Journal of Applied Mathematics

Copyright © 2014 Jian Wang and Cuixia Li. 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

The problem of exponential stability is investigated for a class of stochastic time-delay systems. By using the decomposition technique and Lyapunov stability theory, two improved exponential stability criteria are derived. Finally, a numerical example is given to illustrate the effectiveness and the benefit of the proposed method.

1. Introduction

In fact, time delay constantly occurs in the real world, which results in instability of systems. Thus, the stability problem for time-delay systems has been studied for many years [1–8]. On the other hand, stochastic modelling has come to play an important role in many fields of science or industry. Stability analysis for stochastic systems has become increasingly meaningful. A number of results have appeared in the literature [9–19]. For instance, in [11], the author provided the criteria for the stability of a class of stochastic systems by Lyapunov theory.

It’s worth noting that, the grey systems can be established when parameters are evaluated by grey numbers (see [20]). Until now, there have been a few papers tackling the stability of the systems; some important and innovative results are obtained [20–23]. In [23], the authors provided the delay-dependent criteria for exponential robust stability in the forms of nonlinear matrix inequalities and linear matrix inequalities.

In this paper, we deal with the exponential stability for the time-delay grey stochastic systems. By using the method of [21–23] and Lyapunov stability theory, two improved criteria of mean-square exponential stability are proposed. At last, a numerical example is given to verify the criteria.

Notations. denotes the dimensional Euclidean space, the superscript “” denotes matrix transposition, and the notation , where and are symmetric matrices, means that is positive semidefinite. Let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets). Let be an -dimensional Brownian motion defined on the probability space. Let and be the family of continuous functions from to . Let be the family of all -measurable -valued random variables such that .

2. Problem Formulation

Consider

Definition 1. If there is at least a grey matrix among matrices , and of system (1), then (1) is called grey stochastic time-delay system.

Hence where and are grey matrices and , .

Clearly, if matrices and are replaced by the deterministic matrices and , the grey system (2) becomes system (1). The equations are said to be the continuous matrix-covered sets of and .

Definition 2 (see [20]). System (2) is said to be robustly exponentially stable in the mean square, if, for all and arbitrary matrices , , there exist scalars and , such that In addition, the following assumptions are made on the system (2). (H1). (H2)Supposing there exist scalars , , such that, for , the inequality holds, .

Before giving the main results, we first present Lemmas 3 and 4, which are important for the proof of main theorems.

Lemma 3 (see [20]). For arbitrary whitened matrix , it follows that(i)(ii)(iii),where , , , .

Lemma 4 (see [18]). Let be a real matrix of appropriate dimensions; for any vectors , one has .

3. Proof of the Main Theorem

In this section, we discuss the exponential stability for system (2); two improved criteria for robust exponential stability in mean square are proposed.

Theorem 5. System (2) is exponentially robustly stable in mean square. If there exist positive scalars , , and , such that here Then, for all , the following inequality holds: where is the unique positive solution of the following equation:

Proof. First, define Lyapunov-Krasovskii functional as follows: Then, we have Using Lemmas 3 and 4, we derive By assumption (H2), we can obtain Substituting (11)–(12) into (10), we see that Using Itô’s differential formula and integrating both sides, we obtain Moreover, we have Combining (14) with (15) and noting the definitions of , , we see that Furthermore, Since , and .

When (5) holds, (8) must have a unique solution .

Hence, we have This completes the proof of Theorem 5.

By a similar approach in [22], whitened system of (2) can be written as Since then, we can introduce Combining (19) and (21) together, whitened system of (2) can be rewritten as

Lemma 6. For all and , the following inequality holds: where

Proof. First, by the definition of , we can derive Clearly, By assumption (H2), we see that By (25)–(27) and noting the definitions of , , we can obtain By (28) and integrating both sides, we have Moreover, we can obtain Substituting (30) into (29) and noting the definitions of , , and , (23) holds. The proof of Lemma 6 is completed.

By (22) and Lemma 6, another criterion for system (2) will be given.

Theorem 7. System (2) is exponentially robustly stable in mean square. If there exist positive scalars , , and , such that here Then, for all , the following inequality holds: where is the unique positive solution of the following equation:

Proof. Similar to the proof process of Theorem 5, we can derive Using Itô’s differential formula and integrating and by the definition of , we obtain If , then .

Because , and .

If (31) holds, (34) must have a unique solution .

Therefore, we can easily get or equivalently (33) holds; the proof of Theorem 7 is completed.