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Journal of Applied Mathematics
Volume 2008, Article ID 124269, 12 pages
http://dx.doi.org/10.1155/2008/124269
Research Article

The Robustness of Strong Stability of Positive Homogeneous Difference Equations

1Department of Mathematics, University of Pedagogy, 280 An Duong Vuong Street, HoChiMinh City 70000, Vietnam
2Department of Mathematics, Ton Duc Thang University, 98 Ngo Tat To Street, HoChiMinh City 70000, Vietnam

Received 24 October 2007; Revised 15 March 2008; Accepted 23 June 2008

Academic Editor: Patrick De Leenheer

Copyright © 2008 The Anh Bui and Dang Xuan Thanh Duong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the robustness of strong stability of the homogeneous difference equation via the concept of strong stability radii: complex, real and positive radii in this paper. We also show that in the case of positive systems, these radii coincide. Finally, a simple example is given.

1. Introduction

Motivated by many applications in control engineering, problems of robust stability of dynamical systems have attracted a lot of attention of researchers during the last twenty years. In the study of these problems, the notion of stability radius was proved to be an effective tool, see [15]. In this paper, we study the robustness of strong stability of the homogeneous difference equation under parameter perturbations.

The organization of this paper is as follows. In Section 2, we recall some results on nonnegative matrices and present preliminary results on homogeneous equations for later use. In Section 3, we study a complex strong stability radius under multiperturbations. Next, we present some results on strong stability radii of the positive class equations under parameter perturbations. It is shown that complex, real, and positive strong stability radii of positive systems coincide. More important, estimates and computable formulas of these stability radii are also derived. Finally, a simple example is given.

2. Preliminaries

2.1. Nonnegative Matrices

We first introduce some notations. Let be positive integers, a matrix is said to be nonnegative () if all its entries are nonnegative. It is said to be positive () if all its entries are positive. For , means that . The set of all nonnegative -matrices is denoted by . A similar notation will be used for vectors. Let or , then for any and , we define and by . For any matrix the spectral radius and the spectral abscissa of is defined by and , respectively, where is the spectrum of . We recall some useful results, see [6].

A norm on is said to be monotonic if it satisfiesIt can be shown that a vector norm on is monotonic if and only if for all , see [7]. All norms on we use in this paper are assumed to be monotonic. Theorem 2.1 (Perron-Frobenius). Suppose that . Then(i) is an eigenvalue of and there is a nonnegative eigenvector such that .(ii)If and then the algebraic multiplicity of is not greater than the algebraic multiplicity of the eigenvalue .(iii)Given , there exists a nonzero vector such that if and only if .(iv) exists and is nonnegative if and only if .Theorem 2.2. Let . If then

2.2. Homogeneous Difference Equations

Consider the neutral differential difference equation of the following form:where is linear continuous defined byHere each is an -matrix, each is a constant satisfying and and is defined by . Recall that there is a strictly close relation between the asymptotic behavior of solutions of (2.3) and that of associated linear homogeneous difference equationsor equivalently,A study of the asymptotic behavior of solutions of system (2.6) plays a fundamental role in understanding the asymptotic behavior of solutions of linear neutral differential equations of the form (2.3), see [8].

We recall the definition in [8]: the operator or system (2.6) is called stable if the zero solution of (2.6) with is uniformly asymptotically stable.

Associated with system (2.6) we define the quasipolynomialFor , if , then is called a characteristic root of the quasipolynomial matrix (2.7). Then, a nonzero vector satisfying is called an eigenvector of corresponding to the characteristic root . We set , the spectral set of (2.7), and , the spectral abscissa of (2.7). The following lemma is a well-known result in [8]. Theorem 2.3. System (2.6) is stable if and only if .It is well known that is not continuous in the delays , see [9]. One consequence of the noncontinuity is that arbitrarily small perturbations on the delays may destroy stability of the difference equation. This has led to the introduction of the concept of strong stability in Hale and Verduyn Lunel [10]. Definition 2.4. System (2.6) is strongly stable in the delays if it is stable for each .The concept of strong stability has interested many researchers as in [813] and references therein. Now we recall a result in [10]. Theorem 2.5. The following statements are equivalent:
(i)system (2.6) is strongly stable,(ii).
We set and . Since is continuous in , we imply the continuity of the following function defined byMoreover, by the compactness of the set , there exists such thatBy the above result, we can get the following statement: system (2.6) is strongly stable if and only if

3. Main Results

3.1. Complex Strong Stability Radius

Suppose that system (2.6) is strongly stable. Now we assume that each matrix is subjected to the perturbation of the formwhere are given matrices defining the structure of the perturbations and are unknown matrices, . We write the perturbed systemDefinition 3.1. Let system (2.6) be strongly stable. The complex, real, and positive strong stability radii of system (2.6) under perturbations of the form (3.1) are defined byrespectively, we set .If system (2.6) is strongly stable, we define a function by . It is easy to see that is well-defined. For any , we setTheorem 3.2. Let system (2.6) be strongly stable. Then we have
(i)(ii)in particular, if (or ) for all , then we have
Proof. Let be a destabilizing disturbance. Then there exists such that . This means that there exists a nonzero vector satisfyingThis follows thator equivalently,Choose such that . Multiplying the above equation with , we obtainThis implies thatFrom this inequality and the definition of , the left-hand inequality of (i) follows:Now it remains to prove the right-hand inequality of (i): Indeed, for any , and , there exists nonzero vector such that and . By Hahn-Banach theorem, there exists satisfying and . We define a matrix by settingNow we construct the disturbance defined by It is easy to check that . Moreover, we havewhere . This means that is a destabilizing disturbance. Thus,The proof of (i) is complete, and (ii) can be obtained directly from (i).

In general, the complex, real, and positive radius are distinct, see [4, 5]. Theorem 3.2 reduces the computation of the complex strong stability radius to a global optimization problem with many variations while the problem for the real stability radius is much more difficult, see [5]. It is therefore natural to investigate for which kind of systems these three radii coincide. The answer will be found in the next section.

3.2. Strong Stability Radii of Positive Systems

In this section, we restrict system (2.6) to be positive, that is, are nonnegative for all .Lemma 3.3. Let . Then we have
(i);(ii),
Proof. (i) By Theorem 2.2, we have
(ii) the positivity of can be implied by Theorem 2.1. The right-hand inequality can be obtained by the following formula:This completes the proof.

It is important to note from above lemma that under positivity assumptions, system (2.6) is strongly stable if and only if . Lemma 3.4. Suppose that system (2.6) is positive and strongly stable. Then, for any , we haveProof. For any , we have . Thus, for an arbitrary vector ,By Lemma 3.3, we have . Thus, we implyTheorem 3.5. Let system (2.6) be strongly stable and positive. Assume that all are nonnegative matrices. If or , thenwhere .Proof. By Theorem 3.2, we haveMoreover, using Lemma 3.4, we getSince , we only need to prove thatIndeed, for any , since is a nonnegative matrix, there exists nonnegative vector such that and . Using Krein-Rutman theorem, see [14], there exists satisfying and . We define a nonnegative matrix by settingNow we construct the positive disturbance defined by It is easy to check that . Moreover, we havewhere . It means that is a destabilizing disturbance. ThusThe proof is complete.

Now we turn to a different perturbation structure and assume that each matrix is subjected to perturbations of the formwhere are given matrices defining the structure of the perturbations and are unknown scalars representing parameter uncertainties. So we can write the perturbed systemDefinition 3.6. Let system (2.6) be strongly stable. The complex, real, and positive strong stability radii of system (2.6) under perturbations of the form (3.31) are defined byrespectively, we set , and , where .Lemma 3.7. Suppose system (2.6) is strongly stable, positive and . ThenProof. Because , we only need to prove that Indeed, for a destabilizing disturbance , there exist a and a nonzero vector such thatThis yieldsBy Theorem 2.1, we getIt means that is also a destabilizing disturbance. Thus, by the definition of complex and real radii, . The proof is complete.Theorem 3.8. Suppose system (2.6) is strongly stable, positive and . Thenwhere .Proof. By Lemma 3.7, we only need to prove thatTo do it, taking arbitrary destabilizing disturbance , by Lemma 3.3 and Theorem 2.1, there exist a and a nonzero vector such thator equivalently,This yieldsThen, we haveUsing Theorem 2.1 again, we obtainor equivalently,Thus, from the definition of , one hasOn the other hand, setting . Then, by Theorem 2.1, there exists a nonnegative vector satisfyingThis is equivalent toHence,This means that is a destabilizing disturbance and thus, . The proof is complete.

Now we consider the following example to illustrate the obtained results.

Example 3.9. Consider systemwhereThen we haveThus system (3.50) is strongly stable.

Assume that the matrices are subjected to perturbations of the form , whereThenIf is provided with the norm defined by , then by Theorem 3.5, we haveAssume that the given two matrices are subjected to perturbations of the form , whereThenBy Theorem 3.8, we get

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