DISCRETE TIME LINEAR EQUATIONS DEFINED BY POSITIVE OPERATORS ON ORDERED HILBERT SPACES VASILE DRAGAN and TOADER MOROZAN
In this paper the problem of exponential stability of the zero state equilibrium of a discrete-time time-varying linear equation described by a sequence of linear bounded and positive operators acting on an ordered Hilbert space is investigated. The class of linear equations considered in this paper contains as particular cases linear equations described by Lyapunov operators or symmetric Stein operators as well as nonsymmetric Stein operators. Such equations occur in connection with the problem of mean square exponential stability for a class of difference stochastic equations affected by independent random perturbations and Markovian jumping as well as in connection with some iterative procedures which allow us to compute global solutions of discrete time generalized symmetric or nonsymmetric Riccati equations. The exponential stability is characterized in terms of the existence of some globally defined and bounded solutions of some suitable backward affine equations (inequations, respectively) or forward affine equations (inequations, respectively). AMS 2000 Subject Classification: 39A11, 47H07, 93C55, 93E15. Key words: positive operators, discrete time linear equations, exponential stability, ordered Hilbert spaces, Minkovski norm.
1. INTRODUCTION The stabilization problem, together with various control problems for linear stochastic systems, was intensively investigated in the last four decades. We refer the reader to some of the most popular monographies in the field: [1, 6, 9, 25, 30, 40, 41] and references therein. It is well known that the mean square exponential stability or, equivalently, the second moments exponential stability of the zero solution of a linear stochastic differential equation or a linear stochastic difference equation is equivalent to the exponential stability of the zero state equilibrium of a suitable deterministic linear differential equation or a deterministic linear difference equation. Such deterministic differential (difference) equations are defined by REV. ROUMAINE MATH. PURES APPL., 53 (2008), 2–3, 131–166
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the so-called Lyapunov type operators associated to the given stochastic linear differential (difference) equations. Exponential stability in the case of differential equations or difference equations described by Lyapunov operators has been investigated as a problem with interest in itself in a lot of works. In the time-invariant case results concerning the exponential stability of linear differential equations defined by Lyapunov type operators were derived using spectral properties of positive linear operators on an ordered Banach space obtained by Krein and Rutman [29] and Schneider [39]. A significant extension of the results in [29] and [39] to the class of positive resolvent operators was provided by Damm and Hinrichsen [7, 8]. Similar results were derived also for the discrete-time timeinvariant case, see [21, 38]. In [13] the exponential stability of discrete-time time-varying linear equations defined by linear positive operators acting on a finite dimensional ordered Hilbert space, was studied. In that paper different characterizations of exponential stability in terms of the existence of bounded and uniformly positive solutions of some suitable backward affine equations (inequations, respectively) or affine equations were provided. In the case of continuous-time time-varying systems, a class of linear differential equations on the space of n × n symmetric matrices Sn is studied in [11]. Such equations have the property that the corresponding linear evolution operator is positive on Sn . They contain as particular cases linear differential equations of Lyapunov type arising in connection with the problem of investigation of mean square exponential stability. The results of [11] were extended to an abstract framework of a differential equations with positive evolution on a finite dimensional ordered Hilbert space (see [12]). In this paper, the infinite dimensional counterpart of the results proved in [13] is derived. The discrete-time linear equations under consideration in this paper are defined by sequences of positive bounded linear operators on an ordered Hilbert space. The order relation is induced by a closed, solid, selfdual convex cone. The main tool involved in our developments is a Minkovski norm defined by the Minkovski functional associated to a suitable open and convex set. To characterize exponential stability, a crucial role is played by the unique bounded solution of some suitable backward affine equations as well as of some forward affine equations. We show that if the equations considered are described by periodic sequences of operators, then the bounded solution, if any, also is a periodic sequence. Moreover, in the time-invariant case the bounded solutions to both backward affine equation and forward affine equation are constant. Thus, the results concerning the exponential stability for the timeinvariant case are recovered as special cases of the results proved in this paper.
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The outline of the paper is as follows: Section 2 collects some definitions, some auxiliary results in order to display the framework where the main results are proved. Section 3 contains results which characterize the exponential stability of the zero state equilibrium of a discrete-time time-varying linear equation described by a sequence of linear positive operators on a ordered Hilbert space. Section 4 deals with the problem of preservation of exponential stability under an additive perturbation of the sequence of linear operators defining the discrete time equations under consideration. In Section 5 we consider the case of discrete time, time varying linear equations defined by sequences of positive bounded linear operators on ordered Banach spaces. An application to the mean square exponential stability of a discrete time stochastic stochastic system perturbed by a Markov chain with an infinite number of states is provided. The paper ends with an Appendix which collects the usual definitions concerning the convex cones. Also, some useful properties of the Minkovski seminorm are presented. A set of sufficient conditions is given under which a Minkovski seminorm is just a norm. 2. PRELIMINARIES In this section we describe the framework where the discrete time linear equations investigated in this paper are defined. In our approach a crucial role will be played by the Minkovski norm. More details concerning this norm can be found in Appendix. 2.1. Positive linear operators on ordered Hilbert spaces In this subsection as well as in the following X is a real Hilbert space ordered by the ordering relation “ ≤ ” induced by the closed, solid, selfdual convex cone X + . Since X + is a selfdual convex cone, it follows from Remark A.1 and Proposition A.3 (in Appendix) that X + is a pointed cone. By Proposition A.3, | · |2 defined by (1)
1
|x|2 = (hx, xi) 2
is monotone with respect to X + . Let ξ ∈ Int X + be fixed; we associate the Minkovski functional | · |ξ defined by (82). It follows from Theorem A.2 and Proposition A.2 that | · |ξ is a norm on X . Moreover, from Theorem A.1 (v) and Theorem A.2 we deduce that | · |ξ is equivalent to | · |2 defined by (1). Hence (X , | · |ξ ) is a Banach space. Moreover, | · |ξ has the properties below:
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P1 ) If x, y, z ∈ X are such that y ≤ x ≤ z, then |x|ξ ≤ max{|y|ξ , |z|ξ }.
(2)
P2 ) For an arbitrary x ∈ X with |x|ξ ≤ 1 we have −ξ ≤ x ≤ ξ
(3)
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 kT k = sup|x|≤1 |T x| is the corresponding operator norm. Remark 2.1. a) Since | · |ξ and | · |2 are equivalent, k · kξ and k · k2 are also equivalent. This means that there are two positive constants c1 and c2 such that c1 kT kξ ≤ kT k2 ≤ c2 kT kξ for all bounded linear operators T : X → X . b) If T ∗ : X → X is the adjoint operator of T with respect to the inner product on X , then kT k2 = kT ∗ k2 . In general, the equality kT kξ = kT ∗ kξ is not true. However, follows from a) it follows that there are two positive constants e c1 , e c2 such that (4)
e c1 kT kξ ≤ kT ∗ kξ ≤ e c2 kT kξ .
Definition 2.1. Let (X , X + ) and (Y, Y + ) be ordered vector spaces. An operator T : X → Y is called positive if T (X + ) ⊂ Y + . In this case we write T ≥ 0. If T (Int X + ) ⊂ Int Y + we write T > 0. Proposition 2.1. If T : X → X is a bounded linear operator, then (i) T ≥ 0 if and only if T ∗ ≥ 0; (ii) If T ≥ 0 then kT kξ = |T ξ|ξ . Proof. (i) is a direct consequence of the fact that X + is a selfdual cone. (ii) If T ≥ 0 then from (3) we have −T ξ ≤ T x ≤ T ξ. It follows from (2) that |T x|ξ ≤ |T ξ|ξ for all x ∈ X with |x|ξ ≤ 1, which leads to sup |T x|ξ ≤ |T ξ|ξ ≤ sup |T x|ξ , |x|ξ ≤1
|x|ξ ≤1
hence kT kξ = |T ξ|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 linear operators. If T1 ≤ T2 then kT1 kξ ≤ kT2 kξ . Example 2.1. (i) Let X = Rn and X + = Rn+ , where Rn+ = {x = T x1 x2 · · · xn ∈ Rn | xi ≥ 0, 1 ≤ i ≤ n}. In this case, X + is a closed, solid, pointed, selfdual convex cone. The ordering induced on Rn by
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this cone is known as the component wise ordering. If T : Rn → Rn is a linear operator, then T ≥ 0 iff its corresponding matrix A with respect to the canonical basis on Rn has nonnegative entries. For ξ = (1, 1, 1, . . . , 1)T ∈ Int(Rn+ ), the norm | · |ξ is defined by (5)
|x|ξ = max |xi |. 1≤i≤n
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)
hA, Bi = Tr(B T 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 a + + selfdual cone. On Rm×n we also consider the norm | · |ξ defined by (8)
|A|ξ = max |aij |. i,j
Properties P1 and P2 are fulfilled for the norm (8) with 1 1 1 ··· 1 ξ = · · · · · · · · · · · · · · · ∈ Int Rm×n . + 1 1 1 ··· 1 An important class of linear operators on Rm×n is that of the form LA,B : Rm×n → Rm×n with LA,B Y = AY B T for all Y ∈ Rm×n , where A ∈ Rm×m , B ∈ Rn×n are fixed givenmatrices. These operators are often called “nonsymmetric Stein operators”. It can be checked that LA,B ≥ 0 iff aij blk ≥ 0, ∀ i, j ∈ {1, . . . , m}, l, k ∈ {1, . . . , n}. Hence LA,B ≥ 0 iff the matrix A ⊗ B defines a positive operator on the ordered space (Rmn , Rmn + ), where ⊗ is the Kronecker product. (iii) Let Sn ⊂ Rn×n be the subspace of n × n symmetric matrices. Let X = Sn ⊕ Sn ⊕ · · · ⊕ Sn = SnN with N ≥ 1 fixed. On SnN , consider the inner product (9)
hX, Y i =
N X
T r(Yi Xi )
i=1
for arbitrary X = (X1 , X2 , . . . , XN ) and Y = (Y1 , Y2 , . . . , YN ) in SnN . The space SnN is ordered by the convex cone (10)
SnN,+ = {X = (X1 , X2 , . . . , XN ) | Xi ≥ 0, 1 ≤ i ≤ N },
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whose interior Int SnN,+ = {X ∈ SnN | Xi > 0, 1 ≤ i ≤ N } is nonempty. Here Xi ≥ 0 and Xi > 0 means that Xi is a positive semidefinite matrix or a, positive definite matrix, respectively. One may show that SnN,+ is a selfdual cone. Together with the norm | · |2 induced by the inner product (9), on SnN we also consider the norm | · |ξ defined by (11)
(∀) X = (X1 , . . . , XN ) ∈ SnN ,
|X|ξ = max |Xi |, 1≤i≤N
where |Xi | =
max |λ|, σ(Xi ) is the set of eigenvalues of the matrix Xi .
λ∈σ(Xi )
For the norm defined by (11), properties P1 and P2 are fulfilled with ξ = (In , In , . . . , In ) = J ∈ SnN . (iv) For an infinite dimensional case, let us consider X = `2 (Z+ , R) n o ∞ P 2 2 xi < ∞ . On X where ` (Z+ , R) = x = (x0 , x1 , . . . , xn , . . .) | xi ∈ R, i=0
∞ P
we consider the usual inner product hx, yi`2 = xi yi for all x = {xi }i≥0 , y = i=0 n o ∞ P {yi }i≥0 . Set X + = x = {xi }i≥0 | x0 ≥ 0, x2i ≤ x20 . It is easy to see i=1
that X + is a closed, pointed convex cone. In the finite dimensional case the analogue of this cone is known as a circular cone. n o ∞ P The interior Int X + = x = {xi }i≥0 | x0 > 0, x2i < x20 . It remains to prove that X + is selfdual. Let y ∈ (X + )∗ . Hence
i=1
hx, yi`2 ≥ 0
(12) X +.
for all x = {xi }i≥0 ∈ In particular, taking x = {1, 0, 0, . . . , 0} in (12), we obtain 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 + . ∞ P e = {e Suppose now yt2 > 0. Take x xi }i≥0 defined by t=1
(13) with γ = ∞ P k=1
x e0 = y0 , P ∞ k=1
yk2
− 12
x ei = −γyi y0
e ∈ X + . Replacing (13) in (12), one gets . Obviously, x
yk2 ≤ y02 , which shows that y ∈ X + . Thus, it was shown that (X + )∗ ⊂ X + .
Let now y = {yi }i≥0 ∈ X + . We have to show that (12) holds for all 2 P ∞ ∞ P P ∞ x2k yk2 ≤ x20 y02 , x ∈ X + . Indeed, for x ∈ X + we have xk yk ≤ k=1
k=1
k=1
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P ∞ P ∞ xk yk ≤ x0 y0 . This is equivalent to −x0 y0 ≤ xk yk ≤ which leads to k=1
k=1
x0 y0 , 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,
2 ) √2 2 2 )
= {x ∈ `2 (Z+ , R) | |x − ξ|2 ≤
2 2 }.
It is not difficult to check that √
⊂ Moreover, we have = {tx | t ≥ 0, x ∈ B(ξ, 22 )}. If x = (x0 , x1 , x2 , . . . , xn , . . .) ∈ X , we write x = (x0 , x b) with x b = (x1 , x2 , . . . , xn , . . .). The corresponding Minkovski norm is given by |x|ξ = ∞ P x2i . This equality can be obtained taking into |x0 | + |b x|2 , where |b x|22 = B(ξ,
X +.
X+
i=1
account that |x|ξ = inf{t > 0 | |b x|22 − |x0 |2 − 2tx0 − t2 < 0 and |b x|22 − |x0 |2 + 2tx0 − t2 < 0}. 2.2. Discrete-time affine equations Let L = {Lk }k≥k0 be a sequence of bounded linear operators Lk : X → X and f = {fk }k≥k0 a sequence of elements fk ∈ X . These two sequences define on X two affine equations (14)
xk+1 = Lk xk + fk ,
which will be called “the forward” affine equation or “causal affine equation” defined by (L, f ), and (15)
xk = Lk xk+1 + fk
which will be called “the backward affine equation” or “anticausal affine equation” defined by (L, f ). For each k ≥ l ≥ k0 let T c (k, l) : X → X be the causal evolution operator defined by the sequence L, T c (k, l) = Lk−1 Lk−2 . . . Ll if k > 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 evolution operator on X defined by the sequence L, that is, T (k, l)a = Lk Lk+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 x ek = T c (k, l)x, k ≥ l, l ≥ k0 be fixed. One obtains that {e xk }k≥l verifies the forward linear equation (16)
xk+1 = Lk xk
with initial value xl = x. Also, if yk = T a (k, l)y, k0 ≤ k ≤ l, then from the a one obtains that {y } definition of Tkl k k0 ≤k≤l is the solution of the backward
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linear equation (17)
yk = Lk yk+1
with given terminal value yl = y. Obviously, (16) and (17) lead to T c (k+1, l) = Lk T c (k, l) for all k ≥ l ≥ k0 and T a (k, l) = Lk T a (k +1, l) for all k0 ≤ k ≤ l −1. It should be remarked that, unlike the continuous time case, a solution {xk }k≥l of the forward linear equation (16) with given initial values xl = x is well defined for k ≥ l while a solution {yk }k≤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 equations (16), (17) are well defined for all k ≥ k0 . If (T c (k, l))∗ is the adjoint operator of the causal evolution operator c T (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 that the adjoint of the causal evolution operator associated with the sequence L generates an anticausal evolution. Definition 2.2. a) We say that the sequence L = {Lk }k≥k0 defines a positive evolution if for all k ≥ l ≥ k0 the causal linear evolution operator T (k, l)c ≥ 0. b) We say that the sequence L = {Lk }k≥k0 defines an anticausal positive evolution if for all k0 ≤ k ≤ l the anticausal linear evolution operator T a (k, l) ≥ 0. Since T c (l + 1, l) = Ll , T a (l, l + 1) = Ll , respectively, it follows that the sequence {Lk }k≥k0 generates a causal positive evolution or an anticausal positive 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 positive evolutions. Throughout the paper we shall say that a sequence {Lk }k≥0 generates a positive evolution instead of a causal positive evolution every time when no confusion can arise. Also, in this case, we shall write T (k, l) instead of T c (k, l). The following result is straightforward. It will be used in the next sections. Corollary 2.2. Let Li = {Lik }k≥k0 , i = 1, 2 be two sequences of bounded linear operators and Tic (k, l) be the corresponding causal linear evolution operators. Assume that 0 ≤ L1k ≤ L2k for all k ≥ k0 . Under this assumption we have T2c (k, l) ≥ T1c (k, l) for all k ≥ l ≥ k0 .
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At the end of this subsection we recall the representation formulae of the solutions of affine equations (14), (15). Each solution of the forward affine equation (14) has the representation (18)
xk = T c (k, l)xl +
k−1 X
T c (k, i + 1)fi
i=l
for all k ≥ l + 1. Also, any solution of the backward affine equation (15) has a representation yk = T a (k, l)yl +
l−1 X
T a (k, i)fi ,
k0 ≤ k ≤ l − 1.
i=k
3. EXPONENTIAL STABILITY In this section, we deal with the exponential stability of the zero solution of a discrete time linear equation defined by a sequence of positive bounded linear operators. Definition 3.1. We say that the zero solution of the equation (19)
xk+1 = Lk xk
is exponentially stable or, equivalently, that the sequence L = {Lk }k≥k0 generates an exponentially stable evolution (E.S. evolution) if there are β > 0, q ∈ (0, 1) such that (20)
kT (k, l)kξ ≤ βq k−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 k · k2 , too. In the case where Lk = L for all k, if (20) is satisfied, we shall say that the operator L generates a discrete-time exponentially stable evolution. It is well known that L generates a discrete-time exponentially stable evolution if and only if ρ[L] < 1, where ρ[·] is the spectral radius. It must be remarked that if the sequence {Lk }k≥k0 generates an exponentially stable evolution then it is a bounded sequence. In this section we shall derive several conditions which are equivalent to exponential stability of the zero solution of equation (19) in the case {Lk }k≥k0 . Such results can be viewed as an alternative characterization of exponential stability to the one in terms of Lyapunov functions. First, from Proposition 2.1, Corollary 2.1 and Corollary 2.2 we obtain the following result specific to the case of operators which generate positive evolution.
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Proposition 3.1. Let L = {Lk }k≥k0 , and L1 = {L1k }k≥k0 be two sequences of positive bounded linear operators on X . (i) The following are equivalent: a) L(·) defines an E.S. evolution; b) there exist β ≥ 1, q ∈ (0, 1) such that |T (k, l)ξ|ξ ≤ βq k−l for all k ≥ l ≥ k0 . (ii) If L1k ≤ 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 {Lk }k≥0 be a sequence of positive bounded linear operators Lk : X → X . Then the following assertions are equivalent: (i) the sequence {Lk }k≥0 generates an exponentially stable evolution; k P (ii) there exists δ > 0 such that kTk,l kξ ≤ δ for arbitrary k ≥ k0 ≥ 0; (iii) there exists δ > 0 such that
l=k0 k P
T (k, l)ξ ≤ δξ for arbitrary k ≥ k1 ≥
l=k1
0, δ > 0 being independent of k, k1 ; (iv) for an arbitrary bounded sequence {fk }k≥0 ⊂ X , the solution with zero initial value of the forward affine equation xk+1 = Lk xk + fk ,
k≥0
is bounded. Proof. The implication (iv) → (i) is the discrete-time counter part of Perron’s Theorem (see [22, 37]). It remains to prove the implications (i) → (ii) → (iii) → (iv). β . If (i) is true, then (ii) follows immediately from (20) with δ = 1−q Let us prove that (21)
0 ≤ T (k, l)ξ ≤ kT (k, l)kξ ξ
for arbitrary k ≥ l ≥ 0. If T(k,l) ξ = 0 then it follows from Proposition 2.1 (ii) that ||T(k,l) ||ξ = 0 and (21) is fulfilled. If T (k, l)ξ 6= 0 then from (3) applied to 1 x = |T (k,l)ξ| T (k, l)ξ one gets 0 ≤ T (k, l)ξ ≤ |T (k, l)ξ|ξ ξ and (21) follows from ξ Proposition 2.1 (ii). If (ii) holds then (iii) follows from (21). We have to prove that (iii) → (iv). Let {fk }k≥0 ⊂ X be a bounded sequence, that is, |fk |ξ ≤ µ, k ≥ 0. From (3) we obtain −|fl |ξ ξ ≤ fl ≤ |fl |ξ ξ, 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
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and −µ
k−1 X
T (k, l + 1)ξ ≤
l=0
k−1 X
T (k, l + 1)fl ≤ µ
l=0
k−1 X
T (k, l + 1)ξ.
l=0
Using (2) we deduce that k−1 k−1 X X T (k, l + 1)fl ≤ µ T (k, l + 1)ξ . ξ
l=0
ξ
l=0
If (iii) is valid we conclude by using again (2) that k−1 X T (k, l + 1)fl ≤ µδ, l=0
k≥1
ξ
which shows that (iv) is fulfilled by using (18). Thus the proof is complete.
We note that the proof of Theorem 3.1 shows that in the case of a discrete time linear equation (19) defined by a sequence of positive bounded linear operators, the exponential stability is equivalent to the boundedness of the solution with the zero initial value of the forward affine equation xk+1 = Lk xk + ξ. This is in contrast to the general case of a discrete time linear equation, where if we want to use Perron’s Theorem to characterize the exponential stability we have to check the boundedness of the solution with zero initial value of the forward affine equation xk+1 = Lk xk +fk for an arbitrary bounded sequence {fk }k≥0 ⊂ X . Let us now introduce the concept of uniform positivity. Definition 3.2. We say that a sequence {fk }k≥k0 ⊂ X + is uniformly positive if there exists c > 0 such that fk > cξ for all k ≥ k0 . If {fk }k≥k0 ⊂ X + is uniformly positive, we shall write fk 0, k ≥ k0 . If −fk 0, k ≥ k0 , we shall write fk 0, k ≥ k0 . The next result provides a characterization of the exponential stability by using solutions of some suitable backward affine equations. Theorem 3.2. Let {Lk }k≥k0 be a sequence of positive bounded linear operators Lk : X → X . Then the following assertions are equivalent: (i) the sequence {Lk }k≥k0 generates an exponentially stable evolution; (ii) there exist β1 > 0, q ∈ (0, 1) such that kT ∗ (k, l)kξ ≤ β1 q k−l , (∀) k ≥ l ≥ k0 ;
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(iii) for each k ≥ k0 the series
P
12
T ∗ (l, k)ξ is convergent and there exists
l≥k
δ > 0, independent of k, such that
∞ P
T ∗ (l, k)ξ ≤ δξ;
l=k
(iv) the discrete time backward affine equation xk = L∗k xk+1 + ξ
(22)
has a bounded and uniformly positive solution; (v) for an arbitrary bounded and uniformly positive sequence {fk }k≥k0 ⊂ Int X + the backward affine equation xk = L∗k xk+1 + fk , k ≥ k0
(23)
has a bounded and uniformly positive solution. (vi) There exists a bounded and uniformly positive sequence {fk }k≥k0 ⊂ Int X + such that the corresponding backward affine equation (23) has a bounded solution {e xk }k≥k0 ⊂ X + ; (vii) there exists a bounded and uniformly positive sequence {yk }k≥k0 ⊂ Int X + which verifies L∗k yk+1 − yk 0,
(24)
k ≥ k0 .
Proof. The equivalence (i) ↔ (ii) follows immediately from (4). In a similar way to the proof of inequality (21), we obtain 0 ≤ T ∗ (l, k)ξ ≤ kT ∗ (l, k)kξ ξ
(25)
for all l ≥ k ≥ k0 . P ∗ If (ii) holds, then for each k ≥ k0 the series kT (l, k)kξ of real numbers l≥k
is convergent and we have ∞ X
(26)
kT ∗ (l, k)kξ ≤ δ,
l=k
where δ =
β1 1−q
is independent of k. Therefore, the series
P
T ∗ (l, k)ξ is abso-
l≥k
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 ∞ P each k ≥ k0 , set yk = T ∗ (l, k)ξ. If (iii) holds then yk is well defined and l=k
additionally yk ≤ δξ for all k ≥ k0 . Using the definition of the linear evolution ∞ P operator T ∗ (l, k), we may write yk = ξ + L∗k T ∗ (l, k + 1)ξ or, equivalently, l=k+1
yk = ξ + L∗k yk+1 . This shows that {yk }k≥k0 is a solution of (22). Moreover, we have ξ ≤ yk ≤ δξ for all k ≥ k0 . This means that {yk }k≥k0 is a bounded
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and uniformly positive solution of (22). Thus, we obtain that the implication (iii) → (iv) holds. Now, we prove (ii) → (v). Let {fk }k≥k0 ⊂ Int X + be a bounded and uniformly positive sequence. Hence there exist νi > 0, i = 1, 2, such that ν1 ξ ≤ fl ≤ ν2 ξ for all l ≥ k0 . Since T ∗ (l, k) ≥ 0, for all l ≥ k ≥ k0 we may write ν1 T ∗ (l, k)ξ ≤ T ∗ (l, k)fl ≤ ν2 T ∗ (l, k)ξ
(27)
for all l ≥ k ≥ k0 . The monotonicity of the Minkovski norm together with the equality from Proposition 2.1 (ii) allow us to write ν1 kT ∗ (l, k)kξ ≤ |T ∗ (l, k)fl |ξ ≤ ν2 kT ∗ (l, k)kξ .
(28)
If (ii) is fulfilled then (28) shows that for each k ≥ k0 the series
P
|T ∗ (l, k)fl |ξ
l≥k
of the real numbers is convergent and ∞ X
(29)
|T ∗ (l, k)fl |ξ ≤ δ1
l=k 2 β1 for all k ≥ k0 , where δ1 = ν1−q is independent of k. Thus, one gets that the P ∗ T (l, k)fl is absolutely convergent for all k ≥ k0 . series
l≥k
Set zk =
∞ P
T ∗ (l, k)fl , k ≥ k0 . Using again the definition of the linear
l=k
evolution operator T ∗ (l, k) we can write (30)
zk = fk + L∗k
∞ X
T ∗ (l, k + 1)fl = fk + L∗k zk+1 .
l=k+1
From (29) and (30) we deduce that {zk }k≥k0 is a bounded solution of (23). From (30) we also have that zk ≥ fk ≥ ν1 ξ for all k ≥ k0 . This means that zk 0, k ≥ k0 , thus (v) holds. Further, (v) → (iv) → (vi) are straightforward. We now prove the implication (vi) → (ii). Let us assume that there exists a bounded and uniformly positive sequence {fk }k≥k0 ⊂ Int X + such that the corresponding equation (23) has a bounded solution {b xk }k≥k0 ⊂ X + . Therefore, there exist positive constants γi , i ∈ {1, 2, 3}, such that γ1 ξ ≤ fk ≤ γ2 ξ,
γ1 ξ ≤ x bk ≤ γ3 ξ,
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for all k ≥ k0 . Let k1 ≥ k0 be fixed. Define yek = T ∗ (k, k1 )b xk , ∀ k ≥ k1 . Since T ∗ (k, k1 ) ≥ 0, we may write γ1 T ∗ (k, k1 )ξ ≤ T ∗ (k, k1 )fk ≤ γ2 T ∗ (k, k1 )ξ (31)
γ1 T ∗ (k, k1 )ξ ≤ yek ≤ γ3 T ∗ (k, k1 )ξ
for all k ≥ k1 . From x bk = L∗k x bk+1 + fk , as well as from the definitions of yek ∗ and T (k, k1 ), we obtain successively yek = T ∗ (k, k1 )L∗k x bk+1 + T ∗ (k, k1 )fk = T ∗ (k + 1, k1 )b xk+1 + T ∗ (k, k1 )fk = yek+1 + T ∗ (k, k1 )fk . Thus, we obtained yek+1 = yek − T ∗ (k, k1 )fk for all k ≥ k1 . From (31) we deduce that yek+1 ≤ qe yk
(32)
for all k ≥ k1 , where q = 1 − γγ31 . Taking γ3 large enough in (31) we obtain q ∈ (0, 1). From (32) we obtain inductively yek ≤ q k−k1 x bk1 for all k ≥ k1 . Using again (31) together with x bk1 ≤ γ3 ξ, we deduce that 0 ≤ T ∗ (k, k1 )ξ ≤ γγ13 q k−k1 ξ, which by (2) leads to |T ∗ (k, k1 )ξ|ξ ≤ γγ31 q k−k1 , k ≥ k1 . From Proposition 2.1 (ii) we have kT ∗ (k, k1 )kξ ≤ γγ13 q k−k1 , which means that (ii) holds. The implication (iv) → (vii) follows immediately since a bounded and uniform by positive solution of (22) is a solution with the desired properties of (24). To complete the proof we show that (vii) → (vi). Let {zk }k≥k0 ⊂ Int X + be a bounded and uniformly positive solution of (24). Define fbk = zk −L∗k zk+1 . It follows that {fbk }k≥k0 is bounded and uniform by positive, therefore {zk }k≥0 is a bounded and positive solution of (23) corresponding to {fbk }k≥k0 , thus the proof is complete. The next result provides more information about the bounded solution of the discrete time backward affine equations. Theorem 3.3. Let {Lk }k≥k0 be a sequence of linear operators which generates an exponentially stable evolution on X . Then the following assertions hold: (i) for each bounded sequence {fk }k≥k0 ⊂ X the discrete-time backward affine equation xk = L∗k xk+1 + fk
(33)
has an unique bounded solution which is given by (34)
x ek =
∞ X l=k
T ∗ (l, k)fl ,
k ≥ k0 ;
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(ii) if there exists an integer θ > 1 such that Lk+θ = Lk , fk+θ = fk for all k then the unique bounded solution of equation (33) also is a periodic sequence with period θ; (iii) if Lk = L, fk = f for all k, then the unique bounded solution of equation (33) is constant and it is given by x e = (IX − L∗ )−1 f,
(35)
with IX the identity operator on X ; (iv) if Lk are positive operators and {fk }k≥k0 ⊂ X + is a bounded sequence, then the unique bounded solution of equation (33) satisfies x ek ≥ 0 for all k ≥ k0 . Moreover, if {fk }k≥k0 ⊂ Int X + is a bounded and uniformly positive sequence, then the unique bounded solution {e xk }k≥k0 of equation (33) also is uniformly positive. Proof. (i) From (i) → (ii) of Theorem 3.2, we deduce that for all k ≥ k0 nP o j the series T ∗ (l, k)fl is absolutely convergent and there exists δ > 0 j≥k
l=k
independent of k and j such that j X ∗ (36) T (l, k)fl ≤ δ. ξ
l=k
Set x ek = lim
j P
j→∞ l=k
T ∗ (l, k),
∗ f = Tl,k l
∞ P
T ∗ (l, k)fl . Taking into account the definition of
l=k
we obtain x ek = fk + L∗k
∞ P l=k+1
ek+1 , which T ∗ (l, k + 1)fl = fk + L∗k x
shows that {e xk }k≥k0 solves (33). It follows from (36) that {e xk } is a bounded solution of (33). Let {b xk }k≥k0 be another bounded solution of equation (33). For each 0 ≤ k < j we may write (37)
x bk = T ∗ (j + 1, k)b xj+1 +
j X
T ∗ (l, k)fl .
l=k
Since {Lk }k≥k0 generates an exponentially stable evolution and {b xk }k≥k0 is a bounded sequence, we have lim T ∗ (j + 1, k)b xj+1 = 0. Letting j → ∞ in (37), j→∞
we conclude that x bk =
∞ P l=k
T ∗ (l, k)fl = x ek , which proves the uniqueness of the
bounded solution of equation (33). (ii) If {Lk }k≥k0 , {fk }k≥k0 are periodic sequences with period θ, then in a standard way, using the representation formula (34), one shows that the
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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 periodic sequences with period θ = 1. Based on the above result (ii) one obtains that the unique bounded solution of equation (33) also is periodic with period θ = 1, so it is constant. In this case, x e will verify the equation x e = L∗ x e + f . Since the operator L generates an exponentially stable evolution, we have ρ(L) < 1. Hence the operator IX − L∗ is invertible and we deduce that x e is given by (35). Finally, if Lk are positive operators, the assertions of (iv) follow immediately from the representation formula (34) and thus the proof is complete. Remark 3.1. From the representation formula (18) one obtains that if the sequence {Lk }k≥k0 generates an exponentially stable evolution and {fk }k≥k0 is a bounded sequence, then all solutions of the discrete time forward affine equation (14) with given initial values at time k = k0 are bounded on the interval [k0 , ∞). On the other hand, it follows from Theorem 3.3 (i) that the discrete time backward equation (15) has a unique bounded solution on the interval [k0 , ∞), which is the solution provided by the formula (34). In the case where k0 = −∞, with the same techniques as in the proof of Theorem 3.3, we may obtain a result concerning the existence and uniqueness of the bounded solution of a forward affine equations similar to that proved for the case of backward affine equations. Theorem 3.4. Assume that {Lk }k∈Z is a sequence of linear operators which generates an exponentially stable evolution on X . Then the following assertions hold: (i) for each bounded sequence {fk }k∈Z the discrete time forward affine equation xk+1 = Lk xk + fk
(38)
has a unique bounded solution {b xk }k∈Z . Moreover, this solution has a representation formula (39)
x bk =
k−1 X
T (k, l + 1)fl ,
∀ k ∈ Z;
l=−∞
(ii) if {Lk }k∈Z , {fk }k∈Z are periodic sequences with period θ, then the unique bounded solution of equation (38) is periodic with period θ; (iii) if Lk = L, fk = f , k ∈ Z then the unique bounded solution of equation (38) is constant and is given by x b = (IX − L)−1 f ;
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(iv) if {Lk }k∈Z are positive operators and {fk }k∈Z ⊂ X + , then the unique bounded solution of equation (38) satisfies x bk ≥ 0 for all k ∈ Z. Moreover, if fk 0, k ∈ Z, then x bk 0, k ∈ Z. If {Lk }k∈Z is a sequence of linear operators on X we may associate a new sequence of linear operators {L# k }k∈Z defined by L# k = L∗−k . Lemma 3.1. Let {Lk }k∈Z be a sequence of bounded linear operators on X . The following assertions hold: (i) if T # (k, l) is the causal linear evolution operator on X defined by the sequence {L# k }k∈Z , then T # (k, l) = T ∗ (−l + 1, −k + 1), where T (i, j) is the causal linear evolution operator defined on X by the sequence {Lk }k∈Z ; (ii) {L# k }k∈Z is a sequence of positive linear operators if and only if {Lk }k∈Z is a sequence of positive linear operators; (iii) the sequence {L# k }k∈Z generates an exponentially stable evolution if and only if the sequence {Lk }k∈Z generates an exponentially stable evolution; (iv) the sequence {xk }k∈Z is a solution of the discrete time backward affine equation (33) if and only if the sequence {yk }k∈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 existence of the bounded solution of some suitable forward affine equation. Theorem 3.5. Let {Lk }k∈Z be a sequence of positive bounded linear operators on X . Then the following assertion are equivalent: (i) the sequence {Lk }k∈Z generates an exponentially stable evolution; P (ii) for each k ∈ Z the series l≤k T (k, l)ξ is convergent and there exists δ > 0, independent of k, such that k X
T (k, l)ξ ≤ δξ,
∀ k ∈ Z;
l=−∞
(iii) the forward affine equation (40)
xk+1 = Lk xk + ξ
has a bounded and uniformly positive solution; (iv) for any bounded and uniformly positive sequence {fk }k∈Z ⊂ Int X + , the corresponding forward affine equation (41)
xk+1 = Lk xk + fk
has a bounded and uniformly positive solution;
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(v) there exists a bounded and uniformly positive sequence {fk }k∈Z ⊂ Int X + such that the corresponding forward affine equation (41) has a bounded solution x ek , k ∈ Z ⊂ X + ; (vi) there exists a bounded and uniformly positive sequence {yk }k∈Z which verifies yk+1 − Lk yk 0. The proof follows immediately by combining the result proved in Theorem 3.2 and Lemma 3.1. 4. SOME ROBUSTNESS RESULTS In this section we prove some results which provide a “measure” of the robustness of the exponential stability in the case of positive linear operators. To state and prove this result some preliminary remarks are needed. So, `∞ (Z, X ) stands for the real Banach space of bounded sequences of elements of X . If x ∈ `∞ (Z, X ), we denote |x| = sup |xk |ξ . k∈Z
Let `∞ (Z, X + ) ⊂ `∞ (Z, X ) be the subset of bounded sequences {xk }k∈Z ⊂ + X . It can be checked that `∞ (Z, X + ) is a solid, closed convex cone. Therefore, `∞ (Z, X ) is an ordered real Banach space for which the assumptions of Theorem 2.11 in [8] are fulfilled. Now we are in a position to prove Theorem 4.1. Let {Lk }k∈Z , {Gk }k∈Z be sequences of positive bounded linear operators such that {Gk }k∈Z is a bounded sequence. Under these assumptions, the following assertions are equivalent: (i) the sequence {Lk }k∈Z generates an exponentially stable evolution and ρ[T ] < 1, where ρ[T ] is the spectral radius of the operator T : `∞ (Z, X ) → `∞ (Z, X ), by (42)
y = T x,
k−1 X
yk =
T (k, l + 1)Gl xl ,
l=−∞
where T (k, l) is the linear evolution operator on X defined by the sequence {Lk }k∈Z ; (ii) the sequence {Lk +Gk }k∈Z generates an exponentially stable evolution on X . Proof. (i) → (ii) If the sequence {Lk }k∈Z defines an exponentially stable evolution, then we define the sequence {fk }k∈Z by (43)
fk =
k−1 X l=−∞
T (k, l + 1)ξ.
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Discrete time linear equations
We have fk = ξ +
k−2 P
149
Tk,l+1 ξ which leads to fk ≥ ξ thus fk ∈ Int X + for all
l=−∞
k ∈ Z. This allows us to conclude that f = {fk }k∈Z ∈ Int `∞ (Z, X + ). Applying Theorem 2.11 [8] with R = −I`∞ and P = T we deduce that there exists x = {xk }k∈Z ∈ Int `∞ (Z, X + ) which verifies the equation (I`∞ − T )(x) = f.
(44)
Here, I`∞ stands for the identity operator on `∞ (Z, X ). Partitioning (44) and taking into account (42)–(43), we obtain that for each k ∈ Z we have xk+1 =
k X
k X
T (k + 1, l + 1)Gl xl +
l=−∞
T (k + 1, l + 1)ξ.
l=−∞
Further, we may write k−1 X
xk+1 = Gk xk +ξ+Lk
k−1 X
T (k, l+1)Gl xl +Lk
l=−∞
T (k, l+1)ξ = Gk xk +ξ+Lk xk .
l=−∞
This shows that {xk }k∈Z verifies the equation xk+1 = (Lk + Gk )xk + ξ.
(45)
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 + Gk }k∈Z generates an exponentially stable evolution. Now, we prove the converse implication. If (ii) holds then using the implication (i) → (iii) of Theorem 3.5, we deduce that equation (45) has a bounded and uniform by positive solution {e xk }k∈Z ⊂ Int X + . Equation (45) verified by x ek may be rewritten as x ek+1 = Lk x ek + fek ,
(46)
where fek = Gk x ek + ξ, k ∈ Z, fek ≥ ξ, k ∈ Z. Using the implication (v) → (i) of Theorem 3.5, we deduce that the sequence Lk generates an exponentially stable evolution. Since equation (46) has unique bounded solution which is k−1 P given by the representation formula (39), we have x ek = T (k, l + 1)fel , l=−∞
k ∈ Z, so that (47)
x ek =
k−1 X l=−∞
T (k, l + 1)Gl x el +
k−1 X l=−∞
T (k, l + 1)ξ.
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Invoking (42), equation (47) may be written as x e=Tx e + ge,
(48) where ge = {e gk }k∈Z , gek =
k−1 P
T (k, l + 1)ξ. It is obvious that gek ≥ ξ for all
l=−∞ Int `∞ (Z, X + ).
k ∈ Z. Hence ge ∈ 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. Assume that 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 operator defined by the sequence {Lt }t∈Z . As a consequence of the above equality, one gets T (nθ, 0) = T (θ, 0) for all n ≥ 0. Thus, if the sequence {Lt }t∈Z generates an E.S. evolution then, ρ[T (θ, 0)] < 1. In this case, an operator valued function is well defined by G : {0, 1, . . . , θ} × {0, 1, . . . , θ − 1} → B(X ), with (49)
G(t, s) = T (t, 0)(IX − T (θ, 0))−1 T (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))−1 T (θ, s + 1),
0 ≤ s ≤ θ − 1,
where χt−1 (s) is the indicator function of the set {1, 2, . . . , t − 1}. It is easy to check that (50)
G(0, s) = G(θ, s),
(∀) 0 ≤ s ≤ θ − 1.
In the special case θ = 1 (i.e., the time invariant case) (49) reduces to (51)
G(1, 0) = G(0, 0) = (IX − L)−1 .
Let X θ = X ⊕ X ⊕ · · · ⊕ X (θ times). The elements of this space are finite sequences of the form x = (x0 , x1 , . . . , xθ−1 ), xi ∈ X , 0 ≤ i ≤ θ − 1. On X θ we introduce the norm |x|θ = max{|xi |ξ , 0 ≤ i ≤ θ − 1}. The space X θ is an ordered Banach space with the norm | · |θ and the ordered relation induced by the closed solid normal convex cone X +θ = X + ⊕ · · · ⊕ X + . Consider the operator Π : X θ → X θ defined by y = Πx, where y = (y0 , y1 , . . . , yθ−1 ), (52)
yt =
θ−1 X
G(t, s + 1)Ps xs ,
s=0
0 ≤ t ≤ θ − 1 for all x = (x0 , x1 , . . . , xθ−1 ) ∈ X θ .
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151
Remark 4.1. a) It is obvious that (52) may be extended to t = θ by θ−1 P yθ = G(θ, s + 1)Ps xs . From (50) we deduce that yθ = y0 . This allows us s=0
to extend the finite sequence defined by (52) to an periodic infinite sequence with period θ. b) In the special case θ = 1, X θ coincides with X . Using (51) and (52) we obtain that in this case Πx = (IX − L)−1 P x, for all x ∈ X . Lemma 4.1. Assume that a) the sequences {Lt }t∈Z , {Pt }t∈Z are periodic with period θ ≥ 1; b) Lt ≥ 0, Pt ≥ 0, t ∈ Z; c) {Lt }t∈Z defines an E.S. evolution. Under these asssumptions, the operator Π defined by (52) is a positive bounded linear operator. Proof. The fact that Π is a bounded linear operator is obvious from its definition. It remains to prove that Π ≥ 0. Let x b = (b x0 , x b1 , . . . , x bθ−1 ) ∈ X +θ and yb = Πb x. Construct sequences {e xt }t∈Z , {e yt }t∈Z by x et = x bs , yet = ybs if t = kθ + s, 0 ≤ s ≤ θ − 1. It is obvious that x et+θ = x et , yet+θ = yet , for all t ∈ Z. If we consider the periodicity of Pt , we get that {e yt }t∈Z is a periodic solution of the equation (53)
yet+1 = Lt yet + zet ,
where zet = Pt x et . Since zt ≥ 0, we conclude via Theorem 3.4 (iv) applied to equation (53), that yet ≥ 0 and the proof is complete. The analogue of Theorem 4.1 in the periodic case is Theorem 4.2. Let {Lt }t∈Z , {Pt }t∈Z be sequences of bounded linear operators on X with the properties a) there exists an integer θ ≥ 1 such that Lt+θ = Lt and Pt+θ = Pt for all t ∈ Z; b) Lt ≥ 0, Pt ≥ 0, t ∈ Z. Under these assumptions the following assertions are equivalent: (i) the sum sequence {Lt + Pt }t∈Z generates an E.S. evolution; (ii) the sequence {Lt }t∈Z generates an E.S. evolution and ρ(Π) < 1, where Π is the linear operator defined by (52). Proof. If (i) holds, then using (i) → (iii) of Theorem 3.5 and (ii) of Theorem 3.4 we deduce that the forward affine equation xt+1 = [Lt + Pt ]xt + ξ has a bounded and uniform positive solution {e xt }t∈Z , which is periodic with period θ.
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It can be seen that {e xt }t∈Z solves the equation x et+1 = Lt x et + ft , where ft = Pt x et + ξ. Invoking the implication (v) → (i) of Theorem 3.5, we conclude that {Lt }t∈Z generates an E.S. evolution. Using the representation formula of a periodic solution of a discrete time affine equation we obtain (54)
x et =
θ−1 X
G(t, s + 1)(Ps x es + ξ),
0 ≤ t ≤ θ − 1.
s=0
Set x b = (b x0 , x b1 , . . . , x bθ−1 ) ∈ X θ , where x bt = x et , 0 ≤ t ≤ θ − 1. From (54) one gets that x b solves the equation (55)
(−IX + Π)b x + gb = 0,
where gb = (b g0 , gb1 , . . . , gbθ−1 ) with gbt =
θ−1 P s=0
G(t, s + 1)ξ. It can be seen that gb is
a restriction of the periodic solution of the forward affine equation (56)
get+1 = Lt get + ξ.
Applying Theorem 3.4 (iv), we conclude that get > 0. Thus we deduce that gb ∈ Int X +θ . Invoking the implication (v) → (vi) of Theorem 2.11 in [8] to equation (55) for R = −IX , P = Π, one concludes that ρ(Π) < 1. So we obtain that (ii) holds. Now, we prove the converse implication. If (ii) holds, then from Theorem 3.4 we deduce that equation (56) has a periodic solution {e gt }t∈Z ⊆ Int X + . If gb = (b g0 , gb1 , . . . , gbθ−1 ) is such that gbt = get , 0 ≤ t ≤ θ − 1, then gb ∈ Int X +θ . Invoking again Theorem 2.11 [8] we conclude that equation (55) has a solution x b = (b x0 , x b1 , . . . , x bθ−1 ), x b ∈ Int X +θ . Construct the sequence {e xt }t∈Z by x et = x es if t = kθ + s, 0 ≤ s ≤ θ − 1, k ∈ Z. One can see that {e xt }t∈Z is a periodic sequence of period θ. Also, one obtains that {e xt }t∈Z is a periodic solution of the equation xt+1 = (Lt +Pt )xt +ξ. Applying Theorem 3.5, we conclude that {Lt + Pt }t∈Z generates an E.S. evolution, thus the proof is complete. Using Remark 4.1 b), we deduce that in the time invariant case the result proved in Theorem 4.2 becomes Corollary 4.1. If L, P are two positive bounded linear operators on X , then the following assertions are equivalent: (i) L + P defines an E.S. evolution; (ii) ρ[L] < 1, ρ[(IX − L)−1 P ] < 1.
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5. THE CASE OF DISCRETE TIME LINEAR EQUATIONS DEFINED BY POSITIVE OPERATORS ON ORDERED BANACH SPACES In this section, X is a real Banach space ordered by the order relation “ ≤ ” induced by the closed, solid, pointed convex cone X + . Throughout this section we assume that the by hypothesis H1 below holds: H1 The norm k · k of the Banach space X is monotone with respect to the cone X + . Let ξ ∈ Int X + be fixed. According to Proposition A2, the set Bξ defined by (57)
Bξ = {x ∈ X | −ξ < x < ξ}
is bounded. Using Proposition A.2 and Theorem A.2 we deduce that the corresponding Minkovski functional | · |ξ is a norm equivalent to the norm k · k of the Banach space X . Properties P1 , P2 stated in Section 2 for an Hilbert space are still valid in the case of a Banach space. Also, Proposition 2.1 (ii) is valid in the context of an ordered Banach space. Example 5.1. Let X = Sn∞ , where Sn∞ 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 Sn∞ equipped with the norm (59)
kSk∞ = sup{|S(i)|, i ≥ 1}
is a real Banach space. On Sn∞ we consider the ordered relation induced by the solid, closed, pointed, convex cone Sn∞+ with Sn∞+ = {(S(1), S(2), . . .) ∈ Sn∞ | S(i) ≥ 0, i ≥ 1}. Obviously, H1 holds. Take ξ = (In , In , . . .) ∈ Sn∞+ . In this case, (57) becomes (60)
Bξ = {S = (S(1), S(2), . . .) ∈ Sn∞ | −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 ∈ Sn∞ . For S ∈ Int Sn∞+ 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 3.2 for the case of constant sequences.
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Let us consider a sequence of positive bounded linear operators {Lt }t∈Z on X . If X is a Banach space, it is not clear if one can characterize the exponential stability of the zero solution of the discrete time linear equation xt+1 = Lt xt
(62)
in terms of the existence of a bounded and uniformly positive solution of some suitable forward affine equations as well as in terms of the existence of some bounded and uniformly positive solution of some suitable backward affine equations, as in the Hilbert space case. Now, we show that in the case of a Banach space one can prove a result as in Theorem 3.5 to characterize the exponential stability in the case of equations (60). Theorem 5.1. Let X be a real Banach space ordered by the ordered relation induced by the closed, solid, pointed convex cone X + . Assume that H1 is fulfilled. Let {Lt }t∈Z be a sequence of positive bounded linear operators on X . Then the following assertions are equivalent: (i) the zero state equilibrium ofP (62) is E.S.; T (t, l)ξ is convergent and there exists (ii) for each t ∈ Z the series l≤t
δ > 0 independent of t such that (63)
t X
T (t, l)ξ ≤ δξ
l=−∞
for all t ∈ Z; (iii) the forward affine equation (64)
xt+1 = Lt xt + ξ
has a bounded and uniform positive solution; (iv) for any bounded and uniformly positive sequence {ft }t∈Z ⊂ Int X + the corresponding forward affine equation: (65)
xt+1 = Lt xt + ft
has a bounded and uniformly positive solution; (v) there exists a bounded and uniformly positive sequence (66)
{ft }t∈Z ⊂ Int X +
such that the corresponding forward affine equation (65) has a bounded solution x et , t ∈ Z ⊂ X + ; (vi) there exists a bounded and uniformly positive sequence {yt }t∈Z which verifies (67)
yt+1 − Lt yt 0,
t ∈ Z.
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The proof follows the same lines as in the proof of Theorem 3.2 and is omitted. As we have seen is Subsection 2.2, a sequence {Lt }t≥t0 may also define a backward discrete time linear equation or, equivalently, an anticausal evolution. This is (68)
xt = Lt xt+1 .
Concerning equation (68) we introduce Definition 5.1. We say that the zero state equilibrium of equation (68) is anticausal exponentially stable (A.E.S., for short) or, equivalently, the sequence {Lt }t≥t0 generates an A.E.S. evolution if there exists β ≥ 1, q ∈ (0, 1) such that (69)
kT a (t, s)kξ ≤ βq s−t
for all t0 ≤ t ≤ s, t, s ∈ Z, where T a (t, s) is the anticausal linear evolution operator associated to (68). It must be remarked that in (69) we may use any norm equivalent to the given norm of the Banach space X . Concerning the characterization of the anticausal exponential stability one proves: Theorem 5.2. Let {Lt }t≥t0 be a sequence of linear bounded and positive operators on the ordered Banach space X . If H1 holds, the following assertions are equivalent: (i) the sequence {Lt }t≥t0 generates P aan A.E.S. evolution; (ii) for each t ≥ t0 the series T (s, t)ξ is convergent and there exists s≥t
δ > 0, independent of t, such that (70)
∞ X
T a (t, s)ξ ≤ δξ
s=t
for all t ≥ t0 ; (iii) the discrete time backward affine equation (71)
xt = Lt xt+1 + ξ
has a bounded and uniformly positive solution; (iv) for an arbitrary bounded and uniform by positive sequence {ft }t≥t0 , the backward affine equation (72)
xt = Lt xt+1 + ft ,
t ≥ t0
has a bounded and uniformly positive solution;
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(v) there exists a bounded and uniformly positive sequence {ft }t≥t0 such that the corresponding backward affine equation (72) has a bounded solution {e xt }t≥t0 ⊂ X + ; (vi) there exists a bounded and uniformly positive sequence {yt }t≥t0 which verifies (73)
Lt yt+1 − yt 0,
t ≥ t0 .
The proof may be done by following step by step the proof of Theorem 3.2, replacing the operator T ∗ (t, s) by T a (t, s) and L∗t by Lt . In the last part of this section we illustrate the applicability of Theorem 5.2 to characterize the exponential stability in mean square in the case of discrete-time stochastic linear systems perturbed by a Markov chain with an infinite number of states. Consider the discrete-time stochastic linear system (74)
xt+1 = A(t, ηt )xt ,
t ≥ 0,
where xt ∈ Rn and {ηt }t≥0 is a Markov chain with an infinite but countable set of states on a given probability space (Ω, F, P) and with transition probability matrices Pt . This means (see [10, 23]) that for each t ≥ 0, t ∈ Z, ηt : Ω → Z1 is a random variable with the property (75)
P{ηt+1 = j|Gt } = pt (ηt , j) a.s.,
where Gt = σ[η0 , η1 , . . . , ηt ] is the σ algebra generated by {ηs }0≤s≤t , and Z1 = {i ∈ Z, i ≥ 1}. Setting Pt = (pt (i, j))i,j∈Z1 , pt (i, j) being the scalars on the right hand side of (75), one obtains a sequence of stochastic matrices with an infinite number of rows and columns. To avoid some complications due to the discrete time context (see [14] for the case of systems perturbed by a Markov chain with a finite number of states) we make the assumption 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) = P{ηt = 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 P πt (j) = 1 for all t ∈ Z+ . One verifies inductively that under assumption j∈Z1
H2 , we have πt (j) > 0 for all t ≥ 1 and j ≥ 1, if π0 (i) > 0 for all i ≥ 1. In what follows we assume that π0 (i) > 0 for all i ≥ 1. Concerning the matrix coefficients A(t, i), t ≥ 0, i ≥ 1, of the system (74) we make the assumption H3 (76)
sup |A(t, i)| < ∞, i≥1
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where | · | is the norm induced by the Euclidean norm on Rn . Using the matrices A(t, i) we define the linear operators Lt : Sn∞ → Sn∞ , Lt S = (Lt S(1), Lt S(2), . . .) by X pt (i, j)AT (t, i)S(j)A(t, i) (77) Lt S(i) = j∈Z1
for all i ≥ 1, S = (S(1), S(2), . . .) ∈ Sn∞ . It is clear that Lt ≥ 0. Using (76) one obtains that kLt kξ = sup |AT (t, i)A(t, i)| < ∞. Therefore, Lt is a bounded i∈Z1
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 solution of (74) verifies x(t) = Φ(t, s)x(s), t ≥ s ≥ 0. The next result establishes a relationship between the fundamental matrix solution of system (74) and the 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), . . .) ∈ Sn∞ , where E[·|ηt = i] is the conditional expectation with respect to the event {ηt = i} and T a (t, s) is the anticausal linear evolution operator defined by Lt . The proof is similar to that for the case of systems perturbed by Markov chains 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 a discrete time stochastic linear system. Here we focus our attention on Definition 5.2. We say that the zero state equilibrium of system (74) is exponentially stable in mean square (ESMS – for short) if there exist β ≥ 1 and q ∈ (0, 1) such that for any Markov chain ({ηt }t≥0 , {Pt }t≥0 , Z1 ) with the above properties we have (80)
E[|Φ(t, s)x|2 |ηs = i] ≤ βq t−s |x|2
for all t ≥ s ≥ 0, i ∈ Z1 , x ∈ Rn .
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Combining Definition 5.2 and Corollary 5.1 we obtain Corollary 5.2. Under assumptions H2 and H3 the following assertions are equivalent: (i) the zero state equilibrium of the system (74) is ESMS; (ii) the sequence {Lt }t≥0 defined by (77) generates an A.E.S. evolution. Finally, combining Corollary 5.2 and Theorem 5.2 we obtain the following characterization of exponential stability in mean square for systems of type (74). Theorem 5.4. Under assumptions H2 and H3 the following assertions are equivalent: (i) the zero state of (74) is ESMS; P equilibrium (ii) Xt (i) = pt (i, j)AT (t, i)Xt+1 (j)A(t, i) + In , i ∈ Z1 , t ≥ 0, the j∈Z1
et = system of backward affine equations has a bounded and positive solution X ∞ e e (Xt (1), Xt (2), . . .) ∈ Sn , t ∈ Z1 ; (iii) there exists a bounded sequence {Xt }t≥0 , Xt = (Xt (1), Xt (2), . . .) ∈ Sn∞ and scalars α > 0, δ > 0 such that Xt (i) ≥ δIn and X
pt (i, j)AT (t, i)Xt+1 (j)A(t, i) − Xt (i) ≤ −αIn ,
i ∈ Z1 , t ≥ 0.
j∈Z1
6. APPENDIX In the first part of this section we recall several definitions concerning convex cones, and ordered linear spaces, and provide some basic results. In the second part of this section we investigate the properties of the Minkovski functional and provide conditions under which such a functional becomes a norm. The Minkovski norm play a crucial role in the developments in this paper. 6.1. Convex cones Let (X , k · k) be e real normed linear space. As usual, if X is a Hilbert space we shall use | · |2 instead of k · k. 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.
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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 often say convex cone when we refer to a cone. A cone C ⊂ X induces an ordering “ ≤ ” on X by x ≤ y (or equivalently y ≥ x) if and only if y − x ∈ C. If C is a solid cone, then x < y (or equivalently y > x) if and only if y − x ∈ Int C. Hence C = {x ∈ X | x ≥ 0} and Int C = {x ∈ X | x > 0}. Definition A2. If C ⊂ X is a cone, then C ∗ ⊂ X ∗ is called the dual cone of C if C ∗ consists of all bounded and linear functionals f ∈ X ∗ with the property that f (x) ≥ 0 for all x ∈ C. Based on Ritz theorem for representation of a bounded linear functional on a Hilbert space one sees that if X is a real Hilbert space, then the dual cone C ∗ of a convex cone C may be defined as C ∗ = {y ∈ X | hy, xi ≥ 0, ∀ x ∈ C}, where h· , ·i 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 convex cone. Then C ∗ is a closed and pointed cone. Proof. Let ϕ ∈ C¯∗ . There exists a sequence {ϕk }k≥1 ⊂ C ∗ such that lim ϕk (x) = ϕ(x) for all x ∈ X . For x ∈ C we have ϕ(x) = lim ϕk (x) ≥ 0.
k→∞
k→
Hence ϕ ∈ C ∗ , so C ∗ is a closed set. To show that C ∗ is a pointed cone, we choose ϕ ∈ C ∗ ∩ (−C ∗ ). This leads to ϕ(x) = 0 for all x ∈ C. We have to show that 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. Since ϕ(ξ) = 0, we conclude that ϕ(x0 ) = 0. Thus we obtain that C ∗ ∩ (−C ∗ ) = {0} and the proof is complete. In the finite dimensional case we have Proposition A1. Let X be a finite dimensional real Banach space and C ⊂ 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 follows from the Lemma A1. It remains to show that Int C ∗ is not empty. Applying Theorem 2.1 [29], we deduce that there exists ϕ0 ∈ C ∗ such that ϕ0 (x) > 0 for all x ∈ C\{0}. Since X is a finite dimensional linear space, S1 = {x ∈ C | kxk = 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 ∗ | kϕ − ϕ0 k ≤ δ}. We show that B(ϕ0 , δ) ⊂ C ∗ . If ϕ ∈ B(ϕ0 , δ) then |ϕ(x) − ϕ0 (x)| ≤ δ, ∀ x ∈ X ,
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with kxk = 1. Particularly, for x ∈ S1 we have ϕ(x) − ϕ0 (x) ≥ −δ. Hence ϕ(x) = ϕ0 (x) + (ϕ(x) − ϕ0 (x)) ≥ 2δ − δ = δ. Thus, we have proved that ϕ(x) ≥ δkxk for all x ∈ C \ {0} and for all ϕ ∈ B(ϕ0 , δ). Hence B(ϕ0 , δ) ⊂ C ∗ , which means that ϕ0 ∈ Int C ∗ , thus the proof 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. Denote by 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, 1t 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 the set Bξ is defined by (82)
|x|ξ = inf T (x)
for every x ∈ X . The next theorem collects several important properties of Minkovski functional Theorem A1. The Minkovski functional introduced in (2.2) has the following 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)kxk, (∀) x ∈ X ; (vi) |x|ξ = 1 if and only if x ∈ ∂Bξ , where ∂Bξ is the border of the set Bξ ; ¯ξ , where B ¯ξ = Bξ ∪ ∂Bξ ; vii) |x|ξ ≤ 1 iff x ∈ B ¯ξ = {x ∈ X | −ξ ≤ x ≤ ξ}; (viii) if C is a closed, solid convex cone, then B (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 from the 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 .
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(ii) Let α > 0 be fixed. It is easy to see that t ∈ T (αx) iff α−1 t ∈ 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 exists t ∈ (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 → 1t x, there exists t1 ∈ (0, 1) such that t11 x ∈ 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 (83) (84)
τ1 + τ2 = τ, τ1 > |x|ξ , τ2 > |y|ξ τ1 τ2 1 (x + y) = x1 + y1 , τ τ τ
where x1 = τ11 x, y1 = τ12 y. From (83) and (iii) we deduce that x1 , y1 ∈ Bξ . Since Bξ is a convex set and ττ1 + ττ2 = 1 using (84), we get, τ1 (x + y) ∈ Bξ . Invoking again (iii) we have | τ1 (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 | kx − ξk ≤ δ(ξ)}. Let x ∈ X , x 6= 0. Since x x ξ ± δ(ξ) kxk > 0, we obtain δ(ξ)x kxk ∈ Bξ . From (iii) we have kδ(ξ) kxk |ξ < 1. Using (ii) we deduce |x|ξ < β(ξ)kxk 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 sequence {tk }k≥1 with tk > 1, lim tk = 1, and t1k x ∈ Bξ . From x = lim t1k x we deduce k→∞ k→∞ ¯ξ . Using (iii), x 6∈ Bξ . It follows that x ∈ ∂Bξ . x∈B To prove the converse inclusion, choose x ∈ ∂Bξ and assume that |x|ξ > ε }, where 1. Set ε = |x|ξ −1. Let V be the open ball, V = {y ∈ X | ky−xk < β(ξ) β(ξ) is the constant from (v). Since x ∈ ∂Bξ there exist y ∈ Bξ ∩ V , which ε means that ky −xk < 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)kx − yk < ε, which is a 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).
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¯ξ . From (vii) we have |x|ξ ≤ 1. If |x|ξ < 1 then x ∈ Bξ , (viii) Let x ∈ B which is equivalent to −ξ < x < ξ implying −ξ ≤ x ≤ ξ. If |x|ξ = 1 then there exists a sequence {tk }k≥1 , tk > 1, lim tk = 1 and t1k x ∈ Bξ . This means that k→∞
ξ ± t1k x ∈ C. Letting k → ∞ and taking into account that C is a closed set we conclude that ξ ± x ∈ C, which is equivalent to −ξ ≤ x ≤ ξ. Conversely, if x ∈ X is such that −ξ ≤ x ≤ ξ, then for all t > 1 we have −tξ < x < tξ. This ¯ξ . means that 1t x ∈ Bξ . Letting t → 1 we get x ∈ B (ix) follows from (vi). (x) Let t ∈ (|x|ξ , ∞). This is equivalent to | 1t x|ξ < 1. Hence 1t x ∈ Bξ . This means that t ∈ T (x). Thus, we have proved that (|x|ξ , ∞) ⊂ T (x). To prove the converse inclusion, we choose t ∈ T (x), that is, 1t 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 that max{|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 < 1t y ≤ 1t x ≤ 1t z < ξ and implies t ∈ T (x). Invoking again (x) one deduces that t > |x|ξ which contradicts (85), thus the proof is complete. It follows from (i), (ii) and (iv) that the Minkovski functional defined by (82) is a semi-norm. The next theorem provides a condition which ensures that the seminorm (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 kxk ≤ αξ |x|ξ for all x ∈ X . Proof. To prove that | · |ξ is a norm, we have to show that if |x|ξ = 0 then x = 0. If |x|ξ = 0 then from (x) of Theorem 2.1 we have T (x) = (0, ∞). Hence for all t > 0 we have 1t x ∈ Bξ . Since Bξ is a bounded set, we have k 1t xk ≤ α, with α > 0 not depending on x and t, but possible, depending on ξ. This leads to kxk ≤ αt. Letting t → 0, one obtains that kxk = 0, hence x = 0. To check the last assertion in the statement, we choose x ∈ X , x 6= 0. Invoking again (x), we obtain that for all t ∈ (1, ∞), 1t |x|xξ ∈ Bξ . From the boundedness of Bξ we deduce that kxk ≤ α|x|ξ , thus the proof is complete. Corollary A1. Assume that X is a finite dimensional real Banach space. Assume also that C ⊂ X is a solid cone, C 6= X . If ξ ∈ Int C then the following assertionsare equivalent: (i) the Minkovski semi-norm |x|ξ is a norm;
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(ii) Bξ is a bounded set. Proof. (ii) → (i) follows from Theorem 2.2. Suppose (i) holds. Since X is a finite dimensional Banach space, there exists α > 0 (depending on ξ) such that kxk ≤ α|x|ξ for all x ∈ X . Now, if x ∈ Bξ then |x|ξ < 1, hence kxk < α, that is, Bξ is a bounded set. In the sequel we provide a sufficient condition which guarantees that | · |ξ is a norm for all ξ ∈ Int C. To this end we introduce Definition A3. We say that the k · k is monotone with respect to the cone C if 0 ≤ x ≤ y implies that kxk ≤ kyk. Proposition A2. If k · k is monotone with respect to the cone C, then for 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 kx + ξk ≤ 2kξk. This leads to kxk ≤ 3kξk, thus the proof is complete. Further we prove Proposition A3. If X is a real Hilbert space and C ⊂ X is a cone, then the 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 + k1 y for all y ∈ C and k ≥ 1. If (i) is fulfilled, then |x|22 ≤ |x + k1 y|22 , which is equivalent to 2hx, yi ≥ − k1 |y|22 . Letting k → ∞, we obtain that hx, yi ≥ 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 hy − x, y + xi ≥ 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 is monotone 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 A3 that h−x, xi ≥ 0, which leads to hx, xi = 0. Hence x = 0. This shows that C ∩ (−C) = {0}. The next two examples show that the monotonicity of the norm k · k is only a sufficient condition that Bξ be a bounded set for all ξ ∈ Int C.
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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 other hand, 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 on R2 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 other hand, 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 Euclidian norm on R3 is not monotone with respect to the cone (2.7). Acknowledgement. This work was partially supported by Project CEx05-1123/2005, CEEX Program of the Romanian Ministry of Education and Research.
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