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 sto- chastic 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 nonsym- metric 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 stabi- lity, 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, equiva- lently, 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
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 prob- lem 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 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 and Hinrichsen [7, 8]. Similar results were derived also for the discrete-time time- invariant 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 dif- ferential equations on the space of n×nsymmetric matrices Sn is studied in [11]. Such equations have the property that the corresponding linear evolution operator is positive onSn. They contain as particular cases linear differential equations of Lyapunov type arising in connection with the problem of investi- gation 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 time- invariant case are recovered as special cases of the results proved in this paper.
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 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 con- vex cone X+. SinceX+is a selfdual convex cone, it follows from Remark A.1 and Proposition A.3 (in Appendix) that X+ is a pointed cone.
By Proposition A.3, | · |2 defined by
(1) |x|2 = (hx, xi)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 is a norm on X. Moreover, from Theorem A.1 (v) and Theorem A.2 we deduce that| · |ξ is equivalent to| · |2defined by (1). Hence (X,| · |ξ) is a Banach space.
Moreover, | · |ξ has the properties below:
P1) Ifx, y, z∈ X are such that y≤x≤z, then (2) |x|ξ≤max{|y|ξ,|z|ξ}.
P2) For an arbitraryx∈ 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 kTk= sup|x|≤1|T x|is the corresponding operator norm.
Remark2.1. a) Since| · |ξ and| · |2 are equivalent,k · kξandk · k2 are also equivalent. This means that there are two positive constants c1 and c2 such that c1kTkξ ≤ kTk2 ≤c2kTkξ for all bounded linear operatorsT :X → X.
b) If T∗ :X → X is the adjoint operator of T with respect to the inner product on X, then kTk2 = kT∗k2. In general, the equality kTkξ = kT∗kξ is not true. However, follows from a) it follows that there are two positive constants ec1,ec2 such that
(4) ec1kTkξ≤ kT∗kξ≤ec2kTkξ.
Definition 2.1. Let (X,X+) and (Y,Y+) be ordered vector spaces. An operator T :X → Y is called positive ifT(X+) ⊂ Y+. In this case we write T ≥0. IfT(IntX+)⊂IntY+ we write T >0.
Proposition 2.1. If T :X → X is a bounded linear operator, then (i) T ≥0if and only if T∗≥0;
(ii)If T ≥0 thenkTkξ =|T ξ|ξ.
Proof. (i) is a direct consequence of the fact thatX+ is a selfdual cone.
(ii) IfT ≥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
|x|ξ≤1
|T x|ξ≤ |T ξ|ξ≤ sup
|x|ξ≤1
|T x|ξ,
hence kTkξ=|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 thenkT1kξ ≤ kT2kξ.
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 a closed, solid, pointed, selfdual convex cone. The ordering induced on Rn by
this cone is known as the component wise ordering. IfT :Rn→Rnis a linear operator, thenT ≥0 iff its corresponding matrixAwith 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 = max
1≤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) hA, Bi= Tr(BTA)
∀A, B ∈Rm×n, Tr(M) denoting as usual the trace of a matrixM. OnRm×n we consider the order relation induced by the coneX+=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
i,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 allY ∈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⊗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×nsymmetric matrices. Let X =Sn⊕ Sn⊕ · · · ⊕ Sn =SnN withN ≥1 fixed. On SnN, consider the inner product
(9) hX, Yi=
N
X
i=1
T r(YiXi)
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},
whose interior
IntSnN,+={X∈ SnN |Xi >0, 1≤i≤N}
is nonempty. HereXi≥0 andXi >0 means thatXi 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|ξ = max
1≤i≤N|Xi|, (∀)X= (X1, . . . , XN)∈ SnN, where |Xi| = max
λ∈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 ∈ SnN.
(iv) For an infinite dimensional case, let us consider X = `2(Z+,R) where `2(Z+,R) =
n
x = (x0, x1, . . . , xn, . . .) |xi ∈ R,
∞
P
i=0
x2i < ∞o
. On X we consider the usual inner product hx,yi`2 =
∞
P
i=0
xiyi for allx={xi}i≥0,y= {yi}i≥0. Set X+ = n
x = {xi}i≥0 | x0 ≥ 0,
∞
P
i=1
x2i ≤ x20o
. It is easy to see that X+ is a closed, pointed convex cone. In the finite dimensional case the analogue of this cone is known as a circular cone.
The interior IntX+ =n
x={xi}i≥0 |x0 >0,
∞
P
i=1
x2i < x20o
. It remains to prove that X+ is selfdual. Let y∈(X+)∗. Hence
(12) hx,yi`2 ≥0
for all x={xi}i≥0 ∈ X+. In particular, takingx={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+. Suppose now
∞
P
t=1
yt2 >0. Takexe={xei}i≥0 defined by (13) xe0 =y0, xei =−γyiy0 with γ =
∞ P
k=1
y2k −12
. Obviously, xe ∈ X+. Replacing (13) in (12), one gets
∞
P
k=1
yk2 ≤y20, which shows thaty∈ X+. Thus, it was shown that (X+)∗ ⊂ X+. Let now y = {yi}i≥0 ∈ X+. We have to show that (12) holds for all x ∈ X+. Indeed, for x ∈ X+ we have
∞
P
k=1
xkyk
2 ≤
∞
P
k=1
x2k
∞
P
k=1
y2k ≤ x20y02,
which leads to
∞
P
k=1
xkyk
≤x0y0. This is equivalent to −x0y0 ≤
∞
P
k=1
xkyk ≤ x0y0, which shows that (12) is fulfilled. Thus it was proved thatX+⊂(X+)∗. Hence X+ is selfdual.
Takeξ= (1,0,0, . . . ,0, . . .)∈ X+ and denote byB(ξ,
√2
2 ) the closed ball B(ξ,
√ 2
2 ) ={x ∈`2(Z+,R) | |x−ξ|2 ≤
√ 2
2 }. It is not difficult to check that B(ξ,
√ 2
2 )⊂ X+. Moreover, we haveX+={tx|t≥0, x∈ B(ξ,
√ 2 2 )}.
If x = (x0, x1, x2, . . . , xn, . . .) ∈ X, we write x = (x0,x) withb bx = (x1, x2, . . . , xn, . . .). The corresponding Minkovski norm is given by |x|ξ =
|x0|+|bx|2, where |bx|22 =
∞
P
i=1
x2i. This equality can be obtained taking into account that |x|ξ = inf{t >0| |x|b22− |x0|2−2tx0−t2 <0 and |bx|22− |x0|2+ 2tx0−t2 <0}.
2.2. Discrete-time affine equations
LetL={Lk}k≥k0 be a sequence of bounded linear operatorsLk :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 =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 eachk≥l≥k0 letTc(k, l) :X → X be the causal evolution operator defined by the sequence L, Tc(k, l) = Lk−1Lk−2. . .Ll if k > l and Tc(k, l) =IX ifk=l,IX being the identity operator on X.
For allk0 ≤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=LkLk+1. . .Ll−1 ifk < l and Ta(k, l) =IX ifk=l.
Often the superscripts a and c will be omitted if any confusion is not possible.
Let xek = Tc(k, l)x, k ≥ l, l ≥ k0 be fixed. One obtains that {exk}k≥l verifies the forward linear equation
(16) xk+1=Lkxk
with initial value xl =x. Also, if yk = Ta(k, l)y, k0 ≤k ≤ l, then from the definition of Tkla one obtains that {yk}k0≤k≤l is the solution of the backward
linear equation
(17) yk=Lkyk+1
with given terminal valueyl=y. Obviously, (16) and (17) lead toTc(k+1, l) = LkTc(k, l) for allk≥l≥k0andTa(k, l) =LkTa(k+1, l) for allk0 ≤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 valuesxl=x is well defined fork≥lwhile a solution{yk}k≤l of the backward linear equation (17) with given terminal condition yl =y is well defined fork0 ≤k≤l.
If for eachk, the operatorsLk are invertible, then all solutions of equa- tions (16), (17) are well defined for all k≥k0.
If (Tc(k, l))∗ is the adjoint operator of the causal evolution operator Tc(k, l), we define
zl= (Tc(k, l))∗z, (∀)k0≤l≤k.
By direct calculation one obtains that zl = L∗lzl+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 pos- itive evolution if for all k0 ≤ k ≤ l the anticausal linear evolution operator Ta(k, l)≥0.
Since Tc(l+ 1, l) = Ll, Ta(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 gen- erates 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 Tc(k, l).
The following result is straightforward. It will be used in the next sec- tions.
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 evolu- tion operators. Assume that 0≤ L1k≤ L2k for all k≥k0. Under this assump- tion we have T2c(k, l)≥T1c(k, l) for allk≥l≥k0.
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 =Tc(k, l)xl+
k−1
X
i=l
Tc(k, i+ 1)fi
for all k≥l+ 1. Also, any solution of the backward affine equation (15) has a representation
yk =Ta(k, l)yl+
l−1
X
i=k
Ta(k, i)fi, k0≤k≤l−1.
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=Lkxk
is exponentially stable or, equivalently, that the sequenceL={Lk}k≥k0 gene- rates an exponentially stable evolution (E.S. evolution) if there are β > 0, q ∈(0,1) such that
(20) kT(k, l)kξ ≤βqk−l, k≥l≥k0,
whereT(k, l) is the causal linear evolution operator defined by the sequenceL.
Remark 2.1 in (20) allows us to consider the normk · 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,whereρ[·] is the spectral radius.
It must be remarked that if the sequence {Lk}k≥k0 generates an expo- nentially 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.
Proposition 3.1. Let L = {Lk}k≥k0, and L1 = {L1k}k≥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 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;
(ii)there existsδ >0 such that Pk l=k0
kTk,lkξ≤δ for arbitrary k≥k0 ≥0;
(iii) there exists δ >0 such that
k
P
l=k1
T(k, l)ξ≤δξ for arbitraryk≥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 =Lkxk+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).
If (i) is true, then (ii) follows immediately from (20) withδ = 1−qβ . Let us prove that
(21) 0≤T(k, l)ξ ≤ kT(k, l)kξξ
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. IfT(k, l)ξ 6= 0 then from (3) applied to x= |T(k,l)ξ|1
ξT(k, l)ξ one gets 0≤T(k, l)ξ≤ |T(k, l)ξ|ξξ and (21) follows from 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|ξξ≤fl≤ |fl|ξξ, which leads to−µξ≤fl≤µξ for all l≥0.
Since for eachk≥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
X
l=0
T(k, l+ 1)ξ≤
k−1
X
l=0
T(k, l+ 1)fl≤µ
k−1
X
l=0
T(k, l+ 1)ξ.
Using (2) we deduce that
k−1
X
l=0
T(k, l+ 1)fl ξ
≤µ
k−1
X
l=0
T(k, l+ 1)ξ ξ
.
If (iii) is valid we conclude by using again (2) that
k−1
X
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 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 = 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 exponential stability we have to check the boundedness of the solution with zero initial value of the forward affine equationxk+1=Lkxk+fkfor an arbitrary bounded sequence {fk}k≥0 ⊂ X.
Let us now introduce the concept of uniform positivity.
Definition3.2. We say that a sequence{fk}k≥k0 ⊂ X+ isuniformly posi- tive if there exists c > 0 such thatfk > 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 fk0,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 ≤β1qk−l, (∀)k≥ l≥k0;
(iii)for each k≥k0 the series P
l≥k
T∗(l, k)ξ is convergent and there exists δ >0, independent of k, such that
∞
P
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{fk}k≥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 {fk}k≥k0 ⊂ IntX+such that the corresponding backward affine equation(23)has a bounded solution {exk}k≥k0 ⊂ X+;
(vii) there exists a bounded and uniformly positive sequence {yk}k≥k0 ⊂ IntX+ which verifies
(24) L∗kyk+1−yk 0, k≥k0.
Proof. The equivalence (i)↔(ii) follows immediately from (4). In a similar way to the proof of inequality (21), we obtain
(25) 0≤T∗(l, k)ξ ≤ kT∗(l, k)kξξ for all l≥k≥k0.
If (ii) holds, then for eachk≥k0 the series P
l≥k
kT∗(l, k)kξof real numbers is convergent and we have
(26)
∞
X
l=k
kT∗(l, k)kξ≤δ,
where δ= 1−qβ1 is independent of k. Therefore, the series P
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 =
∞
P
l=k
T∗(l, k)ξ. If (iii) holds then yk is well defined and additionallyyk≤δξfor allk≥k0. Using the definition of the linear evolution operatorT∗(l, k), we may writeyk=ξ+L∗k
∞
P
l=k+1
T∗(l, k+ 1)ξ or, equivalently, yk = ξ+L∗kyk+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
and uniformly positive solution of (22). Thus, we obtain that the implication (iii)→(iv) holds. Now, we prove (ii)→(v). Let {fk}k≥k0 ⊂ IntX+ be a bounded and uniformly positive sequence. Hence there exist νi >0,i= 1,2, such that ν1ξ ≤fl ≤ν2ξ for alll ≥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) ν1kT∗(l, k)kξ≤ |T∗(l, k)fl|ξ≤ν2kT∗(l, k)kξ. If (ii) is fulfilled then (28) shows that for eachk≥k0 the series P
l≥k
|T∗(l, k)fl|ξ of the real numbers is convergent and
(29)
∞
X
l=k
|T∗(l, k)fl|ξ≤δ1
for all k ≥k0, where δ1 = ν1−q2β1 is independent of k. Thus, one gets that the series P
l≥k
T∗(l, k)fl is absolutely convergent for all k≥k0. Set zk =
∞
P
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
∞
X
l=k+1
T∗(l, k+ 1)fl=fk+L∗kzk+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 ≥ν1ξ for all k ≥k0. This means that zk0,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 uniformly positive sequence {fk}k≥k0 ⊂ IntX+ such that the corresponding equation (23) has a bounded solution {xbk}k≥k0 ⊂ X+. Therefore, there exist positive constants γi,i∈ {1,2,3}, such that
γ1ξ ≤fk≤γ2ξ, γ1ξ ≤xbk ≤γ3ξ,
for all k≥k0. Letk1 ≥k0 be fixed. Define yek =T∗(k, k1)bxk,∀k≥k1. Since T∗(k, k1)≥0, we may write
γ1T∗(k, k1)ξ ≤T∗(k, k1)fk≤γ2T∗(k, k1)ξ γ1T∗(k, k1)ξ ≤yek≤γ3T∗(k, k1)ξ (31)
for all k ≥k1. From xbk =L∗kbxk+1+fk, as well as from the definitions of eyk and T∗(k, k1), we obtain successively yek = T∗(k, k1)L∗kbxk+1 +T∗(k, k1)fk = T∗(k+ 1, k1)bxk+1 +T∗(k, k1)fk = eyk+1 +T∗(k, k1)fk. Thus, we obtained yek+1 =yek−T∗(k, k1)fk for allk≥k1. From (31) we deduce that
(32) yek+1 ≤qeyk
for all k ≥ k1, where q = 1− γγ1
3. Taking γ3 large enough in (31) we obtain q ∈(0,1).
From (32) we obtain inductively yek ≤ qk−k1xbk1 for all k ≥ k1. Using again (31) together withxbk1 ≤γ3ξ, we deduce that 0≤T∗(k, k1)ξ ≤ γγ3
1qk−k1ξ, which by (2) leads to |T∗(k, k1)ξ|ξ ≤ γγ3
1qk−k1,k ≥ k1. From Proposition 2.1 (ii) we have kT∗(k, k1)kξ≤ γγ3
1qk−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 ⊂IntX+ be a bounded and uniformly positive solution of (24). Definefbk =zk−L∗kzk+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 asser- tions hold:
(i) for each bounded sequence {fk}k≥k0 ⊂ X the discrete-time backward affine equation
(33) xk=L∗kxk+1+fk
has an unique bounded solution which is given by
(34) exk=
∞
X
l=k
T∗(l, k)fl, k≥k0;
(ii)if there exists an integerθ >1such thatLk+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
(35) xe= (IX − L∗)−1f,
with IX the identity operator onX;
(iv) if Lk are positive operators and {fk}k≥k0 ⊂ X+ is a bounded se- quence, then the unique bounded solution of equation (33) satisfiesxek ≥0 for all k≥k0.
Moreover, if {fk}k≥k0 ⊂ IntX+ is a bounded and uniformly positive sequence, then the unique bounded solution {xek}k≥k0 of equation (33) also is uniformly positive.
Proof. (i) From (i)→(ii) of Theorem 3.2, we deduce that for allk≥k0 the series
n j P
l=k
T∗(l, k)fl
o
j≥k is absolutely convergent and there exists δ >0 independent of kand j such that
(36)
j
X
l=k
T∗(l, k)fl ξ
≤δ.
Set xek = lim
j→∞
j
P
l=k
Tl,k∗ fl =
∞
P
l=k
T∗(l, k)fl. Taking into account the definition of T∗(l, k), we obtain exk = fk+L∗k
∞
P
l=k+1
T∗(l, k+ 1)fl = fk+L∗kexk+1, which shows that {xek}k≥k0 solves (33).
It follows from (36) that{exk}is a bounded solution of (33). Let{bxk}k≥k0 be another bounded solution of equation (33). For each 0 ≤ k < j we may write
(37) xbk =T∗(j+ 1, k)bxj+1+
j
X
l=k
T∗(l, k)fl.
Since {Lk}k≥k0 generates an exponentially stable evolution and{bxk}k≥k0 is a bounded sequence, we have lim
j→∞T∗(j+ 1, k)bxj+1= 0. Lettingj → ∞in (37), we conclude that xbk =
∞
P
l=k
T∗(l, k)fl=xek, 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
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 periodθ= 1, so it is constant. In this case, xe will verify the equationxe=L∗ex+f. Since the operator Lgenerates an exponentially stable evolution, we haveρ(L)<1.
Hence the operatorIX− L∗ is invertible and we deduce thatexis 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.
Remark3.1. 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 wherek0 =−∞, 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
(38) xk+1=Lkxk+fk
has a unique bounded solution {xbk}k∈Z. Moreover, this solution has a repre- sentation formula
(39) xbk =
k−1
X
l=−∞
T(k, l+ 1)fl, ∀k∈Z;
(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 bx= (IX − L)−1f;
(iv)if{Lk}k∈Zare positive operators and{fk}k∈Z⊂ X+, then the unique bounded solution of equation (38) satisfies xbk ≥0 for all k∈Z. Moreover, if fk 0, k∈Z, then xbk0, 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 byL#k =L∗−k.
Lemma 3.1. Let {Lk}k∈Z be a sequence of bounded linear operators on X. The following assertions hold:
(i) ifT#(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∈Zgenerates 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#kyk+f−k, k∈Z.
Theproof 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;
(ii)for eachk∈Zthe seriesP
l≤kT(k, l)ξ is convergent and there exists δ >0, independent of k, such that
k
X
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 {fk}k∈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 {fk}k∈Z ⊂ IntX+ such that the corresponding forward affine equation(41) has a bounded solution xek, k∈Z⊂ X+;
(vi)there exists a bounded and uniformly positive sequence{yk}k∈Zwhich verifies yk+1− Lkyk0.
Theproof 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 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. Ifx∈`∞(Z,X), we denote|x|= sup
k∈Z
|xk|ξ.
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. There- fore, `∞(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 as- sumptions, 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=Tx, yk=
k−1
X
l=−∞
T(k, l+ 1)Glxl,
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.
We have fk=ξ+
k−2
P
l=−∞
Tk,l+1ξ which leads tofk≥ξ thusfk∈IntX+ for all 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
(44) (I`∞− T)(x) =f.
Here,I`∞ stands for the identity operator on`∞(Z,X). Partitioning (44) and taking into account (42)–(43), we obtain that for each k∈Zwe have
xk+1=
k
X
l=−∞
T(k+ 1, l+ 1)Glxl+
k
X
l=−∞
T(k+ 1, l+ 1)ξ.
Further, we may write xk+1=Gkxk+ξ+Lk
k−1
X
l=−∞
T(k, l+1)Glxl+Lk
k−1
X
l=−∞
T(k, l+1)ξ=Gkxk+ξ+Lkxk. This shows that {xk}k∈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 +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 {xek}k∈Z ⊂IntX+. Equation (45) verified by exk may be rewritten as
(46) xek+1 =Lkxek+fek,
where fek = Gkxek+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 given by the representation formula (39), we have xek =
k−1
P
l=−∞
T(k, l+ 1)fel, k∈Z, so that
(47) exk=
k−1
X
l=−∞
T(k, l+ 1)Glxel+
k−1
X
l=−∞
T(k, l+ 1)ξ.
Invoking (42), equation (47) may be written as
(48) ex=Tex+eg,
where ge={egk}k∈Z, egk =
k−1
P
l=−∞
T(k, l+ 1)ξ. It is obvious that gek ≥ξ for all k∈Z. Hencege∈Int`∞(Z,X+).
Using implication (v)→(vi) of Theorem 2.11 in [8] for R = −I`∞ and P =T we obtain thatρ[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))−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 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 Xθ = 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 Xθ we introduce the norm |x|θ = max{|xi|ξ, 0 ≤i≤θ−1}. The space Xθ 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 :Xθ → Xθ defined by y= Πx, wherey= (y0, y1, . . . , yθ−1),
(52) yt=
θ−1
X
s=0
G(t, s+ 1)Psxs, 0≤t≤θ−1 for all x= (x0, x1, . . . , xθ−1)∈ Xθ.
