Uniform Asymptotic Stability and Robust Stability for Positive Linear Volterra Difference Equations in Banach Lattices
© S. Murakami and Y. Nagabuchi. 2008
Received: 5 August 2008
Accepted: 7 November 2008
Published: 17 November 2008
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.
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,  and the references therein.
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.
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 .
(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 .
Here, we give the definition of the positivity of Volterra difference equations.
Also, we follow the standard definitions for stabilities of the zero solution.
The zero solution of (1.1) is said to be
The zero solution of (1.2) is said to be
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  the equivalence among these three properties without such restrictive conditions; more precisely, we have established the following.
Theorem 2.4 (see [4, Theorem 1]).
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 .
By using (2.13) one can verify the following proposition quite similarly.
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.
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.
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.
The zero solution of (1.1) is uniformly asymptotically stable.
The zero solution of (1.2) is uniformly asymptotically stable.
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,  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.
We need the following lemma to prove the theorem.
which completes the proof.
provided that the assumptions of the theorem are satisfied. In fact, these three radii coincide, that is, we have the following.
which contradicts (4.14). Consequently we must have (4.13), and this completes the proof.
The first author is partly supported by the Grant-in-Aid for Scientific Research (C), no. 19540203, Japan Society for the Promotion of Science.
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