SOME LYAPUNOV TYPE POSITIVE OPERATORS ON ORDERED BANACH SPACES

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1 Ann. Aca. Rom. Sci. Ser. Math. Appl. ISSN Vol. 5, No. 1-2 / 2013 SOME LYAPUNOV TYPE POSITIVE OPERATORS ON ORDERED BANACH SPACES Vasile Dragan Toaer Morozan Viorica Ungureanu Abstract In this paper we investigate several properties of Lyapunov type operators occurring in connection with the characterization of exponential stability in mean square of systems of linear Itô ifferential equations perturbe by a Markov process with an infinite countable number of states. A criterion for exponential stability of linear ifferential equations efine by a Lyapunov operator is erive uner the assumption of property of etectability aequately efine for this type of operator value functions. MSC: 34G10, 47B65, 34D20 keywors: Lyapunov type operators, orere Banach spaces, Minkovski norms, exponential stability. Accepte for publication in revise form on April 2, vasile.ragan@imar.ro; toaer.morozan@imar.ro Institute of Mathematics Simion Stoilow of the Romanian Acaemy, Research Unit Nr.2, P.O.Box 1-764, RO , Bucharest, Romania; Department of Mathematics, Constantin Brancusi University, Tg. Jiu, Bulevarul Republicii, nr. 1, ju. Gorj, Romaina, vio@utgjiu.ro. 65

2 66 Vasile Dragan, Toaer Morozan, Viorica Ungureanu 1 Introuction The Lyapunov type linear operators play a central role in the characterization of exponential stability of linear systems of ifferential equations both in eterministic an stochastic framework, continuous time an iscrete-time cases. For the reaers convenience, we refer to [13, 14] for the eterministic framework an [1, 2, 9, 10, 15] for the stochastic context. In the stochastic case, the ifferential equations efine by the Lyapunov type operator value functions offer a eterministic framework for the characterization of exponential stability in mean square for stochastic linear ifferential equations with multiplicative white noise perturbations an/ or Markovian jumping. The systematic investigation of linear ifferential equations efine by Lyapunov type operator value functions regare as mathematical object with interest in itself, starte with the time invariant case, namely, the ifferential equation is efine by a Lyapunov operator not epening upon the time. In this case, criteria for exponential stability of ifferential equations efine by a Lyapunov operator were erive base on properties of resolvent positive operators [4, 5, 6]. In the time varying case, criteria for exponential stability of ifferential equations efine by a Lyapunov type operator value function were obtaine in [8, 7] for the finite imensional case an in [11] for infinite imensional case. The erive criteria are expresse in terms of existence of some global boune an uniform positive solutions of some suitable affine ifferential equations. It is known that in the case when the operator value function efining the Lyapunov type ifferential equation is either perioic function or constant function, the unique global boune solution of the affine ifferential equations involve in stuy of exponential stability criteria, is also perioic or constant function. Therefore, in this important special cases, the existence of a positive efinite solution may be teste by numerical computations. Usually, to a system of stochastic linear ifferential equations one may associate two kins of Lyapunov type operator value functions which efine two types of ifferential equations on some linear spaces of symmetric matrices: a forwar ifferential equation an a backwar ifferential equation. In the case of finite imensional linear space of symmetric matrices one may introuce in a natural way an inner prouct which inuces a Hilbert space structure. In this case the two types of Lyapunov type operators are interconnecte, one of them being the ajoint operator of the other. A etaile stuy of linear ifferential equations efine by the Lyapunov type operator value functions on

3 Lyapunov operators on orere Banach spaces 67 a finite imensional linear space of symmetric matrices may be foun in Chapter 2 in [9]. The goal of the present paper is to stuy the Lyapunov type operator value functions on an infinite imensional linear space of boune sequences of symmetric matrices. Such linear spaces can be organize as infinite imensional orere Banach spaces. We shall prove that uner some aitional assumptions we may ientify some suitable subspaces which may be organize as infinite imensional orere Hilbert spaces where the two Lyapunov type linear operators are interconnecte. More precisely, the restriction of the Lyapunov type operator efining a backwar ifferential equation coincies with the ajoint of the Lyapunov operator efining a forwar ifferential equation. Finally, we shall introuce a concept of etectability for Lyapunov type operators an we shall erive a new criterion for exponential stability for ifferential equations efine by the Lyaponov type operator value functions. 2 Linear ifferential equations with positive evolution on an orere Banach space 2.1 Linear evolution operators on orere Banach spaces Let (X, ) be a real Banach space. Let I R be an interval of real numbers. Let L : I B(X ) be a strongly continuous operator value function. This means that for each x X the vector value function t L(t)x is continuous on I. We consier the linear ifferential equation on X : x(t) = L(t)x(t). (1) t Base on the evelopments in Chapter 3 of [3] we euce that for each (t 0, x 0 ) I X there exists a unique C 1 -function x( ; t 0, x 0 ) : I X satisfying (1) an the initial conition x(t 0 ; t 0, x 0 ) = x 0. In Chapter 3 of [3] it is shown that there exists an operator value function T L : I I B(X ) with the property that x(t; t 0, x 0 ) = T L (t, t 0 )x 0 for all t, t 0 I an x 0 X. The operator value function (t, τ) T L (t, τ) or T L (t, τ) for shortness is name the linear evolution operator on X efine by the operator value function L( ) or, equivalently, the linear evolution operator efine on X by the linear ifferential equation (1).

4 68 Vasile Dragan, Toaer Morozan, Viorica Ungureanu Often, we shall write T (t, τ) instea of T L (t, τ) if confusions are not possible. Remark 1 A linear evolution operator T (t, τ) on a Banach space X has the properties (see [3] Chapter 3 for etails). (i) t T (t, τ) is the unique solution of the problem with given initial value on B(X ) t X(t) = L(t)X(t), X(τ) = I X where I X is the ientity operator on X. More precisely, T (t, τ) = L(t)T (t, τ), t I (2) t T (τ, τ) = I X. (ii) τ T (t, τ) : I B(X ) satisfies (iii) T (t, τ) = T (t, τ)l(τ), τ I. (3) τ T (t, τ)t (τ, s) = T (t, s), ( ) t, τ, s I. (4) (iv) For each (t, τ) I I, the operator T (t, τ) is invertible an T 1 (t, τ) B(X ). More precisely, we have T 1 (t, τ) = T (τ, t). R t (v) T (t, τ) e L(s) s τ, t, τ I. (vi) If L(t) = L B(X ) then T (t, τ) = e L(t τ), where e Lt = k=0 t k k! Lk. This series is convergent in the topology inuce by the operator norm uniformly on any compact subinterval of R. (vii) If I = R an L(t + θ) = L(t) for all t R an some θ > 0 then T (t + kθ, τ + kθ) = T (t, τ) for any t, τ R, k Z. The strongly continuous operator value function L( ) efines also the linear ifferential equation y(t) + L(t)y(t) = 0. (5) t Applying the results from Chapter 3 of [3] to the operator value function t L(t) we euce that for each (t 0, y 0 ) I X the linear ifferential

5 Lyapunov operators on orere Banach spaces 69 equation (5) has a unique solution y( ; t 0, y 0 ) : I X which satisfy the initial conition y(t 0 ; t 0, y 0 ) = y 0. In what follows we enote by T a L (t, t 0) the causal evolution operator generate by L(t) an we call it the anticausal linear evolution operator on X generate by the operator value function L( ). One proves also that y(t; t 0, y 0 ) = T a L (t, t 0)y 0 for all (t; t 0, y 0 ) I I X an therefore the evolution operator T a L (t, t 0) : I I B(X ) will be also calle the anticausal linear evolution operator on X generate by the linear ifferential equation (5). In the sequel, we shall write T a (t, t 0 ) instea of T a L (t, t 0), if confusions are not possible. Remark 2 Many of the assertions of Remark 1 remain vali if the causal linear evolution operator T (t, τ) is replace by the anticausal linear evolution operator T a (t, τ). In the case of anticausal linear evolution operator, the statements (i), (ii), (vi) from Remark 1 become: (i ) for each τ I, t T a (t, τ) satisfies the linear ifferential equation on B(X ): t T a (t, τ) = L(t)T a (t, τ) (6) an the initial conition T a (τ, τ) = I X. (ii ) for each t I, τ T a (t, τ) satisfies: τ T a (t, τ) = T a (t, τ)l(τ). (7) (vi ) If L(t) = L B(X ), t R, then, T a L (t, τ) = el(τ t), t, τ R. Besie the linear ifferential equations (1) an (5) respectively, we associate the following affine ifferential equations: an x(t) = L(t)x(t) + f(t) (8) t y(t) + L(t)y(t) + g(t) = 0 (9) t where L( ) is an operator value function as before, an f : I X, g : I X are continuous vector value functions.

6 70 Vasile Dragan, Toaer Morozan, Viorica Ungureanu The solutions of (8) an (9) have the following representation formulae: t x(t; t 0, x 0 ) = T (t, t 0 )x 0 + for all t [t 0, ) I, x 0 X an y(t; t 0, y 0 ) = T a (t, t 0 )y 0 + t 0 T (t, s)f(s)s (10) t 0 t T a (t, s)g(s)s (11) for all t ( ; t 0 ] I, y 0 X. In the evelopment from this paper, the affine ifferential equation of type (8) will be calle forwar affine ifferential equation while the affine ifferential equations of type (9) will be name backwar affine ifferential equations. Remark 3 If y(t) is a solution of the backwar affine ifferential equation y(t) + L(t)y(t) + g(t) = 0 t then ˆx(t) efine by ˆx(t) = y( t) is a solution of the forwar affine equation t x(t) = ˆL(t)x(t) + ˆf(t) where ˆL(t) = L( t) an ˆf(t) = g( t). Moreover if T a (t, t 0 ) is the anticausal evolution operator efine by the operator value function L( ) then ˆT (t, t 0 ) efine by ˆT (t, t 0 ) = T a ( t, t 0 ), t, t 0 Î = {t R; t I} (12) is the causal evolution operator efine by the operator value function ˆL : Î B(X ), ˆL(t) = L( t). 2.2 Linear ifferential equations with positive evolutions on orere Banach spaces Let (X, ) be a real Banach space orere by a soli, close, normal, convex cone X +. Let L : I B(X ) be a strongly continuous operator value function.

7 Lyapunov operators on orere Banach spaces 71 Definition 1 We say that the operator value function L( ) generates: (i) a causal positive evolution on X, or a positive evolution (for shortness) if T L (t, t 0 )X + X +, for all t t 0, t, t 0 I. (ii) an anticausal positive evolution on X, if T a L (t, t 0)X + X +, for all t t 0, t, t 0 I. In other wors, the operator value function L( ) generates a positive evolution on X if the solutions of the linear ifferential equation (1) have the property that x(t; t 0, x 0 ) X + for all t t 0, t, t 0 I if x 0 X +. In this case we shall say that the linear ifferential equation (1) efines a positive evolution on X. Similarly the operator value function L( ) generates an anticausal positive evolution on X if an only if the solutions of linear ifferential equation (5) have the property that y(t; t 0, y 0 ) X +, for all t t 0, t, t 0 I, if y 0 X +. In this case, we shall say that the linear ifferential equation (5) efines an anticausal positive evolution on X. Base on (12) we obtain: Corollary 1 Let L : I B(X ) be a strongly continuous operator value function an ˆL(t) = L( t), t Î. Then the operator value function L( ) efines an anticausal positive evolution if an only if the operator value function ˆL( ) generates a causal positive evolution on X. The next result was prove in [8] for the finite imensional case. infinite imensions, the proof can be foun in [11]. In Corollary 2 Let L( ), Π( ) be two strongly continuous operator value functions efine on I, taking values in B(X ). Assume that Π(t) 0 for all t I. Then the following are true: (i) If L( ) generates a positive evolution on X then L( ) + Π( ) generates a positive evolution on X. (ii) If L( ) generates an anticausal positive evolution on X, then L( ) + Π( ) generates an anticausal positive evolution on X. (iii) Π( ) generates both a causal positive evolution an anticausal positive evolution on X. Definition 2 (i) We say that the zero state equilibrium of the linear ifferential equation (1) is exponentially stable, or equivalently, the operator

8 72 Vasile Dragan, Toaer Morozan, Viorica Ungureanu value function L( ) generates an exponentially stable evolution if there exist the constants β 1, α > 0 such that T (t, t 0 ) βe α(t t 0) (13) for all t t 0, t, t 0 I. (ii) We say that the zero state equilibrium of the linear ifferential equation (5) is anticausal exponentially stable, or equivalently, the operator value function L( ) generates an anticausal exponentially stable evolution on X if there exist the constants β 1, α > 0 such that T a (t, t 0 ) βe α(t t 0) (14) for all t t 0, t, t 0 I. Since both (1) an (5) are linear ifferential equations we will often say that the linear ifferential equation (1) is exponentially stable an the linear ifferential equation (5) is anticausal exponentially stable, respectively if (13) an (14), respectively, are fulfille. It is worth mentioning that uner the consiere assumptions for any ξ IntX + the corresponing Minkovski operator norm ξ is equivalent with the norm (see the Appenix for the efinition of ξ ). Therefore, the above efinition may be state in terms of the operator norm ξ for some ξ IntX +. Base on the ientity (12) together with the Definition 2 we obtain: Corollary 3 Let L : I B(X ) be a strongly continuous operator value function an ˆL(t) = L( t), t Î = {t R; t I}. Then the operator value function L( ) efines an anticausal exponentially stable evolution if an only if the operator value function ˆL( ) generates a causal exponentially stable evolution. The above corollary allows us to erive criteria for anticausal exponential stability of a linear ifferential equation efine by an operator value function L( ) irectly from the criteria for causal exponential stability for the linear ifferential equation efine by the operator value function ˆL( ). Criteria for exponential stability of ifferential equations with positive evolution on an orere Banach space are erive in [11].

9 Lyapunov operators on orere Banach spaces The case of ifferential equations with positive evolution on orere Hilbert spaces Throughout this subsection (X ;, ) is a real Hilbert space, orere by the orering inuce by the close, soli, selfual, convex cone. Base on Proposition 2.4 in [10] we euce that the norm, inuce by the inner prouct is monotone with respect to the cone X +. So, X + is a normal cone with a constant b = 1 (see Definition 5 from the Appenix). Let ξ IntX + be fixe an ξ be the corresponing Minkovski norm. As we can see, applying Proposition 4 an Proposition 5 one euces that ξ is equivalent with the norm of the Hilbert space X. It is known that, if T B(X ) an T is its ajoint operator, then, T = T. The equality T ξ = T ξ is not, in general, true. However, one can prove, via the equivalence of the operator norms an ξ, that there exist positive constants, c 1, c 2 such that c 1 T ξ T ξ c 2 T ξ, T B(X ). (15) Let L : I B(X ) be a continuous operator value function, I R being a right unboune interval. In this case t L (t) : I B(X ) is also a continuous operator value function. It is known that if T (t, τ), t, τ I, is the linear evolution operator efine by the linear ifferential equation then, τ T (t, τ) verifies So we have: x(t) = L(t)x(t), t I (16) t τ T (t, τ) = L (τ)t (t, τ) (17) T (t, t) = I X. T (t, τ) = TL a (τ, t) (18) t, τ I, where TL a (τ, t) is the anticausal linear evolution operator efine by the operator value function L ( ). This means that, TL a (τ, t) is a linear evolution operator associate to the linear ifferential equation τ y(τ) = L (τ)y(τ). (19) Combining (15), (18) one obtains the following result:

10 74 Vasile Dragan, Toaer Morozan, Viorica Ungureanu Proposition 1 If L : I B(X ) is a continuous operator value function, then the following statements are true: (i) The operator value function L( ) efines a causal positive evolution on X, iff the operator value function L ( ) efines an anticausal positive evolution on X. (ii) The operator value function L( ) efines an exponentially stable evolution on X iff the operator value function L ( ) efines an anticausal exponentially stable evolution on X. Using criteria for the anticausal exponential stability of the linear ifferential equation efine by the operator value function L ( ) an taking into account the equality (18) an Proposition 1 one obtains a set of criteria for the causal exponential stability of the linear ifferential equation (16). For etails see [11]. Such criteria are specific to the linear ifferential equations with positive evolution on orere Hilbert spaces. 3 Orere vector spaces of sequences of symmetric matrices This section collects several examples of real orere Banach spaces. As usual x stans for the eucliian norm of a vector x R n, that is, x = (x T x) 1/2. For a matrix A R m n, A stans for the matrix norm inuce by the eucliian norm, that is A = sup Ax (20) x 1 Also, we shall use the notation A 2 for the Frobenius norm of the matrix A, i.e. A 2 = ( T r[a T A] ) 1/2 (21) where T r[ ] stans for the trace operator. Besie the two norms introuce before, we shall use also the norm [ A 1 = T r (A T A) 1/2] (22) where (A T A) 1/2 is the unique positive semiefinite matrix X such that X 2 = A T A.

11 Lyapunov operators on orere Banach spaces 75 Let S n R n n be the linear subspace of symmetric matrices of size n n, that is S S n iff S = S T. The restrictions of the norm (20)-(22) to the subspace S n take the equivalent form: S = max{ λ ; λ Λ(S)} = sup { x T Sx } (23) x 1 ( n ) 1/2 S 2 = λ 2 i (24) i=1 S 1 = n λ i (25) i=1 where λ 1,..., λ n Λ(S) with Λ(S) is the spectrum of the matrix S. For a matrix S S n the following hol: S S 2 S 1 n S. (26) Throughout this paper D enotes either the set {1, 2,..., } or the set Z The space S D n Let X = Sn D = l {D, S n } be the linear space of the boune sequences of symmetric matrices, that is { } l {D, S n } = X = {X(i)} i D X(i) S n, i D, sup X(i) < + i D. The space S D n equippe with the norm X = sup X(i) (27) i D is a real Banach space. On Sn D we consier the orering inuce by the cone X + = Sn+ D = l {D, S n+ } where l {D, S n+ } = {X = {X(i)} i D ; X(i) 0, i D}. (28)

12 76 Vasile Dragan, Toaer Morozan, Viorica Ungureanu Here X(i) 0 means that X(i) is positive semiefinite. One verifies that X + is a close, soli, convex cone. Its interior IntX + consists of the subset of the sequences X = {X(i), i D; X(i) δi n, i D for some δ > 0 inepenent of i}. Base on the monotonicity of the norm on S n one obtains that introuce by (27) is monotone with respect to the cone X +. Hence X + is a normal cone. Further we shall use Sn instea of Sn D an Sn+ for Sn+ D when D = {1, 2,.., }, while Sn is use for Sn D when D = Z +. In the last case, Sn+ stans for the convex cone l (Z +, S n+ ). It is obvious that (Sn, ) is a finite imensional real orere Banach space, while (Sn, ) is an infinite imensional real orere Banach space. Specializing the results from Theorem 4, Proposition 4, Proposition 5, to the orere Banach space Sn D we obtain: Corollary 4 In the case of the Banach space (Sn D, ), the following hol: (i) If D = {1, 2,..., } an J = (I n, I n,..., I n ) }{{} IntSn+ then the Minkovski norm efine by (112) for ξ = J is: X J = X (29) for all X = (X(1), X(2),..., X()) S n. (ii) If D = Z + an J = (I n, I n,..., I n,...) IntS n+ then the Minkovski norm introuce by (112) for ξ = J is given by for all X = {X(i)} i Z+. X J = X (30) Remark 4 For the sake of brevity we shall use X an X, respectively, instea of X J an X J, respectively, for the Minkovski norms efine by (29) an (30), respectively, if no confusions are possible. 3.2 The space l 1 (D, S n ) Let X = l 1 (D, S n ), where l 1 (D, S n ) = {X = {X(i)} i D S n ; i D X(i) 1 < }.

13 Lyapunov operators on orere Banach spaces 77 Taking X 1 = i D X(i) 1 (31) one obtains that (X, 1 ) is a real Banach space. On the Banach space (X, 1 ) we consier the orer relation inuce by the convex cone X + = l 1 (D, S n+ ) = {X l 1 (D, S n ); X = {X(i)} i D, X(i) 0}. It is a close convex cone. In the case D = {1, 2,..., }, l 1 (D, S n ) coincies with Sn an l 1 (D, S n+ ) coincies with Sn+. In the case D = Z +, X = l 1 (Z +, S n ) Sn. The convex cone l 1 (Z +, S n+ ) has an empty interior. Finally, let us remark that base on (26) we may introuce a new norm on X, by X 1 = X(i). (32) i D Base on (26), (32) we euce that the norms 1 an 1 are equivalent, more precisely we have: for all X = {X(i)} i D l 1 (D, S n ). X 1 X 1 n X 1 (33) 3.3 The space l 2 (D, S n ) Let X = l 2 (D, S n ) = {X = {X(i)} i D l 2 (D, S n ) we introuce the inner prouct: S n ; ( X(i) 2 ) 2 < }. On i D X, Y 2 = i D T r[x(i)y (i)] (34) for all X = {X(i)} i D, Y = {Y (i)} i D from l 2 (D, S n ). To show that the sum from the right han sie of (34) is convergent, let us remark that T r[x(i)y (i)] = 1 { X(i) + Y (i) X(i) Y (i) 2 } 2 = i D i D { 1 X(i) + Y (i) 2 2 } X(i) Y (i) 2 2 R 4 i D i D

14 78 Vasile Dragan, Toaer Morozan, Viorica Ungureanu because X(i) + Y (i) 2 2 < + i D X(i) Y (i) 2 2 < +. i D One may check that the inner prouct, 2 inuces a real Hilbert space structure on l 2 (D, S n ). Set X 2 = X, X 1/2 2. (35) On the space l 2 (D, S n ) we consier the orer relation inuce by the convex cone X + = l 2 (D, S n+ ) = {X = {X(i)} i D l 2 (D, S n ); X(i) 0, i D}. The cone l 2 (D, S n+ ) is a close cone. If D = Z + its interior is empty. Remark 5 In the case D ={1, 2,..., }the linear spaces l (D, S n ), l 1 (D, S n ), l 2 (D, S n ) coincie with Sn = S n S n... S }{{ n. } On Sn we have three norms: introuce via (27), 1 efine in (31) an 2 introuce by (35) for D = {1, 2,..., }. We have S S 2 S 1 n S for all S Sn. The convex cone l 2 (D, S n+ ) coincies with the convex cone Sn+ = S n+ S n+...s }{{ n+ if } D = {1, 2,..., }. The cone Sn+ is a close, soli, selfual convex cone. It is selfual with respect to the inner prouct X, Y = T r[x(i)y (i)] (36) i=1 for all X = (X(1), X(2),..., X()), Y = (Y (1), Y (2),..., Y ()) S n which is the special form of (34) for the case D = {1, 2,..., }. In the case D = Z + we have the following result. Proposition 2 If l 1 (Z +, S n ) an l 2 (Z +, S n+ ) are the linear spaces introuce in a previous subsections for D = Z + then l 1 (Z +, S n ) l 2 (Z +, S n+ ). Proof is similar to the one given in [19] for the case of sequences of nuclear operators an Hilbert-Schmith operators.

15 Lyapunov operators on orere Banach spaces 79 4 Lyapunov type linear ifferential equations on the space S D n In this section we emphases several properties of an important class of operator value functions on the Banach spaces Sn D an l 1 (Z +, S n ), respectively. These operators exten to this framework the well known Lyapunov operators an they will play an important role in the characterization of the exponential stability in mean square of stochastic linear ifferential equation. 4.1 Extene Lyapunov operators Let M D mn := l (D, R m n ) be the space of the boune sequences of matrices A = {A(i)} i D where A(i) R m n. We introuce the norm A = A(i) where A(i) is efine by (20). One obtains that (M D mn, ) sup i D is a real Banach space. If m = n we shall write M D n instea of M D nn. If D = Z +, M mn an M n, respectively stan for M D mn an M D n. It is obvious that Sn D M D n. We make the following convention of notation: (a) If A = {A(i)} i D M D mn, X = {X(i)} i D M D np, by Y = AX we unerstan the sequence Y = {Y (i)} i D M D mp, Y (i) = A(i)X(i), i D. (b) If A = {A(i)} i D M D mn then A T = {A T (i)} i D M D nm. Let A : I M D n be a continuous function. This means that A(t) = {A(t, i)} i D, where t A(t, i) are matrix value functions which are continuous on I uniformly with respect to i D. The extene Lyapunov operators associate to A(t): are efine as follows L A (t) : S D n S D n, L A (t) : S D n S D n, L A (t)x = A(t)X + XA T (t) (37) for all X = {X(i)} i D S D n. L A (t)x = A T (t)x + XA(t) (38)

16 80 Vasile Dragan, Toaer Morozan, Viorica Ungureanu Accoring to the notation introuce at the beginning of this subsection the i th component of (37) an (38) respectively, is: [L A (t)x](i) = A(t, i)x(i) + X(i)A T (t, i) [L A (t)x](i) = A T (t, i)x(i) + X(i)A(t, i) i D, t I. We euce that L A (t)x 2 A(t) X an L A (t)x 2 A(t) X. Hence, L A (t), L A (t) B(Sn D ). Moreover t L A (t) an t L A (t) are continuous functions in the topology inuce by the operator norm. Remark 6 To be sure that the linear ifferential equations (39), (44), respectively, efine by L A (t) an L A (t) on Sn D have nice properties, woul be sufficient to assume that t L A (t) an t L A (t) are strongly continuous operator value functions. This means that for each X Sn D, t L A (t)x an t L A (t)x are continuous vector value functions. If we take X = {X(i)} i D with X(i) = I n, i D one obtains that t A T (t) + A(t) must be continuous. This conition is not far from our assumption that t A(t) is a continuous function. Let us consier the extene Lyapunov equation t X(t) = L A(t)X(t), t I. (39) Let T A (t, t 0 ) t, t 0 I be the linear operator efine by (T A (t, t 0 )X)(i) = Φ i (t, t 0 )X(i)Φ T i (t, t 0 ) (40) i D an X = {X(i)} i D S D n, where Φ i (t, t 0 ) is the funamental matrix solution of the linear ifferential equation on R n : This means that t Φ i (t, t 0 ) verifies x(t) = A(t, i)x(t). t t Φ i(t, t 0 ) = A(t, i)φ i (t, t 0 ) (41) Φ i (t 0, t 0 ) = I n.

17 Lyapunov operators on orere Banach spaces 81 Base on the convention of notations introuce before we may write (40) in the compact form: T A (t, t 0 )X = Φ(t, t 0 )XΦ T (t, t 0 ) (42) for all t, t 0 I, where Φ(t, t 0 ) = {Φ i (t, t 0 )} i D. If D = {1, 2,..., } one may check that t Φ(t, t 0 ) is a ifferentiable map an it satisfies: t Φ(t, t 0) = A(t)Φ(t, t 0 ), Φ(t 0, t 0 ) = J = (I n...i n ). By irect calculations one obtains from (42) that t T A(t, t 0 )X = L A (t)t A (t, t 0 )X (43) T A (t 0, t 0 )X = X for all t, t 0 I, X S n. Therefore T A (t, t 0 ) efine by (40), or equivalently by (42) is just the linear evolution operator on S n efine by the linear ifferential equation (39). It remains to show that (40) efines also the linear evolution operator generate by (39) on S n. To this en, let us notice that Φ i (t, s) e γ(t s) for all i Z +, t, s I, where γ = sup t I A(t). Using also the fact that t A(t, i) are continuous functions uniformly with respect to i Z + we euce that lim h 0 1 h Φ i(t + h, t 0 ) Φ i (t, t 0 ) ha(t, i)φ i (t, t 0 ) = 0 uniformly with respect to i Z +. This shows that t Φ(t, t 0 ) : I M n satisfies: is a ifferentiable map an it t Φ(t, t 0) = L A (t)φ(t, t 0 ), Φ(t 0, t 0 ) = J = (I n...i n...) S n. Thus we may obtain that T A (t, t 0 ) efine by (42) for D = Z + is ifferentiable an satisfies (43).

18 82 Vasile Dragan, Toaer Morozan, Viorica Ungureanu Remark 7 From (40) one sees that T A (t, t 0 )X S D n+ if X S D n+. This shows that the operator value function L A ( ) generates a positive evolution on the Banach space S D n. Changing A(t, i) with A T (t, i) in (40), (41) one obtains that the operator value function L A ( ) generates also positive evolution on the Banach space S D n. However, concerning the operator value function L A ( ) we are intereste by the anticausal evolution operator T a A (t, t 0) efine by the linear ifferential equation t Y (t) + L A(t)Y (t) = 0. (44) Reasoning as in the case of the equation (39) we may conclue that (T a A(t, t 0 )Y )(i) = Φ T i (t 0, t)y (i)φ i (t 0, t) (45) for all i D, 0 t t 0, Y = {Y (i)} i D S D n. From (45) one euces that the operator value function L A ( ) generates an anticausal positive evolution on the Banach space S D n. 4.2 Lyapunov-type ifferential equations on the space S n Let I R be an interval an A k : I M n, k = 0,..., r be continuous functions A k (t) = (A k (t, 1),... A k (t, )), k {0,..., r}, t I. Denote by Q R a matrix which elements q ij verify the conition q ij 0 if i j. (46) For each t I we efine the linear operator L (t), L (t) : S n S n by (L (t) S) (i) = A 0 (t, i) S (i) + S (i) A T 0 (t, i) (47) r + A k (t, i) S (i) A T k (t, i) + q ji S (j), k=1 j=1 (L (t) S) (i) = A T 0 (t, i) S (i) + S (i) A 0 (t, i) (48) r + A T k (t, i) S (i) A k (t, i) + q ij S (j) k=1 j=1

19 Lyapunov operators on orere Banach spaces 83 i D, S S n. It is easy to see that t L (t) an t L (t) are continuous operator value functions. One can see that (47) an (48) may be written as: L(t)S = L A (t)s + Π(t)S L(t)S = L A (t)s + Π(t)S where L A (t) an L A (t) are the extene Lyapunov operators introuce via (37) an (38), respectively, for D = {1, 2,..., } an A(t, i) = A 0 (t, i)+ 1 2 q iii n, (Π(t)S)(i) = r A k (t, i)s(i)a T k (t, i) + k=1 j=1,j i q ji S(j), ( Π(t)S)(i) = r A T k (t, i)s(i)a k(t, i) + k=1 j=1,j i q ij S(j). Base on (46) one obtains that Π(t)S 0, Π(t)S 0 if S 0. Therefore, combining Remark 7 an Corollary 2 (i) we conclue that the operator value function L( ) introuce via (47) generates a positive evolution on Sn. Also, combining (45) an Corollary 2 (ii) we infer that the operator value function L( ) efines an anticausal positive evolution on Sn. Moreover, by irect calculation one obtains that L(t) = L (t) where L (t) is the ajoint operator with respect to the inner prouct (36) of L(t). The Lyapunov operator L (t) efines the following linear ifferential equation on Sn: S (t) = L (t) S (t), t I (49) t while, the linear operator L(t) efines the following backwar ifferential equation S (t) + L (t) S (t) = 0, t I. (50) t Criteria for exponential stability of the zero solution of ifferential equations (49) an (50) may be foun in Chapter 2 in [9].

20 84 Vasile Dragan, Toaer Morozan, Viorica Ungureanu 4.3 Lyapunov-type ifferential equations on the space S n Definition an basic properties Let A k : I M n, 0 k r be continuous an boune functions. This means that A k (t) = {A k (t, i)} i Z+ are such that t A k (t, i) are continuous functions on I uniformly with respect to i Z + an sup t I A k (t) <. Let Q = (q ij ) i,j Z+ be an infinite real matrix whose elements satisfy the conitions: q ij 0, if i j (51) an sup ( q ii + i Z +,j i q ij ) = ν <. (52) It is worth mentioning that the conitions (51) an (52) are satisfie by the generator matrix of a stanar homogeneous Markov process with an infinite countable number of states (η(t), P, Z + ) (see Section 7 in [11] for more etails). Base on the functions t A k (t, i) an the elements q ij of the matrix Q, one constructs the operators L an L by: (L(t)X)(i)=A 0 (t, i)x(i)+x(i)a T 0 (t, i)+ (L(t)X)(i)=A T 0 (t, i)x(i)+x(i)a 0 (t, i)+ for all sequences X = {X(i)} i Z+. r A k (t, i)x(i)a T k (t, i)+ q ji X(j) k=1 (53) r A T k (t, i)x(i)a k(t, i)+ q ij X(j) k=1 (54) Lemma 1 If the real numbers q ij satisfy conitions (51) an (52) then for each t I, L(t) B(l 1 (Z +, S n )) an L(t) B(S n ). Proof. If X l 1 (Z +, S n ) then one obtains via (32), (51)-(53) that: L(t)X 1 = (L(t)X)(i) γ(t) X 1

21 Lyapunov operators on orere Banach spaces 85 where γ(t) = 2 A 0 (t) + r A k (t) 2 + ν. (55) Base on (33) we may write L(t)X 1 n L(t)X 1 nγ(t) X 1 which yiels L(t)X 1 nγ(t) X 1. This shows that L(t) introuce by (53) efines a linear an boune operator on l 1 (Z +, S n ) an L(t) 1 nγ(t), t 0. Similarly, if X Sn one obtains via (51), (52), (54) that k=1 L(t)X γ(t) X where γ(t) is efine by (55). This completes the proof. In the evelopments of this paper the linear operator L(t) introuce via (53) will be name the Lyapunov type operator on the space l 1 (Z +, S n ) efine by the system (A 0, A 1,..., A r ; Q) while L(t) will be name the Lyapunov type operator on the space Sn efine by the system (A 0, A 1,..., A r ; Q). Proposition 3 [11] Uner the consiere assumptions, the operator value function L( ) introuce by (53) efines a positive evolution on l 1 (Z +, S n ) while, the operator value function L( ) introuce by (54) efines an anticausal positive evolution on the Banach space S n. Let T (t, τ), (t, τ) I I be the linear evolution operator on l 1 (Z +, S n ) efine by the linear ifferential equation X(t) = L(t)X(t). (56) t This means that tt (t, τ) = L(t)T (t, τ), T (τ, τ) = I l 1 (Z +,S n). Consier, also T a (t, τ) the anticausal linear evolution operator on Sn efine by the backwar linear ifferential equation This means that X(t) + L(t)X(t) = 0. (57) t t T a (t, τ) = L(t)T a (t, τ), T a (τ, τ) = I S n. (58)

22 86 Vasile Dragan, Toaer Morozan, Viorica Ungureanu Remark 8 Uner the consiere assumptions the operator value functions t L(t) an t L(t) are continuous in the topology inuce by the norms of Banach algebras B(l 1 (Z +, S n )) an B(S n ), respectively. In the previous subsection we saw that in the case D = {1, 2,..., } the operator efine by (48) which is the analogous of the operator L(t) (introuce by (54)) coincies with the ajoint L (t) of the operator L(t). In the case D = Z +, such an equality is not possible because the operator L(t) an L(t) act on ifferent linear spaces. In the next evelopments we shall see that uner some aitional assumptions the restriction of the operator L(t) to the Hilbert space (l 2 (Z +, S n ), 2 ) coincies with the ajoint operator of L(t). To this en we nee the following auxiliary result which coul be also of interest in itself. Lemma 2 If A, M R n n are given matrices, then AM 2 min{ A M 2, A 2 M } where an 2 are the norms introuce by (20) an (21). Theorem 1 Assume that besie the conitions (51)-(52) the real numbers q ij satisfy the conition: sup i Z + q ji = q < +. (59) Let L(t) = L(t) l 2 (Z +,S n) be the restriction of the operator L(t) to l 2 (Z +, S n ) S n. Uner these conitions, for each t I, the following hol: (i) L(t) B(l 2 (Z +, S n )). (ii) L(t) B(l 2 (Z +, S n )). (iii) L(t) = L (t). Proof. (i) Let X = {X(i)} i Z+ l 2 (Z +, S n ) be arbitrary but fixe. Base on (54) we obtain (L(t)X)(i) 2 2 4[ A T 0 (t, i)x(i) X(i)A 0 (t, i) 2 2+ r A T k (t, i)x(i)a k(t, i) q ij X(j) 2 2]. k=1

23 Lyapunov operators on orere Banach spaces 87 Base on Lemma 2 we euce 2 (L(t)X)(i) γ 1 (t) X(i) q ij X(j) 2 (60) where γ 1 (t) = 2 A 0 (t) 2 + r r A k (t) 4. k=1 Let N Z +, N 1 be arbitrary but fixe. We have 2 N N N q ij X(j) 2 q ij q ij X(j) 2 2. (61) Using (52) we obtain: N q ij X(j) 2 Further we have N 1 2 ν N q ij X(j) 2 2. (62) ( ) 2 N q ij X(j) 2 ν N N 1 Z +, N 1 1. Using (59) one gets: N 1 N q ij X(i) 2 2 ( N1 ν q X 2 2 for all N 1, N Z +. Taking the limit for N, N 1 one obtains 2 q ij X(j) 2 q ij X(j) 2 2 q ij X(j) 2 2 ) for all ν q X 2 2 (63) for all i Z +. So, we have shown that the right han sie of (60) is finite. Further, from (60)-(63) we euce: (L(t)X)(i) 2 2 4(γ 1 (t) + ν q) X 2 2.

24 88 Vasile Dragan, Toaer Morozan, Viorica Ungureanu This shows that (L(t)X) l 2 (Z +, S n ) if X l 2 (Z +, S n ). Furthermore we have L(t)X 2 γ 2 (t) X 2 X l 2 (Z +, S n ), with γ 2 (t) = 2(γ 1 (t) + ν q) 1 2. (64) Thus (i) is prove. Further we show that (53) is well efine if X = {X(i)} i Z+ l 2 (Z +, S n ). Proceeing as in the proof of (i), we show that 2 (L(t)X)(i) γ 1 (t) X(i) q ji X(i) 2 (65) i Z +, γ 1 (t) being as in (60). For each N 1 we have ( N which yiels ( N q ji X(j) 2 ) 2 q N q ji X(j) 2 2. Further we obtain N 1 q ji X(j) 2 ) 2 N q ji N q ji X(j) 2 2 ( ) 2 N q ji X(j) 2 ν q X 2 2. Taking the limits for N an N 1 we euce q ji X(j) 2 2 ( q ji X(j) 2 ) 2 ν q X 2 2 for all i Z +, X l 2 (Z +, S n ). This shows that the right han sie of (65) is finite for all i Z +. Furthermore we obtain that (L(t)X)(i) 2 2 γ 2 (t) X 2 2, ( ) X l 2 (Z +, S n ) where γ 2 (t) is efine as in (64). Thus we have prove that L(t) B(l 2 (Z +, S n ). In orer to prove (iii) one employs (34), (53), (54) to show that the equality L(t)X, Y 2 = X, L(t)Y 2 hols for all X, Y l 2 (Z +, L n ). Thus the proof is complete.

25 Lyapunov operators on orere Banach spaces 89 Remark 9 Uner the assumptions of Lemma 1, the conition (59) is satisfie if there exist h 1 0, h 2 0 such that q ij = 0 if i < j h 1 or i > j + h 2. In this case (59) is satisfie with q = (h 1 + h 2 + 1)ν where ν is the constant from (52). By irect calculation one shows that L : I B(l 2 (Z +, S n )) is a strongly continuous operator value function. This function efines the linear ifferential equation: t Y (t) + L(t)Y (t) = 0 (66) t I on the space (l 2 (Z +, S n ), 2 ). Let T ã (t, τ), t, τ I, be the anticausal linear evolution operator on L l 2 (Z +, S n ) efine by the linear ifferential equation (66). Corollary 5 Uner the assumptions of the Theorem 1 we have: T ã L (τ, t) = T (t, τ), t, τ I. T (t, τ) being the linear evolution operator efine by L(t) B(l 2 (Z +, S n )). Proof follows from Theorem 1 (iii) an the equality (18). Let L(t) = L A (t)+ Π(t) be the partition of the linear operator L(t) where ( Π(t)X)(i) = We prove: r A T k (t, i)x(i)a k(t, i) + k=1,j i q ij X(j). (67) Lemma 3 For any monotone an boune sequence {X k } k Z+ S n we have: (i) lim k ( Π(t)[X k ])(i) = ( Π(t)[X])(i) for all i Z +, t I. (ii) lim k (T a (t, t 0 )X k )(i) = (T a (t, t 0 )X)(i) for all i Z +, t t 0, t, t 0 I, where X = {X(i)} i Z+ S n is efine by X(i) = lim k X k(i), i Z +.

26 90 Vasile Dragan, Toaer Morozan, Viorica Ungureanu Proof. Without loss of generality we may assume that {X k } k Z is an increasing an boune sequence. This means that there exist µ j R, j = 1, 2 such that µ 1 I n X k (i) X k+1 (i) µ 2 I n, (k, i) Z + Z + (68) Therefore, for each i Z +, X(i) S n is well efine by X(i) = lim k X k(i). (69) Base on (68) we infer that X = {X(i)} i Z+ S n. From (67) we obtain ( Π(t)X k )(i) ( Π(t)X)(i)= r A T l (t, i)(x k(i) X(i))A l (t, i)+ l=1 + q ij (X k (j) X(j)). (70) j i First, from (68) we obtain lim k k=1 r A T l (t, i)(x k(i) X(i))A l (t, i) = 0 (71) On the other han, applying Corollary 7 from the Appenix for a k (j) = q ij X k (j) X(j), we euce that lim k,j i q ij X k (j) X(j) = 0 which leas to lim q ij (X k (j) X(j)) = 0. (72) k,j i Combining (70)-(72) we obtain that (i) is true. Let us now prove that (ii) hols. To this en, let us enote Y k (t) = T a (t, t 0 )X k, t (, t 0 ] I, t 0 I being fixe. Since T a (t, t 0 ) is a positive operator, if t t 0 the inequalities (68) yiel µ 1 (T a (t, t 0 )J )(i) Y k (t, i) Y k+1 (t, i) µ 2 (T a (t, t 0 )J )(i) (73) for all i Z +, t t 0, t I. From (73) we obtain that the matrices Z(t, i) are well efine by Z(t, i) = lim k Y k(t, i), i Z +, t ( ; t 0 ] I.

27 Lyapunov operators on orere Banach spaces 91 Furthermore (73) yiels Z(t, i) µ 3 T a (t, t 0 )J = µ 3 T a (t, t 0 ) for all i Z +. This leas to Z(t) µ 3 T a (t, t 0 ), t I, t t 0, where Z(t) = {Z(t, i)} i Z+. Since t L(t) is a boune function we euce that T a (t, t 0 ) e c(t 0 t), for all t I, t t 0. This leas to Reasoning in the same way we obtain from (73) Z(t) µ 3 e c(t 0 t). (74) Y k (t) µ 3 e c(t 0 t) (75) for all t I, t t 0, k Z +. Let TA a(t, s) be the anticausal linear evolution operator on S n efine by the extene Lyapunov operator L A (t). We have the representation formula Y k (t) = T a A(t, t 0 )X k + t 0 t T a A(t, s) Π(s)Y k (s)s for all t t 0, t I. Base on (45) written for A(t, i) replace by A 0 (t, i) q iii n, we obtain the component wise representation formula Y k (t, i) = Φ T i (t 0, t)x k (i)φ i (t 0, t) + t 0 t Φ T i (s, t)( Π(s)Y k (s))(i)φ i (s, t)s (76) for all i Z +, t t 0, t I, where Φ i (s, t) is the funamental matrix solution of the ifferential equation t x(t) = (A 0(t, i) q iii n )x(t). Using the result prove in the part (i) of the lemma, we obtain that lim k ΦT i (s, t)( Π(s)Y k (s))(i)φ i (s, t) = Φ T i (s, t)( Π(s)Z(s))(i)Φ i (s, t) (77) for all i Z +, t s t 0.

28 92 Vasile Dragan, Toaer Morozan, Viorica Ungureanu We recall that the bouneness of the function s A 0 (s) together with (52) allow us to euce that Φ i (s, t) e c 1(s t), (78) t s t 0, t I, where c 1 > 0 is a constant not epening upon s, t. Further, from (76), (78) together with the bouneness of the functions s A l (s), 0 l r yiel x T Φ T i (s, t)( Π(s)Y k (s))(i)φ i (s, t)x βe c(s t) x 2 (79) for all t s t 0, where β, c are positive constants. Applying Lebesque s Theorem we obtain via (77) an (79) that t 0 lim k t x T Φ T i (s, t)( Π(s)Y k (s))(i)φ i (s, t)xs = t 0 t x T Φ T i (s, t)( Π(s)Z(s))(i)Φ i (s, t)xs for all x R n. By a stanar proceure, one obtains finally that t 0 lim k t Φ T i (s, t)( Π(s)Y k (s))(i)φ i (s, t)s = t 0 t Φ T i (s, t)( Π(s)Z(s))(i)Φ i (s, t)s for all t t 0, t I. Taking the limit for k in (76) we obtain that Z(t, i) = Φ T i (t 0, t)x(i)φ i (t 0, t) + t 0 t Φ T i (s, t)( Π(s)Z(s))(i)Φ i (s, t)s for all i Z +, t t 0, t I. The above equality may be rewritten in a compact form: Z(t) = T a A(t, t 0 )X + t 0 t T a A(t, s) Π(s)Z(s)s. (80)

29 Lyapunov operators on orere Banach spaces 93 Uner the consiere assumptions the ientity (80) allows us to euce that t Z(t) is ifferentiable an aitionally it solves the problem with given terminal conition: t Z(t) + L(t)Z(t) = 0, t t 0 (81) Z(t 0 ) = X. From the uniqueness of the solution of the problem (81) we conclue that Z(t, i) = (T a (t, t 0 )X)(i) (82) for all i Z +, t t 0, t I. The conclusion follows now from (82). So, the proof is complete. Lemma 4 Assume that the assumptions of Theorem 1 are fulfille. H x i = {Hx i (j)} j Z + be efine by { Hi x 0, if j i (j) = xx T, if j = i. Let (83) where x R n an i Z + are arbitrary but fixe. Uner the consiere assumptions we have: for all t τ, t, τ I. T (t, τ)h x i 1 = x T [(T a (τ, t)j )(i)]x (84) Proof. First we notice that H x i l1 (Z +, S n ) an H x i 1 = x 2. Therefore T (t, τ)h x i is well efine an we have T (t, τ)h x i 1 = T r[(t (t, τ)hi x )(j)] (85) for all t, τ I. For each k Z + we consier J k = {J k (j)} j Z + where J k (j) = { In, if 0 j k 0, if j > k. It is obvious that Jk l 2 (Z +, S n ) Sn an we have Jk 2 = (k + 1) n an Jk = 1. Also we have Jk Jk+1 J for all k Z +. This

30 94 Vasile Dragan, Toaer Morozan, Viorica Ungureanu yiels: T a (τ, t)j k T a (τ, t)j k+1 T a (τ, t)j, for all k Z +, t τ, t, τ I because T a (τ, t) 0 for all t τ This allows us to obtain x T [(T a (τ, t)j k )(i)]x xt [(T a (τ, t)j k+1 )(i)]x xt [(T a (τ, t)j )(i)]x (86) for all k Z +. Moreover, applying Lemma 3 (ii) for X k = Jk we obtain that lim k xt (T a (τ, t)jk )(i)x = xt (T a (τ, t)j )(i)x. (87) On the other han, from Theorem 1 (i) we euce that T a (τ, t)jk l 2 (Z +, S n ). Therefore we may write: = x T [(T a (τ, t)j k )(i)]x = T r[(t a (τ, t)j k )(i)xxt ] = T r[(t a (τ, t)jk )(j)hx i (j)] = T a (τ, t)jk, Hx i 2. Further, the equality prove in Theorem 1 (iii) together with (18) yiel: Thus we obtain Base on (88) we get T (t, τ)h x i 1 = lim T a (τ, t)j k, Hx i 2 = T (t, τ)j k, Hx i 2 = = J k, T (t, τ)hx i 2 = x T [(T a (τ, t)j k )(i)]x = k k k T r[(t (t, τ)h x i )(j)]. k T r[(t (t, τ)hi x )(j)]. (88) T r[(t (t, τ)h x i )(j)] = lim k xt [(T a (τ, t)j k )(i)]x. (89) The conclusion follows from (89) an (87). Thus the proof is complete. Theorem 2 Assume that the assumptions of Theorem 1 are fulfille. Then we have T (t, τ) 1 T a (τ, t) t τ, t, τ I. (90)

31 Lyapunov operators on orere Banach spaces 95 Proof. Let i Z + be arbitrary but fixe an ψ i : l 1 (Z +, S n ) l 1 (Z +, S n ) be efine by { 0, if j i ψ i (X)(j) = (91) X(i), if j = i for any X = {X(j)} j Z+ l 1 (Z +, S n ). We have X k ψ i (X) = 1 X(i) 1 which leas to lim X k ψ i (X) = 0. 1 i=k+1 k Hence X = ψ i (X) for all X l 1 (Z +, S n ). Further we have T (t, τ)x = T (t, τ)ψ i (X) (92) because T (t, τ) B(l 1 (Z +, S n )). Let λ i1, λ i2,..., λ in be real numbers an e i1, e i2,..., e in R n be orthogonal vectors such that e ij = 1, 1 j n an X(i) = (83) an (91) we euce: ψ i (X) = n j=1 n j=1 where H e ij i is efine as in (83) with e ij instea of x. For each k 1 we write k T (t, τ)ψ i (X) 1 k T (t, τ)ψ i (X) 1 Applying Lemma 4 we obtain k T (t, τ)ψ i (X) 1 k j=1 k j=1 λ ij e ij e T ij. Combining λ ij H e ij i (93) k n j=1 λ ij T (t, τ)h e ij i 1. n λ ij e T ij[(t a (τ, t)j )(i)]e ij h λ ij (T a (τ, t)j )(i).

32 96 Vasile Dragan, Toaer Morozan, Viorica Ungureanu Invoking (25) we infer k T (t, τ)ψ i (X) 1 T a (τ, t)j X(i) 1 for all k 1. Hence we have shown that T (τ, t)ψ i (X) 1 T a (τ, t)j X 1. (94) Further, from (92)-(94) we get: T (t, τ)x 1 T a (τ, t)j X 1, X l 1 (Z +, S n ), t τ, t, τ I. So, we may conclue that T (t, τ) 1 T a (τ, t)j for all t τ, t, τ I. To show that the last inequality coincies with (90) we apply Theorem 5 in the special case of the positive operator T a (τ, t) together with Corollary 4 (ii) an obtain that T a (τ, t)j = T a (τ, t). This ens the proof Detectability an exponential stability Let L an L be the Lyapunov type operators efine by (53) an (54), respectively. In this section we iscuss the exponential stability of the anticausal evolution generate by L uner etectability conitions an uner the assumptions of Theorem 1. Since the cone l 1 (Z +, S n+ ) has empty interior we cannot apply the evelopments from Section 4 in [11] in orer to erive criteria for exponential stability of the linear ifferential equation (56) on l 1 (Z +, S n ). The next corollary shows that the criteria for anticausal exponential stability of (57) coul be use as necessary an sufficient conitions for the exponential stability of (56). Corollary 6 Uner the assumptions of Theorem 1 the following are equivalent: (i) the operator value function L( ) efines an exponentially stable anticausal evolution on S n ; (ii) the operator value function L( ) efines an exponentially stable evolution on l 1 (Z +, S n ). Proof. (i) (ii). If (i) hols then there exist β 1, α > 0 such that T a (τ, t) βe α(t τ) for all τ, t I, t τ. Then from Theorem 2 we get T (t, τ) 1 βe α(t τ) (95)

33 Lyapunov operators on orere Banach spaces 97 for all t τ, t, τ I. This shows that (ii) is true. Let us prove now that (ii) (i). If (ii) hols, then there exist β 1, α > 0 such that (95) is true. From Lemma 4 we have x T (T a (τ, t)j )(i)x T (t, τ) 1 H x i 1 which yiels x T (T a (τ, t)j )(i)x βe α(t τ) x 2 for all t τ I, x R n, i Z +. Therefore (T a (τ, t)j )(i) βe α(τ t), ( ) i Z +, which leas to T a (τ, t)j βe α(τ t). (96) Applying Theorem 5 to the positive operator T a (τ, t) an using Corollary 4 (ii) we obtain from (96) that T a (τ, t) βe α(τ t), for all t τ, t, τ I. This confirms that the implication (ii) (i) is true. So the proof is complete. Definition 3 Let C : I M pn be a continuous, boune function. We say that the pair (L, C) is etectable if there is a continuous an boune function F : I M np such that L F (t) (X) = L (t) (X) + F (t) C (t) X + XC (t) T F (t) T, X l 1 (Z +, S n ) (97) generates a positive an exponentially stable causal evolution on l 1 (Z +, S n ). A stanar computation shows that t I L F (t) B ( l 1 (Z +, S n ) ) is a well efine, strongly continuous mapping, which generates a causal evolution operator T LF (t, s) on l 1 (Z +, S n ). Since L F an L have the same form, we can apply Proposition 3 to euce that T LF (t, s) is a positive operator. Moreover, T LF (t, s) is exactly the restriction to l 1 (Z +, S n ) of the positive causal evolution operator T LF (t, s) generate by L F on l 2 (Z +, S n ) ( [18], [3]). Let L F (t) (X) = L (t) (X) + C (t) T F (t) T X + XF (t) C (t), for all X Sn an t I. Obviously, the conclusions of Corollary 6 remain true if we replace the operators L an L with L F an L F, respectively. Therefore L F generates an anticausal exponentially stable evolution on Sn if an only if L F generates a causal exponentially stable evolution on l 1 (Z +, S n ).

34 98 Vasile Dragan, Toaer Morozan, Viorica Ungureanu Remark 10 The pair (L, C) is etectable if an only if there is F : I M n p a continuous an boune function such that L F (t), t I generates a positive an exponentially stable anticausal evolution on S n. Let I = R + := [0, ). A mapping P : I S n+ will be calle nonnegative. The next theorem is the main result of this section. It gives sufficient conitions for the exponential stability of the anticausal evolution generate by the Lyapunov type operator L, in terms of global solvability of an associate affine equation. Theorem 3 Assume that (L, C) is etectable an the backwar ifferential equation P (t) + L (t) P (t) + C T (t) C (t) = 0 (98) t has a nonnegative solution in the class of all boune C 1 -mappings P : I S n. Then L generates an exponentially stable anticausal evolution on S n. Proof. If i Z + an x R n are fixe, then the unique solution in l 1 (Z +, S n ) of the equation Z (t) t = L (t) (Z (t)), t s 0 (99) Z (s) = Hi x, i Z +, (100) exists an is given by Z (t, s; (Hi x)) = T L (t, s) (Hi x ) 0, t s. Let us prove that there is γ > 0 such that s Z (τ, s; H x i ) 1 τ γx T x (101) for all s R +,i Z + an x H. Then, Lemma 4 shows that s x T [(T a L (s, τ)j )(i)]xτ γx T x. From Theorem 4.4 from [11] it follows that L generates an anticausal exponentially stable evolution on Sn an the proof is complete.

35 Lyapunov operators on orere Banach spaces 99 It remains to prove (101). First, we establish a sufficient conition for (101). From the etectability hypothesis, there is a boune an continuous function F : I M np such that the causal evolution operator T LF (t 0, t) is exponentially stable on l 1 (Z +, S n ). We efine Ω F,ε (t) (X) = L F (t) (X) + ε 2 X, X l 1 (Z +, S n ). Using Gronwall s Lemma an a stanar computation we euce that there is ε 0 (0, 1) such that the causal evolution operator T Ω F,ε (t, s), generate by Ω F,ε (t), t R +, is exponentially stable for all 0 < ε < ε 0. Then there are β 1 1 an α 1 (0, 1) such that T Ω F,ε (t, s) 1 β 1 α1 t s for all t s. Moreover, by Proposition 3.3 from [11] (see also Corollary 2), T Ω F,ε (t, s) is a positive operator. Now, for such an ε (0, ε 0 ), we consier the equation Y (t) t = Ω F,ε (t) (Y (t)) + 1 ε 2 F (t) C (t) Z (t) C (t)t F (t) T, t s 0, where Z (t) is the solution of (99). We obtain (Y (t) Z (t)) t (102) Y (s) = H x i, (103) = Ω F,ε (Y (t) Z (t)) + Ψ (t), t s, t N, (104) Y (s) Z (s) = 0, (105) where Ψ (t) = (ε + 1ε ) ( F C Z ε + 1 ) T ε F C (t). By a stanar way it follows that Y (t) Z (t) 0 for all t R + an, consequently, Y (t) 1 Z (t) 1. Therefore the existence of γ > 0 such that s Y (t, s; H x i ) 1 t γx T x, (106) is a sufficient conition for (101) to hol. Then, let us prove (106). Applying (10) for (102), (103) we obtain Y (t) = T Ω F,ε (t, s) (H x i ) + 1 ε 2 t s ( ) T Ω F,ε (t, r) (F CZ) (r) (F C) (r) T r,