# Stabilizing Solution for a Discrete-Time Modified Algebraic Riccati Equation in Infinite Dimensions

Authors: Viorica Mariela Ungureanu
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

We provide necessary and sufficient conditions for the existence of stabilizing solutions for a class of modified algebraic discrete-time Riccati equations (MAREs) defined on ordered Banach spaces of sequences of linear and bounded operators. These MAREs arise in the study of linear quadratic (LQ) optimal control problems for infinite-dimensional discrete-time linear systems (DTLSs) affected simultaneously by multiplicative white noise (MN) and Markovian jumps (MJs). Unlike most of the previous works, where the detectability and observability notions are key tools for studying the global solvability of MAREs, in this paper the conditions of existence of mean-square stabilizing solutions are given directly in terms of system parameters. The methods we have used are based on the spectral theory of positive operators and the properties of trace class and compact operators. Our results generalise similar ones obtained for finite-dimensional MAREs associated with stochastic DTLSs without MJs. Also they complete and extend (in the autonomous case) former investigations concerning the existence of certain global solutions (as minimal, maximal, and stabilizing solutions) for generalized discrete-time Riccati type equations defined on infinite-dimensional ordered Banach spaces.


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## Body

## 1. Introduction

In recent years, the study of optimal control problems associated with stochastic systems with Markovian regime switching is of particular interest to researchers due to their various applications in finance, biology, engineering, and so forth. Even in the case of linear systems, the optimization problems become considerably hard, when we are in infinite dimensions and/or the Markovian process has infinite state space (see, e.g., [1–5] and the references therein). As we know from the LQ optimal control theory of DTLSs with MN and MJs (see, e.g., [5–7]), the design of the optimal control is closely related to the existence of a stabilizing solution (SS) for an associated generalized discrete-time Riccati equation (GDTRE). In this paper, we investigate the solution properties for a class of MAREs of control associated with autonomous infinite-dimensional DTLSs with MN and infinite MJs. These MAREs are time-invariant versions of the infinite-dimensional GDTREs studied in [8]. For a detailed treatment of finite-dimensional GDTREs, the reader can consult [6, 7] and the references therein. The problem of existence of SSs for infinite-dimensional GDTREs associated with stabilizable DTLSs with infinite MJs was investigated in [3, 5, 9, 10] under either stochastic detectability or observability hypotheses. In [8], a set of necessary and sufficient conditions for the existence of SSs is expressed in terms of feasibility of some linear matrix inequalities (LMIs) system.The general theory of GDTREs applies to our MAREs but new special results are strongly expected in the autonomous case. For example, in finite dimensions it is proved (see [11]) that necessary and sufficient conditions for the existence of stabilizing solutions for MAREs can be given directly in terms of system parameters. They consist in verifying whether the stochastic system is stabilizable and λ=1 is not an unobservable eigenvalue for a pair of associated operators. Although it is not always possible to give an infinite-dimensional version of a finite-dimensional result (as an example, we recall the researches proving that Hautus Lemma does not work in infinite dimensions [12, 13]), in this paper we have tried to do this for Theorem 12 from [11].The approach we propose is based on operator theory tools and the properties of positive operators and their adjoints. Using the results from [8], we prove that conditions similar to those in [11] are necessary for the existence of SSs for infinite-dimensional MAREs (see Theorem 10). To obtain sufficient conditions, we extend the class of eigenvalues λ that must not be unobservable to the set [1,∞) and we impose additional constraints for the coefficients of the DTLSs with MN and MJs (see Theorems 11 and 13). When being applied to finite-dimensional DTLSs with MN, our results recover the ones in [11]. However, Corollary 14 seems to be new for finite-dimensional DTLSs with finite Markovian jumps. As we have shown in Example 1, it provides necessary and sufficient conditions for the existence of SSs for MAREs associated with DTLSs with MJs which are neither stochastically observable nor stochastically detectable. In this case, the results like those from [2, 5, 10] do not work, while a LMIs approach would lead to a large number of operations for systems of large dimensions.The paper is organized as follows. In Section2, we introduce some basic notation and terminology and we formulate the problem. Section 3 briefly reminds us about certain properties of nuclear (trace class) and compact operators and extends to a more general framework, the well-known result: “the dual space of trace-class operators space is isomorphic to the set of bounded operators.” Also, we define the notions of maximal and stabilizing solution for a MARE and we show how to apply the results from [8] to the MAREs discussed in this paper. In Section 4, we obtain the main results. We first introduce the notion of unobservable eigenvalue for a pair of operators, by using spectral properties of positive operators. Then, we provide necessary and sufficient conditions for the existence of stabilizing solutions to MAREs (see Theorems 10, 11, and 13). To obtain the sufficient conditions from Theorems 11 and 13, we assume additional hypotheses like compactness of the coefficients or trace-class membership. Corollary 14 shows that the finite-dimensional results from [11] are direct consequences of Theorems 10, 11, and 13.Section5 is devoted to numerical examples. We study three examples, which cover the cases of MAREs associated with finite-dimensional DTLSs with finite MJs and infinite-dimensional DTLSs with infinite MJs. As mentioned above, Example 1 shows the advantages of Corollary 14 over previously published results [2, 5, 10].Finally, the last section provides conclusions and further research lines.


## 2. Notations and Statement of the Problem

LetH, U, and V be real separable Hilbert spaces. We denote by LH,U the real Banach space of linear and bounded operators from H into U and by SH the Banach subspace of L(H)≔LH,H, formed by all self-adjoint operators. As usual ∗ denotes either the adjoint of a linear and bounded operator or the dual of a Banach space. We will write ·,· for the inner product and · for norms of elements and operators, unless otherwise is indicated. An operator A∈L(H) is callednonnegative and we write A≥0, if A is self-adjoint and Ax,x≥0 for all x∈H.LetZ be an interval of integers, which may be finite or infinite. If B,· is a real Banach space, then lBZ={g={gi∈B}i∈Z,gZ=supi∈Zgi<∞} is a real linear space with the usual termwise addition and (real) scalar multiplication. Moreover, lBZ is a Banach space when endowed with the norm ·Z. If B is LU,H or SH, then lBZ will be denoted by lLU,HZ or lSHZ. Let N be the set of natural numbers and N∗=N-{0}. We denote by lN,lSHZ the Banach space llSHZI obtained from lBZ by replacing Z and B by N and lSHZ, respectively.An elementX∈lSHZ is said to benonnegative (we write X⪰0) iff Xi≥0 for all i∈Z. The cone KHZ of all nonnegative elements of lSHZ introduces the following order on lSHZ: (1)X⪰YiffX-Y∈KHZ.LetIH be the identity operator on H and let ΦH be the element {ΦHi=IH}i∈Z of lSHZ. We say that X∈lSHZ ispositive and we write X≻≻0, if there is γ>0 such that X⪰γΦH. A sequence {Xn}n∈N⊂lSHZ is called nonnegative if Xn∈KHZ for all n∈N anduniformly positive (we write Xn≻≻0) if there is γ>0 such that Xn⪰γΦH for all n∈N.A linear and bounded operatorΓ∈LlSHZ,lSUZ is calledpositive iff ΓKHZ⊂KUZ. For A∈lLU,HZ and B∈lLH,UZ, we define the product AB∈lLHZ by ABi=AiBi, i∈Z.Now let us introduce two special notations on spaceslBZ. If P∈lLH,UZ, then P∗ denotes the element of lLU,HZ defined by P∗(i)=P(i)∗, i∈Z (i.e., the ith component of P∗ is the adjoint operator of P(i)). Also, if P∈lSHZ is such that P(i)-1 exists for all i∈Z, then P-1={P(i)-1,i∈Z}. Note that P-1 may not belong to lSHZ.Let(Ω,F,P) be a probability space. For any integrable random variable ξ on (Ω,F,P), we write Eξ for its expectation (mean). We will denote by L2Ω,H the Hilbert space of all H valued random variables ξ with the property Eξ2<∞. If ξ and η are integrable real valued random variables, we denote by Eξ∣η=x the conditional expectation on the event η=x. We recall that, by definition,(2)Eξ∣η=x=1Pηω=x∫χηω=xξωPdω,if P{ω/η(ω)=x}>0, and Eξ∣η=x=0 otherwise.Let{ηt}t∈N be a homogeneous Markov chain on Ω,F,P with the state space Z, the transition probability matrix(3)P=pi,j≔Pηt+1=jηt=ii,j∈Z×Z,t∈N,and the property that Pηt=i>0, t∈N, i∈Z. The last condition ensures the nontrivial computation of the conditional mean Eξ∣ηt=i, ξ∈L2Ω,H.Throughout this paper, we assume that the following hypotheses hold.(P1)
(i)
r∈N∗ is fixed, Ap∈lLHZ, Bp∈lLU,HZ, p=0,…,r, and M∈lSHZ, L∈lLU,HZ, and R∈lSUZ are such that(4)R≻≻0,Q=MLL∗R⪰0.(ii)
{wpt}t∈N, p=1,…,r, are sequences of real valued, zero mean, square integrable random variables on Ω,F,P having the properties that wt=(w1t,w2t,…,wrt)T, t∈N, is a sequence of independent random vectors, EwtwtT=Ir, n∈N and {ηt}t∈N, and {wt}t∈N are independent. (Here T denotes the transposition.)LetFt and Gt be the σ-algebras generated by {w1k,w2k,…,wrk,0≤k≤t} and {ηk,0≤k≤t}. We introduce the σ-algebra Ht,t∈N defined by H0=G0 and Ht=Ft-1∨Gt if t≥1.We consider the stochastic system with MNs and MJs(5)xt+1=A0ηt+∑p=1rwptApηtxt+B0ηt+∑p=1rwptBpηtut,t≥t0,t∈N(6)xt0=x∈Hdenoted A,B and the cost functional(7)Ju;t0,x,i=∑t=t0∞EMηtxt,xt+2Lηtut,xt+Rηtut,ut∣ηt=i.An optimal control problemO associated with (5) and (7) consists in minimizing the cost functional Ju;t0,x,i, i∈Z, subject to (5), over the set Ut0,x of admissible controls u={ut}t≥t0,n∈N with the property that ut, t≥t0, are U-valued, Ht-measurable random variables such that (8)supt≥t0Eut2ηt0=i<∞and Ju;t0,x,i<∞ for all i∈Z.LetE:lSHZ→lSHZ, and(9)EXi=∑j∈ZpijXj,i∈Z.As we know from [5, 7], O has a solution uopt if the following modified algebraic Riccati equation (MARE):(10)Xi=∑p=0rAp∗iEXiApi+Mi-Li+∑p=0rAp∗iEXiBpi·Ri+∑p=0rBp∗iEXiBpi-1·L∗i+∑p=0rBp∗iEXiApi,i∈Z,has a mean-square stabilizing solution (see Definition 8).Our problem is to provide a set of necessary and sufficient conditions for the existence of astabilizing solution for MARE (10).


## 3. Preliminary Results

### 3.1. Sequences of Nuclear and Compact Operators

As in [9], we denote by S1(H) the Banach space of all nuclear operators from SH. It is known that S1(H) is a Banach space when endowed with thenuclear normT1=TrT∗T. Here, Tr· is the trace operator. It is well know (see, e.g., [14, 15]) that(11)·≤·1.For further properties of nuclear operators, the reader is referred to [9, 14–16]. It is known that the linear subspace NH={T∈lSHZ,T1=∑i∈ZTi1<∞} of lSHZ is a Banach space (see [9]) when endowed with the norm ·1. An easy computation (see [9]) shows that(12)·Z≤·1.In the sequel we sometimes use a special element of NH, defined for any i∈Z and x∈H by(13)P^i,xj=0ifj≠i,P^i,xi=x⊗x.Here,x⊗x∈S1(H) is defined by x⊗xu=u,xx.LetCH denote the set all compact operators from SH. Then, CHZ will be the subset of lSHZ formed by sequences of compact operators; that is, CHZ={X∈lSHZ∣Xi∈CH for all i∈Z}.It is well known thatS1(H) is dense in CH in the uniform operator topology (see, e.g., Problem 5.69 from [15]). The following lemma, whose proof is a simple exercise for the reader, states that a similar result remains true for finite sequences of nuclear and compact operators.Lemma 1.
IfZ is finite, then NH is dense in CHZ with respect to ·Z.Remark 2.
In the case whenZ is infinite, the above result is not true as it is proved by the following counter example. Let H=R and let Z=N. Then, CHZ=lSRZ=l∞Z and NH=l1Z. Since l1Z is separable and l∞Z is not, it follows that l1Z is not dense in l∞Z. Thus, NH is not dense in CHZ.Proposition 3.
The dual spaceNH∗ of the Banach space NH is isometrically isomorphic with lSHZ. The isomorphism ψ:lSHZ→NH∗ is defined by ψY=ϕY, where ϕY is the linear functional:(14)ϕYT=∑i∈ZTrTiYi,T∈NH.Proof.
LetY∈lSHZ and T∈NH be arbitrarily chosen. The following inequalities:(15)ϕYT≤∑i∈ZTrTiYi≤∑i∈ZTiYi1≤∑i∈ZYiTi1≤YZ∑i∈ZTi1≤YZT1,show that ϕY∈NH∗ and(16)ϕY≤YZ.We will prove that Y→ϕY is the isometric isomorphism that maps lSHZ into NH∗. Let ϕ∈NH∗. For any i∈Z, we define Yi∈SH by(17)Yix,x=ϕP^i,x,x∈H.Obviously, Yix,x≤ϕP^i,x1=ϕx2. Hence Yi≤ϕ, i∈Z. It follows that Y={Yi}i∈Z∈lSHZ and(18)YZ≤ϕ.On the other hand, a simple computation shows that(19)Yix,x=∑j∈ZTrP^i,xjYjand (17) and (14) imply that(20)ϕP^i,x=ϕYP^i,xfor all x∈H, i∈Z. It is well known that every X∈S1H can be written as a sum ∑n∈Nλnxn⊗xn, where the series is ·1-convergent and xn∈H, n∈N [15]. Hence, the element X~i∈lSHZ, defined by X~ii=X and Xi~j=0, for j≠i, j∈Z, can be written as ∑n∈NλnP^i,xn where the series is ·1-convergent. Then, (20) and the ·1-continuity of ϕ and ϕY ensure that(21)ϕX~i=ϕYX~ifor all X∈S1H. Now, for an arbitraryX∈NH, we define the sequence Xn∈NH, n∈N, Xni≔Xi, if i≤n, i∈Z and Xni≔0 for i≥n, i∈Z. It is not difficult to see that Xn, n∈N converges toX in ·1. From (21), it follows easily that ϕXn=ϕYXn for all n∈N. Using again the ·1-continuity of ϕ and ϕY, we get ϕ(X)=ϕY(X). Therefore, ϕ=ϕY and combining (16) and (18), it follows that ϕY=YZ. We conclude that the map ψ is an isometric isomorphism from NH∗ to lSHZ. The proof is complete.Remark 4.
A direct consequence of (17) and the above lemma is the following property of the isomorphism ψ.
ϕ
∈
N
H
∗ is a positive functional (i.e., ϕX≥0 for any X∈NH, X≥0), if and only if ψ-1ϕ≔Y⪰0.


### 3.2. Maximal and Stabilizing Solutions of MARE

In this section, we recall some definitions and results which we frequently use in the rest of the paper. To be consistent with previous publications (see, e.g., [7, 8] and the references therein) concerning discrete-time Riccati equations associated with DTLSs with MN and MJs, we introduce the following notation. For all X∈lSHZ, let(22)Π1Xi≔∑p=0rAp∗iEXiApi,Π2Xi≔∑p=0rAp∗iEXiBpi,Π3Xi≔∑p=0rBp∗iEXiBpi,(23)ΠX=Π1XΠ2XΠ2X∗Π3X.We see that Π1∈LlSHZ, Π2∈LlSHZ,lLU,HZ, Π3∈LlSHZ,lSUZ, and Π∈LlSHZ,lSH×UZ is a positive operator. Then, MARE (10) can be equivalently rewritten as(24)X=RX,where, for all X∈DomR=X∈lSHZ∣R+Π3Xiis invertible for alli∈Z,(25)RX=Π1X+M-L+Π2XR+Π3X-1L+Π2X∗.Following [8], we define the dissipation operator(26)DΣ:lN,lSHZ⟶lN,lSHZ,DΣXt=Π1Xt+1+M-XtL+Π2Xt+1L+Π2Xt+1∗R+Π3Xt+1and the subsets ΓΣ and Γ~Σ of lN,lSHZ(27)ΓΣ=X∈lN,lSHZ∣R+Π3Xt+1≻≻0,DΣXt⪰0,t∈NΓ~Σ=X∈lN,lSHZ∣DΣXt≻≻0.Lemma 28 from [8] (which is an operatorial version of the Schur complement Lemma) ensures that X∈Γ~Σ iff (if and only if) RX-X≻≻0.Definition 5.
(a) We say thatX∈lSHZ is a solution of MARE (10) if R+Π3Xi is invertible for all i∈Z and X satisfies (10).
(b) We say that a solutionX of MARE (10) is maximal if X⪰X^t, t∈N, for arbitrary X^tt∈N∈ΓΣ.Now, for anyW∈lLH,UZ, we define(28)ΠWX=∑p=0rApi+BpWi∗EXi·Api+BpWi,(29)QW=M+W∗L∗+LW+W∗RW.Obviously,ΠW∈LlSHZ is a positive operator.As in [8], let TWt,t0=ΠWt-t0, t≥t0, t,t0∈N, bethe evolution operator defined byΠW. We say that TWt,t0, isexponentially stable iff there are β≥1, α∈(0,1) such that(30)TWt,t0≤βαt-t0for all t≥t0, t,t0∈N or, equivalently, iff ρΠW<1, where ρΠW denotes the spectral radius of ΠW.In view of Theorem 14 from [8], we have the following.Definition 6.
The linear and positive operatorΠ is stabilizable if there is W∈lLH,UZ such that the evolution operator TWt,t0, t≥t0, t,t0∈N, is exponentially stable.Since (4) and (22) imply that 0∈ΓΣ, the following proposition is a direct consequence of Theorem 9 from [8].Proposition 7.
Assume (P1). If there isW∈lLH,UZ such that TWt,t0 is exponentially stable, then MARE (10) has a maximal solution Xmax which has the property that(31)R+Π3Xmax≻≻0.Theorems 9 and 13 from [8] ensure that, in the time invariant case, the notion of stabilizing solution of MARE (10) from [8] is equivalent with the one introduced by the following.Definition 8.
By a stabilizing solution of MARE (10), we mean a solution X∈lSHZ with the property that the evolution operator TFXt,t0, t≥t0, t,t0∈N, where(32)FX=-R+Π3X-1L+Π2X∗,is exponentially stable.Finally, we recall (see (4.8) in [8]) that mapping R can be equivalently rewritten as(33)RX=ΠWX+QW-FX-W∗R+Π3X-1FX-Wfor all W∈lLH,UZ. We will use later this formula for different values of W.


## 3.1. Sequences of Nuclear and Compact Operators

As in [9], we denote by S1(H) the Banach space of all nuclear operators from SH. It is known that S1(H) is a Banach space when endowed with thenuclear normT1=TrT∗T. Here, Tr· is the trace operator. It is well know (see, e.g., [14, 15]) that(11)·≤·1.For further properties of nuclear operators, the reader is referred to [9, 14–16]. It is known that the linear subspace NH={T∈lSHZ,T1=∑i∈ZTi1<∞} of lSHZ is a Banach space (see [9]) when endowed with the norm ·1. An easy computation (see [9]) shows that(12)·Z≤·1.In the sequel we sometimes use a special element of NH, defined for any i∈Z and x∈H by(13)P^i,xj=0ifj≠i,P^i,xi=x⊗x.Here,x⊗x∈S1(H) is defined by x⊗xu=u,xx.LetCH denote the set all compact operators from SH. Then, CHZ will be the subset of lSHZ formed by sequences of compact operators; that is, CHZ={X∈lSHZ∣Xi∈CH for all i∈Z}.It is well known thatS1(H) is dense in CH in the uniform operator topology (see, e.g., Problem 5.69 from [15]). The following lemma, whose proof is a simple exercise for the reader, states that a similar result remains true for finite sequences of nuclear and compact operators.Lemma 1.
IfZ is finite, then NH is dense in CHZ with respect to ·Z.Remark 2.
In the case whenZ is infinite, the above result is not true as it is proved by the following counter example. Let H=R and let Z=N. Then, CHZ=lSRZ=l∞Z and NH=l1Z. Since l1Z is separable and l∞Z is not, it follows that l1Z is not dense in l∞Z. Thus, NH is not dense in CHZ.Proposition 3.
The dual spaceNH∗ of the Banach space NH is isometrically isomorphic with lSHZ. The isomorphism ψ:lSHZ→NH∗ is defined by ψY=ϕY, where ϕY is the linear functional:(14)ϕYT=∑i∈ZTrTiYi,T∈NH.Proof.
LetY∈lSHZ and T∈NH be arbitrarily chosen. The following inequalities:(15)ϕYT≤∑i∈ZTrTiYi≤∑i∈ZTiYi1≤∑i∈ZYiTi1≤YZ∑i∈ZTi1≤YZT1,show that ϕY∈NH∗ and(16)ϕY≤YZ.We will prove that Y→ϕY is the isometric isomorphism that maps lSHZ into NH∗. Let ϕ∈NH∗. For any i∈Z, we define Yi∈SH by(17)Yix,x=ϕP^i,x,x∈H.Obviously, Yix,x≤ϕP^i,x1=ϕx2. Hence Yi≤ϕ, i∈Z. It follows that Y={Yi}i∈Z∈lSHZ and(18)YZ≤ϕ.On the other hand, a simple computation shows that(19)Yix,x=∑j∈ZTrP^i,xjYjand (17) and (14) imply that(20)ϕP^i,x=ϕYP^i,xfor all x∈H, i∈Z. It is well known that every X∈S1H can be written as a sum ∑n∈Nλnxn⊗xn, where the series is ·1-convergent and xn∈H, n∈N [15]. Hence, the element X~i∈lSHZ, defined by X~ii=X and Xi~j=0, for j≠i, j∈Z, can be written as ∑n∈NλnP^i,xn where the series is ·1-convergent. Then, (20) and the ·1-continuity of ϕ and ϕY ensure that(21)ϕX~i=ϕYX~ifor all X∈S1H. Now, for an arbitraryX∈NH, we define the sequence Xn∈NH, n∈N, Xni≔Xi, if i≤n, i∈Z and Xni≔0 for i≥n, i∈Z. It is not difficult to see that Xn, n∈N converges toX in ·1. From (21), it follows easily that ϕXn=ϕYXn for all n∈N. Using again the ·1-continuity of ϕ and ϕY, we get ϕ(X)=ϕY(X). Therefore, ϕ=ϕY and combining (16) and (18), it follows that ϕY=YZ. We conclude that the map ψ is an isometric isomorphism from NH∗ to lSHZ. The proof is complete.Remark 4.
A direct consequence of (17) and the above lemma is the following property of the isomorphism ψ.
ϕ
∈
N
H
∗ is a positive functional (i.e., ϕX≥0 for any X∈NH, X≥0), if and only if ψ-1ϕ≔Y⪰0.


## 3.2. Maximal and Stabilizing Solutions of MARE

In this section, we recall some definitions and results which we frequently use in the rest of the paper. To be consistent with previous publications (see, e.g., [7, 8] and the references therein) concerning discrete-time Riccati equations associated with DTLSs with MN and MJs, we introduce the following notation. For all X∈lSHZ, let(22)Π1Xi≔∑p=0rAp∗iEXiApi,Π2Xi≔∑p=0rAp∗iEXiBpi,Π3Xi≔∑p=0rBp∗iEXiBpi,(23)ΠX=Π1XΠ2XΠ2X∗Π3X.We see that Π1∈LlSHZ, Π2∈LlSHZ,lLU,HZ, Π3∈LlSHZ,lSUZ, and Π∈LlSHZ,lSH×UZ is a positive operator. Then, MARE (10) can be equivalently rewritten as(24)X=RX,where, for all X∈DomR=X∈lSHZ∣R+Π3Xiis invertible for alli∈Z,(25)RX=Π1X+M-L+Π2XR+Π3X-1L+Π2X∗.Following [8], we define the dissipation operator(26)DΣ:lN,lSHZ⟶lN,lSHZ,DΣXt=Π1Xt+1+M-XtL+Π2Xt+1L+Π2Xt+1∗R+Π3Xt+1and the subsets ΓΣ and Γ~Σ of lN,lSHZ(27)ΓΣ=X∈lN,lSHZ∣R+Π3Xt+1≻≻0,DΣXt⪰0,t∈NΓ~Σ=X∈lN,lSHZ∣DΣXt≻≻0.Lemma 28 from [8] (which is an operatorial version of the Schur complement Lemma) ensures that X∈Γ~Σ iff (if and only if) RX-X≻≻0.Definition 5.
(a) We say thatX∈lSHZ is a solution of MARE (10) if R+Π3Xi is invertible for all i∈Z and X satisfies (10).
(b) We say that a solutionX of MARE (10) is maximal if X⪰X^t, t∈N, for arbitrary X^tt∈N∈ΓΣ.Now, for anyW∈lLH,UZ, we define(28)ΠWX=∑p=0rApi+BpWi∗EXi·Api+BpWi,(29)QW=M+W∗L∗+LW+W∗RW.Obviously,ΠW∈LlSHZ is a positive operator.As in [8], let TWt,t0=ΠWt-t0, t≥t0, t,t0∈N, bethe evolution operator defined byΠW. We say that TWt,t0, isexponentially stable iff there are β≥1, α∈(0,1) such that(30)TWt,t0≤βαt-t0for all t≥t0, t,t0∈N or, equivalently, iff ρΠW<1, where ρΠW denotes the spectral radius of ΠW.In view of Theorem 14 from [8], we have the following.Definition 6.
The linear and positive operatorΠ is stabilizable if there is W∈lLH,UZ such that the evolution operator TWt,t0, t≥t0, t,t0∈N, is exponentially stable.Since (4) and (22) imply that 0∈ΓΣ, the following proposition is a direct consequence of Theorem 9 from [8].Proposition 7.
Assume (P1). If there isW∈lLH,UZ such that TWt,t0 is exponentially stable, then MARE (10) has a maximal solution Xmax which has the property that(31)R+Π3Xmax≻≻0.Theorems 9 and 13 from [8] ensure that, in the time invariant case, the notion of stabilizing solution of MARE (10) from [8] is equivalent with the one introduced by the following.Definition 8.
By a stabilizing solution of MARE (10), we mean a solution X∈lSHZ with the property that the evolution operator TFXt,t0, t≥t0, t,t0∈N, where(32)FX=-R+Π3X-1L+Π2X∗,is exponentially stable.Finally, we recall (see (4.8) in [8]) that mapping R can be equivalently rewritten as(33)RX=ΠWX+QW-FX-W∗R+Π3X-1FX-Wfor all W∈lLH,UZ. We will use later this formula for different values of W.


## 4. Main Results

In this section we establish necessary (see Theorem10) and sufficient conditions (see Theorems 11 and 13) for the existence of a stabilizing solution to MARE (10). They are related to the nonexistence of certain unobservable eigenvalues for a pair of operators which are defined with the coefficients of (10). For finite-dimensional MAREs, we recover the results from [17] (see also [11] for the continuous time case). Unlike [17], our results apply to stochastic systems with Markovian jumps.Assume (P1) and letW∈lLH,UZ and ΠW, QW be defined by (28) and (29), respectively. Since ΠW is a positive operator on lSHZ, we know from [18] (see Theorem 3.2.3 therein) that there is y∗∈lSHZ∗ such that y∗≥0, y∗≠0, and(34)ΠW∗y∗=ρΠWy∗,where ρΠW is the spectral radius of ΠW. Therefore, ΠW∗ has nonnegative eigenvalues.Definition 9.
We say thatλ≥0 is an unobservable eigenvalue for the pair ΠW∗,QW, W∈lLH,UZ, iff there is an y∗∈lSHZ∗ such that y∗≥0, y∗≠0, and(35)ΠW∗y∗=λy∗,y∗QW=0.
Otherwise, we will say thatλ is not an unobservable eigenvalue for the pair ΠW∗,QW.The next theorem gives necessary conditions for the existence of a stabilizing solution to MARE (10).Theorem 10.
If the algebraic Riccati equation (10) has a stabilizing solution, then(a)
Π is stabilizable;(b)
λ=1 is not an unobservable eigenvalue for the pair Π-R-1L∗∗,Q-R-1L∗.Proof.
Statement (a) follows from Definition8. Let us prove (b). If (10) has a stabilizing solution X, then it coincides with the maximal solution, according to Theorem 13 from [8]. Since 0∈ΓΣ, we deduce that X≥0 (see Definition 5(b)). Using Theorem 13 from [8] again, we see that Γ~Σ≠⌀. A slight modification of the proof of Theorem 13 from [8] leads us to the conclusion that, in the time invariant case, Γ~Σ contains a constant sequence Xt=X∈lSHZ, t∈N. For such X we have RX-X≻≻0 and, consequently, there is a δ>0 such that(36)RX-X≥δΦH.Assume by contradiction that (b) is not satisfied. This means that there is y∗≥0, y∗≠0, such that(37)Π-R-1L∗∗y∗=y∗,y∗Q-R-1L∗=0.Using successively (36), (33) (with -R-1L∗ replacing W), and (37), we get (38)0≤y∗RX-X=y∗Π-R-1L∗X-X+Q-R-1L∗-FX+R-1L∗∗R+Π3X·FX+R-1L∗=Π-R-1L∗∗y∗-y∗X+y∗Q-R-1L∗-y∗FX+R-1L∗T·R+Π3XFX+R-1L∗=-y∗FX+R-1L∗TR+Π3X·FX+R-1L∗≤0.Hence y∗RX-X=0. From (36) and the positiveness of y∗, we deduce that (39)0=y∗RX-X≥y∗δΦH≥0.It follows that y∗δΦH=0⇔y∗ΦH=0. Since y∗X≤Xy∗ΦH for all X∈lSHZ, it follows easily that y∗=0. We have obtained a contradiction. Therefore, (b) holds and λ=1 is not an unobservable eigenvalue for the pair Π-R-1L∗∗,Q-R-1L∗. The proof is complete.Now let us introduce a new hypothesis(P2)
(i)
Api∈LH and Bpi∈LU,H, p=0,…,r, i∈Z, are compact operators.In the case whenZ is finite, we have the following converse of Theorem 10.Theorem 11.
Assume thatZ is finite and (P1) and (P2) hold. If(a)
Π is stabilizable and(b)
the pairΠ-R-1L∗∗,Q-R-1L∗ has not an unobservable eigenvalue λ≥1, then the algebraic Riccati equation (10) has a stabilizing solution.Proof.
From Proposition7, we know that the algebraic Riccati equation (10) has a maximal solution Xmax. Assume by contradiction that Xmax is not stabilizing; that is, TFXmaxt,t0 is not exponentially stable. Then, ρΠFXmax≥1 and there is y∗∈lSHZ∗, y∗≥0,y∗≠0, such that ΠFXmax∗y∗=ρΠFXmaxy∗. Hence, y∗ΠFXmaxX=ρΠFXmaxy∗X for all X∈lSHZ. For the sake of simplicity, let us denote ρΠFXmax by ρ¯.
Let us prove thaty∗NH≠{0}. Indeed, if y∗NH={0}, then y∗N¯H=0. Since N¯H=CHZ, according to Lemma 1, we deduce that y∗CHZ={0}. On the other hand, hypothesis (P2) implies that ΠFXmaxX∈CHZ for all X∈lSHZ and, therefore, ρ¯y∗X=y∗ΠFXmaxX=0 for all X∈lSHZ. Hence, y∗=0 that is a contradiction. It follows that y∗NH≠{0}.
Thus,y∗NH is a nonzero linear functional. Obviously, y∗X≤XZy∗≤X1y∗, X∈NH, and y∗NH∈NH∗. Moreover, y∗NH≥0, y∗NH≠0. By Proposition 3, we know that there is Y∈lSHZ, Y≥0, Y≠0, such that(40)y∗NHX=∑i∈ZTrXiYifor all X∈NH. Rewriting (33) for W replaced by FXmax, we see that the maximal solution Xmax of the algebraic Riccati equation (10) satisfies the following equation:(41)Xmax=ΠFXmaxXmax+QFXmax.Hence, QFXmax=-ΠFXmaxXmax-Xmax. From (P1)(i) and (29), QFXmax⪰0. Using the properties of y∗, we obtain(42)0≤y∗QFXmax=-y∗ΠFXmaxXmax-Xmax=-ΠFXmax∗y∗-y∗Xmax=-ρ¯-1y∗Xmax≤0.It follows that y∗QFXmax=0. An easy computation shows that(43)QFXmax=M-LR-1L∗+R-1L∗+FXmax∗RR-1L∗+FXmax.Lemma 28 from [8] (a Schur complement result for sequences of operators) and (P1)(i) imply that M-LR-1L∗⪰0. Since(44)R-1L∗+FXmax∗RR-1L∗+FXmax⪰0and Q-R-1L∗=M-LR-1L∗, it follows easily that(45)y∗Q-R-1L∗=0and y∗R-1L∗+FXmax∗RR-1L∗+FXmax=0.
We know that, for anyX∈SH and {en}n∈N an orthonormal basis of H, the increasing sequence Xn=∑n=1∞Xen⊗Xen∈S1H is weakly convergent to X. It is not difficult to see that this property remains valid for each component Xi of any X∈lSHZ. So there is a nonnegative, increasing sequence Rn∈NH, n∈N which is componentwise weakly convergent to R. The positiveness of y∗ and the inequality R⪰Rn⪰0 imply that y∗R-1L∗+FXmax∗RnR-1L∗+FXmax=0. Since(46)R-1L∗+FXmax∗RnR-1L∗+FXmax∈NH,we can apply (40) and we get(47)∑i∈ZTrYiR-1L∗+FXmaxi∗Rni·R-1L∗+FXmaxi=0⟺∑i∈ZTrYiR-1L∗+FXmaxi∗·RniR-1L∗+FXmaxiYi=0for all i∈Z. Now let us recall the following property of nonnegative operators: TrAA∗=0⇒A=0. From (47), we obtain(48)YiR-1L∗+FXmaxi∗Rni=0⟹YiR-1L∗+FXmaxi∗Rni=0.Passing to the limit as n→∞ in the above equality, we get(49)YiR-1L∗+FXmaxi∗Ri=0.Taking into account that Ri is invertible for all i∈Z, we deduce that Yi(R-1L∗+FXmax)i∗=0, for all i∈Z or, equivalently,(50)YR-1L∗+FXmax∗=0.Thus, YA+BFXmax∗=YA-BR-1L∗∗ for all Y∈lSHZ and(51)TrYΠFXmaxX~=TrYΠ-R-1L∗X~for all X~∈NH. In view of (40), we get(52)y∗NHΠFXmaxX~=y∗NHΠ-R-1L∗X~for all X~∈NH. Let us prove that the above equality remains true if we replace X~ with an arbitrary X∈lSHZ. As mentioned above, there is a sequence X~n∈NH that converges componentwise and weakly to X. That is, X~ni→n→∞Xi, weakly, for all i∈Z. From hypothesis (P2), it follows that Ap+BpFXmax∗iEX~niAp+BpFXmaxi is norm convergent to Ap+BpFXmax∗iEXiAp+BpFXmaxi for all i∈Z and p=0,…,r. This is because the multiplication to the right and to the left with compact operators promotes the weak convergence of operators to the uniform convergence.
Therefore,ΠFXmaxX~n is ·Z-convergent to ΠFXmaxX. Similarly, we deduce that Π-R-1L∗X~n is ·Z-convergent to Π-R-1L∗X. From (52) we have(53)y∗ΠFXmaxX~n=y∗Π-R-1L∗X~nand passing to the limit for n→∞ we obtain(54)y∗ΠFXmaxX=y∗Π-R-1L∗X,X∈lSHZ.So ρ¯y∗=ΠFXmax∗y∗=Π-R-1L∗∗y∗. In view of (45), we just have obtained a contradiction of (b). Therefore, Xmax is a stabilizing solution and the conclusion follows.Remark 12.
If the maximal solution would have the property thatρΠFXmax≤1, the proof of the above theorem could be modified such that condition λ≥1 is to be replaced by λ=1.Before stating the next result, we note thatAX,XA∈NH for all A∈NH and X∈lSHZ; that is, NH is a two-sided ideal of lSHZ.In the case whenZ is infinite, we have the following version of Theorem 11.Theorem 13.
Assume thatZ is infinite and Ap,Bp∈NH, p=0,…,r. If(a)
Π is stabilizable and(b)
the pairΠ-R-1L∗∗,Q-R-1L∗ has not an unobservable eigenvalue λ≥1, then the algebraic Riccati equation (10) has a stabilizing solution.Proof.
Arguing as in the proof of the above theorem, we see that (10) has a maximal solution Xmax. Since (55)ΠFXmaxXi=∑p=0rApi+BpFXmaxi∗EXi·Api+BpFXmaxifor all X∈lSHZ and NH is a two-sided ideal of lSHZ, we deduce easily that(56)ΠFXmaxX∈NHfor all X∈lSHZ. Assuming by contradiction that Xmax is not a stabilizing solution and reasoning as in the proof of the above theorem, we deduce that ρΠFXmax≥1 and there is y∗∈lSHZ∗, y∗≥0, y∗≠0, such that ΠFXmax∗y∗=ρΠFXmaxy∗. Property (56) ensures that y∗NH≠{0}. Otherwise, (56) implies that ρΠFXmax∗y∗X=y∗ΠFXmax∗X=0 for all X∈lSHZ, which contradicts the assumption y∗≠0. As in the proof of Theorem 11, we obtain y∗Q-R-1L∗=0 and(57)y∗R-1L∗+FXmax∗RR-1L∗+FXmax=0.Letting Y=ψ-1y∗NH and repeating step by step the arguments in the proof of Theorem 11, we obtain successively Yi(R-1L∗+FXmax)i∗=0, i∈Z, and YA+BFXmax∗=YA-BR-1L∗∗ for all Y∈lSHZ. Now, property (56) implies that TrYΠFXmaxX=TrYΠ-R-1L∗X for all X∈lSHZ and (54) holds. Therefore, ΠFXmax∗y∗=Π-R-1L∗∗y∗ and Π-R-1L∗∗y∗=ρTA-BFXmax∗y∗. Since y∗Q-R-1L∗=0, we have obtained a contradiction of statement (b). Consequently, Xmax is a stabilizing solution and the proof is complete.Now, let us apply our results to finite-dimensional MAREs associated with stochastic systems of the forms (5) and (6). We will prove that conditions (a) and (b) from Theorem 13 are necessary and sufficient for the existence of a stabilizing solution to MARE (10).Corollary 14.
Assume thatZ is finite and the Hilbert spaces H and U are finite-dimensional. The algebraic Riccati equation (10) has a stabilizing solution if and only if(a)
Π is stabilizable,(b)
λ=1 is not an unobservable eigenvalue for the pair Π-R-1L∗∗,Q-R-1L∗.Proof.
We first recall that, in finite dimensions, all linear and bounded operators are compact and nuclear. Also, ifΠ is stabilizable, the maximal solution of MARE (10) has the property ρΠFXmax≤1. Indeed, from the proof of Theorem 9 from [8], we know that Xmaxi is the strong limit of an increasing sequence Xni, i∈Z, of solutions of certain associated Lyapunov equations. Denoting by Π1n the Lyapunov operators associated with these equations, we also know that Π1n converges to ΠFXmax and ρΠ1n<1 (see [7, 8], e.g.,). Since the eigenvalues of a matrix and consequently its spectral radius depend continuously on the matrix coefficients, we pass to the limit as n→∞ in the last inequality and we get ρΠFXmax≤1. The conclusion follows.
Further, we observe that the hypotheses of Theorems10 and 11 are fulfilled. The necessity part of the corollary follows from Theorem 10 while the sufficiency part is a direct consequence of Theorem 11 and Remark 12.Remark 15.
(a) Under the hypotheses of the above corollary, the operatorΠ-R-1L∗∗ coincides with the Lyapunov operator associated with the close-loop DTLS from [17, Theorem 12] for the stabilizing feedback controller F=-R-1L∗.
(b) In view of (a), Corollary14 shows that Theorems 10, 11, and 13 are infinite-dimensional extensions of Theorem 12 from [17].


## 5. Numerical Examples

In this section, we provide some numerical examples which show the efficiency of our theory. Even in the finite-dimensional case our results (see Corollary14) seem to be new when applied to DTLSs with Markovian jumps. So, let us begin with a finite-dimensional example which proves that Corollary 14 is a viable alternative when stochastic detectability and stochastic observability conditions fail to hold.Example 1.
Consider (5) and (7) in the special case where the multiplicative noise is missing (r=1, A1=0, B1=0) and(P3)
H=U=R, N∈N∗, N>3 is fixed, Z={1,…,N}, lSHZ=RN, A0i=B0i=3(i-1/i), i∈Z, M=C∗C, C=0,1,0,1,0,1,…∈RN, Ri=1, Li=0 for all i∈Z, and pi,i=pi,i+1=1/2 for i=1,…,N-1, pN,N=1, and pi,j=0 otherwise.
Let(58)yt=Cηtxtbe the output of (5). Note that, in its operation, system (5) switches between the modes(59)xt+1=3i-1ixt+3i-1iut,ifηt=i∈Z,according to the law of the Markov chain ηt. For example, if the initial state is ηt0=i, it obeys (59) till time t1 when the Markov chain switches to ηt1=j∈Z from ηt0=i and system (5) obeys (59) with j replacing i.
Let us solve the optimization problemO defined in Section 1.
In this special case, problemO is exactly a linear quadratic control problem associated with (5), (58). The associated MARE (10) has the following particular form:(60)Xi=4.5i-1i2Xi+Xi+1+Mi-20.25i-1/i4Xi+Xi+121+4.5i-1/i2Xi+Xi+1,i∈Z-N,(61)XN=9N-1N2XN+MN-81N-1/N4XN21+9N-1/N2XN.
We know [7] that if MAREs (60) and (61) have a stabilizing solution X, then the optimal control problem O has a solution ut=FXηtxt, where FX is defined by (32). So we only have to find a stabilizing solution of (60) and (61).
LetA be the stochastic system (5) without control (i.e., Bp=0, p=0,…,r) and let A,C be the system defined by A and the output (58). The existing literature results (see [7] and the references therein) show that (10) has a stabilizing solution if the stochastic system with control is stabilizable and if A,C is either stochastically observable or stochastically detectable. We will establish that A,C is neither stochastically uniformly observable nor stochastically detectable. (Note that stochastic observability does not imply stochastic detectability [7].) In this case, we will see that Corollary 14 ensures the existence of the stabilizing solution.
(I) System
A
,
C
Is Not Stochastically Uniformly Observable. According to Theorem 8 from [10], A,C is stochastically uniformly observable iff there are n0∈N and δ>0 such that(62)∑k=0n0Π1kC∗Ci>δ,i∈Z,where Π1 is defined by (22) with the coefficients from (P3). A direct computation shows that (63)Π1Xi=4.5i-1i2Xi+Xi+1,i∈Z-N,Π1XN=9N-1N2XN,for all X∈RN. Using (P3), we see that Π1kC∗C1=0 for all k∈N and (62) fails to hold for i=1. Consequently, A,C is not stochastically uniformly observable.
(II) System
A
,
C
Is Not Stochastically Detectable. Following [5] (see Definition 2 and Proposition 16 therein), the system A,C isdetectable in conditional mean iff there is F∈RN such that the nonnegative operator ΠFC defined by(64)ΠFCXi=3i-1i+FiCi2Xi+Xi+12,i∈Z-N,ΠFCXN=3N-1N+FNCN2XN,X∈RNgenerates an exponentially stable evolution operator. We observe that the matrix ai,j associated with ΠFC is upper triangular with the main diagonals aii=(1/2)3(i-1/i)+FiCi2, i∈Z-{N}, and aNN=3((N-1)/N)+FNCN2. Since a33=2, we deduce that λ=2 is an eigenvalue of ΠFC and ρΠFC≥1. So ΠFC cannot generate an exponentially stable evolution operator and A,C is not detectable in conditional mean.
Now let us show that Corollary14 can be used to prove that MARE (10) has a stabilizing solution in the special case when (P3) holds.
(III) Stabilizing Solution for the Algebraic Riccati Equation. We will prove that conditions (a) and (b) of Corollary 14 hold and, consequently, (60) has a stabilizing solution X. Setting Wi=-5/2, i∈Z, we have (see (28))(65)ΠWXi=14i-1i2Xi+Xi+12,i∈Z-N,ΠWXN=14N-1N2XNfor all X∈RN. The matrix associated with ΠW is upper triangular. Since all the elements of the main diagonal are less than 1/4, it follows that ρΠW<1/4 and ΠW is exponentially stable. Hence, Π is stabilizable, according to Definition 8, and condition (a) of Corollary 14 is satisfied. Let us prove that (b) holds, too. From (P3), we see that Q-R-1L∗=M-LR-1L∗=M and Π-R-1L∗, equal to Π1, are given by (63). The matrix bi,j associated with Π1 is also upper triangular with the main diagonal bi,i=4.5(i-1)/i2 for i∈Z-{N} and bN,N=9(N-1)/N2. It is not difficult to see that λ=1 is not an eigenvalue of Π1 because 1∉{bi,i,i∈Z}. Consequently, λ=1 is not an eigenvalue of Π1∗=Π-R-1L∗∗ and it cannot be an unobservable eigenvalue for the pair (Π-R-1L∗∗,Q-R-1L∗). Condition (b) of Corollary 14 is satisfied and (60) has a stabilizing solution X.
Proposition 14 from [10] ensures that this stabilizing solution is nonnegative and maximal among all nonnegative solutions of (60). Then, solving (60) and choosing the maximal solution, we get the stabilizing solution X.
ForN=4, the stabilizing solution is(66)X1=0,X2=1.7446,X3=0.84627,X4=1.9061and the feedback stabilizing gain is (67)FX1=0,FX2=-0.74455,FX3=0.84627,FX4=0.9061.Then, the optimal control problem O has a solution ut=FXηtxt, with the optimal cost being Ju;t0,x,i=Xix,x, i∈{1,2,3,4}.In the following example, we apply Theorem13 to prove the existence of stabilizing solutions for infinite-dimensional MAREs.Example 2.
Consider MARE (10) under the following additional hypotheses. The Hilbert space H is infinite-dimensional, U=H, Z=N∗, r=1, A1=B1=0, and A0i=B0i=8(i-1/i3)Ki, i∈N∗, where K∈lS1HZ, K≠0. The other coefficients of MARE (10) are L=0, R=ΦH, M1=0, Mi=IH, i∈Z, pi,i=pi,i+1=1/2, i∈Z, and pi,j=0 otherwise. Let us show that the corresponding MARE (10) has a stabilizing solution.
ObviouslyA0,B0∈NH, Q-R-1L∗=M, and Π-R-1L∗=Π1. Assume that λ≥1 is an unobservable eigenvalue of (Π-R-1L∗∗,Q-R-1L∗) and let y∗∈lSHZ∗ be a positive eigenvector of λ. We have(68)λy∗IH,0,0,…,=Π-R-1L∗∗y∗IH,0,0,…,=y∗Π1IH,0,0,…,=y∗0,0,…,=0.It follows that y∗M=y∗ΦH-y∗IH,0,0,…,=y∗ΦH. From Lemma 6 from [10], we know that y∗ΦH=y∗≠0. Hence, y∗Q-R-1L∗=y∗M≠0 and we get a contradiction. Thus, λ≥1 cannot be an unobservable eigenvalue for the pair (Π-R-1L∗∗,Q-R-1L∗). Condition (a) of Theorem 13 holds.
Further, we defineWi=-(1/(2k))2k+2IH, i∈Z, where k=supi∈ZKi1. Then, ΠWΦHi=(1/(2k2))8(i-1/i3)2Ki∗Ki, i∈N∗, and ΠWΦHZ≤k2supi∈N∗(1/2k2)8(i-1/i3)2=1/2. Therefore, ΠW=ΠWΦHZ≤1/2 and ΠW generates an exponentially stable evolution operator. Thus, Π is stabilizable, according to Definition 8. Conditions (a) and (b) of Theorem 13 are satisfied and (10) has a stabilizing solution. Theorem 9 from [8] provides an iterative algorithm for the computation of this stabilizing solution.The following example is an application of Theorem10.Example 3.
Assume that, exceptA0, B0, and M, the other coefficients of MARE (10) satisfy (P3) (see Example 1) with N replaced by ∞. Let A0i=B0i=1, Mi=0, for all i∈N∗, and ϕ=1,…,1,…∈l∞N∗. We recall that Π-R-1L∗=Π1 and Q-R-1L∗=M and Π1 is a positive operator. Since Π1ϕ=ϕ, we can apply Lemma 6 from [10] to deduce that Π1=Π1ϕZ=ϕZ=1. On the other hand, λ=1 is an eigenvalue of Π1 and, consequently, ρΠ1=1. Theorem 3.2.3 from [18] ensures that there is y∗∈lSHZ∗, y∗≥0, y∗≠0, such that Π-R-1L∗∗y∗=y∗. Obviously, y∗M=0 and now it is clear that λ=1 is an unobservable eigenvalue for (Π-R-1L∗∗,Q-R-1L∗). From Theorem 10, it follows that MARE (10) has not a stabilizing solution.


## 6. Conclusions and Further Research

In this paper we have obtained necessary and sufficient conditions for the existence of stabilizing solutions for MARE (10) in infinite dimensions. These conditions are similar to those given in [11, 17] for finite-dimensional MAREs and do not involve other detectability and observability conditions. They are viable alternatives to existing results (see, e.g., [3, 5, 7] and the references therein) obtained under detectability or observability hypotheses.The main difficulties in obtaining a “perfect” analog of the results from [17] (or [11]) are related to the fact that Lemma 1 is not true for infinite Z and we cannot ensure the existence of a nonnegative linear functional y∗ satisfying (34) and condition y∗NH≠{0}. To compensate these gaps, we have assumed a compactness hypothesis in Theorem 11 and the condition Ap,Bp∈NH, p=0,…,r, in Theorem 13. A natural question is how to relax these conditions.Another open problem is whether the assertionρΠFXmax≤1, where Xmax is the maximal solution of MARE (10), remains true in infinite dimensions or for infinite Z. The validity of this assertion will improve Theorems 11 and 13, as indicated in Remark 12.Also, further research is required to study the existence of stabilizing solutions for MARE (10) in the case when the state space of the Markov chain is a general Borel Space as in [1].



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