discrete time linear equations defined by positive operators

DISCRETE TIME LINEAR EQUATIONSDEFINED BY POSITIVE OPERATORS

ON ORDERED HILBERT SPACES

VASILE DRAGAN and TOADER MOROZAN

In this paper the problem of exponential stability of the zero state equilibriumof a discrete-time time-varying linear equation described by a sequence of linearbounded and positive operators acting on an ordered Hilbert space is investigated.The class of linear equations considered in this paper contains as particular caseslinear equations described by Lyapunov operators or symmetric Stein operatorsas well as nonsymmetric Stein operators. Such equations occur in connectionwith the problem of mean square exponential stability for a class of difference sto-chastic equations affected by independent random perturbations and Markovianjumping as well as in connection with some iterative procedures which allow usto compute global solutions of discrete time generalized symmetric or nonsym-metric Riccati equations. The exponential stability is characterized in terms ofthe existence of some globally defined and bounded solutions of some suitablebackward 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 stabi-lity, ordered Hilbert spaces, Minkovski norm.

1. INTRODUCTION

The stabilization problem, together with various control problems forlinear 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, equiva-lently, the second moments exponential stability of the zero solution of a linearstochastic differential equation or a linear stochastic difference equation isequivalent to the exponential stability of the zero state equilibrium of a suitabledeterministic linear differential equation or a deterministic linear differenceequation. Such deterministic differential (difference) equations are defined by

REV. ROUMAINE MATH. PURES APPL., 53 (2008), 2–3, 131–166

132 Vasile Dragan and Toader Morozan 2

the so-called Lyapunov type operators associated to the given stochastic lineardifferential (difference) equations.

Exponential stability in the case of differential equations or differenceequations described by Lyapunov operators has been investigated as a prob-lem with interest in itself in a lot of works. In the time-invariant case resultsconcerning the exponential stability of linear differential equations defined byLyapunov type operators were derived using spectral properties of positivelinear operators on an ordered Banach space obtained by Krein and Rut-man [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 andHinrichsen [7, 8]. Similar results were derived also for the discrete-time time-invariant case, see [21, 38]. In [13] the exponential stability of discrete-timetime-varying linear equations defined by linear positive operators acting on afinite dimensional ordered Hilbert space, was studied. In that paper differentcharacterizations of exponential stability in terms of the existence of boundedand 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 dif-ferential equations on the space of n× n symmetric matrices Sn is studied in[11]. Such equations have the property that the corresponding linear evolutionoperator is positive on Sn. They contain as particular cases linear differentialequations of Lyapunov type arising in connection with the problem of investi-gation of mean square exponential stability. The results of [11] were extendedto an abstract framework of a differential equations with positive evolution ona finite dimensional ordered Hilbert space (see [12]).

In this paper, the infinite dimensional counterpart of the results provedin [13] is derived. The discrete-time linear equations under consideration inthis paper are defined by sequences of positive bounded linear operators onan 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 definedby the Minkovski functional associated to a suitable open and convex set.

To characterize exponential stability, a crucial role is played by the uniquebounded solution of some suitable backward affine equations as well as ofsome forward affine equations. We show that if the equations considered aredescribed by periodic sequences of operators, then the bounded solution, if any,also is a periodic sequence. Moreover, in the time-invariant case the boundedsolutions to both backward affine equation and forward affine equation areconstant. Thus, the results concerning the exponential stability for the time-invariant case are recovered as special cases of the results proved in this paper.

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 re-sults are proved. Section 3 contains results which characterize the exponentialstability of the zero state equilibrium of a discrete-time time-varying linearequation described by a sequence of linear positive operators on a orderedHilbert space. Section 4 deals with the problem of preservation of exponentialstability under an additive perturbation of the sequence of linear operatorsdefining the discrete time equations under consideration.

In Section 5 we consider the case of discrete time, time varying linearequations defined by sequences of positive bounded linear operators on orderedBanach spaces. An application to the mean square exponential stability of adiscrete time stochastic stochastic system perturbed by a Markov chain withan infinite number of states is provided. The paper ends with an Appendixwhich collects the usual definitions concerning the convex cones. Also, someuseful properties of the Minkovski seminorm are presented. A set of sufficientconditions is given under which a Minkovski seminorm is just a norm.

2. PRELIMINARIES

In this section we describe the framework where the discrete time linearequations investigated in this paper are defined. In our approach a crucial rolewill be played by the Minkovski norm. More details concerning this norm canbe found in Appendix.

2.1. Positive linear operators on ordered Hilbert spaces

In this subsection as well as in the following X is a real Hilbert spaceordered by the ordering relation “≤ ” induced by the closed, solid, selfdual con-vex cone X+. Since X+ is a selfdual convex cone, it follows from Remark A.1and Proposition A.3 (in Appendix) that X+ is a pointed cone.

By Proposition A.3, | · |2 defined by

(1) |x|2 = (〈x, x〉)12

is monotone with respect to X+.Let ξ ∈ IntX+ be fixed; we associate the Minkovski functional | · |ξ

defined by (82). It follows from Theorem A.2 and Proposition A.2 that | · |ξ isa norm on X . Moreover, from Theorem A.1 (v) and Theorem A.2 we deducethat | · |ξ is equivalent to | · |2 defined by (1). Hence (X , | · |ξ) is a Banach space.Moreover, | · |ξ has the properties below:


P1) If x, y, z ∈ X are such that y ≤ x ≤ z, then

(2) |x|ξ ≤ max|y|ξ, |z|ξ.

P2) 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|Tx| is the correspondingoperator norm.

Remark 2.1. a) Since | · |ξ and | · |2 are equivalent, ‖ · ‖ξ and ‖ · ‖2 are alsoequivalent. This means that there are two positive constants c1 and c2 suchthat c1‖T‖ξ ≤ ‖T‖2 ≤ c2‖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 innerproduct 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 positiveconstants c1, c2 such that

(4) c1‖T‖ξ ≤ ‖T ∗‖ξ ≤ c2‖T‖ξ.

Definition 2.1. Let (X ,X+) and (Y,Y+) be ordered vector spaces. Anoperator T : X → Y is called positive if T (X+) ⊂ Y+. In this case we writeT ≥ 0. If T (IntX+) ⊂ IntY+ 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 = |Tξ|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 −Tξ ≤ Tx ≤ Tξ. It follows from (2)

that |Tx|ξ ≤ |Tξ|ξ for all x ∈ X with |x|ξ ≤ 1, which leads to

sup|x|ξ≤1

|Tx|ξ ≤ |Tξ|ξ ≤ sup|x|ξ≤1

|Tx|ξ,

hence ‖T‖ξ = |Tξ|ξ, thus the proof is complete.

From (ii) of Proposition 2.1 we obtain

Corollary 2.1. Let Tk : X → X , k = 1, 2, be positive bounded linearoperators. If T1 ≤ T2 then ‖T1‖ξ ≤ ‖T2‖ξ.

Example 2.1. (i) Let X = Rn and X+ = Rn+, where Rn

+ = x =(x1 x2 · · · xn

)T ∈ Rn | xi ≥ 0, 1 ≤ i ≤ n. In this case, X+ is aclosed, solid, pointed, selfdual convex cone. The ordering induced on Rn by


this cone is known as the component wise ordering. If T : Rn → Rn is a linearoperator, then T ≥ 0 iff its corresponding matrix A with respect to the canoni-cal basis on Rn has nonnegative entries. For ξ = (1, 1, 1, . . . , 1)T ∈ Int(Rn

+),the norm | · |ξ is defined by

(5) |x|ξ = max1≤i≤n

|xi|.

Properties P1 and P2 are fulfilled for the norm defined by (5).(ii) Let X = Rm×n be the space of m × n real matrices, endowed with

the inner product

(6) 〈A,B〉 = Tr(BT A)

∀A,B ∈ Rm×n, Tr(M) denoting as usual the trace of a matrix M . On Rm×n

we consider the order relation induced by the cone X+ = Rm×n+ , where

(7) Rm×n+ = A ∈ Rm×n | A = aij, aij ≥ 0, 1 ≤ i ≤ m, 1 ≤ j ≤ n.

The interior of the cone Rm×n+ is not empty. It can be seen that Rm×n

+ is aselfdual cone. On Rm×n we also consider the norm | · |ξ defined by

(8) |A|ξ = maxi,j

|aij |.

Properties P1 and P2 are fulfilled for the norm (8) with

ξ =

1 1 1 · · · 1· · · · · · · · · · · · · · ·1 1 1 · · · 1

∈ IntRm×n+ .

An important class of linear operators on Rm×n is that of the form LA,B :Rm×n → Rm×n with LA,BY = AY BT for all Y ∈ Rm×n, where A ∈ Rm×m,B ∈ Rn×n are fixed givenmatrices. These operators are often called “non-symmetric Stein operators”. It can be checked that LA,B ≥ 0 iff aijblk ≥ 0,∀ i, j ∈ 1, . . . ,m, l, k ∈ 1, . . . , n. Hence LA,B ≥ 0 iff the matrix A ⊗ Bdefines a positive operator on the ordered space (Rmn,Rmn

+ ), where ⊗ is theKronecker product.

(iii) Let Sn ⊂ Rn×n be the subspace of n × n symmetric matrices. LetX = Sn ⊕ Sn ⊕ · · · ⊕ Sn = SN

n with N ≥ 1 fixed. On SNn , consider the inner

product

(9) 〈X, Y 〉 =N∑

i=1

Tr(YiXi)

for arbitrary X = (X1, X2, . . . , XN ) and Y = (Y1, Y2, . . . , YN ) in SNn . The

space SNn is ordered by the convex cone

(10) SN,+n = X = (X1, X2, . . . , XN ) | Xi ≥ 0, 1 ≤ i ≤ N,


whose interior

IntSN,+n = X ∈ SN

n | Xi > 0, 1 ≤ i ≤ Nis nonempty. Here Xi ≥ 0 and Xi > 0 means that Xi is a positive semidefinitematrix 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 = max1≤i≤N

|Xi|, (∀) X = (X1, . . . , XN ) ∈ SNn ,

where |Xi| = maxλ∈σ(Xi)

|λ|, σ(Xi) is the set of eigenvalues of the matrix Xi.

For the norm defined by (11), properties P1 and P2 are fulfilled with ξ =(In, In, . . . , In) = J ∈ SN

n .(iv) For an infinite dimensional case, let us consider X = `2(Z+,R)

where `2(Z+,R) =x = (x0, x1, . . . , xn, . . .) | xi ∈ R,

∞∑i=0

x2i < ∞

. On X

we consider the usual inner product 〈x,y〉`2 =∞∑i=0

xiyi for all x = xii≥0,y =

yii≥0. Set X+ =x = xii≥0 | x0 ≥ 0,

∞∑i=1

x2i ≤ x2

0

. It is easy to see

that X+ is a closed, pointed convex cone. In the finite dimensional case theanalogue of this cone is known as a circular cone.

The interior IntX+ =x = xii≥0 | x0 > 0,

∞∑i=1

x2i < x2

0

. It remains

to prove that X+ is selfdual. Let y ∈ (X+)∗. Hence

(12) 〈x,y〉`2 ≥ 0

for all x = xii≥0 ∈ X+. In particular, taking x = 1, 0, 0, . . . , 0 in (12), weobtain y0 ≥ 0. It is easy to verify that if y0 = 0 then yt = 0 for all t ≥ 1.Since y0 ≥ 0, it is obvious that if yt = 0 for all t ≥ 1, then we have y ∈ X+.

Suppose now∞∑

t=1y2

t > 0. Take x = xii≥0 defined by

(13) x0 = y0, xi = −γyiy0

with γ =( ∞∑

k=1

y2k

)− 12 . Obviously, x ∈ X+. Replacing (13) in (12), one gets

∞∑k=1

y2k ≤ y2

0, which shows that y ∈ X+. Thus, it was shown that (X+)∗ ⊂ X+.

Let now y = yii≥0 ∈ X+. We have to show that (12) holds for all

x ∈ X+. Indeed, for x ∈ X+ we have∣∣∣ ∞∑

k=1

xkyk

∣∣∣2 ≤ ∞∑k=1

x2k

∞∑k=1

y2k ≤ x2

0y20,


which leads to∣∣∣ ∞∑

k=1

xkyk

∣∣∣ ≤ x0y0. This is equivalent to −x0y0 ≤∞∑

k=1

xkyk ≤

x0y0, 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,√

22 ) the closed ball

B(ξ,√

22 ) = x ∈ `2(Z+,R) | |x − ξ|2 ≤

√2

2 . It is not difficult to check thatB(ξ,

√2

2 ) ⊂ X+. Moreover, we have X+ = tx | t ≥ 0, x ∈ B(ξ,√

22 ).

If x = (x0, x1, x2, . . . , xn, . . .) ∈ X , we write x = (x0, x) with x =(x1, x2, . . . , xn, . . .). The corresponding Minkovski norm is given by |x|ξ =

|x0| + |x|2, where |x|22 =∞∑i=1

x2i . This equality can be obtained taking into

account that |x|ξ = inft > 0 | |x|22 − |x0|2 − 2tx0 − t2 < 0 and |x|22 − |x0|2 +2tx0 − t2 < 0.

2.2. Discrete-time affine equations

Let L = Lkk≥k0 be a sequence of bounded linear operators Lk : X → Xand f = fkk≥k0 a sequence of elements fk ∈ X . These two sequences defineon X two affine equations

(14) xk+1 = Lkxk + fk,

which will be called “the forward” affine equation or “causal affine equation”defined by (L, f), and

(15) xk = Lkxk+1 + fk

which will be called “the backward affine equation” or “anticausal affine equa-tion” defined by (L, f). For each k ≥ l ≥ k0 let T c(k, l) : X → X be the causalevolution operator defined by the sequence L, T c(k, l) = Lk−1Lk−2 . . .Ll ifk > l and T c(k, l) = IX if k = l, IX being the identity operator on X .

For all k0 ≤ k ≤ l, T (k, l)a : X → X stands for the anticausal evolutionoperator on X defined by the sequence L, that is, T (k, l)a = LkLk+1 . . .Ll−1

if k < l and T a(k, l) = IX if k = l.Often the superscripts a and c will be omitted if any confusion is not

possible.Let xk = T c(k, l)x, k ≥ l, l ≥ k0 be fixed. One obtains that xkk≥l

verifies the forward linear equation

(16) xk+1 = Lkxk

with initial value xl = x. Also, if yk = T a(k, l)y, k0 ≤ k ≤ l, then from thedefinition of T a

kl one obtains that ykk0≤k≤l is the solution of the backward


linear equation

(17) yk = Lkyk+1

with given terminal value yl = y. Obviously, (16) and (17) lead to T c(k+1, l) =LkT

c(k, l) for all k ≥ l ≥ k0 and T a(k, l) = LkTa(k+1, l) for all k0 ≤ k ≤ l−1.

It should be remarked that, unlike the continuous time case, a solutionxkk≥l of the forward linear equation (16) with given initial values xl = x iswell defined for k ≥ l while a solution ykk≤l of the backward linear equation(17) with given terminal condition yl = y is well defined for k0 ≤ k ≤ l.

If for each k, the operators Lk are invertible, then all solutions of equa-tions (16), (17) are well defined for all k ≥ k0.

If (T c(k, l))∗ is the adjoint operator of the causal evolution operatorT c(k, l), we define

zl = (T c(k, l))∗z, (∀) k0 ≤ l ≤ k.

By direct calculation one obtains that zl = L∗l zl+1. This shows thatthe adjoint of the causal evolution operator associated with the sequence Lgenerates an anticausal evolution.

Definition 2.2. a) We say that the sequence L = Lkk≥k0 defines apositive evolution if for all k ≥ l ≥ k0 the causal linear evolution operatorT (k, l)c ≥ 0.

b) We say that the sequence L = Lkk≥k0 defines an anticausal pos-itive evolution if for all k0 ≤ k ≤ l the anticausal linear evolution operatorT a(k, l) ≥ 0.

Since T c(l + 1, l) = Ll, T a(l, l + 1) = Ll, respectively, it follows thatthe sequence Lkk≥k0 generates a causal positive evolution or an anticausalpositive evolution if and only if for each k ≥ k0, Lk 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 positiveevolutions. Throughout the paper we shall say that a sequence Lkk≥0 gen-erates a positive evolution instead of a causal positive evolution every timewhen no confusion can arise. Also, in this case, we shall write T (k, l) insteadof T c(k, l).

The following result is straightforward. It will be used in the next sec-tions.

Corollary 2.2. Let Li = Likk≥k0, i = 1, 2 be two sequences of

bounded linear operators and T ci (k, l) be the corresponding causal linear evolu-

tion operators. Assume that 0 ≤ L1k ≤ L2

k for all k ≥ k0. Under this assump-tion we have T c

2 (k, l) ≥ T c1 (k, l) for all k ≥ l ≥ k0.


At the end of this subsection we recall the representation formulae of thesolutions of affine equations (14), (15). Each solution of the forward affineequation (14) has the representation

(18) xk = T c(k, l)xl +k−1∑i=l

T c(k, i + 1)fi

for all k ≥ l + 1. Also, any solution of the backward affine equation (15) hasa representation

yk = T a(k, l)yl +l−1∑i=k

T a(k, i)fi, k0 ≤ k ≤ l − 1.

3. EXPONENTIAL STABILITY

In this section, we deal with the exponential stability of the zero solutionof a discrete time linear equation defined by a sequence of positive boundedlinear operators.

Definition 3.1. We say that the zero solution of the equation

(19) xk+1 = Lkxk

is exponentially stable or, equivalently, that the sequence L = Lkk≥k0 gene-rates an exponentially stable evolution (E.S. evolution) if there are β > 0,q ∈ (0, 1) such that

(20) ‖T (k, l)‖ξ ≤ βqk−l, k ≥ l ≥ k0,

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 Lk = L for all k, if (20) is satisfied, we shall say that the operator Lgenerates a discrete-time exponentially stable evolution.

It is well known that L generates a discrete-time exponentially stableevolution if and only if ρ[L] < 1, where ρ[·] is the spectral radius.

It must be remarked that if the sequence Lkk≥k0 generates an expo-nentially stable evolution then it is a bounded sequence.

In this section we shall derive several conditions which are equivalent toexponential stability of the zero solution of equation (19) in the case Lkk≥k0 .Such results can be viewed as an alternative characterization of exponentialstability to the one in terms of Lyapunov functions.

First, from Proposition 2.1, Corollary 2.1 and Corollary 2.2 we obtainthe following result specific to the case of operators which generate positiveevolution.


Proposition 3.1. Let L = Lkk≥k0 , and L1 = L1kk≥k0 be two se-

quences 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 ≤ βqk−l for all

k ≥ l ≥ k0.(ii) If L1

k ≤ Lk for all k ≥ k0 and L generates an E.S. evolution, then L1

also generates an E.S. evolution.

Further, we shall prove:

Theorem 3.1. Let Lkk≥0 be a sequence of positive bounded linearoperators Lk : X → X . Then the following assertions are equivalent:

(i) the sequence Lkk≥0 generates an exponentially stable evolution;

(ii) there exists δ > 0 such thatk∑

l=k0

‖Tk,l‖ξ ≤ δ for arbitrary k ≥ k0 ≥ 0;

(iii) there exists δ > 0 such thatk∑

l=k1

T (k, l)ξ ≤ δξ for arbitrary k ≥ k1 ≥

0, δ > 0 being independent of k, k1;(iv) for an arbitrary bounded sequence fkk≥0 ⊂ X , the solution with

zero initial value of the forward affine equation

xk+1 = Lkxk + fk, k ≥ 0

is bounded.

Proof. The implication (iv) → (i) is the discrete-time counter partof 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 = β1−q .

Let us prove that

(21) 0 ≤ T (k, l)ξ ≤ ‖T (k, l)‖ξξ

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 6= 0 then from (3) applied tox = 1

|T (k,l)ξ|ξ T (k, l)ξ one gets 0 ≤ T (k, l)ξ ≤ |T (k, l)ξ|ξξ and (21) follows fromProposition 2.1 (ii).

If (ii) holds then (iii) follows from (21). We have to prove that (iii) → (iv).Let fkk≥0 ⊂ X be a bounded sequence, that is, |fk|ξ ≤ µ, k ≥ 0. From (3)we obtain −|fl|ξξ ≤ fl ≤ |fl|ξξ, which leads to −µξ ≤ fl ≤ 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)fl ≤ µT (k, l + 1)ξ


and

−µ

k−1∑l=0

T (k, l + 1)ξ ≤k−1∑l=0

T (k, l + 1)fl ≤ µ

k−1∑l=0

T (k, l + 1)ξ.

Using (2) we deduce that

∣∣∣∣ k−1∑l=0

T (k, l + 1)fl

∣∣∣∣ξ

≤ µ

∣∣∣∣ k−1∑l=0

T (k, l + 1)ξ∣∣∣∣ξ

.

If (iii) is valid we conclude by using again (2) that

∣∣∣∣ k−1∑l=0

T (k, l + 1)fl

∣∣∣∣ξ

≤ µδ, k ≥ 1

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 discretetime linear equation (19) defined by a sequence of positive bounded linearoperators, the exponential stability is equivalent to the boundedness of thesolution with the zero initial value of the forward affine equation xk+1 =Lkxk + 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 exponentialstability we have to check the boundedness of the solution with zero initialvalue of the forward affine equation xk+1 = Lkxk+fk for an arbitrary boundedsequence fkk≥0 ⊂ X .

Let us now introduce the concept of uniform positivity.

Definition 3.2. We say that a sequence fkk≥k0 ⊂ X+ is uniformly posi-tive if there exists c > 0 such that fk > cξ for all k ≥ k0. If fkk≥k0 ⊂ X+

is uniformly positive, we shall write fk 0, k ≥ k0. If −fk 0, k ≥ k0, weshall write fk 0, k ≥ k0.

The next result provides a characterization of the exponential stabilityby using solutions of some suitable backward affine equations.

Theorem 3.2. Let Lkk≥k0 be a sequence of positive bounded linearoperators Lk : X → X . Then the following assertions are equivalent:

(i) the sequence Lkk≥k0 generates an exponentially stable evolution;(ii) there exist β1 > 0, q ∈ (0, 1) such that ‖T ∗(k, l)‖ξ ≤ β1q

k−l, (∀) k ≥l ≥ k0;


(iii) for each k ≥ k0 the series∑l≥k

T ∗(l, k)ξ is convergent and there exists

δ > 0, independent of k, such that∞∑

l=k

T ∗(l, k)ξ ≤ δξ;

(iv) the discrete time backward affine equation

(22) xk = L∗kxk+1 + ξ

has a bounded and uniformly positive solution;(v) for an arbitrary bounded and uniformly positive sequence fkk≥k0 ⊂

IntX+ the backward affine equation

(23) xk = L∗kxk+1 + fk, k ≥ k0

has a bounded and uniformly positive solution.(vi) There exists a bounded and uniformly positive sequence fkk≥k0 ⊂

IntX+ such that the corresponding backward affine equation (23) has a boundedsolution xkk≥k0 ⊂ X+;

(vii) there exists a bounded and uniformly positive sequence ykk≥k0 ⊂IntX+ which verifies

(24) L∗kyk+1 − yk 0, k ≥ k0.

Proof. The equivalence (i) ↔ (ii) follows immediately from (4). In asimilar way to the proof of inequality (21), we obtain

(25) 0 ≤ T ∗(l, k)ξ ≤ ‖T ∗(l, k)‖ξξ

for all l ≥ k ≥ k0.If (ii) holds, then for each k ≥ k0 the series

∑l≥k

‖T ∗(l, k)‖ξ of real numbers

is convergent and we have

(26)∞∑

l=k

‖T ∗(l, k)‖ξ ≤ δ,

where δ = β1

1−q is independent of k. Therefore, the series∑l≥k

T ∗(l, k)ξ is abso-

lute 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 ≥ k0, set yk =∞∑

l=k

T ∗(l, k)ξ. If (iii) holds then yk is well defined and

additionally yk ≤ δξ for all k ≥ k0. Using the definition of the linear evolution

operator T ∗(l, k), we may write yk = ξ+L∗k∞∑

l=k+1

T ∗(l, k+1)ξ or, equivalently,

yk = ξ + L∗kyk+1. This shows that ykk≥k0 is a solution of (22). Moreover,we have ξ ≤ yk ≤ δξ for all k ≥ k0. This means that ykk≥k0 is a bounded


and uniformly positive solution of (22). Thus, we obtain that the implication(iii) → (iv) holds. Now, we prove (ii) → (v). Let fkk≥k0 ⊂ IntX+ be abounded and uniformly positive sequence. Hence there exist νi > 0, i = 1, 2,such that ν1ξ ≤ fl ≤ ν2ξ for all l ≥ k0. Since T ∗(l, k) ≥ 0, for all l ≥ k ≥ k0

we may write

(27) ν1T∗(l, k)ξ ≤ T ∗(l, k)fl ≤ ν2T

∗(l, k)ξ

for all l ≥ k ≥ k0.The monotonicity of the Minkovski norm together with the equality from

Proposition 2.1 (ii) allow us to write

(28) ν1‖T ∗(l, k)‖ξ ≤ |T ∗(l, k)fl|ξ ≤ ν2‖T ∗(l, k)‖ξ.

If (ii) is fulfilled then (28) shows that for each k ≥ k0 the series∑l≥k

|T ∗(l, k)fl|ξ

of the real numbers is convergent and

(29)∞∑

l=k

|T ∗(l, k)fl|ξ ≤ δ1

for all k ≥ k0, where δ1 = ν2β1

1−q is independent of k. Thus, one gets that theseries

∑l≥k

T ∗(l, k)fl is absolutely convergent for all k ≥ k0.

Set zk =∞∑

l=k

T ∗(l, k)fl, k ≥ k0. Using again the definition of the linear

evolution operator T ∗(l, k) we can write

(30) zk = fk + L∗k∞∑

l=k+1

T ∗(l, k + 1)fl = fk + L∗kzk+1.

From (29) and (30) we deduce that zkk≥k0 is a bounded solution of (23).From (30) we also have that zk ≥ fk ≥ ν1ξ for all k ≥ k0. This means thatzk 0, k ≥ k0, thus (v) holds.

Further, (v) → (iv) → (vi) are straightforward. We now prove the impli-cation (vi) → (ii). Let us assume that there exists a bounded and uniformlypositive sequence fkk≥k0 ⊂ IntX+ such that the corresponding equation(23) has a bounded solution xkk≥k0 ⊂ X+. Therefore, there exist positiveconstants γi, i ∈ 1, 2, 3, such that

γ1ξ ≤ fk ≤ γ2ξ, γ1ξ ≤ xk ≤ γ3ξ,


for all k ≥ k0. Let k1 ≥ k0 be fixed. Define yk = T ∗(k, k1)xk, ∀ k ≥ k1. SinceT ∗(k, k1) ≥ 0, we may write

γ1T∗(k, k1)ξ ≤ T ∗(k, k1)fk ≤ γ2T

∗(k, k1)ξ

γ1T∗(k, k1)ξ ≤ yk ≤ γ3T

∗(k, k1)ξ(31)

for all k ≥ k1. From xk = L∗kxk+1 + fk, as well as from the definitions of yk

and T ∗(k, k1), we obtain successively yk = T ∗(k, k1)L∗kxk+1 + T ∗(k, k1)fk =T ∗(k + 1, k1)xk+1 + T ∗(k, k1)fk = yk+1 + T ∗(k, k1)fk. Thus, we obtainedyk+1 = yk − T ∗(k, k1)fk for all k ≥ k1. From (31) we deduce that

(32) yk+1 ≤ qyk

for all k ≥ k1, where q = 1 − γ1

γ3. Taking γ3 large enough in (31) we obtain

q ∈ (0, 1).From (32) we obtain inductively yk ≤ qk−k1 xk1 for all k ≥ k1. Using

again (31) together with xk1 ≤ γ3ξ, we deduce that 0 ≤ T ∗(k, k1)ξ ≤ γ3

γ1qk−k1ξ,

which by (2) leads to |T ∗(k, k1)ξ|ξ ≤ γ3

γ1qk−k1 , k ≥ k1. From Proposition 2.1

(ii) we have ‖T ∗(k, k1)‖ξ ≤ γ3

γ1qk−k1 , which means that (ii) holds.

The implication (iv) → (vii) follows immediately since a bounded anduniform by positive solution of (22) is a solution with the desired properties of(24). To complete the proof we show that (vii) → (vi). Let zkk≥k0 ⊂ IntX+

be a bounded and uniformly positive solution of (24). Define fk = zk−L∗kzk+1.It follows that fkk≥k0 is bounded and uniform by positive, therefore zkk≥0

is a bounded and positive solution of (23) corresponding to fkk≥k0 , thus theproof is complete.

The next result provides more information about the bounded solutionof the discrete time backward affine equations.

Theorem 3.3. Let Lkk≥k0 be a sequence of linear operators whichgenerates an exponentially stable evolution on X . Then the following asser-tions hold:

(i) for each bounded sequence fkk≥k0 ⊂ X the discrete-time backwardaffine equation

(33) xk = L∗kxk+1 + fk

has an unique bounded solution which is given by

(34) xk =∞∑

l=k

T ∗(l, k)fl, k ≥ k0;


(ii) if there exists an integer θ > 1 such that Lk+θ = Lk, fk+θ = fk for allk then the unique bounded solution of equation (33) also is a periodic sequencewith period θ;

(iii) if Lk = L, fk = f for all k, then the unique bounded solution ofequation (33) is constant and it is given by

(35) x = (IX − L∗)−1f,

with IX the identity operator on X ;(iv) if Lk are positive operators and fkk≥k0 ⊂ X+ is a bounded se-

quence, then the unique bounded solution of equation (33) satisfies xk ≥ 0 forall k ≥ k0.

Moreover, if fkk≥k0 ⊂ IntX+ is a bounded and uniformly positivesequence, then the unique bounded solution xkk≥k0 of equation (33) also isuniformly positive.

Proof. (i) From (i) → (ii) of Theorem 3.2, we deduce that for all k ≥ k0

the series j∑

l=k

T ∗(l, k)fl

j≥k

is absolutely convergent and there exists δ > 0

independent of k and j such that

(36)∣∣∣∣ j∑

l=k

T ∗(l, k)fl

∣∣∣∣ξ

≤ δ.

Set xk = limj→∞

j∑l=k

T ∗l,kfl =∞∑

l=k

T ∗(l, k)fl. Taking into account the definition of

T ∗(l, k), we obtain xk = fk + L∗k∞∑

l=k+1

T ∗(l, k + 1)fl = fk + L∗kxk+1, which

shows that xkk≥k0 solves (33).It follows from (36) that xk is a bounded solution of (33). Let xkk≥k0

be another bounded solution of equation (33). For each 0 ≤ k < j wemay write

(37) xk = T ∗(j + 1, k)xj+1 +j∑

l=k

T ∗(l, k)fl.

Since Lkk≥k0 generates an exponentially stable evolution and xkk≥k0 is abounded sequence, we have lim

j→∞T ∗(j +1, k)xj+1 = 0. Letting j →∞ in (37),

we conclude that xk =∞∑

l=k

T ∗(l, k)fl = xk, which proves the uniqueness of the

bounded solution of equation (33).(ii) If Lkk≥k0 , fkk≥k0 are periodic sequences with period θ, then in

a standard way, using the representation formula (34), one shows that the


unique bounded solution of the equation (33) is also periodic with period θ.In this case we may take that k0 = −∞.

(iii) If Lk = L, fk = f for all k, then they may be viewed as periodicsequences with period θ = 1. Based on the above result (ii) one obtains thatthe 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 . Sincethe operator L generates an exponentially stable evolution, we have ρ(L) < 1.Hence the operator IX −L∗ is invertible and we deduce that x is given by (35).Finally, if Lk are positive operators, the assertions of (iv) follow immediatelyfrom the representation formula (34) and thus the proof is complete.

Remark 3.1. From the representation formula (18) one obtains that if thesequence Lkk≥k0 generates an exponentially stable evolution and fkk≥k0

is a bounded sequence, then all solutions of the discrete time forward affineequation (14) with given initial values at time k = k0 are bounded on theinterval [k0,∞). On the other hand, it follows from Theorem 3.3 (i) that thediscrete time backward equation (15) has a unique bounded solution on theinterval [k0,∞), which is the solution provided by the formula (34).

In the case where k0 = −∞, with the same techniques as in the proof ofTheorem 3.3, we may obtain a result concerning the existence and uniquenessof the bounded solution of a forward affine equations similar to that provedfor the case of backward affine equations.

Theorem 3.4. Assume that Lkk∈Z is a sequence of linear operatorswhich generates an exponentially stable evolution on X . Then the followingassertions hold:

(i) for each bounded sequence fkk∈Z the discrete time forward affineequation

(38) xk+1 = Lkxk + fk

has a unique bounded solution xkk∈Z. Moreover, this solution has a repre-sentation formula

(39) xk =k−1∑

l=−∞T (k, l + 1)fl, ∀ k ∈ Z;

(ii) if Lkk∈Z, fkk∈Z are periodic sequences with period θ, then theunique bounded solution of equation (38) is periodic with period θ;

(iii) if Lk = L, fk = f , k ∈ Z then the unique bounded solution ofequation (38) is constant and is given by x = (IX − L)−1f ;


(iv) if Lkk∈Z are positive operators and fkk∈Z ⊂ X+, then the uniquebounded solution of equation (38) satisfies xk ≥ 0 for all k ∈ Z. Moreover, iffk 0, k ∈ Z, then xk 0, k ∈ Z.

If Lkk∈Z is a sequence of linear operators on X we may associate anew sequence of linear operators L#

kk∈Z defined by L#k = L∗−k.

Lemma 3.1. Let Lkk∈Z be a sequence of bounded linear operators onX . The following assertions hold:

(i) if T#(k, l) is the causal linear evolution operator on X defined by thesequence L#

kk∈Z, then T#(k, l) = T ∗(−l + 1,−k + 1), where T (i, j) is thecausal linear evolution operator defined on X by the sequence Lkk∈Z;

(ii) L#kk∈Z is a sequence of positive linear operators if and only if

Lkk∈Z is a sequence of positive linear operators;(iii) the sequence L#

kk∈Z generates an exponentially stable evolution ifand only if the sequence Lkk∈Z generates an exponentially stable evolution;

(iv) the sequence xkk∈Z is a solution of the discrete time backwardaffine equation (33) if and only if the sequence ykk∈Z defined by yk = x−k+1

is a solution of the discrete time forward equation yk+1 = L#k yk +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 existenceof the bounded solution of some suitable forward affine equation.

Theorem 3.5. Let Lkk∈Z be a sequence of positive bounded linearoperators on X . Then the following assertion are equivalent:

(i) the sequence Lkk∈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 thatk∑

l=−∞T (k, l)ξ ≤ δξ, ∀ k ∈ Z;

(iii) the forward affine equation

(40) xk+1 = Lkxk + ξ

has a bounded and uniformly positive solution;(iv) for any bounded and uniformly positive sequence fkk∈Z ⊂ IntX+,

the corresponding forward affine equation

(41) xk+1 = Lkxk + fk

has a bounded and uniformly positive solution;


(v) there exists a bounded and uniformly positive sequence fkk∈Z ⊂IntX+ such that the corresponding forward affine equation (41) has a boundedsolution xk, k ∈ Z ⊂ X+;

(vi) there exists a bounded and uniformly positive sequence ykk∈Z whichverifies yk+1 − Lkyk 0.

The proof follows immediately by combining the result proved in Theo-rem 3.2 and Lemma 3.1.

4. SOME ROBUSTNESS RESULTS

In this section we prove some results which provide a “measure” of therobustness of the exponential stability in the case of positive linear operators.To state and prove this result some preliminary remarks are needed.

So, `∞(Z,X ) stands for the real Banach space of bounded sequences ofelements of X . If x ∈ `∞(Z,X ), we denote |x| = sup

k∈Z|xk|ξ.

Let `∞(Z,X+) ⊂ `∞(Z,X ) be the subset of bounded sequences xkk∈Z ⊂X+. It can be checked that `∞(Z,X+) is a solid, closed convex cone. There-fore, `∞(Z,X ) is an ordered real Banach space for which the assumptions ofTheorem 2.11 in [8] are fulfilled.

Now we are in a position to prove

Theorem 4.1. Let Lkk∈Z, Gkk∈Z be sequences of positive boundedlinear operators such that Gkk∈Z is a bounded sequence. Under these as-sumptions, the following assertions are equivalent:

(i) the sequence Lkk∈Z generates an exponentially stable evolution andρ[T ] < 1, where ρ[T ] is the spectral radius of the operator T : `∞(Z,X ) →`∞(Z,X ), by

(42) y = T x, yk =k−1∑

l=−∞T (k, l + 1)Glxl,

where T (k, l) is the linear evolution operator on X defined by the sequenceLkk∈Z;

(ii) the sequence Lk+Gkk∈Z generates an exponentially stable evolutionon X .

Proof. (i) → (ii) If the sequence Lkk∈Z defines an exponentially stableevolution, then we define the sequence fkk∈Z by

(43) fk =k−1∑

l=−∞T (k, l + 1)ξ.


We have fk = ξ +k−2∑

l=−∞Tk,l+1ξ which leads to fk ≥ ξ thus fk ∈ IntX+ for all

k ∈ Z. This allows us to conclude that f = fkk∈Z ∈ Int `∞(Z,X+).Applying Theorem 2.11 [8] with R = −I`∞ and P = T we deduce that

there exists x = xkk∈Z ∈ Int `∞(Z,X+) which verifies the equation

(44) (I`∞ − T )(x) = f.

Here, I`∞ stands for the identity operator on `∞(Z,X ). Partitioning (44) andtaking into account (42)–(43), we obtain that for each k ∈ Z we have

xk+1 =k∑

l=−∞T (k + 1, l + 1)Glxl +

k∑l=−∞

T (k + 1, l + 1)ξ.

Further, we may write

xk+1 = Gkxk+ξ+Lk

k−1∑l=−∞

T (k, l+1)Glxl+Lk

k−1∑l=−∞

T (k, l+1)ξ= Gkxk+ξ+Lkxk.

This shows that xkk∈Z verifies the equation

(45) xk+1 = (Lk + Gk)xk + ξ.

Since Lk and Gk are positive operators and x ≥ 0, (45) shows that xk ≥ ξ.Thus, we get that equation (40) associated with the sum operator Lk + Gk

has a bounded and uniform positive solution. Using implication (iii) → (i)of Theorem 3.5 we conclude that the sequence Lk + Gkk∈Z generates anexponentially stable evolution.

Now, we prove the converse implication. If (ii) holds then using theimplication (i) → (iii) of Theorem 3.5, we deduce that equation (45) has abounded and uniform by positive solution xkk∈Z ⊂ IntX+. Equation (45)verified by xk may be rewritten as

(46) xk+1 = Lkxk + fk,

where fk = Gkxk + ξ, k ∈ Z, fk ≥ ξ, k ∈ Z. Using the implication (v) → (i)of Theorem 3.5, we deduce that the sequence Lk generates an exponentiallystable evolution. Since equation (46) has unique bounded solution which is

given by the representation formula (39), we have xk =k−1∑

l=−∞T (k, l + 1)fl,

k ∈ Z, so that

(47) xk =k−1∑

l=−∞T (k, l + 1)Glxl +

k−1∑l=−∞

T (k, l + 1)ξ.


Invoking (42), equation (47) may be written as

(48) x = T x + g,

where g = gkk∈Z, gk =k−1∑

l=−∞T (k, l + 1)ξ. It is obvious that gk ≥ ξ for all

k ∈ Z. Hence g ∈ Int `∞(Z,X+).Using implication (v) → (vi) of Theorem 2.11 in [8] for R = −I`∞ 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. Assumethat there exists θ ≥ 1 such that Lt+θ = Lt and Pt+θ = Pt 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 operatordefined by the sequence Ltt∈Z. As a consequence of the above equality, onegets T (nθ, 0) = T (θ, 0) for all n ≥ 0. Thus, if the sequence Ltt∈Z generatesan E.S. evolution then, ρ[T (θ, 0)] < 1. In this case, an operator valued functionis well defined by G : 0, 1, . . . , θ × 0, 1, . . . , θ − 1 → B(X ), with

(49) G(t, s) = T (t, 0)(IX − T (θ, 0))−1T (θ, s + 1) + T (t, s + 1)χt−1(s)

if 1 ≤ t ≤ θ, 0 ≤ s ≤ θ − 1 and

G(0, s) = (IX − T (θ, 0))−1T (θ, s + 1), 0 ≤ s ≤ θ − 1,

where χt−1(s) is the indicator function of the set 1, 2, . . . , t − 1. It is easyto 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) = (IX − L)−1.

Let X θ = X ⊕ X ⊕ · · · ⊕ X (θ times). The elements of this space arefinite sequences of the form x = (x0, x1, . . . , xθ−1), xi ∈ X , 0 ≤ i ≤ θ − 1. OnX θ we introduce the norm |x|θ = max|xi|ξ, 0 ≤ i ≤ θ − 1. The space X θ isan ordered Banach space with the norm | · |θ and the ordered relation inducedby the closed solid normal convex cone X+θ = X+ ⊕ · · · ⊕ X+. Consider theoperator Π : X θ → X θ defined by y = Πx, where y = (y0, y1, . . . , yθ−1),

(52) yt =θ−1∑s=0

G(t, s + 1)Psxs,

0 ≤ t ≤ θ − 1 for all x = (x0, x1, . . . , xθ−1) ∈ X θ.


Remark 4.1. a) It is obvious that (52) may be extended to t = θ by

yθ =θ−1∑s=0

G(θ, s + 1)Psxs. From (50) we deduce that yθ = y0. This allows us

to extend the finite sequence defined by (52) to an periodic infinite sequencewith 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 = (IX − L)−1Px, for all x ∈ X .

Lemma 4.1. Assume thata) the sequences Ltt∈Z , Ptt∈Z are periodic with period θ ≥ 1;b) Lt ≥ 0, Pt ≥ 0, t ∈ Z;c) Ltt∈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 itsdefinition. It remains to prove that Π ≥ 0.

Let x = (x0, x1, . . . , xθ−1) ∈ X+θ and y = Πx. Construct sequencesxtt∈Z , ytt∈Z by xt = xs, yt = ys if t = kθ + s, 0 ≤ s ≤ θ− 1. It is obviousthat xt+θ = xt, yt+θ = yt, for all t ∈ Z. If we consider the periodicity of Pt,we get that ytt∈Z is a periodic solution of the equation

(53) yt+1 = Ltyt + zt,

where zt = Ptxt. Since zt ≥ 0, we conclude via Theorem 3.4 (iv) applied toequation (53), that yt ≥ 0 and the proof is complete.

The analogue of Theorem 4.1 in the periodic case is

Theorem 4.2. Let Ltt∈Z, Ptt∈Z be sequences of bounded linear op-erators on X with the properties

a) there exists an integer θ ≥ 1 such that Lt+θ = Lt and Pt+θ = Pt forall t ∈ Z;

b) Lt ≥ 0, Pt ≥ 0, t ∈ Z.Under these assumptions the following assertions are equivalent:(i) the sum sequence Lt + Ptt∈Z generates an E.S. evolution;(ii) the sequence Ltt∈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) ofTheorem 3.4 we deduce that the forward affine equation xt+1 = [Lt +Pt]xt + ξhas a bounded and uniform positive solution xtt∈Z, which is periodic withperiod θ.


It can be seen that xtt∈Z solves the equation xt+1 = Ltxt + ft, whereft = Ptxt + ξ. Invoking the implication (v) → (i) of Theorem 3.5, we concludethat Ltt∈Z generates an E.S. evolution. Using the representation formula ofa periodic solution of a discrete time affine equation we obtain

(54) xt =θ−1∑s=0

G(t, s + 1)(Psxs + ξ), 0 ≤ t ≤ θ − 1.

Set x = (x0, x1, . . . , xθ−1) ∈ X θ, where xt = xt, 0 ≤ t ≤ θ− 1. From (54)one gets that x solves the equation

(55) (−IX + Π)x + g = 0,

where g = (g0, g1, . . . , gθ−1) with gt =θ−1∑s=0

G(t, s + 1)ξ. It can be seen that g is

a restriction of the periodic solution of the forward affine equation

(56) gt+1 = Ltgt + ξ.

Applying Theorem 3.4 (iv), we conclude that gt > 0. Thus we deduce thatg ∈ IntX+θ. Invoking the implication (v) → (vi) of Theorem 2.11 in [8] toequation (55) for R = −IX , P = Π, one concludes that ρ(Π) < 1. So weobtain 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 solutiongtt∈Z ⊆ IntX+. If g = (g0, g1, . . . , gθ−1) is such that gt = gt, 0 ≤ t ≤ θ − 1,then g ∈ IntX+θ. Invoking again Theorem 2.11 [8] we conclude that equation(55) has a solution x = (x0, x1, . . . , xθ−1), x ∈ IntX+θ. Construct the sequencextt∈Z by xt = xs if t = kθ + s, 0 ≤ s ≤ θ − 1, k ∈ Z. One can see thatxtt∈Z is a periodic sequence of period θ. Also, one obtains that xtt∈Z is aperiodic solution of the equation xt+1 = (Lt+Pt)xt+ξ. Applying Theorem 3.5,we conclude that Lt + Ptt∈Z generates an E.S. evolution, thus the proof iscomplete.

Using Remark 4.1 b), we deduce that in the time invariant case the resultproved in Theorem 4.2 becomes

Corollary 4.1. If L, P are two positive bounded linear operators onX , then the following assertions are equivalent:

(i) L+ P defines an E.S. evolution;(ii) ρ[L] < 1, ρ[(IX − L)−1P ] < 1.


5. THE CASE OF DISCRETE TIME LINEAR EQUATIONSDEFINED 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 thissection we assume that the by hypothesis H1 below holds:

H1 The norm ‖ · ‖ of the Banach space X is monotone with respect tothe cone X+.

Let ξ ∈ IntX+ be fixed. According to Proposition A2, the set Bξ de-fined by

(57) Bξ = x ∈ X | −ξ < x < ξ

is bounded. Using Proposition A.2 and Theorem A.2 we deduce that the cor-responding Minkovski functional | · |ξ is a norm equivalent to the norm ‖ · ‖of the Banach space X . Properties P1,P2 stated in Section 2 for an Hilbertspace are still valid in the case of a Banach space.

Also, Proposition 2.1 (ii) is valid in the context of an ordered Banachspace.

Example 5.1. Let X = S∞n , where S∞n consists of all sequences S =(S(1), S(2), . . . , S(k), . . .) with S(i) ∈ Sn 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 S∞n we consider the ordered relation induced by the solid, closed,

pointed, convex cone S∞+n with S∞+

n = (S(1), S(2), . . .) ∈ S∞n | S(i) ≥ 0,i ≥ 1. Obviously, H1 holds.

Take ξ = (In, In, . . .) ∈ S∞+n . In this case, (57) becomes

(60) Bξ = S = (S(1), S(2), . . .) ∈ S∞n | −In < S(i) < In, i ∈ Z+, i ≥ 1.

Hence Bξ is a bounded set. Using the definition of the Minkovski functional(see Appendix), we obtain that

(61) |S|ξ = |S|∞

for all S ∈ S∞n . For S ∈ IntS∞+n we shall write S 0 if S(i) ≥ εIn for all

i ≥ 1, with ε > 0 not depending upon i. This is a special case of Definition 3.2for the case of constant sequences.


Let us consider a sequence of positive bounded linear operators Ltt∈Z

on X . If X is a Banach space, it is not clear if one can characterize theexponential stability of the zero solution of the discrete time linear equation

(62) xt+1 = Ltxt

in terms of the existence of a bounded and uniformly positive solution ofsome suitable forward affine equations as well as in terms of the existenceof some bounded and uniformly positive solution of some suitable backwardaffine equations, as in the Hilbert space case.

Now, we show that in the case of a Banach space one can prove a resultas in Theorem 3.5 to characterize the exponential stability in the case ofequations (60).

Theorem 5.1. Let X be a real Banach space ordered by the orderedrelation induced by the closed, solid, pointed convex cone X+. Assume thatH1 is fulfilled. Let Ltt∈Z be a sequence of positive bounded linear operatorson 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

(63)t∑

l=−∞T (t, l)ξ ≤ δξ

for all t ∈ Z;(iii) the forward affine equation

(64) xt+1 = Ltxt + ξ

has a bounded and uniform positive solution;(iv) for any bounded and uniformly positive sequence ftt∈Z ⊂ IntX+

the corresponding forward affine equation:

(65) xt+1 = Ltxt + ft

has a bounded and uniformly positive solution;(v) there exists a bounded and uniformly positive sequence

(66) ftt∈Z ⊂ IntX+

such that the corresponding forward affine equation (65) has a bounded solutionxt, t ∈ Z ⊂ X+;

(vi) there exists a bounded and uniformly positive sequence ytt∈Z whichverifies

(67) yt+1 − Ltyt 0, t ∈ Z.


The proof follows the same lines as in the proof of Theorem 3.2 and isomitted.

As we have seen is Subsection 2.2, a sequence Ltt≥t0 may also de-fine a backward discrete time linear equation or, equivalently, an anticausalevolution. This is

(68) xt = Ltxt+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 se-quence Ltt≥t0 generates an A.E.S. evolution if there exists β ≥ 1, q ∈ (0, 1)such that

(69) ‖T a(t, s)‖ξ ≤ βqs−t

for all t0 ≤ t ≤ s, t, s ∈ Z, where T a(t, s) is the anticausal linear evolutionoperator associated to (68).

It must be remarked that in (69) we may use any norm equivalent to thegiven norm of the Banach space X . Concerning the characterization of theanticausal exponential stability one proves:

Theorem 5.2. Let Ltt≥t0 be a sequence of linear bounded and positiveoperators on the ordered Banach space X . If H1 holds, the following assertionsare equivalent:

(i) the sequence Ltt≥t0 generates an A.E.S. evolution;(ii) for each t ≥ t0 the series

∑s≥t

T a(s, t)ξ is convergent and there exists

δ > 0, independent of t, such that

(70)∞∑s=t

T a(t, s)ξ ≤ δξ

for all t ≥ t0;(iii) the discrete time backward affine equation

(71) xt = Ltxt+1 + ξ

has a bounded and uniformly positive solution;(iv) for an arbitrary bounded and uniform by positive sequence ftt≥t0,

the backward affine equation

(72) xt = Ltxt+1 + ft, t ≥ t0

has a bounded and uniformly positive solution;


(v) there exists a bounded and uniformly positive sequence ftt≥t0 suchthat the corresponding backward affine equation (72) has a bounded solutionxtt≥t0 ⊂ X+;

(vi) there exists a bounded and uniformly positive sequence ytt≥t0 whichverifies

(73) Ltyt+1 − yt 0, t ≥ t0.

The proof may be done by following step by step the proof of Theo-rem 3.2, replacing the operator T ∗(t, s) by T a(t, s) and L∗t by Lt.

In the last part of this section we illustrate the applicability of Theo-rem 5.2 to characterize the exponential stability in mean square in the case ofdiscrete-time stochastic linear systems perturbed by a Markov chain with aninfinite number of states.

Consider the discrete-time stochastic linear system

(74) xt+1 = A(t, ηt)xt, t ≥ 0,

where xt ∈ Rn and ηtt≥0 is a Markov chain with an infinite but countable setof states on a given probability space (Ω,F ,P) and with transition probabilitymatrices Pt. This means (see [10, 23]) that for each t ≥ 0, t ∈ Z, ηt : Ω → Z1

is a random variable with the property

(75) Pηt+1 = j|Gt = pt(ηt, j) a.s.,

where Gt = σ[η0, η1, . . . , ηt] is the σ algebra generated by ηs0≤s≤t, and Z1 =i ∈ Z, i ≥ 1. Setting Pt = (pt(i, j))i,j∈Z1 , pt(i, j) being the scalars on theright hand side of (75), one obtains a sequence of stochastic matrices with aninfinite number of rows and columns. To avoid some complications due to thediscrete time context (see [14] for the case of systems perturbed by a Markovchain with a finite number of states) we make the assumption

H2 For each t ∈ Z+, Pt is an nondegenerate stochastic matrix.We recall that a stochastic matrix Pt is called “nondegenerate” if for each

j ∈ Z1 there exists i ∈ Z1 such that pt(i, j) > 0.Define πt(j) = Pηt = j and set πt = (πt(1), πt(2), . . .); πt is the

distribution of the random variable ηt. We have πt(j) ≥ 0, ∀j ∈ Z1 and∑j∈Z1

πt(j) = 1 for all t ∈ Z+. One verifies inductively that under assumption

H2, we have πt(j) > 0 for all t ≥ 1 and j ≥ 1, if π0(i) > 0 for all i ≥ 1. Inwhat follows we assume that π0(i) > 0 for all i ≥ 1. Concerning the matrixcoefficients A(t, i), t ≥ 0, i ≥ 1, of the system (74) we make the assumption

H3

(76) supi≥1

|A(t, i)| < ∞,


where | · | is the norm induced by the Euclidean norm on Rn. Using thematrices A(t, i) we define the linear operators Lt : S∞n → S∞n , LtS = (LtS(1),LtS(2), . . .) by

(77) LtS(i) =∑j∈Z1

pt(i, j)AT (t, i)S(j)A(t, i)

for all i ≥ 1, S = (S(1), S(2), . . .) ∈ S∞n . It is clear that Lt ≥ 0. Using (76) oneobtains that ‖Lt‖ξ = sup

i∈Z1

|AT (t, i)A(t, i)| < ∞. Therefore, Lt is a bounded

linear operator. For each t ≥ s ≥ 0 we define Φ(t, s) = A(t − 1, ηt−1)A(t −2, ηt−2) . . . A(s, ηs) if t ≥ s+1 and Φ(t, s) = In for t = s. In the sequel, Φ(t, s)will be called the fundamental matrix solution of system (74). Each solutionof (74) verifies x(t) = Φ(t, s)x(s), t ≥ s ≥ 0. The next result establishesa relationship between the fundamental matrix solution of system (74) andthe anticausal linear evolution operator T a(t, s) defined by the operator Lt

introduced by (77).

Theorem 5.3 (representation formula). Under the assumptions made,we have

(78) [T a(t, s)S](i) = E[ΦT (s, t)S(ηs)φ(s, t)|ηt = i]

for all i ∈ Z1, 0 ≤ t ≤ s, S = (S(1), S(2), . . .) ∈ S∞n , where E[·|ηt = i] is theconditional expectation with respect to the event ηt = i and T a(t, s) is theanticausal linear evolution operator defined by Lt.

The proof is similar to that for the case of systems perturbed by Markovchains with a finite number of states (see e.g. [14, 33]).

From (78) for S = ξ = (In, In, . . .) we obtain

Corollary 5.1. Under the assumptions made we have

(79) xT (T a(t, s)ξ)(i)x = E[|Φ(s, t)x|2|ηt = i]

for all i ≥ 1, x ∈ Rn.It is known (see e.g. [14]–[19], [24, 28, 30, 31] that there exist several

ways to introduce the concept of exponential stability in mean square for adiscrete time stochastic linear system. Here we focus our attention on

Definition 5.2. We say that the zero state equilibrium of system (74) isexponentially stable in mean square (ESMS – for short) if there exist β ≥ 1and q ∈ (0, 1) such that for any Markov chain (ηtt≥0, Ptt≥0,Z1) with theabove properties we have

(80) E[|Φ(t, s)x|2|ηs = i] ≤ βqt−s|x|2

for all t ≥ s ≥ 0, i ∈ Z1, x ∈ Rn.


Combining Definition 5.2 and Corollary 5.1 we obtain

Corollary 5.2. Under assumptions H2 and H3 the following asser-tions are equivalent:

(i) the zero state equilibrium of the system (74) is ESMS;(ii) the sequence Ltt≥0 defined by (77) generates an A.E.S. evolution.

Finally, combining Corollary 5.2 and Theorem 5.2 we obtain the follow-ing characterization of exponential stability in mean square for systems oftype (74).

Theorem 5.4. Under assumptions H2 and H3 the following assertionsare equivalent:

(i) the zero state equilibrium of (74) is ESMS;(ii) Xt(i) =

∑j∈Z1

pt(i, j)AT (t, i)Xt+1(j)A(t, i) + In, i ∈ Z1, t ≥ 0, the

system of backward affine equations has a bounded and positive solution Xt =(Xt(1), Xt(2), . . .) ∈ S∞n , t ∈ Z1;

(iii) there exists a bounded sequence Xtt≥0, Xt = (Xt(1), Xt(2), . . .) ∈S∞n and scalars α > 0, δ > 0 such that Xt(i) ≥ δIn and∑

j∈Z1

pt(i, j)AT (t, i)Xt+1(j)A(t, i)−Xt(i) ≤ −αIn, i ∈ Z1, t ≥ 0.

6. APPENDIX

In the first part of this section we recall several definitions concerningconvex cones, and ordered linear spaces, and provide some basic results.

In the second part of this section we investigate the properties of theMinkovski functional and provide conditions under which such a functionalbecomes a norm.

The Minkovski norm play a crucial role in the developments in this paper.

6.1. Convex cones

Let (X , ‖ · ‖) be e real normed linear space. As usual, if X is a Hilbertspace we shall use | · |2 instead of ‖ · ‖.

Definition A1. a) A subset C ⊂ X is called a cone if(i) C + C ⊂ C;(ii) αC ⊂ C for all α ∈ R, α ≥ 0;

b) A cone C is called a pointed cone if C ∩ (−C) = 0;c) A cone C is called a solid cone if its interior Int C is not empty.


We recall that if A,B are two subsets of X and α ∈ R, then A + B =x + y | x ∈ A, y ∈ B and αA = αx | x ∈ A.

It is easy to see that a cone C is a convex subset and thus we shall oftensay convex cone when we refer to a cone.

A cone C ⊂ X induces an ordering “≤ ” on X by x ≤ y (or equivalentlyy ≥ x) if and only if y−x ∈ C. If C is a solid cone, then x < y (or equivalentlyy > x) if and only if y − x ∈ Int C. Hence C = x ∈ X | x ≥ 0 andInt C = x ∈ X | x > 0.

Definition A2. If C ⊂ X is a cone, then C∗ ⊂ X ∗ is called the dualcone of C if C∗ consists of all bounded and linear functionals f ∈ X ∗ with theproperty that f(x) ≥ 0 for all x ∈ C.

Based on Ritz theorem for representation of a bounded linear functionalon a Hilbert space one sees that if X is a real Hilbert space, then the dual coneC∗ of a convex cone C may be defined as C∗ = y ∈ X | 〈y, x〉 ≥ 0, ∀x ∈ C,where 〈· , ·〉 is the inner product on X .

If X is a real Hilbert space, a cone C is called selfdual if C∗ = C.

Lemma A1. Let X be a real Banach space and C ⊂ X a solid convexcone. Then C∗ is a closed and pointed cone.

Proof. Let ϕ ∈ C∗. There exists a sequence ϕkk≥1 ⊂ C∗ such thatlim

k→∞ϕk(x) = ϕ(x) for all x ∈ X . For x ∈ C we have ϕ(x) = lim

k→ϕk(x) ≥ 0.

Hence ϕ ∈ C∗, so C∗ is a closed set. To show that C∗ is a pointed cone, wechoose ϕ ∈ C∗ ∩ (−C∗). This leads to ϕ(x) = 0 for all x ∈ C. We have to showthat ϕ(x) = 0 for all x ∈ X . Let x0 ∈ X be arbitrary. Let ξ ∈ Int C be fixed.For ε > 0 small enough we have ξ + εx0 ∈ C. Hence ϕ(ξ + εx0) = 0. Sinceϕ(ξ) = 0, we conclude that ϕ(x0) = 0. Thus we obtain that C∗ ∩ (−C∗) = 0and the proof is complete.

In the finite dimensional case we have

Proposition A1. Let X be a finite dimensional real Banach space andC ⊂ X a closed, pointed, solid convex cone. Then the dual cone C∗ is a closed,pointed and solid convex cone.

Proof. The fact that C∗ is a closed and pointed convex cone followsfrom the Lemma A1. It remains to show that Int C∗ is not empty. ApplyingTheorem 2.1 [29], we deduce that there exists ϕ0 ∈ C∗ such that ϕ0(x) > 0 forall x ∈ C\0. Since X is a finite dimensional linear space, S1 = x ∈ C | ‖x‖ =1 is a compact set. Hence there exists δ > 0, such that ϕ0(x) ≥ 2δ,∀x ∈ S1.Consider the closed ball B(ϕ0, δ) = ϕ ∈ X ∗ | ‖ϕ − ϕ0‖ ≤ δ. We showthat B(ϕ0, δ) ⊂ C∗. If ϕ ∈ B(ϕ0, δ) then |ϕ(x) − ϕ0(x)| ≤ δ, ∀x ∈ X ,


with ‖x‖ = 1. Particularly, for x ∈ S1 we have ϕ(x) − ϕ0(x) ≥ −δ. Henceϕ(x) = ϕ0(x) + (ϕ(x)− ϕ0(x)) ≥ 2δ − δ = δ.

Thus, we have proved that ϕ(x) ≥ δ‖x‖ for all x ∈ C \ 0 and for allϕ ∈ B(ϕ0, δ). Hence B(ϕ0, δ) ⊂ C∗, which means that ϕ0 ∈ Int C∗, thus theproof is complete.

6.2. Minkovski semi-norms and Minkovski norms

Let X be a real normed linear space. Let C ⊂ X be a solid convex cone.Assume that C 6= X . This means that 0 6∈ Int C. Let ξ ∈ Int C be fixed. Denoteby Bξ the set defined by Bξ = x ∈ X | − ξ < x < ξ. It is easy to see that

(81) Bξ = (ξ − Int C) ∩ (−ξ + Int C).

From (81) we deduce that Bξ is an open and convex set. For each x ∈ X ,denote T (x) = t ∈ R | t > 0, 1

t x ∈ Bξ. Since Bξ is an open set and 0 ∈ Bξ,T (x) is not empty for all x ∈ X . The Minkovski functional associated to theset Bξ is defined by

(82) |x|ξ = inf T (x)

for every x ∈ X .

The next theorem collects several important properties of Minkovskifunctional

Theorem A1. The Minkovski functional introduced in (2.2) has the fol-lowing properties:

(i) |x|ξ ≥ 0 and |0|ξ = 0;(ii) |αx|ξ = |α| |x|ξ for all α ∈ R, x ∈ X ;(iii) |x|ξ < 1 if and only if x ∈ Bξ;(iv) |x + y|ξ ≤ |x|ξ + |y|ξ for all x, y ∈ X ;(v) there exists β(ξ) > 0 such that |x|ξ ≤ β(ξ)‖x‖, (∀) x ∈ X ;(vi) |x|ξ = 1 if and only if x ∈ ∂Bξ, where ∂Bξ is the border of the set Bξ;vii) |x|ξ ≤ 1 iff x ∈ Bξ, where Bξ = Bξ ∪ ∂Bξ;(viii) if C is a closed, solid convex cone, then Bξ = x ∈ X | −ξ ≤ x ≤ ξ;(ix) |ξ|ξ = 1;(x) the set T (x) coincides with (|x|ξ,∞);(xi) if x, y, z ∈ X are such that y ≤ x ≤ z, then |x|ξ ≤ max|y|ξ, |z|ξ.

Proof. Let us remark that assertions (i)–(iv), (vi) and (vii) follow fromthe properties of the Minkovski functional in linear topological spaces (see[16]). Here, for completeness we shall give a complete proof of this result.(i) follows immediately from the definition of | · |ξ.


(ii) Let α > 0 be fixed. It is easy to see that t ∈ T (αx) iff α−1t ∈ T (x).This leads to T (αx) = αT (x). Taking the infimum, we conclude that |αx|ξ =α|x|ξ = |α| |x|ξ.

On the other hand, x ∈ Bξ iff −x ∈ Bξ. This allows us to deduce that|−x|ξ = |x|ξ. Let α < 0 be fixed. We have |αx|ξ = |−|α|x|ξ = ||α|x|ξ = |α| |x|ξ,thus (ii) holds.

(iii) Let x ∈ X be such that |x|ξ < 1. This means that there existst ∈ (0, 1) such that −tξ < x < tξ, hence −ξ < x < ξ, therefore x ∈ Bξ.Conversely, let x ∈ Bξ. By the continuity at t = 1 of the function t → 1

t x,there exists t1 ∈ (0, 1) such that 1

t1x ∈ Bξ. Hence, T (x) ∩ (0, 1) 6= ∅. This

leads to |x|ξ < 1, thus (iii) holds.To prove (iv) it is enough to show that if τ > |x|ξ + |y|ξ, then τ > |x+y|ξ.

Let τ > |x|ξ+|y|ξ and define ε = 12(τ−|x|ξ−|y|ξ). Let τ1 = ε+|x|ξ, τ2 = ε+|y|ξ.

We have

τ1 + τ2 = τ, τ1 > |x|ξ, τ2 > |y|ξ(83)1τ(x + y) =

τ1

τx1 +

τ2

τy1,(84)

where x1 = 1τ1

x, y1 = 1τ2

y. From (83) and (iii) we deduce that x1, y1 ∈ Bξ.Since Bξ is a convex set and τ1

τ + τ2τ = 1 using (84), we get, 1

τ (x + y) ∈ Bξ.Invoking again (iii) we have | 1τ (x + y)|ξ < 1. Using (ii) we have |x + y|ξ < τ ,thus (iv) is proved.

(v) Since ξ ∈ Int C, there exists δ(ξ) > 0 such that the ball B(ξ, δ(ξ)) ⊂Int C, with B(ξ, δ(ξ)) = x ∈ X | ‖x − ξ‖ ≤ δ(ξ). Let x ∈ X , x 6= 0. Sinceξ ± δ(ξ) x

‖x‖ > 0, we obtain δ(ξ)x‖x‖ ∈ Bξ. From (iii) we have ‖δ(ξ) x

‖x‖ |ξ < 1.Using (ii) we deduce |x|ξ < β(ξ)‖x‖ for all x ∈ X with β(ξ) = δ−1(ξ), thus(v) is proved.

(vi) Let x ∈ X with |x|ξ = 1. This means that there exists a sequencetkk≥1 with tk > 1, lim

k→∞tk = 1, and 1

tkx ∈ Bξ. From x = lim

k→∞1tk

x we deduce

x ∈ Bξ. Using (iii), x 6∈ Bξ. It follows that x ∈ ∂Bξ.To prove the converse inclusion, choose x ∈ ∂Bξ and assume that |x|ξ >

1. Set ε = |x|ξ−1. Let V be the open ball, V = y ∈ X | ‖y−x‖ < εβ(ξ), where

β(ξ) is the constant from (v). Since x ∈ ∂Bξ there exist y ∈ Bξ ∩ V , whichmeans that ‖y−x‖ < ε

β(ξ) . From (iii) we get |y|ξ < 1. Combining (ii), (iv), (v),one successively obtains ε < |x|ξ − |y|ξ ≤ |x− y|ξ ≤ β(ξ)‖x− y‖ < ε, which isa contradiction. Hence |x|ξ ≤ 1. On the other hand, since Bξ is an open set,we have Bξ ∩ ∂Bξ = ∅. This means that x 6∈ Bξ. Hence |x|ξ ≥ 1. Therefore,|x|ξ = 1 and (vi) is proved.

(vii) follows from (iii) and (vi).


(viii) Let x ∈ Bξ. From (vii) we have |x|ξ ≤ 1. If |x|ξ < 1 then x ∈ Bξ,which is equivalent to −ξ < x < ξ implying −ξ ≤ x ≤ ξ. If |x|ξ = 1 then thereexists a sequence tkk≥1, tk > 1, lim

k→∞tk = 1 and 1

tkx ∈ Bξ. This means that

ξ ± 1tk

x ∈ C. Letting k → ∞ and taking into account that C is a closed setwe conclude that ξ ± x ∈ C, which is equivalent to −ξ ≤ x ≤ ξ. Conversely, ifx ∈ X is such that −ξ ≤ x ≤ ξ, then for all t > 1 we have −tξ < x < tξ. Thismeans that 1

t x ∈ Bξ. Letting t → 1 we get x ∈ Bξ.(ix) follows from (vi).(x) Let t ∈ (|x|ξ,∞). This is equivalent to |1t x|ξ < 1. Hence 1

t x ∈ Bξ.This means that t ∈ T (x). Thus, we have proved that (|x|ξ,∞) ⊂ T (x). Toprove the converse inclusion, we choose t ∈ T (x), that is, 1

t x ∈ Bξ. From (ii)and (iii) we have |x|ξ < t, i.e. t ∈ (|x|ξ,∞).

(xi) Let x, y, z ∈ X be such that y ≤ x ≤ z. Let us assume thatmax|y|ξ, |z|ξ < |x|ξ. Let t be such that:

(85) max|y|ξ, |z|ξ < t < |x|ξ.

It follows from (x) that t ∈ T (y) ∩ T (z). This means that −ξ < 1t y < ξ and

−ξ < 1t z < ξ. This leads to −ξ < 1

t y ≤1t x ≤

1t z < ξ and implies t ∈ T (x).

Invoking again (x) one deduces that t > |x|ξ which contradicts (85), thus theproof is complete.

It follows from (i), (ii) and (iv) that the Minkovski functional definedby (82) is a semi-norm.

The next theorem provides a condition which ensures that the semi-norm (82) is a norm.

Theorem A2. If Bξ is a bounded set, then the Minkovski semi-norm| · |ξ defined by (2.2) is a norm. Moreover, there exists αξ > 0 such that‖x‖ ≤ αξ|x|ξ for all x ∈ X .

Proof. To prove that | · |ξ is a norm, we have to show that if |x|ξ = 0 thenx = 0. If |x|ξ = 0 then from (x) of Theorem 2.1 we have T (x) = (0,∞). Hencefor all t > 0 we have 1

t x ∈ Bξ. Since Bξ is a bounded set, we have ‖1t x‖ ≤ α,

with α > 0 not depending on x and t, but possible, depending on ξ. Thisleads to ‖x‖ ≤ αt. Letting t → 0, one obtains that ‖x‖ = 0, hence x = 0. Tocheck the last assertion in the statement, we choose x ∈ X , x 6= 0. Invokingagain (x), we obtain that for all t ∈ (1,∞), 1

tx|x|ξ ∈ Bξ. From the boundedness

of Bξ we deduce that ‖x‖ ≤ α|x|ξ, thus the proof is complete.

Corollary A1. Assume that X is a finite dimensional real Banachspace. Assume also that C ⊂ X is a solid cone, C 6= X . If ξ ∈ Int C then thefollowing assertionsare equivalent:

(i) the Minkovski semi-norm |x|ξ is a norm;


(ii) Bξ is a bounded set.

Proof. (ii) → (i) follows from Theorem 2.2. Suppose (i) holds. Since Xis a finite dimensional Banach space, there exists α > 0 (depending on ξ) suchthat ‖x‖ ≤ α|x|ξ for all x ∈ X . Now, if x ∈ Bξ then |x|ξ < 1, hence ‖x‖ < α,that is, Bξ is a bounded set.

In the sequel we provide a sufficient condition which guarantees that | · |ξis a norm for all ξ ∈ Int C. To this end we introduce

Definition A3. We say that the ‖ ·‖ is monotone with respect to the coneC if 0 ≤ x ≤ y implies that ‖x‖ ≤ ‖y‖.

Proposition A2. If ‖ · ‖ is monotone with respect to the cone C, thenfor all ξ ∈ Int C the set Bξ is bounded.

Proof. Let ξ ∈ Int C be fixed. If x ∈ Bξ we have −ξ < x < ξ or,equivalently, 0 < ξ +x < 2ξ. Hence ‖x+ ξ‖ ≤ 2‖ξ‖. This leads to ‖x‖ ≤ 3‖ξ‖,thus the proof is complete.

Further we prove

Proposition A3. If X is a real Hilbert space and C ⊂ X is a cone, thenthe following assertions are equivalent:

(i) the norm | · |2 is monotone with respect to C;(ii) C ⊂ C∗.

Proof. (i) → (ii). Let x ∈ C. It is easy to see that 0 ≤ x ≤ x + 1ky for all

y ∈ C and k ≥ 1. If (i) is fulfilled, then |x|22 ≤ |x + 1ky|22, which is equivalent

to 2〈x, y〉 ≥ − 1k |y|

22. Letting k → ∞, we obtain that 〈x, y〉 ≥ 0 for all y ∈ C.

This means that x ∈ C∗, hence C ⊂ C∗.

To prove the converse implication, let x, y ∈ X be such that 0 ≤ x ≤ y.This means that both y − x and y + x are in C. If (ii) is fulfilled, then〈y − x, y + x〉 ≥ 0. This is equivalent to (|y|2)2 ≥ (|x|2)2, which shows that(ii) → (i) holds. Thus the proof is complete.

Remark A1. If X is a real Hilbert space such that the norm |x|2 ismonotone with respect to the cone C ⊂ X , then C is a pointed cone. Indeed,if both x ∈ C and −x ∈ C, then it follows from (i) → (ii) of Proposition A3that 〈−x, x〉 ≥ 0, which leads to 〈x, x〉 = 0. Hence x = 0. This shows thatC ∩ (−C) = 0.

The next two examples show that the monotonicity of the norm ‖ · ‖ isonly a sufficient condition that Bξ be a bounded set for all ξ ∈ Int C.


Example A1. Let X = R2 and the cone

(86) C = (x, y)T ∈ R2 | x ≥ 0, y ≤ x.

It is easy to verify that Bξ is a bounded set for all ξ ∈ Int C. On the otherhand, the dual cone is given by C∗ = (u, v) ∈ R2 | u ≥ 0, −u ≤ v ≤ 0.Hence C∗ ⊂ C. From Proposition A3 we deduce that the Euclidian norm onR2 is not monotone with respect to the cone C defined by (2.6).

Example A2. Let X = R3 and the cone C be defined by

(87) C = (x, y, z)T ∈ R3 | x ≥ 0, |y| ≤ x, |z| ≤ x.

It can be verified that Bξ is a bounded set for each ξ ∈ Int C. On the otherhand, the dual cone of C is C∗ = (u, v, w)T ∈ R3 | u ≥ 0, |v| + |w| ≤ u.Obviously, C∗ ⊂ C. Using again Proposition A3, we deduce that the Euclidiannorm on R3 is not monotone with respect to the cone (2.7).

Acknowledgement. This work was partially supported by Project CEx05-11-23/2005, CEEX Program of the Romanian Ministry of Education and Research.

REFERENCES

[1] L. Arnold, Stochastic Differential Equations; Theory and Applications. Wiley, New York,1974.

[2] A. Berman, M. Neumann and R.J. Stern, Nonnegative Matrices in Dynamic Systems.Wiley, New York, 1989.

[3] A. El Bouhtouri, D. Hinrichsen and A.J. Pritchard, H∞ type control for discrete-timestochastic systems. Int. J. Robust Nonlinear Control 9 (1999), 923–948.

[4] O.L.V. Costa and M.D. Fragoso, Necessary and sufficient conditions for mean squarestability of discrete-time linear systems subject to Markovian jumps. In: Proc. 9th In-ternat. Sympos. on Math. Theory of Networks and Systems, Kobe, Japan, 1991, pp.85–86.

[5] O.L.V. Costa and M.D. Fragoso, Stability results for discrete-time linear systems withMarkovian jumping parameters. J. Math. Anal. Appl. 179 (1993), 2, 154–178.

[6] R.F. Curtain, Stability of Stochastic Dynamical Systems. Lecture Notes in Math. 294.Springer Verlag, 1972.

[7] T. Damm and D. Hinrichsen, Newton’s method for a rational matrix equation occuringin stochastic control. Linear Algebra Appl. 332/334 (2001), 81–109.

[8] T. Damm and D. Hinrichsen, Newton’s method for concave operators with resolventpositive derivatives in ordered Banach spaces. Linear Algebra Appl. 363 (2003), 43–64.

[9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions. CambridgeUniversity Press, Cambridge, 1992.

[10] J.L. Doob, Stochastic Processes. Wiley, New York, 1967.[11] V. Dragan, G. Freiling, A. Hochhaus, and T. Morozan, A class of nonlinear differential

equations on the space of symmetric matrices. Electron. J. Diff. Eqns. 96 (2004), 1–48.


[12] V. Dragan, T. Damm, G. Freiling and T. Morozan, Differential equations with positiveevolutions and some applications. Result. Math. 48 (2005), 206–236.

[13] V. Dragan and T. Morozan, Exponential stability for discrete time linear equationsdefined by positive operators. Integral Equ. Oper. Theory 54 (2006), 465–493, (Onlinefirst 2005 Birkhauser Verlag Basel/Switzerland DOI 10.1007/s00020-005-1371-7, p. 1–29, http://dx.doi.org/10.1007/s00020-005-1371-7)

[14] V. Dragan and T. Morozan, Mean square exponential stability for some stochastic lineardiscrete time systems. European Journal of Control, 4, 12 (2006), 373–395.

[15] V. Dragan and T. Morozan, Mean Square Exponetial Stability for Discrete-Time Time-Varying Linear Systems with Markovian Switching. In: Proc. 17th Internat. Sympos.Math. Theory of Networks and Systems, Kyoto, Japan, 2006, CD-Rom.

[16] N. Dunford and J. Schwartz, Linear Operators: Part I. Interscience Publishers, NewYork, 1958.

[17] Y. Fang and K. Loparo, Stochastic stability of jump linear systems. IEEE Trans. Aut.Control 47 (2002), 7, 1204–1208.

[18] X. Feng and K. Loparo, Stability of linear Markovian jump systems. Proc. 29th IEEEConf. Decision Control, 1408, Honolulu, Hawaii, 1990.

[19] X. Feng, K. Loparo, Y. Ji and H.J. Chizeck, Stochastic stability properties of jump linearsystems. IEEE Trans. Aut. Control 37 (1992), 1, 38–53.

[20] M.D. Fragoso, O.L.V. Costa and C.E. de Souza, A new approach to linearly perturbedRiccati equations arising in stochastic control. Appl. Math. Optim. 37 (1998), 99–126.

[21] G. Freiling and A. Hochhaus, Properties of the solutions of rational matrix differenceequations. Advances in difference equations, IV. Comput. Math. Appl. 45 (2003), 1137–1154.

[22] A. Halanay and D. Wexler, Qualitative Theory of Systems with Impulses. PublishingHouse of the Romanian Academy 1968 (Russian translation MIR, Moskow, 1971).

[23] M. Iosifescu, Finite Markov Chains and Applications. Ed. Tehnica, Bucharest, 1977.(Romanian)

[24] Y. Ji, H.J. Chizeck, X. Feng and K. Loparo, Stability and control of discrete-time jumplinear systems. Control Theory and Advances Tech. 7 (1991), 2, 247–270.

[25] R.Z. Khasminskii, Stochastic Stability of Differential Equations. Sythoff and Noordhoff:Alpen aan den Ryn, 1980.

[26] M.A. Krasnosel’skij, Positive solutions of operator equations. Fizmatgiz, Moskow, 1962.(Russian)

[27] M.A. Krasnosel’skij, J.A. Lifshits and A.V. Sobolev, Positive Linear Systems- TheMethod of Positive Operators. Sigma Series in Applied Mathematics 5. HeldermannVerlag, Berlin, 1989.

[28] R. Krtolica, U. Ozguner, H. Chan, H. Goktas, J. Winkelman and M. Liubakka, Stabilityof linear feedback systems with random communication delays. Proc. 1991 ACC, Boston,MA, June 26–28, 1991.

[29] M.G. Krein, and R. Rutman, Linear operators leaving invariant a cone in a Banachspace. Amer. Math. Soc. Translations Ser. 1 10 (1962), 199–325 (Russian version: UspehiMat. Nauk (N.S.) 1(23) (1948), 1–95)

[30] M. Mariton, Jump Linear Systems in Automatic Control. Marcel Dekker, New York,1990.

[31] T. Morozan, Stability and control of some linear discrete-time systems with jump Markovdisturbances. Rev. Roumaine Math. Pures Appl. 26 (1981), 1, 101–119.

[32] T. Morozan, Optimal stationary control for dynamic systems with Markovian perturba-tions. Stochastic Anal. and Appl. 1 (1983), 3, 299–325.


[33] T. Morozan, Stability and control for linear discrete-time systems with Markov pertur-bations. Rev. Roumaine Math. Pures Appl. 40 (1995), 5-6, 471–494.

[34] T. Morozan, Stabilization of some stochastic discrete-time control systems. StochasticAnal. and Appl. 1 (1983), 1, 89–116.

[35] T. Morozan, Stability radii of some discrete-time systems with independent randomperturbations. Stochastic Anal. Appl. 15 (1997), 3, 375–386.

[36] T. Morozan, Discrete-time Riccati equations connected with quadratic control for linearsystems with independent random pertrubations Rev. Roumaine Math. Pures Appl. 37(1992), 3, 233–246.

[37] K.M. Przyluski, Remarks on the stability of linear infinite dimensional discrete timesystems. J. Differential Equations 72 (1989), 189–200.

[38] A.C.M. Ran and M.C.B. Reurings, The symmetric linear matrix equation. Electronic J.Linear Algebra 9 (2002), 93–107.

[39] H. Schneider, Positive operators and an Inertia Theorem. Numerische Mathematik 7(1965), 11–17.

[40] W.H. Wonham, Random Differential Equations in Control Theory. In: A.T. Braucha-Reid (Ed.), Probabilistic Methods in Applied Mathematics 2, pp. 131–212. AcademicPress, New York, 1970.

[41] J. Yong and X.Y. Zhou, Stochastic Controls. Hamiltonian Systems and HJB Equations.Springer-Verlag, New York, 1999.

[42] J. Zabczyk, Stochastic Control of Discrete-time Systems. Control Theory and Topics inFunct. Analysis 3. IAEA, Vienna, 1976.

Received 5 June 2007 Romanian AcademyInstitute of Mathematics

P.O. Box 1–764014700 Bucharest, Romania

[email protected]@imar.ro

discrete time linear equations defined by positive operators

Documents