remark that (10.1) can be written as an equation on ~(A~ in which all the terms have their range in the space Lr_l = (5fl)*(~-1 was introduced at the


313

beginning of Section 6),

A*X + J(A ~ + C*QCAw = - C * N * F ~ Proof. As before, u ~ x ~ and yOpt are the functions from (8.6), with the additional assumption that x0 ~ ~(A~ Let ya,opt, Xa,opt, and u a,~ be the corresponding functions for the anticausal dual system E d, as in (8.7). Using (9.3) and (9.6), this means that, for all t > 0, ya'~ = (QCAw + N*F~176 (10.2) Independently of all the functions introduced so far, we also consider the state trajectory and the output function of X corresponding to the initial state z0 ~ ~(A) and the input u = 0, that is,

x(t) = Ttzo,

y(t) = Cx(t).

(10.3)

We apply Proposition 8.1 for the above u, x, y and u a,~ x ~,~ ya,opt, which yields

(x(t),xa'~

=

(y(a),ya'~

da.

t By Propositions 8.3 and 9.1 and by formulas (10.2) and (10.3), this becomes

('I~tZO,X~'tPtxo) -~ l; o do-.

(10.7)

From Propositions 8.3 and 9.1 we have that x~ = Ht ~opt z0 and xa'~ = X"]~~ 0. The functions u ~ and yOpt can be expressed via (9.3) and (9.6). For ya,Opt we have formula (10.2), but with X~ corresponding to the initial state x0. By (8.2) and by Proposition 8.3 we have that ua'~ = B*AwXx~ again with X~ corresponding to x0. Substituting all these formulas into (10.7) and differentiating at t = 0, we obtain (10.6). 9

Proposition 10.5. If C ~ s

~t) and R is invertible, then X satisfies the followin9 equation." for every xo, zo ~ ~(A~ (zo, A*Xxo) + (A*Xzo, xo) + (QCzo, Cxo) = (R-X(B*X + NC)zo, (B*X + NC)xo). This proposition was obtained in [FLT] and [LT1]. Notice that if B is bounded (i.e., B e ~(q/,~r)), then ~(A ~ = ~(A) and the above equation reduces to (10.4).

Proof.

First we prove the formula (A ~

Xxo ) : (zo, A *Xxo ) + ( F~

B* Xxo ) .

( 10. 8)

We have, using (9.8), (A~

XXo ) : ( (A -'k BF~

Xxo >

---- lira (2(21 -- A)-I(A + BF~ 2--~+oo = lim ( z 0 , 2 ( 2 I - A * ) - l A * X x 0 ) + lim (F~ 2~+oo

Xxo)

B'2(2I - A*)-IXxo).

Since, according to Remark 10.3, Xxo e ~(A*), we obtain (10.8). If we transform the first two terms in (10.6) using (10.8), and then use formula (10.5) to express F ~ then after some simple algebra we obtain the equation in the last proposition. 9


315

If E and E d are regular, then we can replace CAw and B~w by CA and B~, in all the formulas of this section.

11. Spectral Factor Systems We use the assumptions and the notation of Section 2. Our first theorem concerns a surprising way to generate new extended output maps for the semigroup qr (i.e., operators tp satisfying the functional equation (4.1)).

Theorem 11.1. Let F be a shift-invariant bounded operator from L2([0, oo), d//) to L2([O, ~ ) , ~/). Then the operator Wnew = ~:.W is an extended output map for qL Let G e H~176 ~1)) be the transfer function o f f and assume that G is weakly regular. Let E = (T, ~, ~P, IF) be a realization of C; as in Theorem 4.2, i.e., such that Cp__is bounded from the state space of E,_ denoted ~, to L2([0, oo), ~J). We denote by A, B, C, D the generating operators of E. Then the observation operator o f t I jnew is cnewx0

=

(B~wL+D C)xo,

forall

xo e N ( A ) ,

(11.1)

where L = Cp*tp (so that L e ~(~Y, ~f)) and L maps N(A) mto" N(B~w ).-* According to Proposition 6.2, the operator L = ~*W can be expressed by

J;

Lxo = lirnoo Proof.

T-*~* , C CAT~Xo da,

for all

x0 e 3f.

The functional equation (4.1) can be rewritten equivalently in the form S z* ~

new

x0 = ~I/new]Fzx0

(11.2)

(we wrote ~pnew in place of W). Indeed, the functions on the left- and right-hand side of (4.1) and obviously equal for t e [0, z), and for t > z equality (4.1) reduces to (11.2). We check (11.2). We have (using that W satisfies (11.2)) * new = s

*F*v

F*s

*v

= l~,~p~ = ~ w ~ . It remains to prove (11.1). By formula (4.6) we have that, for every x0 e N(A),

cn~

= (v~176

=

where w = Wx0. Both w and fi~ are in L2([0, oo), ~ ) (since w = tPAxo). We use the representation of v = IF*w given in Proposition 6.4 (with I~ in place of Z). Thus, we define q: [0, oo) --~ ~ by q(t) = @*Stw and then, according to (6.6),

v(t) =/)~wq(t) + D*w(t) holds for all t > 0 (in particular, q(t) e N(/)~w) for all t > 0). Taking t = 0, we

316


obtain formula (11.1):

CneWxo= v(O) = B-* ; ~ w V "* w + b*w(O) = (BAwtF -* ~ * tF +

3*

C)xo.

9

Still working with the notation of Theorem 11.1, we introduce the space ~-1, which is the analogue of Lr_l from the beginning of Section 6, but for the system ~,. Thus, ~e_l is the completion of Y" with respect to the norm [[z][~ = II(ill - A*)-lz[[, where f l e p(.4*). Equivalently, ~-1 is the dual of N(A).

Proposition 11.2. We use the notation of Theorem 11.1. The operator L = 6/*~g from (11.1) is a solution of the Sylvester equation (11.3)

i]*L + LA = - C ' C , where all the terms are in 5e(Y'l, ;:~e1).

Proof.

Take x0 e N(A) and z0 e ~(A). The functions Wx0 and ~z0 and their derivatives are in L2([0, oo), ~/), which implies in particular that ~

lim (q/xo)(t) = tl~m (q?zo)(t) = O.

t--*+~

We have, for almost every t > O,

~((Vxo)(t), (Cezo)(t)) = ((VAx0)(t), (q'z0)(t) ) + ((Vx0)(t), (q'A~0)(t)). Integrating from 0 to 0% we get

- ( Cxo, Czo) = ('FAxo, Cezo) + (~gxo, ~Azo), which is equivalent to the Sylvester equation in the proposition. We recall the concept of spectral factorization. Let out" be a separable Hilbert space and let (o e L~(ilR, s be such that (0(ico) _> eI holds for almost every co e ~ , where e > 0. Then {0 admits a spectral factorization: there exists a e H~176 such that ~-1 e H~176 (&~ and ~0(ico) = ~(ico)*~(ico),

for almost every

co e IR.

(11.4)

This follows from Theorem 3.7 in [RR]. In the above formula, ((ico) is defined like G(ico) in (2.11), see the comment after (2.11). The function ( is called a spectral factor of ~0. The spectral factor ~ is unique up to left multiplication by a unitary operator U e ~(~ut~ Let ~-- e Aa(L2([0, oo), A")) be the Toeplitz operator corresponding to fp, as in (7.2). Then it follows from (7.2) and (11.4) that


317

where ~ is the shift-invariant bounded operator on L2([0, ~ ) , ~ ) whose transfer function is ~. Thus, we have the factorization ~-- = ~ - * ~ ,

(11.5)

which allows us to compute ~---1 = o~-1~--, (in general, ~--1 is not a Toeplitz operator). Here, ~ - 1 is the shift-invariant operator with transfer function ~-1. Notice that we write ~--* in place of (o~-1)*. In the following theorem we show that if E is a spectral factor of the Popov function II from (2.11), i.e., H(io)) = E(ia~)*E(io)), then there exists a well-posed linear system E 4 which has the same semigroup and the same control operator as Y,, and whose transfer function is E. We call such a system a spectral factor system corresponding to the open-loop system E and the cost function J. Theorem 11.3. Let "~ be a spectral factor of H and let IF~ be the corresponding shift-invariant operator on L2([0, ~ ) , o//). I f we define ~ : 5~ -+ L2([0, oo), 0-//) by V ~ = (IFr

+ N)V,

(11.6)

then Xr = (T, tg, ~r IFr is a well-posed linear system.

Proof.

We define the shift-invariant and bounded operator iF by IF = (QF + N*)(IF~) -1.

By the first part of Theorem 11.1, W~ = IF*W is an extended output map for T. To get all the properties listed in Definition 4.1, it remains to be proved that the functional equation (4.2) holds, i.e., for every u, v ~ L2([0, ~ ) , d//) and every z > 0,

Fr

(> v) = IFr ~ ( v r T

+ IF*v).

T

This equality of functions is satisfied on the interval [0, z), because of the causality of IFr (see (4.9)). It remains to prove it on [z, ~ ) , which is the same as S*Fr

~ v) = V r

+ ]FCv.

(11.7)

Since (T, ~, ~t', F) is a well-posed linear system, from (4.2) we have

S*IF(u O v) = q ' ~ u + lFv. Applying IF* to both sides above, we get S*IF*IF(u ~ v) = V ~ r u + HT*IFv.

(11.8)

T

Note that (11.8) resembles (11.7), only that we have IF*F in place of IF~. The idea of the remaining part of the proof is to show that, in fact, the causal part of F ' I F is F ~ and the anticausal part does not count. Using Proposition 7.1, formula (11.5), and the definition of IF, we have (]F~)*IF~ = ~l = (IF*Q + N ) F + IF*N* + R

-- (IF~)*IF*IF + IF*N* + R.

318


This readily implies that ]Fg - ]F'IF = (IFg)-*(IF*N * + R), so that 1Fr - IF*IF commutes with S~ (in particular, it is anticausal). Therefore, -

v) = ( F r

-

T

Adding this last relation to (11.8), we get (11.7).

Remark 11.4.

Except for formula (11.1) (and its proof), everything in this section remains true for well-posed linear systems, i.e., by omitting all the regularity assumptions. If ~d is regular, then J~w in (11.1) may be replaced b y / ~ . Example 11.5. We give an example of a regular linear system and of a cost function for which the spectral factors of the Popov function are not regular. This system satisfies the assumptions of Section 2 and it is single-input single-output, i.e., qg = ~t = C. For our construction we need a function ~ 9 H ~ such that ~(2 k) = (-1) k,

for all

k E N.

Such ~ exist because (2k) is a universal interpolation sequence in IE0. For the theory of universal interpolation sequences on the disk we refer to Chapter 9 of [D], and the results there can be translated easily to the haif-ptane. Denote m = I[~[[oo and define E(s) = 2m + {(s), so that E 9 H m, • is not regular, and [E(s)] > m,

for all

s 9 leo.

Clearly .~-1 9 H ~. Define the function ~09 L•(ilR) by ~o(iw) = [~(i09)12 - ira2. Since fp(ia~) >_ 89m z for almost every co c IR, ~0has a spectral factofization ~0(io9) = [G0(ico)12,

for almost every

ico E 1R,

where Go 9 H ~176 Go may be nonregular, but G(s) = e-SG0(s) is regular. Let Z be a realization of G, as in Theorem 4.2. Consider the optimal control problem for the regular linear system Z with the cost function from (2.8), where R = lm2, Q = 1, and N = 0. We see from (2.11) that the corresponding Popov function is Yl(ico) = 21-m2 + ]G0(iog)l2 = [E(ico)[2. Thus, a spectral factor of FI is E, which is nonregular.

12. Regular Spectral Factorization We use the assumptions and the notation of Section 2, together with B~o, from (6.2), X from (7.4), and A ~ and F ~ from Section 9.


319

Definition 12.1. Let E be a spectral factor of the Popov function H from (2.11). We say that II satisfies the regular spectral factorization assumption, abbreviated RSFA, if the following two conditions hold: (i) E is regular, i.e., there is a D s Z~o(q/) such that lim E(2)v = Dv,

2~+oe

for all

v e q/.

(12.1)

(ii) D is invertible. The regularity assumption (12.1) is not always satisfied, see the example at the end of the previous section. Note that the above definition is independent of the choice of the spectral factor. Indeed, if E is a spectral factor of 17, then (as already mentioned) every other spectral factor is of the form UE, where U E ~ ( q / ) is unitary. In fact, if the R S F A holds, then we may select a "canonical" spectral factor E by requiring that its feedthrough operator D be positive, D > 0 (this is achievable via the polar factorization of an initial D). Note that the operator D*D (which will play an important role) does not depend on the choice of the spectral factor. The following two propositions are consequences of results in Section 4 of [W4], by considering unity feedback for the transfer function I - E.

Proposition 12.2.

Suppose that ~ is a spectral factor of II and (12.1) holds. Then

D is left invertible. It follows that if ~//is finite-dimensional, then the R S F A is equivalent to (12.1). Recall that if E is a transfer function, then its dual is defined by Ed(s) = E(g)* (see Proposition 6.1).

Proposition 12.3.

Suppose that E is a spectral factor of 17 and (12.1) holds. Then the following two statements are equivalent." (I) D is invertible. (II) E -1 is regular.

I f the above statements are true, then the feedthrough operator o f E -1 is D -1. I f E d is regular, then (I) and (II) hold. An immediate consequence of Theorem 11.3 is that any spectral factor of 17 has a realization with the same semigroup generator A and the same control operator B as the original system Z. The following theorem gives the expression in terms of X of the observation operator of the spectral factor system, in the case that the R S F A holds.

Theorem 12.4. Let E be a spectral factor of 17, assume that the RSFA holds and let D be the feedthrough operator of E. Let Z ~ be the corresponding spectral factor

320


system, as in Theorem 11.3. Then the observation operator o f Z ~ is - D F , where FXo = - ( D ' D ) -1 (B*AwX + NC)xo,

for all

xo ~ ~ ( A ) ,

(12.2)

in particular, X m a p s ~ ( A ) into ~(B~w ).

Proofi We denote by C r the observation operator of Xr so that Cr ~(A) ~ q/. We have to show that C r - - D F . From (11.6) we have IFr

r = (F*Q + N)q J.

(12.3)

According to Theorem 11.1, IFr r is an output map for the semigroup T and its observation operator is B~wL1 + D * C ~, where L1 = ~IJ4*tIj~. In particular, L1 maps ~(A) into ~(B~w ). By the same theorem, (IF*Q + N)~P is also an output map for T and its observation operator is B~wL2 + NC, where L2 = qJ*QqJ and L2~(A) c ~(B~w ). From (12.3) and the uniqueness of the observation operator, we obtain that B'AwL1 + D* C ~ = B*AwL2+ NC, or /

C ~ = D-*(B*Aw(L2 - L1) + N C ) ,

(12.4)

where (L2 - L1)~(A) c ~(B~w ). We compute L2 - L1 using (11.6): L2 - L1 = qJ*QUL - tIJ~*~Ij~ = qJ* QqJ - W*(QIF + N*)(IF~)-I (IFr

(F*Q + N)W.

Since ~ = IFr r the last expression coincides with X as given by (7.4). Thus, L2 - L1 = X and (12.4) is equivalent to (12.2). 9 Combining Theorems 11.3 and 12.4, we conclude that, with the RSFA, 7~(s) = D(I - FA(SI -- A)-IB).

(12.5)

Comparing this with the similar expression known from the finite-dimensional theory, it should not be surprising that F^ and F ~ coincide, in a sense which is made explicit in the following theorem. Theorem 12.5.

With the assumptions and the notation of Theorem 12.4,

A ~ = A + BFA on its natural domain

~(A ~

----{x0 ~ ~(FA)I(A + BFA)XO E 3{'),

and

F~ Proofi

-- FAx0,

for all

xo ~ ~(A~

We denote by A r the operator defined as ACxo = (A + BFA)xo


321

on its natural domain ~ ( A r = {x0 E ~(FA)I(A + BFA)XO ~ ~}. We have to show that A~ = A ~ (in particular, these operators have equal domains). Since the regular system with generators A, B, - D F , D has the transfer function =, which is invertible in H ~ (~(q/)), we can use the results in Section 7 of [W4] to conclude that Ar BD -1, F, D -~ are the generators of a regular linear system with transfer function E -1. In particular, A ~ generates a semigroup and ,~-1 (S) = ( I + F A ( S I

-- A ~ ) - I B ) D

-1 .

By (7.5) and (11.6), we have that, for every initial state zo e s uOpt = __~-1 (F*Q + N)Vezo = _ (IF~)-I (IF~)-. (IF*O + N)Wzo

= -(IF~)-l~eCzo. Taking here the Laplace transformation, we get fi~

-1 [ - D F ( s I - A)-l]z0

= - ( I + FA(sI - Ar

= r ( s I - A)-lzo + FA(sI - A ~ ) - I B F ( s I - A)-lz0.

Now using Proposition 6.6 of [W4] we obtain ft~

= rA(sI -- Ar

Combining this with (9.9), we get that, for every z0 e X, FA (SI -

A~)-lz0 =

F~

-

(12.6)

A~

We apply B to both sides of this identity and use the following facts: by the definition of A ~, BFA = A ~ - A on N(A~), while, by (9.8), BF~ = A ~ A on ~(A~ Thus we deduce that (A ~ - A)(sI - A~)-lz0 = (A ~ - A)(sI

-

A~

0.

Using simple manipulations in this equality, we arrive at ( S I --

A~)-lz0 --

(sI -

A~

From this it follows directly that A ~ = A~ and, using (12.6), we also obtain that F ~ = FA on ~(A~ 9

Proposition 12.6. I f the R S F A holds, then F from (12.2) is the optimal state feedback operator for E with the cost function J, as defined in Section 3. Proof. Since A, B, - D F , and D are the generating operators of E ~ (by Theorem 12.4), it follows that A, B, F, and 0 are also the generators of a regular linear system (shown as a block in Fig. 1), so that (1) of Definition 3.1 holds. By (12.5) we have that I - F A ( S I - A ) - I B = D-1E(s), which is invertible in H~(LP(q/)), so that (2) of Definition 3.1 holds as well. Finally, (3) of Definition 3.1 holds according to Theorem 12.5. 9

322


Remark 12.7. We return to the optimal feedback system shown in Fig. 1, ~opt = (][opt, (i)opt, ~i/opt ]FOpt), whose generators were derived in Section 3. This tpopt is the same as the one in (9.4), and F ~ (]Fr where F r is as in (11.6). It is interesting to compute the cost J(xo, u) for arbitrary x0 ~5f and v e L2([0, ~ ) , q/) (u being the corresponding output of E~ J(xo, u) is a quadratic form in x0 and v, which for each given x0 attains its unique minimum (Xxo,xo) for v = 0 (because then u = u~ Hence q(v)= J(xo, u ) - (Xxo,xo) must be a quadratic form in v alone. To compute q(v), we take x0 = 0 in (2.9) and we obtain q(v) = ( ~ u , u) = ( ~ ~ F~ Using the formula of IF~ and the factorization ~ = (1Fr r we obtain that q(v) = (Dr, Dr). Hence

J(xo, u) = (Xxo, xo~ + IlOvll 2 For a more general result in tiffs direction see Theorem 2.6 in [$8]. The next theorem shows that the operator X satisfies, as expected, an operator Riccati equation. Recall that the space ~-1 is the analogue of ~r_l for the dual system Ed, or, equivalently, ~-1 = ~(A)*, see the beginning of Section 6. Theorem 12.8. With the assumptions and the notation of Theorem 12.4, the optimal cost operator X satisfies the Riccati equation

A*X + XA + C * Q C = (B*AwX+ NC)*(D*D)-I(B*AwX + NC),

(12.7)

where the terms are in ~ ( ~ 1 , .-~(-1). Here, B~xwX + NC is regarded as an operator from ~r1 to all, so that its adjoint maps ql into ~-1. In the proof of Theorem 12.4, we have shown that X = L2 - L1, where L2 = tp*Qtp satisfies, by Proposition 11.2, the Lyapunov equation

Proof.

(12.8)

A*L2 + LzA + C*QC = O, and L1 = ~r162 satisfies, by the same result, the Lyapunov equation

(12.9)

A*La + L1A + C~*Cr = O. Subtracting (12.9) from (12.8), we deduce that X satisfies

A*X + XA + C*QC = Cr

r

If we use here formula (12.4) for C ~, then (12.7) follows. Remark 12.9. With the assumptions and the notation of Theorem 12.4, we have the following interesting formula due to Staffans: for any v e q/,

D*Dv = Rv + lim B*AwX(2I 2---,-~o

-

A)-IBv.

This is essentially contained in Corollary 7.2 of [$8], although it is not entirely obvious how to "translate" his notation into ours. We said "essentially c o n -

Optimal Control of Stable WeaklyRegular Linear Systems

323

tained" because Staffans in this corollary assumes that both the original system X and its dual are regular, while we assume only weak regularity--but this is just a minor technical detail (the limit above is taken in the strong sense). Staffans also claims (without proof) that if X > 0, then D*D - R > O, and similarly if X < 0, then D*D - R < O. The above formula is equivalent to the following one: for all complex s with Re s sufficiently large, D*E(s) : R + B*awX(SI - A ) - I B .

Proposition 12.10.

With the notation o f Section 2, if qJ is finite-dimensional, B is infinite-time admissible (as defined in Section 5), and C is bounded, i.e., C ~ Aa(gf, ~t), then any spectral factor E o f the Popovfunction 17 is regular and its feedthrough operator D satisfies D * D = R.

Sketch of the Proof.

For any v ~ q/, the corresponding impulse response (defined as the inverse Laplace transform of G(s)v) is in L2([0, ~ ) , Yr This can be shown by considering the impulse response of the dual system. Using the techniques of Salamon [$2], we can find a realization ~, = (It, ~, 6L, IF) of G such that the state space of~, is 5~ = L2([0, ~ ) , Y/), ~? is the left shift semigroup on 2 , the observation operator is C x = x(O), and the control operator B is bounded:/~ e ~ ( q / , 2 ) . We formulate the optimal control problem for X with the same R, Q, and N, so that the corresponding Popov function 17 remains the same as before. According to Theorem 11.3, any spectral factor E has a realization with the same semigroup T and the same control operator/~. However, any system with a bounded control operator is regular, as is easy to see. The identity D*D = R follows from Remark 12.9. [] It would be useful to obtain strengthenings of Proposition 12.10. If we drop the assumption that ~ is finite-dimensional, then it is possible to have D * D # R, see [WZ]. But is it possible to have E not regular, with bounded C?

13. An Example Consider the finite-dimensional discrete-time system axa = Aaxa + Baud,

(13.1)

Yd = CdXd,

where Ad, Bd, and Ca are matrices of dimensions n x n, n x m, and p x n, respectively. The sequences Xd = ( x d ( k ) ) ~ N, Ud = (ud(k))ic~ N, and Yd = (yd(k))gE~ have values in IEn, •m, and ~P, respectively and a is the unit left shift, i.e., (axd)(k) = x a ( k + 1). We associate with (13.1) a continuous-time infinite-dimensional system by replacing the inverse shift a -1 by a unit time vector delay line, as illustrated in

324

M. Weissand G. Weiss x

d

(

k

)

~

~d(k)

.

J

(a)

Fig. 3. The infinite-dimensionalsystem(b) associatedwith the discrete-timefinite-dimensionalsystem (a). In (b), two parallel delaylines are shown.

Fig. 3. In this way, we obtain an interconnection of n (scalar delay lines and several (scalar) constant multiplication blocks. Notice that any system built as an interconnection of a finite number of delay lines of commensurable times (i.e., they are all integer multiples of a number) and of constant multiplication blocks, with feedthrough zero, can be obtained from a discrete-time system like (13.1) in the way described above (after rescaling the time). Thus, we are dealing here with a fairly large class of delay systems. These systems were also considered in a nice paper by Staffans [$5], who obtained related results. Earlier, Grabowski [G2] had investigated a particular system of this type by the spectral factorization approach, and had anticipated the problems with the Riccati equation approach. Since the transfer function of the discrete-time system (13.1) is Ca(z) = Ca(z1 - Ad)-IB

,

it is easy to see that the transfer function of the system in Fig. 3(b) is G(s) : Ga(e s) = Cd(eSI - Ad)-IBd.

(13.2)

If A d is power-stable, i.e., all its eigenvalues are inside the open unit disk, then G d is in H ~ of the outside of the unit disk and from (13.2) it follows that G is in H ~ of the right half-plane. We show that, actually, if Ad is power stable, then the infinite-dimensional system in Fig. 3(b) has a realization which is an exponentially stable, regular linear system. First, regularity is no problem in this case, since from (13.2), we have limz~+~G(2) = 0. It can be checked easily that the system in Fig. 3(b) is described by the following first-order partial differential equation with boundary


325

control and boundary observation: ax Ox [0, 8t - 8~' x(t, 1) = Adx(t, O) + BdU(t),

~

(0, 1), t e [0, o0),

(13.3)

y(t) = Cdx(t, 0),

x(0, ~) = X0(~),

~ E (0, 1).

We notice that it is sufficient to find a realization of this system for the special case n = rn = p, Bd = Cd = I. Indeed, if A, B, C, and 0 are the generating operators for (13.3) in this particular case, then the generators for the general case are ..4, BBd, CdC, O. This is particularly easy to see from Fig. 3(b). We simply construct the realization for the system with input v(t) and output 97(t). Therefore, we assume that n = m = p and Bd = Cd = I. In this case system (13.3) consists of an n-dimensional unit time delay line with an output feedback Ad. Using this observation, we find the realization of (13.3) by considering a realization of the delay line and by applying the feedback theory from [W4]. A suitable realization for the n-dimensional delay line (see Section 2 of [W3]), that we already used in Example 7.3, is defined on the state space ~r = L2([0, 1], •n) by the generating operators A6, B,~, Ca, and 0, where

{

0)

is the generator of the left shift semigroup on X, B6~ = 5~fi = 051,

fie II]n,

G x = 6ox = x(O),

x e ~(A,~).

If Ad is power stable, then (I - e-~Ad) -1 is in H ~176 of some open right half-plane including R e s > 0. According to [W4], Aa is an admissible feedback operator for the delay line e -s and the closed-loop system is generated by A = A6 + B6Adca,

B = B6,

C = Ca,

D = O.

It also follows from the theory in [W4] that ~1 = ~ ( A ) =

x~

~-~ ~ r , x ( 1 ) = A d x ( O

.

The operator A is simply the differentiation on ~rl, and the semigroup generated by A is exponentially stable. Returning to the general case, the generating operators for the system (13.3) are A, B, C, and 0, where A was already defined, B: Cm ---* ,~r_l,

Bu = (BdU)t~l,

C: ~ 1 ----+(~-~P,

C x = Cdx(O ) .

326


We consider, with system (13.3), the quadratic cost function

J(xo, u)= I o ( [ Q

N*I[Y(t) ] [ y ( t ) ] ) d t .

(13.4)

L u(t) I' L u(t) J

The associated Popov function is given by (2.11), and, using (13.2), we have

H( ico) = rla(ei~) ,

(13.5)

where

I-ld(ei~~ = R + NGa(e i~) + Ga(ei~~

* + Ga(ei~

i~

is the discrete-time Popov function (see [IW]) associated with the discrete-time system (13.1) and the discrete-time quadratic cost

Lug(k)] L ud(k) I)

R ] Fyd(k) k=0

Iyd(k)

'

(13.6)

We solve the quadratic optimal control problem for the infinite-dimensional system (13.3) with the cost (13.4) by relating it to the solution of the finitedimensional discrete-time LQ-problem for system (13.1) with the cost (13.6). Notice that if we assume (as in (2.12) that I-I(ico) > el, then from (13.5) we obtain that Hd(e i~~ > ~I. According to classical discrete-time LQ theory (see, for example, [IW]) this last fact implies that the discrete-time Riccati equation

A~XdAd -- Xd -1- CjQCd

= (B~tXdAd + NCa)*(R + B~XdBd) -1 (B~XdAd + NCd)

(13.7)

has a stabilizing solution Xd and the LQ-problem is solved by the feedback law Ud(k) = F~Ptxd(k), where f~ pt = - ( R + B~iXdBd)-l(B~lXdAd + NCd). Moreover, we have the spectral factorization identity

na(e i~

= S~(ei~~

+ B~XaBu)Sa(ei~~

where Sd(Z) = I -- F~Pt(zI - Ad)-IBg is invertible in the space H ~176 of the outside of the unit disk. If we define S(s) = Sa(d), then S is invertible in the space H ~ of the right half-plane and, using(13.5), H(ico) = S*(im)(R + B~XdBd)S(iog).

(13.8)

Moreover, lim,~+~S(2)= lim,~_~+~Sd(e;~ ----I. Comparing this and (13.8) with (12.1), we see that the operator D*D which we need to write the Riccati equation (called the indicator in [$5]) is

D*D = R + B~XdBd.

(13.9)

Comparing the expression of S with (12.5) and using Theorem 12.5, we deduce that the optimal state feedback operator F for (13.3) and (13.4) is

Fx = (60, F2Ptx)

= FdPtx(0),

X ~ ~V1.

(13.10)

OptimalControlofStableWeaklyRegularLinearSystems

327

This actually gives the required solution for our LQ-problem. We notice that all we needed in this case was to solve a fmite-dimensional discrete-time Riccati equation. It is interesting to determine the solution of the infinite-dimensional Riccati equation (12.7). For this, we recall that, by (7.3), J(xo, u ~ = (Xxo, xo)~. We compute

J(xo, uOpt)=j;l[Q

]~, 1 [yOpt(/)[uOpt 1,(/)[[y~ ]uOpt(t)J )dt

yopt,k+,]ryopt,+,l) ,

R

+ r)

u~

[.u~

+ r)

dr

=J2k=0 Now, for a fixed r e [0, 1], y~ + r) and u~ + r), k e N, are respectively the optimal output and the optimal input for the discrete-time LQ-problem (13.1) and (13.6) with the initial state Xod= x0(r). Therefore,

~:o(I Q iV*] [y~~

+ r) ]a

[y~

+ r)

Combining the last two formulas, we obtain

J(xo,/./opt) : (Xx0,x0)~ :

J'o (XdX0(r), xo("g))dr,

from which we deduce the relation between the infinite-dimensional Rieeati operator and the finite-dimensional discrete-time Riccati matrix (see also [S5]):

(Xxo)(r)=XdXo(r),

re[0,1],

forall

x0e~.

(13.11)

The LQ-problem considered has the remarkable property that the Riccati operator can be found by solving a finite-dimensional discrete-time Riccati equation. We check that, indeed, the operator X defined by (13.11) satisfies (12.2), and, of course, the Riccati equation (12.7). For this, we compute F from (12.2), using (13.9) and the expressions o r B and C: for x e N(A),

Fx = -(R + B~XdBd)-1 (NCd + BdXdAd)x(O). We have used the fact that, for x e N(A), x(1) = Aax(O). The above coincides, as can be readily seen, with Fx as given by (13.10).

expression

To check the Riccati equation (12.7), we use the fact that the extension of A* to 3f, acting on functions x e H 1[0, 1], is

A'x--

dx d~ + [A~x(1) -

x(0)]fi0.

This can be shown by approximating x e H 1[0, 1] by functions x~ from

~(A*) = {x ~ 5fix(0 ) =

A~x(1)}.

(13.12)

328


We can choose, for example, { x(~), x,(~) =

for ~ e [e, 1],

A~x(1) + ~ x(e) - A~x(1),

for ~ e [0, e).

Then it can be easily seen that x~ ~ x in Y" (as e --+ 0), so that A*x~ ~ A*x in ~-1.

Using (13.12), we have that, for x e ~(A), A*Xx =

dx

+

[A~XdX(1) - Xdx(O)]c~o,

dx = --Xd -~ + [A~XdAd -- Xd]x(O)Oo.

(13.13)

Further we have, still for x ~ ~(A), dx X A x = X d -'~,

(13.14)

(B*AX + N C ) x -= (B~XdAd + NCd)x(O).

From the last relation we can deduce that (B*AX + UC)*u = (B~XdAd + UCd)*Ufio.

Combining this formula with the previous one and with (13.9), we obtain (B*AX + NC)*(D*D) -1 (B~X + S C ) x = (B~XdAd + NCd)(R + B~XdBd)-I(B*dXdAd + NCd)x(O)c~o.

(13.15)

Finally, for x e ~(A), C* QCx = C~QCax(O)6o .

(13.16)

Adding relations (13.13)-(13.16) and comparing with the discrete-time Riccati equation (13.7) that is satisfied by Xd, we deduce that X satisfies (12.7). Acknowledgments. We have had very stimulating discussions on optimal control with Frank Callier, Olof Staffans, Ruth Curtain, and Hans Zwart. The latter put at our disposal his hand-written notes on the subject, now published [Z]. References [BDDM] A. Bensoussan, G. Da Prato, M. C. Delfour, and S. K. Mitter: Representation and Control of Infinite Dimensional Systems, Vol. II, Birkh/iuser, Boston, 1993. [CW1] F. Callier and J. Winkin: Spectral factorization and LQ-optimal regulation for multivariable distributed systems, Internat. J. Control, 52:55-75, 1990. [CW2] F. CaUier and J. Winkin: LQ-optimal control of infinite-dimensional systems by spectral factorization, Automatiea, 28:757-770, 1992.


329

[cz] R. F. Curtain and H. J. Zwart: An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, New York, 1995.

[D] P. L. Duren: Theory of liP Spaces, Academic Press, New York, 1970. [FLT] F. Flandoli, I. Lasiecka, and R. Triggiani: Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-BernouUi boundary control problems, Ann. Mat. Pura Appl., 153:307-382, 1988. [G1] P. Grabowski: The LQ controller synthesis problem, IMA Y. Math. Control Inform., 10:131148, 1993. [G2] P. Grabowski: The LQ controller problem: an example, IMA J. Math. Control Inform., 11:355-368, 1994. [G3] P. Grabowski: Admissibility of observation functionals, Internat. ,1. Control. 62:1161-1173, 1995. [HW] S. Hansen and G. Weiss: New results on the operator Carleson measure criterion, IMA J. Math. Control Inform., 14:3-32, 1997. [IW] V. Ionescu and M. Weiss: Continuous and discrete-time Riccati theory: A Popov function approach, Linear Algebra Appl., 193:173-209, 1993. [K] V. Ku6era: New results in state estimation and regulation, Automatica, 17:745-748, 1981. [LT1] I. Lasiecka and R. Triggiani: Riccati equations for hyperbolic partial differential equations with Lz(0, T; L2(F))-Dirichlet boundary terms, SIAM J. Control Optim., 24:884-925, 1986. [LT2] I. Lasiecka and R. Triggiani: Differential and Algebraic Riccati Equations with Applications to Boundary/Point Control Problems: Continuous Theory and Approximation Theory, Lecture Notes in Control and Information Sciences, vol. 164, Springer-Verlag, Berlin, 1991. [M] B. P. Molinari: Equivalence relation for the algebraic Riccati equation, SIAM J. Control, 11:272-285, 1973. [R] R. Rebarber: Conditions for the equivalence of internal and external stability for distributed parameter systems, IEEE Trans. Automat. Control, 38:994-998, 1993. [RR] M. Rosenblum and J. Rovnyak: Hardy Classes and Operator Theory, Oxford University Press, New York, 1985. [S1] D. Salamon: Infinite dimensional linear systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Sot., 300:383-431, 1987. [$2] D. Salamon: Realization theory in Hilbert space, Math. Systems Theory, 21:147-164, 1989. [$3] O. J. Staffans: Quadratic optimal control of stable systems through spectral factorization, Math. Control Signals Systems, 8:167-197, 1995. [$4] O. J. Staffans: Quadratic optimal control through coprime and spectral factorizations, Reports on Computer Science and Mathematics, Series A, no. 178, Abo Akademi University, Abo, Finland, 1996. [$5] O. J. Staffans: On the discrete and continuous-time infinite-dimensional algebraic Riccati equations, Systems Control Lett. , 29:131-138, 1996. [$6] O. J. Staffans: Quadratic optimal control of stable well-posed linear systems, Trans. Amer. Math. Soc., to appear. [$7] O. J. Staffans: Coprime factorizations and well-posed linear systems, SIAM J. Control Optim., to appear. [$8] O. J. Staffans: Quadratic optimal control of well-posed linear systems, SIAM J. Control Optim., to appear. [W1] G. Weiss. Weak LP-stability of a linear semigroup on a Hilbert space implies exponential stability, J. Differential Equations, 76:269-285, 1988. [W2] G. Weiss: Admissibility observation operators for linear semigroups, Israel J. Math., 65:1743, 1989. [W3] G. Weiss. Transfer functions of regular linear systems, Part I: Characterizations of regularity, Trans. Amer. Math. Soe., 342:827-854, 1994. [W4] G. Weiss: Regular linear systems with feedback, Math. Control Signals Systems, 7:23-57, 1994. [W5] G. Weiss: Transfer functions of regular linear systems, Part II: Duality, in preparation.

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M. Weiss and G. Weiss [WZ] G. Weiss and H. Zwart: An example in linear quadratic optimal control, Systems Control Lett., to appear. [W6] M. Weiss: Riccati Equations in Hilbert Spaces: A Popov Function Approach, Ph.D. thesis, Mathematics Department, University of Groningen, 1994. [Z] H. J. Zwart: Linear quadratic optimal control for abstract linear systems, in Modellin9 and Optimization. of Distributed Parameter Systems; Applications to En#ineering, K. Malanowski, Z. Nahorski and M. Peszynska, eds., Chapman & Hall, London, 1996.

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