DISCRETE TIME LINEAR EQUATIONS DEFINED BY POSITIVE OPERATORS ON ORDERED HILBERT SPACES

Transcription

1 DISCRETE TIME LINEAR EQUATIONS DEFINED BY POSITIVE OPERATORS ON ORDERED HILBERT SPACES VASILE DRAGAN and TOADER MOROZAN In this paper the problem of exponential stability of the zero state equilibrium of a discrete-time time-varying linear equation described by a sequence of linear bounded and positive operators acting on an ordered Hilbert space is investigated. The class of linear equations considered in this paper contains as particular cases linear equations described by Lyapunov operators or symmetric Stein operators as well as nonsymmetric Stein operators. Such equations occur in connection with the problem of mean square exponential stability for a class of difference stochastic equations affected by independent random perturbations and Markovian jumping as well as in connection with some iterative procedures which allow us to compute global solutions of discrete time generalized symmetric or nonsymmetric Riccati equations. The exponential stability is characterized in terms of the existence of some globally defined and bounded solutions of some suitable backward affine equations (inequations, respectively) or forward affine equations (inequations, respectively). AMS 2000 Subject Classification: 39A11, 47H07, 93C55, 93E15. Key words: positive operators, discrete time linear equations, exponential stability, ordered Hilbert spaces, Minkovski norm. 1. INTRODUCTION The stabilization problem, together with various control problems for linear stochastic systems, was intensively investigated in the last four decades. We refer the reader to some of the most popular monographies in the field: [1, 6, 9, 25, 30, 40, 41] and references therein. It is well known that the mean square exponential stability or, equivalently, the second moments exponential stability of the zero solution of a linear stochastic differential equation or a linear stochastic difference equation is equivalent to the exponential stability of the zero state equilibrium of a suitable deterministic linear differential equation or a deterministic linear difference equation. Such deterministic differential (difference) equations are defined by REV. ROUMAINE MATH. PURES APPL., 53 (2008), 2 3,

2 132 Vasile Dragan and Toader Morozan 2 the so-called Lyapunov type operators associated to the given stochastic linear differential (difference) equations. Exponential stability in the case of differential equations or difference equations described by Lyapunov operators has been investigated as a problem with interest in itself in a lot of works. In the time-invariant case results concerning the exponential stability of linear differential equations defined by Lyapunov type operators were derived using spectral properties of positive linear operators on an ordered Banach space obtained by Krein and Rutman [29] and Schneider [39]. A significant extension of the results in [29] and [39] to the class of positive resolvent operators was provided by Damm and Hinrichsen [7, 8]. Similar results were derived also for the discrete-time timeinvariant case, see [21, 38]. In [13] the exponential stability of discrete-time time-varying linear equations defined by linear positive operators acting on a finite dimensional ordered Hilbert space, was studied. In that paper different characterizations of exponential stability in terms of the existence of bounded and uniformly positive solutions of some suitable backward affine equations (inequations, respectively) or affine equations were provided. In the case of continuous-time time-varying systems, a class of linear differential equations on the space of n n symmetric matrices S n is studied in [11]. Such equations have the property that the corresponding linear evolution operator is positive on S n. They contain as particular cases linear differential equations of Lyapunov type arising in connection with the problem of investigation of mean square exponential stability. The results of [11] were extended to an abstract framework of a differential equations with positive evolution on a finite dimensional ordered Hilbert space (see [12]). In this paper, the infinite dimensional counterpart of the results proved in [13] is derived. The discrete-time linear equations under consideration in this paper are defined by sequences of positive bounded linear operators on an ordered Hilbert space. The order relation is induced by a closed, solid, selfdual convex cone. The main tool involved in our developments is a Minkovski norm defined by the Minkovski functional associated to a suitable open and convex set. To characterize exponential stability, a crucial role is played by the unique bounded solution of some suitable backward affine equations as well as of some forward affine equations. We show that if the equations considered are described by periodic sequences of operators, then the bounded solution, if any, also is a periodic sequence. Moreover, in the time-invariant case the bounded solutions to both backward affine equation and forward affine equation are constant. Thus, the results concerning the exponential stability for the timeinvariant case are recovered as special cases of the results proved in this paper.

3 3 Discrete time linear equations 133 The outline of the paper is as follows: Section 2 collects some definitions, some auxiliary results in order to display the framework where the main results are proved. Section 3 contains results which characterize the exponential stability of the zero state equilibrium of a discrete-time time-varying linear equation described by a sequence of linear positive operators on a ordered Hilbert space. Section 4 deals with the problem of preservation of exponential stability under an additive perturbation of the sequence of linear operators defining the discrete time equations under consideration. In Section 5 we consider the case of discrete time, time varying linear equations defined by sequences of positive bounded linear operators on ordered Banach spaces. An application to the mean square exponential stability of a discrete time stochastic stochastic system perturbed by a Markov chain with an infinite number of states is provided. The paper ends with an Appendix which collects the usual definitions concerning the convex cones. Also, some useful properties of the Minkovski seminorm are presented. A set of sufficient conditions is given under which a Minkovski seminorm is just a norm. 2. PRELIMINARIES In this section we describe the framework where the discrete time linear equations investigated in this paper are defined. In our approach a crucial role will be played by the Minkovski norm. More details concerning this norm can be found in Appendix Positive linear operators on ordered Hilbert spaces In this subsection as well as in the following X is a real Hilbert space ordered by the ordering relation induced by the closed, solid, selfdual convex cone X +. Since X + is a selfdual convex cone, it follows from Remark A.1 and Proposition A.3 (in Appendix) that X + is a pointed cone. By Proposition A.3, 2 defined by (1) x 2 = ( x, x ) 1 2 is monotone with respect to X +. Let ξ Int X + be fixed; we associate the Minkovski functional ξ defined by (82). It follows from Theorem A.2 and Proposition A.2 that ξ is a norm on X. Moreover, from Theorem A.1 (v) and Theorem A.2 we deduce that ξ is equivalent to 2 defined by (1). Hence (X, ξ ) is a Banach space. Moreover, ξ has the properties below:

4 134 Vasile Dragan and Toader Morozan 4 P 1 ) If x, y, z X are such that y x z, then (2) x ξ max{ y ξ, z ξ }. P 2 ) For an arbitrary x X with x ξ 1 we have (3) ξ x ξ and ξ ξ = 1. We recall that if Y is a Banach space, T : Y Y a bounded linear operator and a norm on Y, then T = sup x 1 T x is the corresponding operator norm. Remark 2.1. a) Since ξ and 2 are equivalent, ξ and 2 are also equivalent. This means that there are two positive constants c 1 and c 2 such that c 1 T ξ T 2 c 2 T ξ for all bounded linear operators T : X X. b) If T : X X is the adjoint operator of T with respect to the inner product on X, then T 2 = T 2. In general, the equality T ξ = T ξ is not true. However, follows from a) it follows that there are two positive constants c 1, c 2 such that (4) c 1 T ξ T ξ c 2 T ξ. Definition 2.1. Let (X, X + ) and (Y, Y + ) be ordered vector spaces. An operator T : X Y is called positive if T (X + ) Y +. In this case we write T 0. If T (Int X + ) Int Y + we write T > 0. Proposition 2.1. If T : X X is a bounded linear operator, then (i) T 0 if and only if T 0; (ii) If T 0 then T ξ = T ξ ξ. Proof. (i) is a direct consequence of the fact that X + is a selfdual cone. (ii) If T 0 then from (3) we have T ξ T x T ξ. It follows from (2) that T x ξ T ξ ξ for all x X with x ξ 1, which leads to sup T x ξ T ξ ξ sup T x ξ, x ξ 1 x ξ 1 hence T ξ = T ξ ξ, thus the proof is complete. From (ii) of Proposition 2.1 we obtain Corollary 2.1. Let T k : X X, k = 1, 2, be positive bounded linear operators. If T 1 T 2 then T 1 ξ T 2 ξ. Example 2.1. (i) Let X = R n and X + = R n +, where R n + = {x = ( x1 x 2 x n ) T R n x i 0, 1 i n}. In this case, X + is a closed, solid, pointed, selfdual convex cone. The ordering induced on R n by

5 5 Discrete time linear equations 135 this cone is known as the component wise ordering. If T : R n R n is a linear operator, then T 0 iff its corresponding matrix A with respect to the canonical basis on R n has nonnegative entries. For ξ = (1, 1, 1,..., 1) T Int(R n +), the norm ξ is defined by (5) x ξ = max 1 i n x i. Properties P 1 and P 2 are fulfilled for the norm defined by (5). (ii) Let X = R m n be the space of m n real matrices, endowed with the inner product (6) A, B = Tr(B T A) A, B R m n, Tr(M) denoting as usual the trace of a matrix M. On R m n we consider the order relation induced by the cone X + = R m n +, where (7) R m n + = {A R m n A = {a ij }, a ij 0, 1 i m, 1 j n}. The interior of the cone R m n + is not empty. It can be seen that R m n + is a selfdual cone. On R m n we also consider the norm ξ defined by (8) A ξ = max a ij. i,j Properties P 1 and P 2 are fulfilled for the norm (8) with ξ = Int R m n An important class of linear operators on R m n is that of the form L A,B : R m n R m n with L A,B Y = AY B T for all Y R m n, where A R m m, B R n n are fixed givenmatrices. These operators are often called nonsymmetric Stein operators. It can be checked that L A,B 0 iff a ij b lk 0, i, j {1,..., m}, l, k {1,..., n}. Hence L A,B 0 iff the matrix A B defines a positive operator on the ordered space (R mn, R mn + ), where is the Kronecker product. (iii) Let S n R n n be the subspace of n n symmetric matrices. Let X = S n S n S n = Sn N with N 1 fixed. On Sn N, consider the inner product N (9) X, Y = T r(y i X i ) for arbitrary X = (X 1, X 2,..., X N ) and Y = (Y 1, Y 2,..., Y N ) in Sn N. space Sn N is ordered by the convex cone (10) S N,+ n i=1 = {X = (X 1, X 2,..., X N ) X i 0, 1 i N}, The

6 136 Vasile Dragan and Toader Morozan 6 whose interior Int S N,+ n = {X S N n X i > 0, 1 i N} is nonempty. Here X i 0 and X i > 0 means that X i is a positive semidefinite matrix or a, positive definite matrix, respectively. One may show that Sn N,+ is a selfdual cone. Together with the norm 2 induced by the inner product (9), on Sn N we also consider the norm ξ defined by (11) X ξ = max 1 i N X i, ( ) X = (X 1,..., X N ) S N n, where X i = max λ, σ(x i) is the set of eigenvalues of the matrix X i. λ σ(x i ) For the norm defined by (11), properties P 1 and P 2 are fulfilled with ξ = (I n, I n,..., I n ) = J Sn N. (iv) For an infinite dimensional case, let us consider X = l 2 (Z +, R) { } where l 2 (Z +, R) = x = (x 0, x 1,..., x n,...) x i R, x 2 i <. On X we consider the usual inner product x, y l 2 = x i y i for all x = {x i } i 0, y = i=0 { } {y i } i 0. Set X + = x = {x i } i 0 x 0 0, x 2 i x 2 0. It is easy to see that X + is a closed, pointed convex cone. In the finite dimensional case the analogue of this cone is known as a circular cone. { } The interior Int X + = x = {x i } i 0 x 0 > 0, x 2 i < x2 0. It remains i=1 to prove that X + is selfdual. Let y (X + ). Hence (12) x, y l 2 0 for all x = {x i } i 0 X +. In particular, taking x = {1, 0, 0,..., 0} in (12), we obtain y 0 0. It is easy to verify that if y 0 = 0 then y t = 0 for all t 1. Since y 0 0, it is obvious that if y t = 0 for all t 1, then we have y X +. Suppose now yt 2 > 0. Take x = { x i } i 0 defined by t=1 (13) x 0 = y 0, x i = γy i y 0 ( ) 1 2 with γ =. Obviously, x X +. Replacing (13) in (12), one gets yk 2 k=1 yk 2 y2 0, which shows that y X +. Thus, it was shown that (X + ) X +. k=1 Let now y = {y i } i 0 X +. We have to show that (12) holds for all x X +. Indeed, for x X + we have 2 x k y k yk 2 x2 0 y2 0, k=1 i=1 i=0 x 2 k k=1 k=1

7 7 Discrete time linear equations 137 which leads to x k y k x0 y 0. This is equivalent to x 0 y 0 x k y k k=1 k=1 x 0 y 0, which shows that (12) is fulfilled. Thus it was proved that X + (X + ). Hence X + is selfdual. Take ξ = (1, 0, 0,..., 0,...) X + and denote by B(ξ, B(ξ, 2 2 ) = {x l 2(Z +, R) x ξ ) the closed ball }. It is not difficult to check that 2 2 )}. B(ξ, 2 ) X +. Moreover, we have X + = {tx t 0, x B(ξ, If x = (x 0, x 1, x 2,..., x n,...) X, we write x = (x 0, x) with x = (x 1, x 2,..., x n,...). The corresponding Minkovski norm is given by x ξ = x 0 + x 2, where x 2 2 = x 2 i. This equality can be obtained taking into i=1 account that x ξ = inf{t > 0 x 2 2 x 0 2 2tx 0 t 2 < 0 and x 2 2 x tx 0 t 2 < 0} Discrete-time affine equations Let L = {L k } k k0 be a sequence of bounded linear operators L k : X X and f = {f k } k k0 a sequence of elements f k X. These two sequences define on X two affine equations (14) x k+1 = L k x k + f k, which will be called the forward affine equation or causal affine equation defined by (L, f), and (15) x k = L k x k+1 + f k which will be called the backward affine equation or anticausal affine equation defined by (L, f). For each k l k 0 let T c (k, l) : X X be the causal evolution operator defined by the sequence L, T c (k, l) = L k 1 L k 2... L l if k > l and T c (k, l) = I X if k = l, I X being the identity operator on X. For all k 0 k l, T (k, l) a : X X stands for the anticausal evolution operator on X defined by the sequence L, that is, T (k, l) a = L k L k+1... L l 1 if k < l and T a (k, l) = I X if k = l. Often the superscripts a and c will be omitted if any confusion is not possible. Let x k = T c (k, l)x, k l, l k 0 be fixed. One obtains that { x k } k l verifies the forward linear equation (16) x k+1 = L k x k with initial value x l = x. Also, if y k = T a (k, l)y, k 0 k l, then from the definition of T a kl one obtains that {y k} k0 k l is the solution of the backward

8 138 Vasile Dragan and Toader Morozan 8 linear equation (17) y k = L k y k+1 with given terminal value y l = y. Obviously, (16) and (17) lead to T c (k+1, l) = L k T c (k, l) for all k l k 0 and T a (k, l) = L k T a (k+1, l) for all k 0 k l 1. It should be remarked that, unlike the continuous time case, a solution {x k } k l of the forward linear equation (16) with given initial values x l = x is well defined for k l while a solution {y k } k l of the backward linear equation (17) with given terminal condition y l = y is well defined for k 0 k l. If for each k, the operators L k are invertible, then all solutions of equations (16), (17) are well defined for all k k 0. If (T c (k, l)) is the adjoint operator of the causal evolution operator T c (k, l), we define z l = (T c (k, l)) z, ( ) k 0 l k. By direct calculation one obtains that z l = L l z l+1. This shows that the adjoint of the causal evolution operator associated with the sequence L generates an anticausal evolution. Definition 2.2. a) We say that the sequence L = {L k } k k0 defines a positive evolution if for all k l k 0 the causal linear evolution operator T (k, l) c 0. b) We say that the sequence L = {L k } k k0 defines an anticausal positive evolution if for all k 0 k l the anticausal linear evolution operator T a (k, l) 0. Since T c (l + 1, l) = L l, T a (l, l + 1) = L l, respectively, it follows that the sequence {L k } k k0 generates a causal positive evolution or an anticausal positive evolution if and only if for each k k 0, L k is a positive operator. Hence, in contrast with the continuous time case, in the discrete time case, only sequences of positive operators define equations which generate positive evolutions. Throughout the paper we shall say that a sequence {L k } k 0 generates a positive evolution instead of a causal positive evolution every time when no confusion can arise. Also, in this case, we shall write T (k, l) instead of T c (k, l). The following result is straightforward. It will be used in the next sections. Corollary 2.2. Let L i = {L i k } k k 0, i = 1, 2 be two sequences of bounded linear operators and Ti c (k, l) be the corresponding causal linear evolution operators. Assume that 0 L 1 k L2 k for all k k 0. Under this assumption we have T2 c(k, l) T 1 c(k, l) for all k l k 0.

9 9 Discrete time linear equations 139 At the end of this subsection we recall the representation formulae of the solutions of affine equations (14), (15). Each solution of the forward affine equation (14) has the representation k 1 (18) x k = T c (k, l)x l + T c (k, i + 1)f i for all k l + 1. Also, any solution of the backward affine equation (15) has a representation i=l l 1 y k = T a (k, l)y l + T a (k, i)f i, k 0 k l 1. i=k 3. EXPONENTIAL STABILITY In this section, we deal with the exponential stability of the zero solution of a discrete time linear equation defined by a sequence of positive bounded linear operators. Definition 3.1. We say that the zero solution of the equation (19) x k+1 = L k x k is exponentially stable or, equivalently, that the sequence L = {L k } k k0 generates an exponentially stable evolution (E.S. evolution) if there are β > 0, q (0, 1) such that (20) T (k, l) ξ βq k l, k l k 0, where T (k, l) is the causal linear evolution operator defined by the sequence L. Remark 2.1 in (20) allows us to consider the norm 2, too. In the case where L k = L for all k, if (20) is satisfied, we shall say that the operator L generates a discrete-time exponentially stable evolution. It is well known that L generates a discrete-time exponentially stable evolution if and only if ρ[l] < 1, where ρ[ ] is the spectral radius. It must be remarked that if the sequence {L k } k k0 generates an exponentially stable evolution then it is a bounded sequence. In this section we shall derive several conditions which are equivalent to exponential stability of the zero solution of equation (19) in the case {L k } k k0. Such results can be viewed as an alternative characterization of exponential stability to the one in terms of Lyapunov functions. First, from Proposition 2.1, Corollary 2.1 and Corollary 2.2 we obtain the following result specific to the case of operators which generate positive evolution.

10 140 Vasile Dragan and Toader Morozan 10 Proposition 3.1. Let L = {L k } k k0, and L 1 = {L 1 k } k k 0 be two sequences of positive bounded linear operators on X. (i) The following are equivalent: a) L( ) defines an E.S. evolution; b) there exist β 1, q (0, 1) such that T (k, l)ξ ξ βq k l for all k l k 0. (ii) If L 1 k L k for all k k 0 and L generates an E.S. evolution, then L 1 also generates an E.S. evolution. Further, we shall prove: Theorem 3.1. Let {L k } k 0 be a sequence of positive bounded linear operators L k : X X. Then the following assertions are equivalent: (i) the sequence {L k } k 0 generates an exponentially stable evolution; k (ii) there exists δ > 0 such that T k,l ξ δ for arbitrary k k 0 0; l=k 0 k (iii) there exists δ > 0 such that T (k, l)ξ δξ for arbitrary k k 1 l=k 1 0, δ > 0 being independent of k, k 1 ; (iv) for an arbitrary bounded sequence {f k } k 0 X, the solution with zero initial value of the forward affine equation is bounded. x k+1 = L k x k + f k, k 0 Proof. The implication (iv) (i) is the discrete-time counter part of Perron s Theorem (see [22, 37]). It remains to prove the implications (i) (ii) (iii) (iv). If (i) is true, then (ii) follows immediately from (20) with δ = β 1 q. Let us prove that (21) 0 T (k, l)ξ T (k, l) ξ ξ for arbitrary k l 0. If T (k,l) ξ = 0 then it follows from Proposition 2.1 (ii) that T (k,l) ξ = 0 and (21) is fulfilled. If T (k, l)ξ 0 then from (3) applied to x = 1 T (k,l)ξ ξ T (k, l)ξ one gets 0 T (k, l)ξ T (k, l)ξ ξ ξ and (21) follows from Proposition 2.1 (ii). If (ii) holds then (iii) follows from (21). We have to prove that (iii) (iv). Let {f k } k 0 X be a bounded sequence, that is, f k ξ µ, k 0. From (3) we obtain f l ξ ξ f l f l ξ ξ, which leads to µξ f l µξ for all l 0. Since for each k l + 1 0, T (k, l + 1) is a positive operator, we have: µt (k, l + 1)ξ T (k, l + 1)f l µt (k, l + 1)ξ

11 11 Discrete time linear equations 141 and k 1 T (k, l + 1)ξ k 1 µ l=0 l=0 k 1 T (k, l + 1)f l µ l=0 T (k, l + 1)ξ. Using (2) we deduce that k 1 k 1 ξ T (k, l + 1)f l µ T (k, l + 1)ξ. ξ l=0 l=0 If (iii) is valid we conclude by using again (2) that k 1 ξ T (k, l + 1)f l µδ, k 1 l=0 which shows that (iv) is fulfilled by using (18). Thus the proof is complete. We note that the proof of Theorem 3.1 shows that in the case of a discrete time linear equation (19) defined by a sequence of positive bounded linear operators, the exponential stability is equivalent to the boundedness of the solution with the zero initial value of the forward affine equation x k+1 = L k x k + ξ. This is in contrast to the general case of a discrete time linear equation, where if we want to use Perron s Theorem to characterize the exponential stability we have to check the boundedness of the solution with zero initial value of the forward affine equation x k+1 = L k x k +f k for an arbitrary bounded sequence {f k } k 0 X. Let us now introduce the concept of uniform positivity. Definition 3.2. We say that a sequence {f k } k k0 X + is uniformly positive if there exists c > 0 such that f k > cξ for all k k 0. If {f k } k k0 X + is uniformly positive, we shall write f k 0, k k 0. If f k 0, k k 0, we shall write f k 0, k k 0. The next result provides a characterization of the exponential stability by using solutions of some suitable backward affine equations. Theorem 3.2. Let {L k } k k0 be a sequence of positive bounded linear operators L k : X X. Then the following assertions are equivalent: (i) the sequence {L k } k k0 generates an exponentially stable evolution; (ii) there exist β 1 > 0, q (0, 1) such that T (k, l) ξ β 1 q k l, ( ) k l k 0 ;

12 142 Vasile Dragan and Toader Morozan 12 (iii) for each k k 0 the series l k T (l, k)ξ is convergent and there exists δ > 0, independent of k, such that T (l, k)ξ δξ; l=k (iv) the discrete time backward affine equation (22) x k = L k x k+1 + ξ has a bounded and uniformly positive solution; (v) for an arbitrary bounded and uniformly positive sequence {f k } k k0 Int X + the backward affine equation (23) x k = L k x k+1 + f k, k k 0 has a bounded and uniformly positive solution. (vi) There exists a bounded and uniformly positive sequence {f k } k k0 Int X + such that the corresponding backward affine equation (23) has a bounded solution { x k } k k0 X + ; (vii) there exists a bounded and uniformly positive sequence {y k } k k0 Int X + which verifies (24) L k y k+1 y k 0, k k 0. Proof. The equivalence (i) (ii) follows immediately from (4). In a similar way to the proof of inequality (21), we obtain (25) 0 T (l, k)ξ T (l, k) ξ ξ for all l k k 0. If (ii) holds, then for each k k 0 the series l k T (l, k) ξ of real numbers is convergent and we have (26) T (l, k) ξ δ, l=k where δ = β 1 1 q is independent of k. Therefore, the series l k T (l, k)ξ is absolute convergent. From (25) and (26) we deduce that the inequality from (iii) is fulfilled. Thus, the validity of the implication (ii) (iii) is confirmed. For each k k 0, set y k = T (l, k)ξ. If (iii) holds then y k is well defined and l=k additionally y k δξ for all k k 0. Using the definition of the linear evolution operator T (l, k), we may write y k = ξ +L k T (l, k +1)ξ or, equivalently, l=k+1 y k = ξ + L k y k+1. This shows that {y k } k k0 is a solution of (22). Moreover, we have ξ y k δξ for all k k 0. This means that {y k } k k0 is a bounded

13 13 Discrete time linear equations 143 and uniformly positive solution of (22). Thus, we obtain that the implication (iii) (iv) holds. Now, we prove (ii) (v). Let {f k } k k0 Int X + be a bounded and uniformly positive sequence. Hence there exist ν i > 0, i = 1, 2, such that ν 1 ξ f l ν 2 ξ for all l k 0. Since T (l, k) 0, for all l k k 0 we may write (27) ν 1 T (l, k)ξ T (l, k)f l ν 2 T (l, k)ξ for all l k k 0. The monotonicity of the Minkovski norm together with the equality from Proposition 2.1 (ii) allow us to write (28) ν 1 T (l, k) ξ T (l, k)f l ξ ν 2 T (l, k) ξ. If (ii) is fulfilled then (28) shows that for each k k 0 the series T (l, k)f l ξ l k of the real numbers is convergent and (29) T (l, k)f l ξ δ 1 l=k for all k k 0, where δ 1 = ν 2β 1 1 q is independent of k. Thus, one gets that the series T (l, k)f l is absolutely convergent for all k k 0. l k Set z k = T (l, k)f l, k k 0. Using again the definition of the linear l=k evolution operator T (l, k) we can write (30) z k = f k + L k l=k+1 T (l, k + 1)f l = f k + L k z k+1. From (29) and (30) we deduce that {z k } k k0 is a bounded solution of (23). From (30) we also have that z k f k ν 1 ξ for all k k 0. This means that z k 0, k k 0, thus (v) holds. Further, (v) (iv) (vi) are straightforward. We now prove the implication (vi) (ii). Let us assume that there exists a bounded and uniformly positive sequence {f k } k k0 Int X + such that the corresponding equation (23) has a bounded solution { x k } k k0 X +. Therefore, there exist positive constants γ i, i {1, 2, 3}, such that γ 1 ξ f k γ 2 ξ, γ 1 ξ x k γ 3 ξ,

14 144 Vasile Dragan and Toader Morozan 14 for all k k 0. Let k 1 k 0 be fixed. Define ỹ k = T (k, k 1 ) x k, k k 1. Since T (k, k 1 ) 0, we may write (31) γ 1 T (k, k 1 )ξ T (k, k 1 )f k γ 2 T (k, k 1 )ξ γ 1 T (k, k 1 )ξ ỹ k γ 3 T (k, k 1 )ξ for all k k 1. From x k = L k x k+1 + f k, as well as from the definitions of ỹ k and T (k, k 1 ), we obtain successively ỹ k = T (k, k 1 )L k x k+1 + T (k, k 1 )f k = T (k + 1, k 1 ) x k+1 + T (k, k 1 )f k = ỹ k+1 + T (k, k 1 )f k. Thus, we obtained ỹ k+1 = ỹ k T (k, k 1 )f k for all k k 1. From (31) we deduce that (32) ỹ k+1 qỹ k for all k k 1, where q = 1 γ 1 γ 3. Taking γ 3 large enough in (31) we obtain q (0, 1). From (32) we obtain inductively ỹ k q k k1 x k1 for all k k 1. Using again (31) together with x k1 γ 3 ξ, we deduce that 0 T (k, k 1 )ξ γ 3 γ 1 q k k 1 ξ, which by (2) leads to T (k, k 1 )ξ ξ γ 3 γ 1 q k k 1, k k 1. From Proposition 2.1 (ii) we have T (k, k 1 ) ξ γ 3 γ 1 q k k 1, which means that (ii) holds. The implication (iv) (vii) follows immediately since a bounded and uniform by positive solution of (22) is a solution with the desired properties of (24). To complete the proof we show that (vii) (vi). Let {z k } k k0 Int X + be a bounded and uniformly positive solution of (24). Define f k = z k L k z k+1. It follows that { f k } k k0 is bounded and uniform by positive, therefore {z k } k 0 is a bounded and positive solution of (23) corresponding to { f k } k k0, thus the proof is complete. The next result provides more information about the bounded solution of the discrete time backward affine equations. Theorem 3.3. Let {L k } k k0 be a sequence of linear operators which generates an exponentially stable evolution on X. Then the following assertions hold: (i) for each bounded sequence {f k } k k0 X the discrete-time backward affine equation (33) x k = L k x k+1 + f k has an unique bounded solution which is given by (34) x k = T (l, k)f l, k k 0 ; l=k

15 15 Discrete time linear equations 145 (ii) if there exists an integer θ > 1 such that L k+θ = L k, f k+θ = f k for all k then the unique bounded solution of equation (33) also is a periodic sequence with period θ; (iii) if L k = L, f k = f for all k, then the unique bounded solution of equation (33) is constant and it is given by (35) x = (I X L ) 1 f, with I X the identity operator on X ; (iv) if L k are positive operators and {f k } k k0 X + is a bounded sequence, then the unique bounded solution of equation (33) satisfies x k 0 for all k k 0. Moreover, if {f k } k k0 Int X + is a bounded and uniformly positive sequence, then the unique bounded solution { x k } k k0 of equation (33) also is uniformly positive. Proof. (i) From (i) (ii) of Theorem 3.2, we deduce that for all k k 0 { j the series T (l, k)f l is absolutely convergent and there exists δ > 0 l=k }j k independent of k and j such that j ξ (36) T (l, k)f l δ. Set x k = lim j j l=k T l,k f l = l=k l=k T (l, k), we obtain x k = f k + L k T (l, k)f l. Taking into account the definition of l=k+1 T (l, k + 1)f l = f k + L k x k+1, which shows that { x k } k k0 solves (33). It follows from (36) that { x k } is a bounded solution of (33). Let { x k } k k0 be another bounded solution of equation (33). For each 0 k < j we may write j (37) x k = T (j + 1, k) x j+1 + T (l, k)f l. Since {L k } k k0 generates an exponentially stable evolution and { x k } k k0 is a bounded sequence, we have lim T (j + 1, k) x j+1 = 0. Letting j in (37), j we conclude that x k = T (l, k)f l = x k, which proves the uniqueness of the l=k bounded solution of equation (33). (ii) If {L k } k k0, {f k } k k0 are periodic sequences with period θ, then in a standard way, using the representation formula (34), one shows that the l=k

16 146 Vasile Dragan and Toader Morozan 16 unique bounded solution of the equation (33) is also periodic with period θ. In this case we may take that k 0 =. (iii) If L k = L, f k = f for all k, then they may be viewed as periodic sequences with period θ = 1. Based on the above result (ii) one obtains that the unique bounded solution of equation (33) also is periodic with period θ = 1, so it is constant. In this case, x will verify the equation x = L x + f. Since the operator L generates an exponentially stable evolution, we have ρ(l) < 1. Hence the operator I X L is invertible and we deduce that x is given by (35). Finally, if L k are positive operators, the assertions of (iv) follow immediately from the representation formula (34) and thus the proof is complete. Remark 3.1. From the representation formula (18) one obtains that if the sequence {L k } k k0 generates an exponentially stable evolution and {f k } k k0 is a bounded sequence, then all solutions of the discrete time forward affine equation (14) with given initial values at time k = k 0 are bounded on the interval [k 0, ). On the other hand, it follows from Theorem 3.3 (i) that the discrete time backward equation (15) has a unique bounded solution on the interval [k 0, ), which is the solution provided by the formula (34). In the case where k 0 =, with the same techniques as in the proof of Theorem 3.3, we may obtain a result concerning the existence and uniqueness of the bounded solution of a forward affine equations similar to that proved for the case of backward affine equations. Theorem 3.4. Assume that {L k } k Z is a sequence of linear operators which generates an exponentially stable evolution on X. Then the following assertions hold: (i) for each bounded sequence {f k } k Z the discrete time forward affine equation (38) x k+1 = L k x k + f k has a unique bounded solution { x k } k Z. Moreover, this solution has a representation formula (39) x k = k 1 l= T (k, l + 1)f l, k Z; (ii) if {L k } k Z, {f k } k Z are periodic sequences with period θ, then the unique bounded solution of equation (38) is periodic with period θ; (iii) if L k = L, f k = f, k Z then the unique bounded solution of equation (38) is constant and is given by x = (I X L) 1 f;

17 17 Discrete time linear equations 147 (iv) if {L k } k Z are positive operators and {f k } k Z X +, then the unique bounded solution of equation (38) satisfies x k 0 for all k Z. Moreover, if f k 0, k Z, then x k 0, k Z. If {L k } k Z is a sequence of linear operators on X we may associate a new sequence of linear operators {L # k} k Z defined by L # k = L k. Lemma 3.1. Let {L k } k Z be a sequence of bounded linear operators on X. The following assertions hold: (i) if T # (k, l) is the causal linear evolution operator on X defined by the sequence {L # k} k Z, then T # (k, l) = T ( l + 1, k + 1), where T (i, j) is the causal linear evolution operator defined on X by the sequence {L k } k Z ; (ii) {L # k} k Z is a sequence of positive linear operators if and only if {L k } k Z is a sequence of positive linear operators; (iii) the sequence {L # k} k Z generates an exponentially stable evolution if and only if the sequence {L k } k Z generates an exponentially stable evolution; (iv) the sequence {x k } k Z is a solution of the discrete time backward affine equation (33) if and only if the sequence {y k } k Z defined by y k = x k+1 is a solution of the discrete time forward equation y k+1 = L # k y k + f k, k Z. The proof is straightforward and it is omitted. The next result is obtained by combining Theorem 3.2 and Lemma 3.1. It provides a characterization of exponential stability in terms of the existence of the bounded solution of some suitable forward affine equation. Theorem 3.5. Let {L k } k Z be a sequence of positive bounded linear operators on X. Then the following assertion are equivalent: (i) the sequence {L k } k Z generates an exponentially stable evolution; (ii) for each k Z the series l k T (k, l)ξ is convergent and there exists δ > 0, independent of k, such that k T (k, l)ξ δξ, k Z; l= (iii) the forward affine equation (40) x k+1 = L k x k + ξ has a bounded and uniformly positive solution; (iv) for any bounded and uniformly positive sequence {f k } k Z Int X +, the corresponding forward affine equation (41) x k+1 = L k x k + f k has a bounded and uniformly positive solution;

18 148 Vasile Dragan and Toader Morozan 18 (v) there exists a bounded and uniformly positive sequence {f k } k Z Int X + such that the corresponding forward affine equation (41) has a bounded solution x k, k Z X + ; (vi) there exists a bounded and uniformly positive sequence {y k } k Z which verifies y k+1 L k y k 0. The proof follows immediately by combining the result proved in Theorem 3.2 and Lemma SOME ROBUSTNESS RESULTS In this section we prove some results which provide a measure of the robustness of the exponential stability in the case of positive linear operators. To state and prove this result some preliminary remarks are needed. So, l (Z, X ) stands for the real Banach space of bounded sequences of elements of X. If x l (Z, X ), we denote x = sup k Z x k ξ. Let l (Z, X + ) l (Z, X ) be the subset of bounded sequences {x k } k Z X +. It can be checked that l (Z, X + ) is a solid, closed convex cone. Therefore, l (Z, X ) is an ordered real Banach space for which the assumptions of Theorem 2.11 in [8] are fulfilled. Now we are in a position to prove Theorem 4.1. Let {L k } k Z, {G k } k Z be sequences of positive bounded linear operators such that {G k } k Z is a bounded sequence. Under these assumptions, the following assertions are equivalent: (i) the sequence {L k } k Z generates an exponentially stable evolution and ρ[t ] < 1, where ρ[t ] is the spectral radius of the operator T : l (Z, X ) l (Z, X ), by (42) y = T x, y k = k 1 l= T (k, l + 1)G l x l, where T (k, l) is the linear evolution operator on X defined by the sequence {L k } k Z ; (ii) the sequence {L k +G k } k Z generates an exponentially stable evolution on X. Proof. (i) (ii) If the sequence {L k } k Z defines an exponentially stable evolution, then we define the sequence {f k } k Z by (43) f k = k 1 l= T (k, l + 1)ξ.

19 19 Discrete time linear equations 149 We have f k = ξ + k 2 l= T k,l+1 ξ which leads to f k ξ thus f k Int X + for all k Z. This allows us to conclude that f = {f k } k Z Int l (Z, X + ). Applying Theorem 2.11 [8] with R = I l and P = T we deduce that there exists x = {x k } k Z Int l (Z, X + ) which verifies the equation (44) (I l T )(x) = f. Here, I l stands for the identity operator on l (Z, X ). Partitioning (44) and taking into account (42) (43), we obtain that for each k Z we have x k+1 = k l= Further, we may write k 1 x k+1 = G k x k +ξ+l k l= T (k + 1, l + 1)G l x l + k l= k 1 T (k, l+1)g l x l +L k l= This shows that {x k } k Z verifies the equation (45) x k+1 = (L k + G k )x k + ξ. T (k + 1, l + 1)ξ. T (k, l+1)ξ = G k x k +ξ+l k x k. Since L k and G k are positive operators and x 0, (45) shows that x k ξ. Thus, we get that equation (40) associated with the sum operator L k + G k has a bounded and uniform positive solution. Using implication (iii) (i) of Theorem 3.5 we conclude that the sequence {L k + G k } k Z generates an exponentially stable evolution. Now, we prove the converse implication. If (ii) holds then using the implication (i) (iii) of Theorem 3.5, we deduce that equation (45) has a bounded and uniform by positive solution { x k } k Z Int X +. Equation (45) verified by x k may be rewritten as (46) x k+1 = L k x k + f k, where f k = G k x k + ξ, k Z, fk ξ, k Z. Using the implication (v) (i) of Theorem 3.5, we deduce that the sequence L k generates an exponentially stable evolution. Since equation (46) has unique bounded solution which is given by the representation formula (39), we have x k = k 1 T (k, l + 1) f l, k Z, so that (47) x k = k 1 l= T (k, l + 1)G l x l + k 1 l= l= T (k, l + 1)ξ.

20 150 Vasile Dragan and Toader Morozan 20 Invoking (42), equation (47) may be written as (48) x = T x + g, where g = { g k } k Z, g k = k 1 l= T (k, l + 1)ξ. It is obvious that g k ξ for all k Z. Hence g Int l (Z, X + ). Using implication (v) (vi) of Theorem 2.11 in [8] for R = I l and P = T we obtain that ρ[t ] < 1, thus the proof is complete. In the second part of this section we consider the periodic case. Assume that there exists θ 1 such that L t+θ = L t and P t+θ = P t for all t Z. Inductively, one obtains that, in this case, we have: T (t + kθ, s + kθ) = T (t, s) for all t s, k 0, t, s, k Z, where T (t, s) is the linear evolution operator defined by the sequence {L t } t Z. As a consequence of the above equality, one gets T (nθ, 0) = T (θ, 0) for all n 0. Thus, if the sequence {L t } t Z generates an E.S. evolution then, ρ[t (θ, 0)] < 1. In this case, an operator valued function is well defined by G : {0, 1,..., θ} {0, 1,..., θ 1} B(X ), with (49) G(t, s) = T (t, 0)(I X T (θ, 0)) 1 T (θ, s + 1) + T (t, s + 1)χ t 1 (s) if 1 t θ, 0 s θ 1 and G(0, s) = (I X T (θ, 0)) 1 T (θ, s + 1), 0 s θ 1, where χ t 1 (s) is the indicator function of the set {1, 2,..., t 1}. It is easy to check that (50) G(0, s) = G(θ, s), ( ) 0 s θ 1. In the special case θ = 1 (i.e., the time invariant case) (49) reduces to (51) G(1, 0) = G(0, 0) = (I X L) 1. Let X θ = X X X (θ times). The elements of this space are finite sequences of the form x = (x 0, x 1,..., x θ 1 ), x i X, 0 i θ 1. On X θ we introduce the norm x θ = max{ x i ξ, 0 i θ 1}. The space X θ is an ordered Banach space with the norm θ and the ordered relation induced by the closed solid normal convex cone X +θ = X + X +. Consider the operator Π : X θ X θ defined by y = Πx, where y = (y 0, y 1,..., y θ 1 ), θ 1 (52) y t = G(t, s + 1)P s x s, s=0 0 t θ 1 for all x = (x 0, x 1,..., x θ 1 ) X θ.

21 21 Discrete time linear equations 151 Remark 4.1. a) It is obvious that (52) may be extended to t = θ by y θ = θ 1 G(θ, s + 1)P s x s. From (50) we deduce that y θ = y 0. This allows us s=0 to extend the finite sequence defined by (52) to an periodic infinite sequence with period θ. b) In the special case θ = 1, X θ coincides with X. Using (51) and (52) we obtain that in this case Πx = (I X L) 1 P x, for all x X. Lemma 4.1. Assume that a) the sequences {L t } t Z, {P t } t Z are periodic with period θ 1; b) L t 0, P t 0, t Z; c) {L t } t Z defines an E.S. evolution. Under these asssumptions, the operator Π defined by (52) is a positive bounded linear operator. Proof. The fact that Π is a bounded linear operator is obvious from its definition. It remains to prove that Π 0. Let x = ( x 0, x 1,..., x θ 1 ) X +θ and ŷ = Π x. Construct sequences { x t } t Z, {ỹ t } t Z by x t = x s, ỹ t = ŷ s if t = kθ + s, 0 s θ 1. It is obvious that x t+θ = x t, ỹ t+θ = ỹ t, for all t Z. If we consider the periodicity of P t, we get that {ỹ t } t Z is a periodic solution of the equation (53) ỹ t+1 = L t ỹ t + z t, where z t = P t x t. Since z t 0, we conclude via Theorem 3.4 (iv) applied to equation (53), that ỹ t 0 and the proof is complete. The analogue of Theorem 4.1 in the periodic case is Theorem 4.2. Let {L t } t Z, {P t } t Z be sequences of bounded linear operators on X with the properties a) there exists an integer θ 1 such that L t+θ = L t and P t+θ = P t for all t Z; b) L t 0, P t 0, t Z. Under these assumptions the following assertions are equivalent: (i) the sum sequence {L t + P t } t Z generates an E.S. evolution; (ii) the sequence {L t } t Z generates an E.S. evolution and ρ(π) < 1, where Π is the linear operator defined by (52). Proof. If (i) holds, then using (i) (iii) of Theorem 3.5 and (ii) of Theorem 3.4 we deduce that the forward affine equation x t+1 = [L t + P t ]x t + ξ has a bounded and uniform positive solution { x t } t Z, which is periodic with period θ.

22 152 Vasile Dragan and Toader Morozan 22 It can be seen that { x t } t Z solves the equation x t+1 = L t x t + f t, where f t = P t x t + ξ. Invoking the implication (v) (i) of Theorem 3.5, we conclude that {L t } t Z generates an E.S. evolution. Using the representation formula of a periodic solution of a discrete time affine equation we obtain θ 1 (54) x t = G(t, s + 1)(P s x s + ξ), 0 t θ 1. s=0 Set x = ( x 0, x 1,..., x θ 1 ) X θ, where x t = x t, 0 t θ 1. From (54) one gets that x solves the equation (55) ( I X + Π) x + ĝ = 0, where ĝ = (ĝ 0, ĝ 1,..., ĝ θ 1 ) with ĝ t = θ 1 s=0 G(t, s + 1)ξ. It can be seen that ĝ is a restriction of the periodic solution of the forward affine equation (56) g t+1 = L t g t + ξ. Applying Theorem 3.4 (iv), we conclude that g t > 0. Thus we deduce that ĝ Int X +θ. Invoking the implication (v) (vi) of Theorem 2.11 in [8] to equation (55) for R = I X, P = Π, one concludes that ρ(π) < 1. So we obtain that (ii) holds. Now, we prove the converse implication. If (ii) holds, then from Theorem 3.4 we deduce that equation (56) has a periodic solution { g t } t Z Int X +. If ĝ = (ĝ 0, ĝ 1,..., ĝ θ 1 ) is such that ĝ t = g t, 0 t θ 1, then ĝ Int X +θ. Invoking again Theorem 2.11 [8] we conclude that equation (55) has a solution x = ( x 0, x 1,..., x θ 1 ), x Int X +θ. Construct the sequence { x t } t Z by x t = x s if t = kθ + s, 0 s θ 1, k Z. One can see that { x t } t Z is a periodic sequence of period θ. Also, one obtains that { x t } t Z is a periodic solution of the equation x t+1 = (L t +P t )x t +ξ. Applying Theorem 3.5, we conclude that {L t + P t } t Z generates an E.S. evolution, thus the proof is complete. Using Remark 4.1 b), we deduce that in the time invariant case the result proved in Theorem 4.2 becomes Corollary 4.1. If L, P are two positive bounded linear operators on X, then the following assertions are equivalent: (i) L + P defines an E.S. evolution; (ii) ρ[l] < 1, ρ[(i X L) 1 P ] < 1.

23 23 Discrete time linear equations THE CASE OF DISCRETE TIME LINEAR EQUATIONS DEFINED BY POSITIVE OPERATORS ON ORDERED BANACH SPACES In this section, X is a real Banach space ordered by the order relation induced by the closed, solid, pointed convex cone X +. Throughout this section we assume that the by hypothesis H 1 below holds: H 1 The norm of the Banach space X is monotone with respect to the cone X +. Let ξ Int X + be fixed. According to Proposition A2, the set B ξ defined by (57) B ξ = {x X ξ < x < ξ} is bounded. Using Proposition A.2 and Theorem A.2 we deduce that the corresponding Minkovski functional ξ is a norm equivalent to the norm of the Banach space X. Properties P 1, P 2 stated in Section 2 for an Hilbert space are still valid in the case of a Banach space. Also, Proposition 2.1 (ii) is valid in the context of an ordered Banach space. Example 5.1. Let X = S n, where S n consists of all sequences S = (S(1), S(2),..., S(k),...) with S(i) S n for all i such that (58) sup{ S(i), i 1} <. The space S n equipped with the norm (59) S = sup{ S(i), i 1} is a real Banach space. On Sn we consider the ordered relation induced by the solid, closed, pointed, convex cone Sn + with Sn + = {(S(1), S(2),...) Sn S(i) 0, i 1}. Obviously, H 1 holds. Take ξ = (I n, I n,...) Sn +. In this case, (57) becomes (60) B ξ = {S = (S(1), S(2),...) S n I n < S(i) < I n, i Z +, i 1}. Hence B ξ is a bounded set. Using the definition of the Minkovski functional (see Appendix), we obtain that (61) S ξ = S for all S Sn. For S Int Sn + we shall write S 0 if S(i) εi n for all i 1, with ε > 0 not depending upon i. This is a special case of Definition 3.2 for the case of constant sequences.

24 154 Vasile Dragan and Toader Morozan 24 Let us consider a sequence of positive bounded linear operators {L t } t Z on X. If X is a Banach space, it is not clear if one can characterize the exponential stability of the zero solution of the discrete time linear equation (62) x t+1 = L t x t in terms of the existence of a bounded and uniformly positive solution of some suitable forward affine equations as well as in terms of the existence of some bounded and uniformly positive solution of some suitable backward affine equations, as in the Hilbert space case. Now, we show that in the case of a Banach space one can prove a result as in Theorem 3.5 to characterize the exponential stability in the case of equations (60). Theorem 5.1. Let X be a real Banach space ordered by the ordered relation induced by the closed, solid, pointed convex cone X +. Assume that H 1 is fulfilled. Let {L t } t Z be a sequence of positive bounded linear operators on X. Then the following assertions are equivalent: (i) the zero state equilibrium of (62) is E.S.; (ii) for each t Z the series l t T (t, l)ξ is convergent and there exists δ > 0 independent of t such that t (63) T (t, l)ξ δξ l= for all t Z; (iii) the forward affine equation (64) x t+1 = L t x t + ξ has a bounded and uniform positive solution; (iv) for any bounded and uniformly positive sequence {f t } t Z Int X + the corresponding forward affine equation: (65) x t+1 = L t x t + f t has a bounded and uniformly positive solution; (v) there exists a bounded and uniformly positive sequence (66) {f t } t Z Int X + such that the corresponding forward affine equation (65) has a bounded solution x t, t Z X + ; (vi) there exists a bounded and uniformly positive sequence {y t } t Z which verifies (67) y t+1 L t y t 0, t Z.

25 25 Discrete time linear equations 155 The proof follows the same lines as in the proof of Theorem 3.2 and is omitted. As we have seen is Subsection 2.2, a sequence {L t } t t0 may also define a backward discrete time linear equation or, equivalently, an anticausal evolution. This is (68) x t = L t x t+1. Concerning equation (68) we introduce Definition 5.1. We say that the zero state equilibrium of equation (68) is anticausal exponentially stable (A.E.S., for short) or, equivalently, the sequence {L t } t t0 generates an A.E.S. evolution if there exists β 1, q (0, 1) such that (69) T a (t, s) ξ βq s t for all t 0 t s, t, s Z, where T a (t, s) is the anticausal linear evolution operator associated to (68). It must be remarked that in (69) we may use any norm equivalent to the given norm of the Banach space X. Concerning the characterization of the anticausal exponential stability one proves: Theorem 5.2. Let {L t } t t0 be a sequence of linear bounded and positive operators on the ordered Banach space X. If H 1 holds, the following assertions are equivalent: (i) the sequence {L t } t t0 generates an A.E.S. evolution; (ii) for each t t 0 the series T a (s, t)ξ is convergent and there exists s t δ > 0, independent of t, such that (70) T a (t, s)ξ δξ s=t for all t t 0 ; (iii) the discrete time backward affine equation (71) x t = L t x t+1 + ξ has a bounded and uniformly positive solution; (iv) for an arbitrary bounded and uniform by positive sequence {f t } t t0, the backward affine equation (72) x t = L t x t+1 + f t, t t 0 has a bounded and uniformly positive solution;

26 156 Vasile Dragan and Toader Morozan 26 (v) there exists a bounded and uniformly positive sequence {f t } t t0 such that the corresponding backward affine equation (72) has a bounded solution { x t } t t0 X + ; (vi) there exists a bounded and uniformly positive sequence {y t } t t0 which verifies (73) L t y t+1 y t 0, t t 0. The proof may be done by following step by step the proof of Theorem 3.2, replacing the operator T (t, s) by T a (t, s) and L t by L t. In the last part of this section we illustrate the applicability of Theorem 5.2 to characterize the exponential stability in mean square in the case of discrete-time stochastic linear systems perturbed by a Markov chain with an infinite number of states. Consider the discrete-time stochastic linear system (74) x t+1 = A(t, η t )x t, t 0, where x t R n and {η t } t 0 is a Markov chain with an infinite but countable set of states on a given probability space (Ω, F, P) and with transition probability matrices P t. This means (see [10, 23]) that for each t 0, t Z, η t : Ω Z 1 is a random variable with the property (75) P{η t+1 = j G t } = p t (η t, j) a.s., where G t = σ[η 0, η 1,..., η t ] is the σ algebra generated by {η s } 0 s t, and Z 1 = {i Z, i 1}. Setting P t = (p t (i, j)) i,j Z1, p t (i, j) being the scalars on the right hand side of (75), one obtains a sequence of stochastic matrices with an infinite number of rows and columns. To avoid some complications due to the discrete time context (see [14] for the case of systems perturbed by a Markov chain with a finite number of states) we make the assumption H 2 For each t Z +, P t is an nondegenerate stochastic matrix. We recall that a stochastic matrix P t is called nondegenerate if for each j Z 1 there exists i Z 1 such that p t (i, j) > 0. Define π t (j) = P{η t = j} and set π t = (π t (1), π t (2),...); π t is the distribution of the random variable η t. We have π t (j) 0, j Z 1 and π t (j) = 1 for all t Z +. One verifies inductively that under assumption j Z 1 H 2, we have π t (j) > 0 for all t 1 and j 1, if π 0 (i) > 0 for all i 1. In what follows we assume that π 0 (i) > 0 for all i 1. Concerning the matrix coefficients A(t, i), t 0, i 1, of the system (74) we make the assumption H 3 (76) sup A(t, i) <, i 1