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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 391461, 6 pages
Sufficient Conditions on the Exponential Stability of Neutral Stochastic Differential Equations with Time-Varying Delays
1School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455002, China
2School of Mathematics and Physics, Anyang Institute of Technology, Anyang, Henan 455002, China
Received 24 January 2014; Revised 26 April 2014; Accepted 26 April 2014; Published 12 May 2014
Academic Editor: Shuping He
Copyright © 2014 Yanwei Tian and Baofeng Chen. 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.
The exponential stability is investigated for neutral stochastic differential equations with time-varying delays. Based on the Lyapunov stability theory and linear matrix inequalities (LMIs) technique, some delay-dependent criteria are established to guarantee the exponential stability in almost sure sense. Finally a numerical example is provided to illustrate the feasibility of the result.
Neutral differential equations are well-known models from many areas of science and engineering, wherein, quite often the future state of such systems depends not only on the present state but also involves derivatives with delays. Deterministic neutral differential equations were originally introduced by Hale and Meyer  and discussed in Hale and Lunel (see ) and Kolmanovskii et al. (for details, see also [3, 4]), among others. Motivated by chemical engineering systems as well as theory of aeroelasticity, stochastic neutral delay systems have been intensively studied over recent year [5–9]. Mao initiated the study of exponential stability of neutral stochastic differential delay equations in , while  incorporated Razumikhinis approach in neutral stochastic functional differential equations to investigate the stability problem. It is pointed out in Section 5  that the conditions imposed in [5, 9] make the theory not applicable to the delay equation.
More recently, Luo et al.  proposed new criteria on exponential stability of neutral stochastic delay differential equations. In [11, 12], Milošević investigated the almost sure exponential stability of a class of highly nonlinear neutral stochastic differential equations with time-dependent delay, and some sufficient conditions were given for the considered systems. However, when the exponential stability of the neutral system with time-delay is considered, one always assumes that the derivative of the delay function is less than 1 (e.g., ). Meanwhile, the delay-independent conditions in [6, 10] are restricted when the delay is small. On the other hand, some results are proposed on stochastic Markovian jumping systems (e.g., [13–20]) and finite-time problems of stochastic systems (e.g., [18–22]), which can provide some useful methods and techniques for the neutral stochastic systems. This paper aims to develop the exponential stability in almost sure sense of the neutral stochastic differential equations with time-varying delays. Under the weaker assumptions that the derivative of time delay is less than some constant, sufficient conditions for the exponential stability are given in terms of linear matrix inequality (LMI) based on Lyapunov stability theory, which can be checked easily by MATLAB LMI Toolbox.
The paper is organized as follows. In the remainder of this section we recall some preliminaries, mainly from . In Section 3 we state the main results on exponential stability. Section 4 will provide numerical examples to illustrate the feasibility and effectiveness of the results, and the conclusion will be made in Section 5.
Throughout this paper, unless otherwise specified, let be a complete probability space with a filtration satisfying the usual conditions (i.e., right continuous and containing all -null sets). Let be -dimensional Wiener process defined on the probability space. Let denote the Euclidean norm in . stands for the transpose of the vector or matrix . If is a matrix, its trace norm is denoted by . denotes . are maximum eigenvalue and minimum eigenvalue, respectively.
Consider the following -dimensional neutral stochastic differential delay equations with time-varying delays: where , , and . The functions are continuously differentiable such that . Let denote the family of continuous functions from to with the norm . Let be the family of all -measurable -valued random variables such that . To guarantee the existence and uniqueness of the solution, we first list the following hypotheses.(H1)Both the functionals and satisfy the uniform Lipschitz conditions. That is, there is a diagonal positive matrix such that for all and those .(H2)There is a constant such that for all It is well known (see, e.g., ) that under hypotheses H1, H2 (1) has a unique continuous solution on .
To obtain sufficient conditions on almost sure exponential stability, the following lemmas and definition are given.
Lemma 1 (see ). For any positive definite constant matrix , scalar , and vector function such that the integrations in the following are well defined, then the following inequality holds: The following semimartingale convergence theorem will play an important role in the later parts.
Lemma 2 (see ). Let and be two continuous adapted increasing processed on with a.s. Let be a real-valued continuous local martingale with a.s. Let be a nonnegative -measurable random variable. Define If is nonnegative, then where a.s. means . In particular, if a.s., then for almost all , that is, both and converge to finite random variables.
3. Main Results
Theorem 4. Let hypotheses H1, H2 hold. System (1) is almost sure exponentially stable, if there exists positive definite matrix such that the following LMI holdswhere
Proof. To confirm that the stochastic neutral differential equation (1) is mean-square exponentially stable with decay rate , we define a Lyapunov-Krasovskii functional as follows:
For simplicity, let , . By generalizing Ito’s formula, we have that
Then, the derivative of along the solution of (1) gives
Note that, from system (1) and Newton-Leibniz formula, we have
By calculation, it is clear that
and then by which, we have
Moreover, by Lemma 2, one can get
Letting and substituting (14)–(17) into (13) yield
Furthermore, from (14), it follows that
where , , , , , and , are matrices with compatible dimensions.
It can be shown that where is defined as By Schur complement, we know that . On the other hand, it follows that Note that is bounded and are continuous; then must be nonnegative bounded. Moreover, can be obtained directly: By applying Lemma 2 to (23), one sees that hence there exists a positive random variable satisfying Since, for any we must have From the above inequality (26), it yields the desired result That completes the proof.
In this section, a numerical example will be given to illustrate that the proposed method is effective.
Example 1. Consider the following system: where the delay function is defined as . It is obvious that (29) satisfies the assumptions H1 and H2, and here , . Moreover, since , then .
Remark 5. Comparing with some existing sufficient criteria for neural stochastic differential equations (e.g., [6, 11, 12]), the obtained result is given in terms of linear matrix inequality (LMI), which can be easily checked by MATLAB LMI Toolbox.
The exponential stability is investigated for a class of neutral stochastic differential equations with time-varying delays. In order to overcome the difficulties, we introduce suitable Lyapunov functionals and employ linear matrix inequalities (LMIs) technique, and then a delay-dependent criteria are given to check the almost sure exponential stability of the concerned equations.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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