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Uniform Asymptotic Stability and Robust Stability for Positive Linear Volterra Difference Equations in Banach Lattices

Advances in Difference Equations20082008:598964

DOI: 10.1155/2008/598964

Received: 5 August 2008

Accepted: 7 November 2008

Published: 17 November 2008

Abstract

For positive linear Volterra difference equations in Banach lattices, the uniform asymptotic stability of the zero solution is studied in connection with the summability of the fundamental solution and the invertibility of the characteristic operator associated with the equations. Moreover, the robust stability is discussed and some stability radii are given explicitly.

1. Introduction

A dynamical system is called positive if any solution of the system starting from nonnegative states maintains nonnegative states forever. In many applications where variables represent nonnegative quantities we often encounter positive dynamical systems as mathematical models (see [1, 2]), and many researches for positive systems have been done actively; for recent developments see, for example, [3] and the references therein.

In this paper we treat the Volterra difference equations
(1.1)
together with
(1.2)
in a (complex) Banach lattice X, where is a sequence of compact linear operators on satisfying the summability condition , and we study stability properties of (1.1) and (1.2) under the restriction that the operators , are positive. In fact, the restriction on yields the positivity for the above equations (whose notion is introduced in Section 2). Also, without the restriction, in [4] the authors characterized the uniform asymptotic stability of the zero solution of (1.1), together with (1.2), in connection with the invertibility of the characteristic operator
(1.3)

of (1.1) for any complex numbers such that . In Section 3, we will prove that under the restriction that the operators , are positive, the invertibility of the characteristic operator reduces to that of the operator , and consequently the uniform asymptotic stability of the zero solution for positive equations is equivalent to the condition which is much easier than the one for the characteristic operator in checking (Theorem 3.6). Moreover, we will discuss in Section 4 the robust stability of (1.1) and give explicit formulae of some stability radii.

2. Preliminaries

Let , , , , , and be the sets of natural numbers, nonnegative integers, nonpositive integers, integers, nonnegative real numbers, real numbers and complex numbers, respectively.

To make the presentation self-contained, we give some basic facts on Banach lattices which will be used in the sequel (see, e.g., [5]). Let be a real vector space endowed with an order relation Then is called an ordered vector space. Denote the positive elements of by . If furthermore the lattice property holds, that is, if for then is called a vector lattice. It is important to note that is generating, that is,
(2.1)
Then, the modulus of is defined by If is a norm on the vector lattice satisfying the lattice norm property, that is, if
(2.2)

then is called a normed vector lattice. If, in addition, is a Banach space, then is called a (real) Banach lattice.

We now extend the notion of Banach lattices to the complex case. For this extension all underlying vector lattices are assumed to be relatively uniformly complete, that is, if for every sequence in satisfying and for every and every sequence in , it holds that
(2.3)
Now, let be a relatively uniformly complete vector lattice. The complexification of is defined by The modulus of is defined by
(2.4)
A complex vector lattice is defined as the complexification of a relatively uniformly complete vector lattice equipped with the modulus (2.4). If is normed, then
(2.5)

defines a norm on satisfying the lattice norm property; in fact, the norm restricted to is equivalent to the original norm in , and we use the same symbol to denote the (new) norm. If is a Banach lattice, then equipped with the modulus (2.4) and the norm (2.5) is called a complex Banach lattice, and is called the real part of .

For Banach spaces and , we denote by the Banach space of all bounded linear operators from to equipped with the operator norm, and use the notation in place of . Let and be Banach lattices with real parts and , respectively. An operator is called real if . A linear operator from to is called positive, denoted by , if holds. Such an operator is necessarily bounded (see [5]) and hence real. Denote by and the sets of real operators and positive operators between and , respectively:
(2.6)
Then we observe that
(2.7)
Indeed, it is clear that the inequality holds true for any , the real part of . Let with and . Since by (2.4), we get
(2.8)
with and . Then, it follows from the modulus (2.4) that
(2.9)
as required. We also emphasize the simple fact that
(2.10)

(see, e.g., [5, page 230]). Moreover, by the symbol we mean for Throughout this paper, is assumed to be a complex Banach lattice with the real part and the positive convex cone .

For any interval we use the same notation meaning the discrete one , for example, for . Also, for an -valued function on a discrete interval , its norm is denoted by . Let and a function be given. We denote by the solution of (1.1) satisfying on . Similarly, for and a function , we denote by the solution of (1.2) satisfying on . We then recall the representation formulae of solutions for initial value problems of (1.1) and (1.2) (see [68]). Let be the fundamental solution of (1.1) (or (1.2)), that is, the sequence in satisfying
(2.11)
for Then, the solution is given by
(2.12)
for arbitrary initial function , and also for arbitrary initial function , the solution of (1.2) is given by
(2.13)

for , where we promise for .

Here, we give the definition of the positivity of Volterra difference equations.

Definition 2.1.

Equation (1.1) is said to be positive if for any and , the solution for . Similarly,(1.2) is said to be positive if for any and , the solution for .

Also, we follow the standard definitions for stabilities of the zero solution.

Definition 2.2.

The zero solution of (1.1) is said to be

  1. (i)

    uniformly stable if for any there exists a such that if and is an initial function on with then for all ;

     
  2. (ii)

    uniformly asymptotically stable if it is uniformly stable, and if there exists a such that, for any there exists an with the property that, if and is an initial function on with then for all .

     

Definition 2.3.

The zero solution of (1.2) is said to be

  1. (i)

    uniformly stable if for any there exists a such that if and is an initial function on with then for all ;

     
  2. (ii)

    uniformly asymptotically stable if it is uniformly stable, and if there exists a such that, for any there exists an with the property that, if and is an initial function on with then for all .

     

Here and subsequently, denotes the Z-transform of ; that is, , which is defined for under our assumption . Then, is called the characteristic operator associated with (1.1) (or (1.2)). In [6, 7], under some restrictive conditions on , we discussed the uniform asymptotic stability of the zero solution of (1.2) in connection with the summability of the fundamental solution , as well as the invertibility of the characteristic operator ; see also [912] for the case that is finite dimensional. Moreover, we have shown in [4] the equivalence among these three properties without such restrictive conditions; more precisely, we have established the following.

Theorem 2.4 (see [4, Theorem 1]).

Let , and assume that are all compact. Then the following statements are equivalent.

  1. (i)

    The zero solution of (1.1) is uniformly asymptotically stable.

     
  2. (ii)

    The zero solution of (1.2) is uniformly asymptotically stable.

     
  3. (iii)

    .

     
  4. (iv)

    For any such that , the operator is invertible in .

     

3. Stability for Positive Volterra Difference Equations

In this section, we will prove that the uniform asymptotic stability of the zero solution of positive Volterra difference equations (1.1) and (1.2) is, in fact, characterized by the invertibility of the operator for . To this end we need some observations on the spectral radius of the -transform of the convolution kernel .

First of all, we show the relation between the positivity of Volterra difference equations and that of the sequence of bounded linear operators .

Proposition 3.1.

Equation (1.1) is positive if and only if all are positive.

Proof.

Suppose that all are positive. Then from (2.11) each element ( of the fundamental solution is also positive; so that by virtue of (2.12), for , where and are arbitrary. Hence, (1.1) is positive. Conversely, suppose (1.1) to be positive. Then we have in particular for , which implies that . Let and for , be any function such that except . Then it follows from (2.12) that
(3.1)

which implies that for . Thus, for since is arbitrary.

By using (2.13) one can verify the following proposition quite similarly.

Proposition 3.2.

Equation (1.2) is positive if and only if all are positive.

In what follows, we assume that and each is compact. For any closed operator on we denote by , , and the spectrum, the point spectrum, and the resolvent set of , respectively. Also denote by the interior of the unit disk of the complex plane. Then for the uniform asymptotic stability of the zero solution of (1.1) we have the following criterion.

Theorem 3.3.

Suppose that (1.1) is positive. If , the zero solution of (1.1) is uniformly asymptotically stable.

Proof.

In view of Theorem 2.4 it is sufficient to show that is invertible for with . Suppose by contradiction that is not invertible for some with . Then and hence since and is compact. Let , be an eigenvector of the operator for the eigenvalue . Then, . We generally get for any . Notice that the spectral radius of is less than by the assumption. Therefore, it follows from the well known Gelfand's formula (see, e.g., [13, Theorem 10.13]) for the spectral radius of bounded linear operators that , which implies . On the other hand, since , are positive by Proposition 3.1, we get
(3.2)
by (2.7); and generally,
(3.3)
for any . Therefore,
(3.4)
and it follows from the lattice norm property that
(3.5)

This is a contradiction, because we must get by (3.5).

The converse of Theorem 3.3 also holds. To see this we need another proposition. Let be the spectral radius of for .

Proposition 3.4.

Suppose that are all positive. Then, is nonincreasing and continuous as a function on

Proof.

Let . Then,
(3.6)
Observe that the resolvent of ( is given by whenever . Then, we deduce from (3.6) that
(3.7)
Note also that under our assumption is a positive operator for since is closed in . In particular, is also positive and hence for ,
(3.8)
by (2.7). Therefore so that
(3.9)

Now, let us assume that holds for some and with . Since is positive, it follows from [5, Chapter 5, Proposition 4.1] that . Observe that if and , then is invertible in with . Since , the above observation leads to the fact that as ; consequently, we get as . On the other hand it follows from for that and the function is continuous on . Hence, we get , which is a contradiction. Consequently, for .

We next show the left continuity of on . If is not left continuous at some , we have
(3.10)

Since , we have . Notice that is an open set. Hence, it follows that if is small enough, that is, for such an . On the other hand, by virtue of [5, Chapter 5, Proposition 4.1] again, the positivity of yields ; this is a contradiction.

is right continuous as well in . Indeed, if it is not so, there exists a such that
(3.11)
In view of the positivity of we have . Also, since is compact, is an isolated point of its spectrum ; in particular, there exists an such that . By the continuity of there corresponds an such that is invertible in for . Moreover, by the fact that for , one can see
(3.12)
from which readily follows because of the positivity of . Therefore, passing to the limit , we deduce that . Let be any number such that . By the same reasoning, we see that . Since
(3.13)
we get
(3.14)
Then, for , it follows from (2.7) that
(3.15)
hence
(3.16)

for any with . This is a contradiction, because as . The proof is now completed.

Theorem 3.5.

Suppose that (1.1) is positive. If the zero solution of (1.1) is uniformly asymptotically stable, then .

Proof.

Set . To prove the theorem it is sufficient to show . Assume that . Since by Proposition 3.4   is continuous, there exists such that , that is, . It follows from the positivity of , together with its compactness, that belongs to ; hence, there exists an such that , or equivalently
(3.17)
Setting , we have
(3.18)

so that is a solution of (1.2). By virtue of Theorem 2.4 and our assumption, (1.2) is uniformly asymptotically stable and therefore as , which is impossible because for all . Thus we must have .

Combining the results above with Theorem 2.4, we have, for positive Volterra difference equations, the equivalence among the uniform asymptotic stability of the zero solution of (1.1) and (1.2), the summability of the fundamental solution and the invertibility of the operator outside the unit disk.

Theorem 3.6.

Let the assumptions in Theorem 2.4 hold. If, in addition, ac.

  1. (i)

    The zero solution of (1.1) is uniformly asymptotically stable.

     
  2. (ii)

    The zero solution of (1.2) is uniformly asymptotically stable.

     
  3. (iii)

     
  4. (iv)

    The operator is invertible in for .

     
Before concluding this section, we will give an example to which our Theorem 3.6 is applicable. In [6, 7], following the idea in [14], we have shown that Volterra difference equations on a Banach space are naturally derived from abstract differential equations on with piecewise continuous delays of type
(3.19)
where denotes the Gaussian symbol and is the inifinitesimal generator of a strongly continuous semigroup , of bounded linear operators on , and , are bounded linear operators on such that
(3.20)
Recall that a function with is called a (mild) solution of (3.19) on , if is continuous on , and satisfies the relation
(3.21)
In case of for some , the relation above yields that
(3.22)
Letting , we get the Volterra difference equation
(3.23)
where are bounded linear operators on defined by
(3.24)
for Conversely, if satisfies (3.23) with , then the function extended to nonintegers by the relation
(3.25)

is a (mild) solution of (3.19). Thus, abstract differential equations of type (3.19) lead to Volterra difference equations on .

Now suppose that the semigroup is compact. Then, is continuous in with respect to the operator norm ([15]) and also , defined by the relation (3.24) are compact operators on (see [6, Proposition 1]). Moreover, it follows from (3.20) that . Moreover in the restricted case where , are given by for some with , we know by [7, Proposition 1] that the spectrum of the characteristic operator of (3.23) is given by the formula:
(3.26)
Hence, in the restricted case, Theorem 2.4 implies that the zero solution of (3.23) is uniformly asymptotically stable if and only if
(3.27)

for all and .

We further assume that is a complex Banach lattice, the compact semigroup on is positive, and that are all nonnegative. Then the sequence defined by (3.24) meets the assumptions in Theorem 3.6. Noticing that , we know by Theorem 3.6 in the further restricted case that the zero solution of (3.23) is uniformly asymptotically stable if and only if
(3.28)

for all .

4. Robust Stability and Some Stability Radii of Positive Volterra Difference Equations

Let (1.1) be uniformly asymptotically stable, that is, the zero solution of (1.1) is uniformly asymptotically stable, and consider a perturbed difference equation of the form
(4.1)

where are given operators corresponding to the structure of perturbations and is an unknown (disturbance) parameter. Here and are also assumed to be complex Banach lattices. Our objective in this section is to determine various stability radii of (1.1) provided that are all positive; for this topic in case that the space is finite dimensional, see, for example, [16] and the references therein. By the stability radius of (1.1) we mean the supremum of positive numbers such that the uniform asymptotic stability of the perturbed (4.1) persists whenever the size of the perturbation , measured by the -norm , is less than (for precise definitions see the paragraph preceding Theorem 4.3).

Here and hereafter we also assume that for any perturbation , , , are all compact, although this assumption is not necessary in the case that at least one of the operators and is compact. In what follows, we define by convention.

Theorem 4.1.

Let , , be positive. Suppose that (1.1) is uniformly asymptotically stable and , are both positive. Then the perturbed (4.1) is still uniformly asymptotically stable if
(4.2)

We need the following lemma to prove the theorem.

Lemma 4.2.

Under the same assumptions as in Theorem 4.1 we have
(4.3)

Proof.

Since (1.1) is uniformly asymptotically stable, , the spectral radius of , is less than by Theorem 3.5. In particular, it follows that is convergent and coincides with . Let , , be given. As in the proof of Theorem 3.3, one can see for and ; hence , so that . Therefore,
(4.4)
The positivity of and then implies
(4.5)
for , and we thus obtain
(4.6)

which completes the proof.

Proof.

Assume that perturbed (4.1) is not uniformly asymptotically stable for some satisfying , that is,
(4.7)
Then, by Theorem 2.4 there exists a with such that is not invertible. So since is compact. Hence, there corresponds an with satisfying . By virtue of the uniform asymptotic stability of (1.1) we know that is invertible; and therefore we get
(4.8)
so that and . In view of Lemma 4.2,
(4.9)

which gives , a contradiction to (4.7). The proof is completed.

Let be the set of all compact operators mapping into . We introduce three classes of perturbations defined as , and . Then the complex, real and positive stability radius of(1.1) under perturbations is defined, respectively, by
(4.10)
where the convention is used. By definition, it is easy to see that . On the other hand Theorem 4.1 yields the estimate
(4.11)

provided that the assumptions of the theorem are satisfied. In fact, these three radii coincide, that is, we have the following.

Theorem 4.3.

Let , , be positive. Suppose that (1.1) is uniformly asymptotically stable and , are both positive. Then
(4.12)

Proof.

By the fact , combined with (4.11), it is sufficient to prove that
(4.13)
We may consider the case since otherwise the theorem is trivial. Suppose by contradiction that (4.13) does not hold. Then there is an , , such that
(4.14)
By the same reasoning as in the proof of Lemma 4.2, one can see that and hence . Also by (2.10) one may choose with such that
(4.15)
Now let . Since , there exists a positive , , satisfying (see [17, Proposition 1.5.7]). Consider a map defined by
(4.16)
Then, it is easy to see that and . Hence,
(4.17)
By setting , we have
(4.18)
Notice that because , and that , or equivalently . Define a perturbation by
(4.19)
Then . Moreover,
(4.20)
and in particular , which means, by Theorem 3.6 (or Theorem 2.4), that the perturbed (4.1) with is not uniformly asymptotically stable. Therefore,
(4.21)

which contradicts (4.14). Consequently we must have (4.13), and this completes the proof.

Declarations

Acknowledgment

The first author is partly supported by the Grant-in-Aid for Scientific Research (C), no. 19540203, Japan Society for the Promotion of Science.

Authors’ Affiliations

(1)
Department of Applied Mathematics, Okayama University of Science
(2)
Department of Applied Science, Okayama University of Science

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Copyright

© S. Murakami and Y. Nagabuchi. 2008

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