- Research Article
- Open Access

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

- Satoru Murakami
^{1}Email author and - Yutaka Nagabuchi
^{2}

**2008**:598964

https://doi.org/10.1155/2008/598964

© S. Murakami and Y. Nagabuchi. 2008

**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.

## Keywords

- Fundamental Solution
- Spectral Radius
- Vector Lattice
- Robust Stability
- Banach Lattice

## 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.

*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

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.

*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,

*lattice norm property*, that is, if

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

*relatively uniformly complete,*that is, if for every sequence in satisfying and for every and every sequence in , it holds that

*complexification*of is defined by The modulus of is defined by

*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

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 .

*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:

(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 , 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

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

- (i)
*uniformly stable*if for any there exists a such that if and is an initial function on with then for all ; - (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 [9–12] 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.

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

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

- (iii)
.

- (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.

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.

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.

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 .

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.

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.

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.

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

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

- (iii)
- (iv)
The operator is invertible in for .

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

for all and .

for all .

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

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.

We need the following lemma to prove the theorem.

Lemma 4.2.

Proof.

which completes the proof.

Proof.

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

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

Theorem 4.3.

Proof.

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

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