Approximation of the Algebraic Riccati Equation in the Hilbert Space of Hilbert–Schmidt Operators

SIAM J. CONTROL AND OPTIMIZATIONVoi. 31, No. 4, pp. 847-874, July 1993

(C) 1993 Society for Industrial and Applied Mathematics002

APPROXIMATION OF THE ALGEBRAIC RICCATI EQUATION IN THEHILBERT SPACE OF HILBERT-SCHMIDT OPERATORS*

A. DE SANTISt, A. GERMANI:I:, AND L. JETTO

Abstract. This paper deals with the problem of approximating the infinite-dimensional algebraic Riccatiequation, considered as an abstract equation in the Hilbert space of Hilbert-Schmidt operators. Two kindsof approximating schemes are proposed. The first scheme exploits the already established approximabilityof the corresponding dynamical Riccati equation together with its time convergence toward the steady state.The second method considers a particular sequence of finite-dimensional linear equations whose solutionsare proved to converge toward the exact steady-state solution of the original problem.

Key words, infinite-dimensional systems, Galerkin approximation, algebraic Riccati equation

AMS subject classifications. Q3C25, Q3Ell, 41A65, 65L60

1. Introduction. Both linear quadratic (LQ) optimal control and optimal linearfiltering problems for linear systems evolving in Hilbert spaces lead to an infinite-dimensional Riccati equation. This has motivated the wide interest that, for at leasttwo decades, has been devoted to establishing conditions for the existence and unique-ness of the solution of this equation [6], [9], [10], [12]. This problem has also beenconsidered in [8], [15], [16], [20], [26], and [32] with particular reference to the LQoptimal control, in [6], [7], [19], [29], and [36] with reference to the optimal linearfiltering, and in [10] and [11] with reference to both cases. In the above papers thetopic is treated in different settings according to the different forms that the infinite-dimensional Riccati equation can take, depending on the structure assumed for thesystem dynamics.

Because ofthe infinite dimensionality, this Riccati equation cannot be instrumentedby the, standard computation techniques used in the classical finite-dimensional caseand requires various approximation methods and truncation techniques. A large numberof papers have been devoted to this problem. See, for instance, [4], [14], [22], [24],[25], [27], and [28]. The methods described in these papers work by projecting theinfinite-dimensional Riccati equation onto a sequence of finite-dimensional subspacesof the original Hilbert space. The exact solution of the actual Riccati equation is soapproximated by a sequence of solutions of finite-dimensional approximate Riccatiequations.

Particularly important is the so-called infinite-horizon problem, which arises whenthe Riccati equation admits a steady-state solution. The corresponding nondynamicalequation is referred to as the algebraic Riccati equation (ARE) [11], [18], [20], and[38]. Unfortunately, in this case the approximation problem tends to be much moredifficult than the finite-horizon problem, for which significant results are available.Such a problem was discussed in [3], [21], and [22] with reference to the LQ optimalcontrol, under the hypothesis that both the approximate semigroups and their adjoints

Received by the editors June 4, 1990; accepted for publication (in revised form) December 3, 1991.t Istituto di Analisi dei Sistemi ed Informatica del Consiglio Nazionale delle Ricerche, Viale Manzoni

30, 00185 Roma, Italy.$ Dipartimento di Ingegneria Elettrica, Universit dell’Aquila, 67100 Monteluco (L’Aquila), Italy, and

Istituto di Analisi dei Sistemi ed Informatica del Consiglio Nazionale delle Ricerche, Viale Manzoni 30,00185 Roma, Italy.

Dipartimento di Elettronica ed Automatica, Universit di Ancona, via Brecce Bianche, 67100 Ancona,Italy.

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848 A. DE SANTIS, A. GERMANI, AND L. JETTO

converge in the Trotter-Kato sense. Moreover, these papers also assumed uniformstability of the approximating semigroups.

The finite-dimensional approximation of the ARE in the space of Hilbert-Schmidt(H.S.) operators has recently been investigated in [33] and [34] under the hypothesisthat the generator of the semigroup governing the system is strongly coercive.

In the present paper the approximation problem of the ARE in Hilbert spaces isconsidered with reference to the optimal-linear-filtering problem. As it is well acknowl-edged (see, e.g., [1], [6], and [7]), such a problem can be formally stated as follows.

Let us consider the following linear infinite-dimensional system on the Hilbertspace H:

(1.1)( t) Ax( t) + Bo)( t),

y(t) Cx( t) + Go)(t),

x(0) Xo,

where the following hold:(a) A is the infinitesimal generator of a strongly continuous semigroup { T(t)} on

H such that T(t)ll <--(b) to L2(0, T; Hn), where Hn is the Hilbert space where the noise takes values;(c) B’H H, C" H-Ho (where H0 is the observation Hilbert space) and

G" H, Ho are bounded linear operators, with B H.S., such that GB* O, GG* IHo.The precise meaning of (1.1) is

x(t)= r(t)Xo+ r(t-s)o(s) as.

If L2(0, T; H.) is equipped with the standard Gauss cylinder measure (whichcorresponds to model to as a white-noise process) and xo is a Gaussian random variablewith mean vector mo and nuclear covariance Po, the filtering problem consists of findingthe best linear estimate (t) of x(t), given {y(s); 0_-< s_-< t}. It evolves according to theequation

( t) A( t) + P( t)C*(y( t) C( t)), (0) mo,

where P(t) is the unique self-adjoint, nonnegative-definite, strongly continuous solutionof the following Riccati equation"

(1.2) P(t)x: T(t)PoT*(t)x+ T(t-s)(BB*-P(s)C*CP(s))T*(t-s)xds,

x H. If (1.2) admits a steady-state solution P, namely,

(1.3) Poox= T(t)PT*(t)x+ T(t-s)(BB*-PC*CPoo)T*(t-s)xds,

then P is the solution of the following ARE:

(1.4) APoo + PooA* PC* CPoo + BB* O.

By arguing as in [20], equation (1.4) can be derived from (1.3) only for x D(A*),but (1.4) is justified because AP+PA* admits the unique bounded extensionPC*CP-BB* to all of H.

The H.S. property of B has been assumed because it corresponds to the physicallymeaningful assumption of a spatially smoothed input noise. This hypothesis impliesthat the covariance operator of Boo(t) is the nuclear operator BB*. On the other hand,also in the Ito formulation of the filtering problem, the distributed Wiener process is

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APPROXIMATION OF THE ALGEBRAIC RICCATI EQUATION 849

defined with an incremental nuclear covariance operator. Regarding the observationnoise, the white-noise approach has the advantage that the covariance operator is notrequired to be nuclear.

If the operator B is assumed to be nuclear, the ARE can be considered as anabstract equation in the Hilbert space of H.S. operators. Therefore, the smoothnessproperty of its solution permits us to prove the convergence of the two approximatingschemes here proposed toward the exact solution, under the unique assumption thatthe approximating semigroups converge in the Trotter-Kato sense. If the strong-smoothness action of H.S. operators [19, pp. 1289-1290] is exploited, no hypothesison the finite-dimensional approximability of the adjoint semigroups is required. Thismakes such a class of approximation schemes much more feasible, because the choiceofthe approximating subspaces is crucial when the domain ofthe infinitesimal generatorof the semigroup governing the system and the domain of its adjoint have a nondenseintersection. This is the case, for instance, for hereditary systems [14], [17], [22], and[24]. In these cases the choice of approximating subspaces appears to be a nontrivialissue, because the simplest way of projecting the evolution equation on a finite-dimensional subspace does not give a convergent sequence for the correspondingsolutions [14], [17], [22], and [24].

In this regard, it has recently been proved [5] that the spline scheme developedin [2] is such that the strong convergence property of the adjoint semigroups does nothold. A modified version of these splines was proposed in [24], where the optimal-control approximation problem was concerned. With reference to the same problem,the averaging approximation scheme was used in [22], where the trace-norm conver-gence of the solution of the resulting Riccati equation was obtained. However, in theabove paper the approximation problem was dealt with on the basis of a conjecture;it assumed that the approximating systems are uniformly exponentially stable forsufficiently large dimensions if the original system is exponentially stable. This conjec-ture was shown to be correct for the averaging projection scheme in [35, 4.2].

The present paper is organized as follows. The discussion of the ARE in theHilbert space of H.S. operators is given in 2. Existence and uniqueness of its solutionare established together with a convergent sequence approximating the solution itself.Two kinds of finite-dimensional approximation schemes for the solution are given in

3. On the basis of recent results [19] concerning finite-dimensional approximabilityfor the finite-horizon dynamical Riccati equation, the first method reduces the approxi-mation problem to finding a large enough horizon time for approximating the steady-state solution. It constitutes an extension of the results in [19] and provides relaxedconditions for the approximability of the solution of the ARE. The second methodcomputes the approximate solution by means of only algebraic linear operations, whichcan be easily implemented. It requires the assumption concerning the exponentialstability ofthe perturbed semigroup. Section 4 contains a numerical example concerningthe filtering problem for a delay system with approximating subspaces generated byfirst-order splines.

2. The algebraic Riccati equation in the space of Hilbert-Schmidt operators. LetH be a real, separable Hilbert space. An H.S. operator S on H is a bounded linearoperator such that

E IlSe, =:= Ilsll.s.< +,i=1

where {ei} is any orthonormal basis. The space N(H) of all H.S. operators on H is a

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Hilbert space with the inner product

[& U]H.S. Z (Se,,i=1

that is independent of the basis {ei} 1, p. 106]. The space of self-adjoint H.S. operatorsis a subspace of N(H), which will be denoted by Ns(H). The cone of nonnegative-definite operators in Ns(H) will be denoted by N-(H). In the following, explicitreference to the space H will be omitted when this is clear from the context.

Let us recall that N is a (left and right) ideal [23, p. 148] of the space L(H) oflinear bounded operators on H such that

(2.1) IlStll,.s.<=llSll..s.llttl, S6N, LL(H),

and, furthermore,

(2.2) S H.S. S* H.S., S N.+In this section the following ARE will be proved to have a unique solution in Ns:

(2.3) AP + PA* PXP + A O,

where(HI) A is the infinitesimal generator of a strongly continuous semigroup of

operators on H, { T(t), _-> 0}, such that T(t)l[ =< Me’’(H2) A is trace,class operator in N, and .S is a bounded self-adjoint nonnegative-

definite operator (not necessarily H.S.).The hypotheses H and H2 will be referred to throughout the paper, and the next

theorem provides a result that will be used later.LEMMA 2.1. Let U(t), >_-0} be a strongly continuous semigroup of operators on

H, and let

(2.4) s( t)x u(t)xu*(t), x N.

Then {S(t), t-0} is a strongly continuous semigroup on N.Proof. Only the strong continuity needs to be shown. Let K be such that for h

sufficiently small

(2.5) IIU(h)ll<-K,and it follows by (2.1), (2.2), and (2.5) that

U(h)XU*(h)- x .. v(h)x( V*(h)- i)/( v(h)- )x ..(2.6)

=< 2K I1( U(h)- t)X* ,.. + 211( U(h)- t)X ...Consider the first term of the last inequality in (2.6).

N

II(u(h)-*)x*ll?,..-- Y II(U(h)- I)X*6,II2+ Z II(U(h)- *)X*,/,,lli-: i--: Ne+l

N,,

E II(u(h)-*)x*,,,ll-+(l + K) 2 IIx*,/,,lli-I i-Nr+l

provided that N. is chosen large enough that the second term is not greater than e/2,whereas the first term can be made as small as needed by choosing h sufficiently small,because of the strong continuity of U(h). The proof is completed by using the samearguments for the second term of (2.6).

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LEMMA 2.2. Let A be the infinitesimal generator ofthe strongly continuous uniformlybounded semigroup { U(t), >- 0}. Then the unique solution of the algebraic equation in N

(2.7) AX + XA* + G O, G N

admits a unique continuous extension on H given by

(. x= s(a,

where S(t) is defined as in (2.4) provided that the integral (2.8) exists.

Proof. First of all, let us observe that S(t) has the associate generator on Ngiven by

(2.9) X AX + XA*,

with

D(g) {X N R(X) D(A)},

which is dense because of the strong continuity of {S(t), t_->0}. Let us consider theequation in C(0, oo; N)

(2.10) R(t) MX(t) + O, X(0) 6 N

that admits the mild solution 11, p. 41]

().. x(=s(x(o+ s( , ->0.

Moreover,

X lim X(t) S(7")G d-t--

satisfies equation (2.7), as can be readily verified.The existence theorem for the solution of equation (2.3) will be proved by

constructing a sequence in Ns whose limit satisfies the ARE (2.3). For this we willuse the following result concerning linear perturbed equations.

THEOREM 2.1 Let P N+s, and let hypotheses H 1 and H2 be satisfied. Moreover,

let us assume that(i) (A- P.,) is the infinitesimal generator of an asymptotically stable Co-semigroup

{ Tp(t), >- 0} such that

T.( t)(A + PP) T* t)z, z) at < o, v z H.

(ii) (A, A 1/2) is a controllable pair (in the sense specified in [10, p. 60]). Then theequation

(2.12) AX + XA* P,X X,P+ P,P + A 0

admits a unique mild solution X N+ such that(a) X is positive definite and(b) (A-X2) generates an asymptotically stable Co-semigroup Tx(t) such that

(2.13) ’( T,( t)(A + X,X) T*( t)z, z) dt <= (Xz, z).

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Proof Equation (2.12) can be rewritten as

(A- P,)X + X(A- PX)* + PXP + A =0,

which is of the kind (2.7), with G P,P + A Ns. Hence from assumption (i) on theperturbed operator and by Lemma 2.1 it follows that the unique mild solution ofequation (2.12) is

(2.14) X=I Tp(t)(A+PZP)T*p(t)dt, XNs.

The H.S. property of X is immediate. To prove the positive definiteness, let usassume the existence of a z e H, z # 0, such that

0 (Xz, z) Tp( t)(A + P2P) T(t)z, z) at

IIA /T(t)zl[ at + Z’/PT(t)z]l at,

from which

(2.15) A 1/2 T*p(t)z O, [0, c],

(2.16) ,/2pT*p t)z O,

Moreover, by a well-known perturbation formula for semigroups [13, p. 69], T*p(t)satisfies the following equation:

Premultiplying both sides by A /2 and taking into account (2.15) and (2.16), we have

(2.7) A/r*(t)z 0,

against the hypothesis of approximate controllability [11, p. 60].To prove assertion (b), let us consider the adjoint Co-semigroup {T*(t), t->0}

generated by (A-XX)*. Let y(t) T*(t)z, e H, and consider the following Lyapunov-like function:

V(y)=(Xy, y).(2.18)

We have

Q(y)=(X(A-X,)*T*(t)z, T*(t)z)+(XT*(t)z, (A-X.,)*T*(t)z)

T,,(t)(XA* XEX +AX X,X) T*( t)z, z),

and by using (2.12) we obtain

(2.19)9"(y)=-(T,,(t)((P-X),(P-X)+A +X,X)T*(t)z, z)

<-- -(IIA ’/2T*( t) ll + II’/XT*( t)zll --< 0.

To obtain inequality (2.13), let us write

0 <= V(y(t)) V(y(O)) + Q(y(z)) d’.

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By (2.18) and (2.19) it follows that

0<= (Xz, z)- Tx(r)(A + X,Y,X) T*(r)z, z) dr, V t>--0,

which gives (2.13) by taking the limit for going to infinity. ONow we can prove the uniqueness theorem for the steady-state solution of the

ARE. For this purpose we need to state in advance the following lemma.LEMMA 2.3. Let { P, be a sequence ofnonnegative definite self-adjoint H.S. operators

such that P, <-Po. Then by denoting y the maximum eigenvalue of Po we have

Proof It is enough to observe that for any x H

(P2,x, x) ,/2 o 01/2., pinX(P.P. x, P’./2x) <= (. o-, ,P. ,. P’Jx) (p.x. x)

<-_ ,(Pox, x).

THEOREM 2.2. Assume that hypotheses H and HE are verified. For a given Po Nlet (A Po2) be the infinitesimal generator of a Co-semigroup { To(t), >- 0} such that

(2.20) (To(t)(A +Po,Y.Po)T*o(t)z,z) dt=(P,z,z)<oo, V ze H.

Then the sequence { P,} defined by

(2.21) P,+, T,(t)(A + P,XP,)T*(t) dt, n=0, 1,...

where T( t) is generated by (A-P,,), converges in the H.S. norm toward the H.S.operator Po, provided that (A, A /2) is a controllable pair. Moreover, ifP is a trace-classoperator, so is P.

Proof. Equation (2.21) is well defined for n =0. Moreover, by Theorem 2.1, P.,n 0, is such that

(Pz,z)= (T_,(t)(A+P_,ZP,_l)T*_lZ, Z) dt<oo.

Hence (2.21) is well defined for each n, and the sequence Q, N given by

(2.22)

can be defined. Now let us write (2.12) for P P, and X P,+ (respectively, P P.+and X P,+2). Then, let us subtract the second equation so obtained from the first.By defining Q,+ P,+- P,+2, after simple calculations we get

AQ.+, + Q.+,A* + Q.Q. P.+,2Q.+, Q.+,ZP.+I O.which is again of the kind (2.12). Hence from (2.14)

(2.23) Q,,+, T.+I(t)Q.XQ.T*.+(t) dt.

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where the integral exists, because from (2.18) and (2.19) we have

(Q,+z,z) <-- Q(y) dt<=(P,+lZ, Z)<O, VzeH.

Moreover, (2.23) implies Q, _>- O, n 1, 2, , so that P, _>- P,+. Let us define

(2.24) Po Po- Y. O,.i=0

Now it will be shown that P,$P in the H.S. norm. First one proves thatliP, II.s.$ Pll.s.. Now let us recall that A >- B, A, B Ns implies that [IAIlu.s._-> IIBIlu.s.,so we get

(2.25)

and

P, II..s. P II..s. Pooll

(2.26) lim IlP,,llH.s.= p >= [IPoo[lH.S..

On the other hand, we have

p2= lim (P.49,. P.4’,)= lim (P4’,.neo i=1 noo i=1

N

=lim E (Pd),b,)+lim E (P4),,4’,)i=l i=Ne+l

Ne-< lim . P2. ( ( + 2 ")/ Po() dD

n--,,oo i= i= N,+I

where the last inequality follows from Lemma 2.3.By the H.S. property of P, we have that for a suitable choice of N

N N

p2_< lim (P4,, b,)+= E lim [IP,ll=/i=1 . 2

Ne EIlP,ll2+-

i=1 2’

and therefore

(2.27) p2 < Pooll .s. + .Since e is arbitrary, from (2.26) and (2.27) it follows that

(2.28) lim IIP.ll..s.-p- IIPll..s..

Now, from the parallelogram law, namely, IIX YII + IIx + Nil == 211xll = + 211 Nil ,by choosing X P, and Y P we can write

(2.29)2 P, .s.- 2 P .s. + 2P .s.- ]1P, +P .s..

Moreover, P, +P2P, so that

2.30) liP, + PII... [[2P[[....

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Consequently, from (2.29) and (2.30) we have

liP, Pll ,.s.--< 2([1P, .s.-which by (2.28) implies that a large enough n can be found such that

P Po .s. < .The nuclearity of Po follows from the fact that Po<= P.Remark 2.1. Note that Theorem 2.2 does not imply that P<-Po, so if /90=0

satisfies (2.20), it can be well assumed to start the iteration.Remark 2.2. Inequality (2.20) is not equivalent to assuming that T0(t) is exponen-

tially stable. This stronger property is guaranteed if it is also assumed that the pair(A, A/2) is exponentially stabilizable [1 p. 246]. In this case the nuclearity of P isalso guaranteed by the nuclearity of A. In the following we shall assume that P1 istrace class.

To prove that Po is the unique solution of the ARE, we need to state somepreliminary results.

LEMMA 2.4 11, Thm. 2.31 ]. Let 0 be the infinitesimal generator ofa Co-semigroup{F(t), t-> 0} on the Banach space B such that

IIr(t)ll--< Me’, 7 # 0, -> 0,

and let P be a bounded linear operator on B. Then (0 + P) is the infinitesimal generatorof a Co-semigroup {Fp(t), > 0} such that

IIr.(t)ll <- Me(’+IIPII)’, > O.

LtMMA 2.5. The following hold"

(2.31) lim sup [[T.(t)-t[0, T]

(2.32) lim sup IJT*,(t)-T*(t)l] H.S. O,t[0,T]

where Tn (t) is defined as in Theorem 2.2 and To(t) is the semigroup generated by AProof By Lemma (2.4) we have

(2.33) T,(t)ll -< Me(X+llP’f’ll)t <= Me(a+llP’llHsllZll)t Meat,

where a A + IIPll,.s.llzll. The same bound (2.33) holds for T(t)ll. From the semi-group perturbation formula [13, p. 69] we have

To( t)x T, t)x + T, r)(P, Po),To(r)x dr,

from which, after some calculations, it follows that

<= M2e2at T2 2To(t) T,, (t) .s. Po Pn ...,which proves (2.31) and (2.32) in light of Theorem 2.2 and equation (2.2). [q

LEMMA 2.6. The operator Po of Theorem 2.2 satifies the following equation"

(2.34) Po To(t)(A + PoZPo)T*o(t) dr+ To(T)PoT*(T), T>0.

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Moreover, the operators in N+s o To(t)(A + PZP) T*(t) dt, and Poo are positivedefinite.

Proof. From

P+= T,(t)(A+P,Y.P,)T*,(t)dt+ T(t)(A+P,2P,)T*,(t)dt

we easily obtain

P.+= T.(t)(A + P.ZP.)T*.(t) dt+ T.(T) T.(t)(A + P.ZP.)T*.(t) dt

T.(t)(A +P.ZP.)T*.(t) dt+ T.(T)P.+T*.(T),

which gives (2.34) by taking the limit for going to infinity. From (2.34) it followsthat ?/" is well defined because necessarily o//.<_ p<_ p. By arguing as in (2.15)-(2.17)the positive definiteness of ?/" follows from the assumed approximate controllabilityof (A, A 1/2). This in turn implies the positive definiteness of P. [3

LEMMA 2.7. Let Q(’) N+ be defined as

(2.35) Q(’r) To( r)(A + PZP) T*(’),

with P and To( t) as in Theorem 2.2 and Lemma 2.5, respectively. Then is positivedefinite if and only if

(2.36) VxH, ::1-=>0: (Q(’)x,x)>O.

Proof. If ?/" > 0, then /x H

0 < Ux, x) Q(’)x, x) d’,

which implies (2.36). On the other hand, if (2.36) is satisfied for a ?> 0, then by thestrong continuity of the semigroup there exists a cr > 0 such that (Q(?+ e)x, x)> O,’de [0, tr], which guarantees the positive definiteness of r. [3

Since Q(r) N, ’ -> 0, ’ x H, it admits the representation

(-) ()(2.37) Q(r)x= E i(T)(X, Ui )"ii=1

with

EA,2.(r) <c, h,(-)_->0 i=l,...,

where {u)} is a complete orthonormal eigenfunctions system.Denoting the set of nonnegative rationals by Z/, we state the following lemma.LEMMA 2.8. With the usual notations,

(2.38) span{u[, i= 1,2,..., rZ+:hi(r)>O} =- H

provided that the pair (A, A 1/2) is a controllable pair.Proof Let us suppose there exists a nonnull h H such that

(2.39) (h,u[)=0, i=1,2,... Vr6Z+, A,(r)>0.

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Hence by (2.38) and (2.39)

E (h, /h,(r) uT)a=E (h, Q’/2(r)uir)2

IIQ/(r)hll--(Q(r)h, h)=0.

This implies (Q(r)h, h)= 0 I r R/ because Z/ is dense in R+, and hence (//’h, h)= 0,against the hypothesis of controllability by Lemma 2.6.

THEOREM 2.3. Let (T(t), >-_ O) be the semigroup generated by A= A-Poo2.Then there exists an orthonormal basis {ut) on H such that

(2.40) lim T(t)u, 0.

Proof. First, let us prove that

(2.41) lim T(t)uTll 0 V u’hi(r)> O.

We have

(Poox, x)>-_ (T(t-r)Q(r)T(t-r)x,x) dt

Q(r) T*( r)x, T( r)x) dt

Ai(r)( T(t-r)x, u)2 dt.

Now, for any orthonormal sequence {bl} on H we have

r)2 dttr Po>-E (PChl, b/)= E Ai(r)(T*(t-r)dl, u,

E A,(r) T(t r)u,r. at.

Since it must be that tr P<, (2.41) is easily seen to hold.Let us denote by {i, i= 1, 2,...} a suitable renumbering of {u, i= 1, 2,...;

rZ/:Ai(r)>O}. Finally, by denoting by {Ul, l= 1,2,...} a basis on H obtained bythe Gram-Schmidt orthogonalization procedure applied to {ff, i= 1, 2,...}, equation(2.40) follows.

At this point the main theorem can be proved.THEOREM 2.4. Let { To(t), >= 0} be defined as before, and suppose there exists

M1 < o such that To(t)II <- M,. Then for any operator 9 N- the following equationholds:

(2.42) lim To(t)lgll.s. 0.

Proof. By definition of H.S. norm one has

T(t)OIl.s.-Y To(/)O4,ll- Y (To(t)Ob,

For {ut} defined as in Theorem 2.3 we have

06, Z o,,, u,) u,,

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and substituting in the above expression we obtain

r(t)t9 H.S.:E E ()i, th)(To(t)Ul, T(t)O)6i)

/:1

d-E E (/96i Ul)(To(t)ui, T(t)Och,)/=N+I

/=1

(2.43) +/=N+I /=N+I

--< E Ou, II. II/gT*(t) T(t)u,/=1

+ IlOu,/=N+I /=N+I

6)T*(t) To( t)Ul, chi)2)1/2

IIII=’M IIT(t)u, II+IIII..s.’M"1=1

Since t9 is an H.S. operator, we can choose N such that1/2

8

2’l=/Qe+l

and as a consequence of Theorem 2.3 we can find T in order to obtain

so that from (2.43)

11 -" M,. T(t)u, <- V > T/=1 2

which proves the theorem.COROLLARY 2.1. The semigroup T(t), _>-0, is strongly stable.Proof For each z H, z 0, let gz be the H.S. projection operator defined as

gzx (x, z)z, and let us choose an orthonormal basis {th,} on n such thatWe have

IIT(t)Ozll.s. IITo=(t)A,,ll== IITo(t)zll ,i=0

which proves the strong stability of To(t).At this point the main theorem of this section can be stated.THEOREM 2.5. Let P, be as in Theorem 2.2, and let P1 be trace class. Then the

trace-class operator Po given by (2.34) is the unique solution of the ARE.

AP+ PoA* Po2P+ A O.

Proof By (2.34) we have

II I0 IIPo- To( t)(A + PoXPoo) T’o(t) dtFIoSo

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where the right-hand side goes to zero for T going to infinity by Theorem 2.4 becauseN. Hence

(2.44) Poo Too( t)(A + Poo,Poo) T( t) dt.

Now, following the reasoning of [22, pp. 557-558], where the dual control problem isconsidered, one has that Poo maps the domain of A* into the domain of A. Hence, bydifferentiating (2.44) we obtain

(2.45) 0= Too(t)[APoo+PooA*+A-Poo,Poo]T(t) ft>-_O.

Equation (2.45) implies that Poo is a solution of the ARE.For the uniqueness let P and S be two such solutions, and let { Tq(t), t->_ 0} and

{Tp(t), t->0} be the semigroups generated by (A, Q2) and (A-P2), respectively.We find that the operator E P-Q satisfies

form which

(A- Q2)E + E(A- Q2)*- E.S,E O,

(2.46) E Tq(t)EXET*q(t) dt.

Considering now the operator F Q-P and arguing as before, we readily obtain

(2.47) F= T(t)FXFT*(t)

From (2.46) and (2.47) it follows that E P- Q < 0 and F 0- P < 0. Hence it mustbe that P O.

3. Approximation methods.3.1. Dynamic approximation. This approximation method is based on hypotheses

H1 and H2 of 2 and the trace-class property of the operator P1 defined in Theorem2.2. Moreover, it is assumed that /x H

(3.1) lim sup Ilrl,T(t)x-T(t)rIxll-o,t[0, T]

where(i) tin is a projection operator on the finite-dimensional subspace Hn c H such

that R(rI*)c D(A) and rlnrIn is strongly convergent to the identity on H. Therefore,IlIo <--M < oo V n and for a suitably chosen M.

(ii) Tn(t) is the semigroup generated by rlnAH*.Necessary and sufficient conditions for (3.1) to hold are established in the Trotter-

Kato theorem (see, e.g., [31, p. 87]).To state the convergence results for this approximation method we need to recall

some results concerning the finite-horizon approximation of the Riccati equation.Let P(t) be the solution of the Riccati equation in C(0, T; N+s(H)),

(3.2) P(t)= T(t)P(O)T*(t)+ T(t-s)[A-P(s).Y.P(s)]T(t-s)* ds,

where A and .Y are as in hypothesis H2, P(0) is trace class, and P"(t) is the solution

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on C(O, T; N+(H,,)) of

Pt"](t) Tn(t)II,P(O)II*T*(t)

(3.3) + T.(t-s)[II.AII*.-Pt"(s)II..,Y, II*.Pt"](s)]T*.(t-s) ds.

Then the following theorem holds [19]:THEOREM 3.1. Under hypothesis (3.1), for any T>0

(3.4) lim sup IIPt"(t)-rI.P(t)rI*ll..=o.t[O,T]

This allows us to prove the following theorem:

THEOREM 3.2. For each e > 0 there exists a T such that for any T> T an integer

nr exists such that for any n > nr the following holds:

(3.5) P["](T) II,,PI-I * ]1H.S. < e,

where p[n( T) is the solution of (3.3) and Po is the solution of the ARE.Proof

(3.6) Ilpt"( T)- rl.Prl.* ]lH.S._--< IIPt’( T)- II,P( T)I’In* [IH.S.+ MIIP( T)- PIIH.S..As in [19, p. 1299], the duality with the infinite-horizon control problem allows us toexploit the results stated in [22, Thm. 4.3] to prove that the following inequality holds:

(3.7) O<-_P(t, O)-P <- To(t)OT*(t), t>0,

where P(t, O) is the solution of (3.2) with 0= P(0), 0> P and is trace class, andT(t) is the semigroup generated by (A- P.Z). By (3.7) and Theorem 2.5 a sufficientlylarge T can be found such that V T> T(3.8) lIP(T, 0)- PoollH.S. < e/2M2.

By Theorem 3.1 it follows that V T> T there exists anr such that V n > nT

Pt"( T)- II,,P( T)rI * ll,-,.s. < /2,

which, together with (3.8), implies (3.5). [3

Remark 3.1. Theorem 3.2 represents an extension of [19, Thm. 5] because of thequite restrictive hypothesis about the semigroup T(t) generated by the operator Aassumed there, i.e.,

,o > IIA’/II IIZ’/llis now removed. In fact, any w > 0 is sufficient to guarantee that P1 is trace class, sothat Theorem 3.2 applies.

3.2. Algebraic approximation. This second approximation method is again basedon hypotheses H1 and H2 of 2, on the nuclearity of P1, and on the Trotter-Katoconvergence of the approximating semigroup (3.1). Moreover, it is assumed that thesemigroup To(t) generated by (A-PZ) is exponentially stable, i.e.,

(3.9) Too(t)ll-<- e-% o- > 0, _-> 0.

By the perturbation formula of semigroups (see Lemma 2.4) it can be shown that (3.9)implies that a constant h exists such that

(3.10) T( t)ll <- e, T_>0,

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which is equivalent to the dissipativity of A-AI. Inequality (3.10) implies that thesemigroup T,(t) generated by 1-I,An,* is such that T,(t)ll--< ex’, ->_ 0, n 1, 2,... [37].Let {P,} be the sequence of H.S. operators, decreasing to P, already computed inTheorem 2.2, and assume that Po is close enough to Poo to verify

Po PII H.S."

By the monotonicity of {P,} if follows that

(3.11 P, Poo H.S." X --< PoNote that such a choice of Po is always possible by assuming as Po the nth term ofthe sequence {P,} for a suitable n.

LEMMA 3.1. The sequence of semigroups { Tk(t), t>= 0} generated by the operators(A PkX), k O, 1, , satisfies the following inequality:

(3.12) Tk(t)l[----< e -=’’, _>-- 0, k 0, 1,...,

where tx is a positive constant defined by

/x 1/2(o--II(Po- P)II Ilxll)> 0.

Proof. The proof is easily achieved with Lemma 2.4 by setting O A- PoX andP=(Po-Pk)X and by considering (3.11).

Now let {II,} be a sequence of finite-dimensional orthoprojectors on H such thatII,H H, c D(A) strongly converges to the identity, and define

(3.13) 2, lI,ZII., A, n,An,, Pk")= rI.PrI..Moreover, let A, be such that

(i) A. A if;(ii) A n,Aono;(iii) A, a II..s.- 0 as n oo;(iv) (A,, A I,/2) is a controllable couple ’q’ n.A possible choice of A satisfying (i)-(iv) is

where

A, 1-I,(A + E,)I-I,,

1 (x, ,)E,x L oh,, xeH

for any orthonormal basis {bi} on H. In fact, E, is a full-rank operator whose H.S.norm is 1/n.

For each n let {/3k")} be a sequence of positive-definite self-adjoint finite-rankoperators converging to /5(-) defined by

IoP(kn2l-- ")(t)[A, +/3")X,/3")] t(")*(t) at, k=0, 1,. .,(3.14)

/5(o") p(o")

where { ’(")(t), >= 0} is the semigroup generated by the operator (A,- fi(’)Z,). Thepositive definiteness of /5(.) follows from (iv) as in Lemma 2.6. By Theorem 2.2 itfollows that if/3(-) is well defined, then the same is true for 5(-) and we havek+l

(3.15) (-) </37)ak+l

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Let { Tn)(t), => 0} and { .n)(t), -> 0} be the semigroups generated byand (An--1-lPkl-l,), respectively. By considering (3.14) we see that

(3.16) o)(t) To)(t).

Note that condition (3.12) implies [37] the following uniform-growth condition onk’)(t)

")(t)ll =< e --’’ t>O.

So by Lemma 2.4

(3.17) Tn)(t) -< e-2"’)" ellrI"PkZn-PZ"’z" t)

provided that we set f2 An IInPk2II, and P IIPk2II pn)2n. Now we can provethe following lemmas.

LEMMA 3.2. Let Pk be defined as in Theorem 2.2, and let P) and n be defined asin (3.13). Then for each e > 0 a n exists such that

(3.18) IIP)- Pk[lH.s.< e V n > nuniformly with respect to k 1, 2,...,

(3.19) IIP(z-n)ll..s.< v n > n

uniformly with respect to n 1, 2,...,

(3.20) IInPn. P).ll...< v >

uniformly with respect to k 1, 2,....Proo After choosing an ohonormal basis {} on H such that {, i= 1,..., n}

is a basis on H,, by simple calculations one obtains

IIP P II..s. 211P( z) II..s.(3.21) ( )1/2 ( + )1/2=2 2 IIPI[ 2 =2 (P, )

i=n+l i=

Taking into account Lemma 2.3 and that P P by Theorem 2.2, one has that theright-hand side of (3.21) is bounded by 2(y,+(P, ))/z, where y is the maximumeigenvalue of P, so from the trace-class propey of P it follows that a n’ can befound such that (3.18) is satisfied.

Analogously, we have

( )1/2P I H. n.s. 2 Pii=n+l

i=n+l

which is less than e for n > n", because X*PIX is trace class.For (3.20) it is enough to observe that

which is less than e for n > n’ as in (3.21). By defining u max{u’, u", n’} the proofis achieved.

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LEMMA 3.3. Let F(t) and F(n)(t) be two semigroups on H and Hn, respectively,such that IIr(t)ll < ’e > 0, and lira(t)II < ’e >--_ O, and satisfying the Trotter-Katoconvergence theorem"

(3.22) lim sup IIrI.r(t)x- F,,(t)I/,xl[- 0 v x H.t[0, T]

Moreover, let Pk be defined as in Theorem 2.2, and let L be a linear operator N(H). Then

(3.23) sup II(n.r(t)-r(t)n.)Pll...< v n>t[0, T]

uniformly with respect to k 1, 2,...,

(3.24) sup II(1-[F(t)-F(t)II,)Ll[..s.<e V n> m(e,L).t[0, T]

Proof. For each P, by definition of H.S. norm we have

Ne,kII(n.r(t)-r.(t)ri.)Pll... E II(n.F(t)-r.(t)Ii.)P,ll

i=1

(3.25)

+ Z II(n.r(t)-r.(t)n.)P,ll,i= Ne, +

where qi is an orthonormal sequence on H. Let us choose N,k such that

E2E Pkq, <- e

i= Ne, + 8

So the second term on the right-hand side of (3.25) is less than e:/4. By (3.22) thefinite summation in (3.25) can be made less than e/4 by choosing n greater than asuitable integer m(e, Pk). Hence for each Pk we have

(3.26) sup II(n.r.(t)-r("(t)H.)Pkll..s.te[0, T]

for n > re(e, P).Now, let K be such that for any K > K, IIe PII < (el4) e-T. Then, for K > K

(n. r(t) r"(t)H.)Pk II..s.

< e V n > m(e/2, P).

By choosing n=max{m(e,P),...,m(e, Pk), m(e/2, P)}, (3.23) is achieved.Inequality (3.24) is a straightforward consequence of (3.26).

LEPTA 3.4. Let { T"( t), 0} and { T( t), 0} be defined as before. en an

no exists such that

(3.27)

Moreover,

(3.28)

IIT)(t)ll<:e-’’ V n> no, V k.

sup IIn.T(t)x- T")( t)rI,,xll <-_ V n > nte[o,r]

uniformly with respect to k 1, 2,....

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Proof. Inequality (3.27) follows from (3.17) if we take into account (3.20) andchoose e _-</x. As far as (3.28) is concerned, by the perturbation formula we have ’ x H

T(k"( t)II,,x T’( t)II,,x T( -)(II,,Pk.Zl-I,,) T(k(-)l-I,x d-

from which

(rI,, T,, (t)- T(k")( t)l-I,)x-< (lI,,T(t) T’(t)H)xl]

+ II(HT(t--)P2T(-)- T)(t--)(IIP2H)T)(-)II)xIId

(3.29) (I-t, T(t) T(t)rIo)x

+ II(rIT(t-)P2,- T(t--)(II,P,rI))T(-)xll d-

The first term on the right-hand side of (3.29) is less than e V n > n,, by (3.1). Forthe second term we have

o’ll(IIT( r)P, T"(t- r)(IIP2.rI.)) T(r)xll dr

+ [IrIT(t--)PX,(I-H)ll,.s.llT(-)xl[ d-

=< JJZlJ. r sup jJ(Hr(t)- r(t)l-I)Pll.s.JJxH[o,r]

+ rellll IIPZ(-no)ll.s.< , v >

uniformly with respect to k 1, 2, , by (3.19) and (3.23) when we take into accountthat T(t) and T(t) satisfy the condition of Lemma 3.3.

For the third term of right-hand side of (3.29) we have, taking into account that

<_-- e IlPollH.S.llll I[(rIT(-)- T"(’)n,)xll dr.

,)So it follows that V n > max{n,, n,(1-IT(t)- Tk(t)rI)x _-< 2 + eT Poll H.s." IlZll

II(rt,T(’)- T"(’)rI,)xll d-.

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Finally, (3.28) follows from the Gronwall inequality if we choose el such that

e 2 exp{ertl PollH.s.lll;ll T}.

Before proving the final convergence results, we observe that from IIPll.s.-<IIPoll.s. and by definition of P") and 2;, it follows that inequality (3.17) implies

(3.30) T()(t)ll--< e ’t, t_--> 0, 2(- / Poll H.s.ll: ),

so by (3.12) it is also true that

(3.31) IIr(t)ll =e t>_--O.

Moreover, if we set O An P(kn)-Zn and P (P(kn)-/5(kn))Xn, Lemma 2.4 and inequality(3.17) give

(3.32) (kn)( t) <= e(-2z+llrI,,Pk’y’rI,,-P(k’’)z,,ll+tlP(kE)-f’(k’’)llz’n)t,

which, after we take into account that /5(kn)-----/5(0")= P(on), implies

(3.33) (kn)(t)<=e-’, t>-_o,

LEMMA 3.5. Assume that for a given k

(3.34) lim IlPk)- P)ll..s.- 0.

Then

(3.35) lim sup II()(t)-T()(t)ll--O.neo te[0,T]

Proof. Again, using the perturbation formula yields

IlIo"(t) T"(t)ll- T(kn(t--’)[((kn)--P(kn))n](kn)(") d"

Moreover, from (3.34), for n large enough

By (3.33) there exists a constant M >0 such that

sup (t)ll M.te[O,T]

Moreover, by (3.27) and (3.34) it follows that

Msup ((t) T(n(t)ll < sup tM e-"(t-’ dr<,

te[o,r] te[O,T]

which proves (3.35). VITHEOREM 3.3. Let P(kn and P(kn) be defined by (3.13) and (3.14), respectively. Then

for any k

(3.36) lim llP"- P"llI-i.s.- o v k.

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Proof. From (3.14) and (3.16) we have

I1- PII..s.T(o")( t)[A. + P(o")Z.P(o")] T(o")*(t) at

II. To( t)[A + PoZPo] T*o(t) dt IInH.S.

0 H.S.

(n) (n) rn*+ T")(t)[A. +-o --o (t) dtr H.S.

+ HTo(t)[A +PoZPo]T(t)H. dtT H.S.

From (3.12) and (3.27) we may choose an n > no such that the above expression isless than or equal to

(n) (n) Tn*H.S.

(3.37)

...llZ. [I)at + e-4"’(llA II...+ Poll..ilzll)dr.

Let us choose T such that

’L(T(on)(t)[A,, + o(,)v (")T(o")*-o "-’-o (t)

o

where

Note that

I-InTo(t)[A + PoZPo] To*(t)II.) dt

-< + T(o)n.LrI T(o )* (t)2

II, To( t)LT*o t)II, llH.S dt,

L A + Po2Po, Ln An + D(n)V10 -,nlO

(3.38) lim L L, H.s. 0

by the H.S. property of Po and by definition of An and P(on).Hence by simple calculation we have

P- Pll..s.-<_/ (llT(o(t)llllT(or(t)llllL-Lll..s.

+ I]( Ton)( t)Hn -HTo( t))Lll H.S." T")*(t)ll+ To(t)ll [IL(H.T"*(t)- T(t)H.)[[...) dt.

H.S.

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Now let us observe that because of (3.28), (3.30), and (3.31) the semigroups Tk(t)and Tkn)(t) satisfy the condition of Lemma 3.3, so by (3.24), property (2.2) of H.S.operators, and (3.38) it follows that the integral on the right-hand side of the lastinequality is less than e/2 for n > n. Hence, given e > 0, there exists a n > nl suchthat n > n

(3.39) IIP")- P")ll..s. < .Now let us suppose that propey (3.36) holds for a given k. We will show the sameis true for k + 1. We have_

r([ +ee]r(H.S.

H.S.

r .s.

By (3.32), taking into account (3.20), the assumption on the kth step, and the uniformboundedness of X,, for n large enough we obtain

II2")(t)lle-2"+2)’=e -"’’ 0< ’= 2-2e.

Now, by taking the norm in the above integral expression, it follows that

")_+ Pl ..s. <=

+ e-"’(ll .s. + Poll

Taking into account that N, we can find a suitable constant q such that V nwe have

a +O--l.s. A A .s. + A .s. + 0 .s.I1 .s. + .s. + (11- e’ll.s. + el.s.)lll q

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because A, is uniformly converging to A in the H.S. norm and by (3.39). Hence a Texists such that

(3.40)

By defining

we have

-II.T(t)[A + PZPJT*k(t)H.) dt I1 H.S.

L. k A. + 5(.)v 5(,,)ak ,r-,na k

Lk A + Pk2Pk,

(3.41) lim t.,k Lk II..s.- 0,

To prove (3.41) it is enough to observe that

L,,,k tk II..s. =< a A, II-.s. + P") P’)II..s. / Pk") Pkand to take into account the convergence of A, to A in the H.S. norm, the assumptionon the kth step, and (3.18). Hence by arguing as in the derivation of (3.39) we have that

-k+, --k+,llH.S. (t)ll i")*(t)ll IIL..,- LkIIH.S.

(3.42) + 11(2")(t)rt. -It.T(t))Lll..s.. 2)*(t)11/ T( t)ll L(n.L")*(t) T*(t)n.) ..) dr.

Moreover,

(3.43)I1( ")( t)II. II.T,( t) L II..s. < I1( ")( t)YI,, T’)( t)H,) L,II..s.

/ I1( T")(t)rI. -n.T(t))L ll..s..By (3.30), (3.33), and (3.35) the semigroups ’"(t) and T"(t) satisfy the conditionof Lemma 3.3. Because the same is true for T"(t) and Tk(t), it follows by (3.24) thatthe right-hand side of (3.43) is less than e/2T for n > re(e, Lk). So by taking intoaccount property (2.2) of H.S. operators and (3.41), an n,k can be found such thatthe integral of (3.42) is less than e/2 for n >

So far we have shown that if property (3.36) holds at the kth step, it still holdsat the (k + 1)th step. Consequently, by (3.39) the theorem is proved by induction. 13

TrtEORE 3.4. Let P be the solution of the ARE, and let Pk") be defined by (3.14).Then, given e > O, we can choose k and n in order to obtain

Proof. By noting that

P P")II..s. P Pk + Pk P(k") + P(k") ff(k" H.S.

-<-IIP- PkllH.S.+ IIP- Pi")ll..s. + IlPl")-/5k")ll n.s.

we see that the proof follows immediately if we exploit Theorem 2.2 for the first term,property (3.18) for the second term, and Theorem 3.3 for the last term. E!

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4. Numerical results. As an example of application of the approximation tech-niques described in 3, let us consider the filtering problem for linear hereditarysystems, defined by the following equations:

(4.i(

=oAz( h) + o(SZ( +s s + oo(,

te[0, T],z(0)=Zo, z(s)=zl(s), -r<-_s<O,

(4.2) y(t) Coz(t) + Ooto2(t),

where 0 ho <" < ha r, z( t) g", Ai (i", R"), 1,. , d, AolL2(-r, 0; (R",")), Bo(P,"), Co(",q), Go(r,lq), and to1 andare assumed to be standard P-valued and r-valued independent white-noise pro-cesses, respectively.

We consider the Hilbert space M2=" L2(-r, 0; ") with inner product

[(Vo, Vl), (Uo, ul)] VUo+ v(s)ul(s)g(s) ds,

where g(s) is a step weighting function such that

g(s) for-hd_i+l<=s<-hd_i,i=l,...,d,

and we define the operator A" (A)--> M2 by

(A) {z (Zo, zl) M2"zl Hl(-r, 0; "), Zo zl(0)},

Az= Aozo+ Aiz(-h)+ Aol(s)zl(s) ds, :ili=1

Then the operator (A-AI)is dissipative in M, i.e., [Az, eand generates a strongly continuous semigroup r(t) on M such thate’llzll, [2]. Moreover, if T((t) is an approximating semigroup, a constant existssuch that Ily((t)[I =<exp{t}.

By setting

[BOO] r TB w =[wl w] L2(O,T;p+) xT=(zO, Zl) T,

C=[Co 0], G=[0 Go],

equations (4.1) and (4.2) can be rewritten as

(4.3) x()= T(t)xo+ T(t-s)Boo(s) ds, re[O, T],

(4.4) y(t) Cx( t) + Goo(

and the solution of (4.1) is given by the first component of (4.3). Without any loss ofgenerality we can always assume that GoG I; moreover, B is H.S. because of thefinite-dimensionality of its range.

Now, given the mean vector and the covariance operator of the initial state xo,the best linear estimate of x(t), and consequently of z(t), can be obtained by theinfinite-dimensional Kalman filter.

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For each n let IIn be the projection operator on the subspace Vn generated bythe following piecewise linear splines"

(n)Vi,o (t)= lt_r/,,Ol(t){1 +(nt/r)}ei,(n)vi,,(t) --lt--,r/,,--(,--l)r/nl(t){m-- 1 +(nt)/r}

lt_(,+l)r/n,_,r/nl{m+l+(nt)/r}ei, m=l,... ,2-1,t) _lt_ (t){n-l+(nt)/r}ei, i=1 2,... n,i,n 1,--l+r/n]

where ei, i= 1, 2,. ., n, is the canonical basis of n, and we are considering @(A)----Hl(--r, 0; n). For such a scheme, the convergence of the approximate semigrouptoward the actual one holds [2].

It is worth recalling that, as shown in [5], the above scheme does not satisfy thestrong convergence property for the adjoint semigroups. Nevertheless, it can be usedin the present filtering problem because both of the approximating metho.ds proposedhere do not require the fulfillment of such a property.

For the matrix representation, with respect to the selected basis, of all the operatorsto be used in the implementation of both the approximation methods, we refer to [2]and [19]. The numerical example proposed consists of the state estimation for thesystem described by the following equations:

(4.5) 2(t) aoX( t) + alx( 1) + bw( t),

(4.6) y(t) cx( t) + gv( t),

with initial conditions x(s) -50s + 50, -1 -<_ s < 0, and with w(t) and v(t) beingindependent standard white-noise processes.

To satisfy all the hypotheses that make the approximation theorems of 3 hold,we imposed the stability condition ao/la,l<0 [30] by choosing ao=-3, al =2.Moreover, we set b 3.5 and d--2. The system dynamic was simulated according tothe following difference equations:

x((k + 1)A)= e%’ax(kA)+- [alx((k + 1)A 1)+ e%Aalx(kA 1)]+ Wk,

y kza x(kA + Vk,

where A =0.025, k=0, 1,...,80, and {Wk} and {Vk} are independent zero-meanGaussian white sequences, with covariances b2 Io exp{2aot} dt and d 2, respectively,which were generated by using NAG FORTRAN subroutine G05DDF.

The filter was initialized with 9(0)= y(0). The estimate )(t) was determined ateach time instant by using the gain operator obtained off-line from the approximatesolution P of the ARE. It was computed by exploiting the two methods proposedin 3, for n 3, 5, 7. We will refer first to the method of obtaining P through thedynamical Riccati equation evolving toward the steady state; the second method is forcomputing P by algebraic linear operations.

The numerical values )n(t), k(2A), k 1, 2,..., 40, have been computed byintegrating the approximate filter equation by means of the NAG FORTRAN sub-routine D02EAF, which uses a variable-order, variable-step Gear method. The samesubroutine has been also used to integrate the approximate dynamical Riccati equationand to compute the integral in equation (3.14) with null initial condition at eachiteration. For both the approximation methods the achievement of numerical stabilityhas been tested within the tolerance limit of 10-5 for 10 successive iterations.

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For each experiment the filter performance was evaluated by computing the errorstatistics

o’p -4-d x kA 2 kA 2

k=l

and by defining the signal-to-noise ratio improvement (SNRI) as

SNRI := 10 lOglo(variance of observation noise/o-).For the first method we obtained SNRI 1.606, 4.153, 4.382 for n 3, 5, 7, respectively,and for the second method we obtained SNRI= 1.609, 4.150, 4.378 for the samerespective values of n.

The whole numerical example was carried out on a VAX 780 computer. Thesimulation results are reported in Figs. 1, 2, and 3, corresponding to schemes of ordern 3, 5, and 7, respectively. In each figure plots (a) and (b) represent the behavior ofthe filter built according to the first and second method, respectively. The goodagreement between the two approximation methods is also evident from a comparisonof the numerical values reported in Table 1.

5. Concluding remarks. The problem of approximating the infinite-dimensionalalgebraic Riccati equation, as an abstract equation in the Hilbert space of H.S. operators,has been considered. Two methods have been proposed. The first one is based on: theresults established in a previous paper, where, by exploiting the approximability ofthe corresponding dynamical Riccati equation and its time convergence toward thesteady state, the problem was reduced to finding a large enough time horizon toapproximate the steady-state solution. Here this approximation scheme has been shownto converge under more general conditions than those in the previous setting, relaxing,in particular, the strong hypothesis concerning the exponential stability of the unper-turbed semigroup.

o’.,o

FIG. 1. Time plots of the noisy-state observation (continuous ragged line), true-state evolution (continuoussmooth line), and approximate-Kalman-state estimation (dashed line) obtained with the (a) first and (b)second method with n 3. (c) Time plot of the difference between the estimate values.

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FIG. 2. Time plots of the noisy-state observation (continuous ragged line), true-state evolution (continuoussmooth line), and approximate-Kalman-state estimation (dashed line) obtained with the (a) first and (b)second method with n 5. (c) Time plot of difference between the estimate values.

FIG. 3. Time plots o/the noisy-state observation (continuous ragged line), true-state evolution (continuoussmooth line), and approximate-Kalman-state estimation (dashed line) obtained with the (a) first and (b)second method with n 7. (c) Time plot qf the difference between the estimate values.

The second method provides a scheme for computing the approximate solutionby means of only algebraic linear operations under the exponential-stability assumptionfor the perturbed semigroup. In fact, the solutions of a sequence of finite-dimensionallinear equations have been proved to converge to the exact steady-state solution ofthe original problem.

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TABLEApproximate steady-state solution of the algebraic Riccati equation P’) computed by using the first and secondmethods with schemes of order n 3, 5, 7.

1st Method 2nd Method

n P) P)

0.0204 0.0108 0.00920.0108 0.0151 0.01250.0092 0.0125 0.0138

0.0186 0.0088 0.00690.0088 0.0128 0.00980.0069 0.0098 0.0107

0.0215 0.0138 0.0092 0.0091 0.01000.0138 0.0195 0.0143 0.0087 0.00860.0092 0.0143 0.0196 0.0146 0.00960.0091 0.0087 0.0146 0.0182 0.01540.0100 0.0086 0.0096 0.0154 0.0168

0.0205 0.0127 0.0080 0.0079 0.00870.0127 0.0182 0.0129 0.0074 0.00710.0080 0.0129 0.0181 0.0131 0.00810.0079 0.0074 0.0131 0.0166 0.01370.0087 0.0071 0.0081 0.0137 0.0151

0.0217 0.0159 0.0117 0.0096 0.0084 0.0091 0.0102 0.0203 0.0144 0.0102 0.0081 0.0068 0.0074 0.00830.0159 0.0202 0.0163 0.0117 0.0093 0.0082 0.0089 0.0144 0.0186 0.0148 0.0101 0.0075 0.0064 0.00690.0117 0.0163 0.0198 0.0165 0.0120 0.0094 0.0087 0.0102 0.0148 0.0182 0.0149 0.0103 0.0076 0.0067

7 0.0096 0.0117 0.0165 0.0195 0.0165 0.0115 0.0095 0.0081 0.0101 0.0149 0.0177 0.0147 0.0096 0.00740.0084 0.0093 0.0120 0.0165 0.0200 0.0165 0.0119 0.0068 0.0075 0.0103 0.0147 0.180 0.0144 0.00970.0091 0.0082 0.0094 0.0115 0.0165 0.0194 0.0170 0.0074 0.0064 0.0076 0.0096 0.0144 0.0171 0.01460.0102 0.0089 0.0087 0.0095 0.0119 0.0170 0.0182 0.0083 0.0069 0.0067 0.0074 0.0097 0.0146 0.01157

Both the solving methods work under quite general conditions that, on the otherhand, do not allow for a uniform convergence of the approximate solution toward theactual one.

Finally, we wish to point out that no assumption on the finite-dimensionalapproximability of the adjoint semigroup is made. Such a hypothesis, which is generallyrequested in the literature, is often difficult to verify when D(A) f) D(A*) is not dense.This is the case, for instance, in hereditary systems.

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Documents

Approximation of the Algebraic Riccati Equation in the Hilbert Space of Hilbert–Schmidt Operators