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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 781983, 6 pages
http://dx.doi.org/10.1155/2013/781983
Research Article

-Stability of Linear Matrix Differential Systems

1College of Electronic and Information Engineering, Hebei University, Baoding, Hebei 071002, China
2College of Mathematics and Computer Science, Hebei University, Baoding, Hebei 071002, China

Received 2 December 2012; Accepted 1 February 2013

Academic Editor: Yong Hong Wu

Copyright © 2013 Peiguang Wang and Xiaowei Liu. 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

This paper investigates the stability problem of linear matrix differential systems and gives some sufficient conditions of -stability for linear matrix system and its associated perturbed system by using the Kronecker product of matrices. An example is also worked out to illustrate our results.

1. Introduction

The theory of stability in the sense of Lyapunov is well known and is used in the real world. It is obvious that, in applications, asymptotic stability is more important than stability because the desirable feature is to know the size of the region of asymptotic stability. However, when we study the asymptotic stability, it is not easy to work with nonexponential types of stability. In recent years, Medina and Pinto [1, 2] extended the study of exponential stability to a variety of reasonable systems called -systems. They introduced the notion of -stability with the intention of obtaining results about stability for a weakly stable system (at least, weaker than those given exponential stability and the uniform Lipschitz stability) under some perturbations. Choi et al. [3] investigated -stability for the nonlinear differential systems by employing the notion of -similarity and the Lyapunov functions. And then, Choi et al. [46] also characterized the -stability in variation for nonlinear difference systems via -similarity and the Lyapunov functions and obtained some results related to stability for the perturbations of nonlinear difference systems.

However, as far as the author's scope, there are few discussions and results for matrix differential systems. In this paper, we shall investigate the -stability problem for linear matrix differential systems by employing the Kronecker product of matrices which can be found in Lakshmikantham and Deo's monograph [7]. Some preliminaries are presented in Section 2. A theorem is given in this section, which is important to complete the main results of this paper. In Section 3, sufficient conditions for the -stability are given for linear matrix system and its associated perturbed system. An example is also worked out at the end of this paper.

2. Preliminaries

Consider the linear matrix differential equation and its associated perturbed system where , , , and .

Now, we introduce the operator which maps an matrix onto the vector composed of the rows of Let us begin by defining the Kronecker product of matrices.

Definition 1 (see [7]). If , , then the Kronecker product of and , , is defined by the matrix

Among the main properties of this product presented in [8], we recall the following useful ones: (1), (2),where , and   and   is an identity matrix.

Then, the equivalent vector differential systems of (1) and (2) can be written as where , , , , and .

In order to investigate -stability of linear matrix equation and its associated perturbed system, we need to consider the following systems and their properties. The techniques and results are similar to those of [7].

Consider the linear differential system where is an continuous matrix and its perturbation where . Suppose that the solution of (7) exists for all . The fundamental matrix solution of (7) is given by [7] and .

We are now in a position to give the Alekseev formula, which connects the solutions of (7) and (8).

Lemma 2 (see [7]). If is the solution of (7) and exists for , any solution of (8), with , satisfies the integral equation for , where .

Lemma 3 (see [7]). Assume that is the solution of (7) through , which exists for , then where .

The following theorem gives an analog of the variation of parameters formula for the solution of (2).

Theorem 4. Assume that is the solution of (5) for , let Then one has the following.
(i) exists and is the fundamental matrix solution of the variational equation such that , and therefore where and are solutions of respectively.
 (ii) Any solution of (2) satisfies the integral equation for .

Proof. (i) It is obvious that exists and is the fundamental matrix solution of the variational equation such that .
Furthermore, we get with the initial value which has the solution where and are the solutions of (15) and (16), respectively, and is the identity matrix.
Therefore,
(ii) Employing Lemma 2 and substituting for the right-hand side of (22), we get for , where is any solution of (6).
Now, we define , , and by , , and . Thus, we have that for , where is the unique solution of (1) for .
The proof is completed.

3. Main Results

We firstly give some notions.

Definition 5. A generalized matrix valued norm from to is a mapping such that (a), if and only if , (b), is a constant, (c).

Definition 6. The zero solution of (1) is said to be (hS)-stability if there exist , , and a positive bounded continuous function on such that for and , . (hSV)-stability in variation if there exist , and a positive bounded continuous function on satisfying provided , where and are given in Theorem 4.

Lemma 7 (see [4]). The linear system is if and only if there exist a constant and a positive continuous bounded function defined on such that for every in , for all , where is an continuous matrix and is a fundamental matrix of (27).

Theorem 8. The solution of (1) is if and only if the solution of (5) is .

Proof. Necessity. Since the solution of (1) is , there exist , , and a positive bounded continuous function on such that for every ,  , where is the solution of (1), satisfying then we obtain It follows that Thus,
Sufficiency. It can be easily proved by the same method. The proof is completed.

Theorem 9. The solution of (1) is if and only if there exist a constant and a positive continuous bounded function defined on such that for every in , for all .

Proof. Sufficiency. Following Lemma 3 and Theorem 4, we have It follows that Hence, Therefore, the solution of (5) is . By Theorem 8, it implies that the solution of (1) is .
Necessity. If the solution of (1) is , then the solution of (5) is using Theorem 8. By Lemma 7, we have From Theorem 4, we obtain that Thus, This completes the proof.

Corollary 10. If the zero solution of (1) is , then the zero solution of (1) is .

Next, we offer sufficient conditions for the -stability of linear matrix differential systems by using the Lyapunov functions.

Defining the Lyapunov functions for and for the solution of (5), Then, it is well known that if is the Lipschitzian in for each .

Theorem 11. Suppose that is a positive bonded continuously differentiable function on . Furthermore, assume that there exists a function satisfying the following properties: (i), and is Lipschitzian in for each , (ii), , , (iii), .Then, the solution of (1) is .

Proof. Let be the solution of (5). As a consequence of (iii), we obtain From the condition (ii), we have By Theorem 8, we can easily get that the solution of (1) is . The proof is completed.

Now, we examine the properties of the perturbed linear matrix differential system.

Lemma 12 (see [9]). Suppose that is strictly increasing in for with the property for . If , then for all .

Theorem 13. Assume that of (1) is with the nonincreasing function and . Consider the scalar differential equation Suppose that where is strictly increasing in for each fixed with .
If is , then the solution of (2) is also , whenever .

Proof. By Theorem 4, the solutions of (1) and (2) with the same initial values are related by Then, we have From Corollary 10, it easily follows that where . Since and are nonincreasing, we obtain By Lemma 12, we have for all . Since of (47) is , This completes the proof.

Theorem 14. Assume that(i)the zero solution of (1) is ,(ii) provided that and for . Then, the solution of (2) is .

Proof. By Theorem 4, the solutions of (1) and (2) with the same initial values are related by The assumptions (i) and (ii) yield Then, by Gronwall's inequality, we get where
The proof is completed.

4. Example

In this section, we give a simple but illustrative example. Consider the matrix differential equation Then, we can obtain the following equations: The solutions of (58) and (59) are respectively.

Then, Thus, we have where , , , , and is an identity matrix. So, from Theorem 9, we can conclude that the solution of (57) is .

Acknowledgments

The authors would like to thank the reviewers for their valuable suggestions and comments. This paper is supported by the National Natural Science Foundation of China (10971045 and 11271106).

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