Abstract

This paper discusses the mean-square exponential stability of uncertain neutral linear stochastic systems with interval time-varying delays. A new augmented Lyapunov-Krasovskii functional (LKF) has been constructed to derive improved delay-dependent robust mean-square exponential stability criteria, which are forms of linear matrix inequalities (LMIs). By free-weight matrices method, the usual restriction that the stability conditions only bear slow-varying derivative of the delay is removed. Finally, numerical examples are provided to illustrate the effectiveness of the proposed method.

1. Introduction

Dynamical systems, such as the distributed networks, electric power systems, and communication systems, can be efficiently modeled by neutral stochastic functional differential equations, which have been extensively studied in recent years, see [1–8] and references therein. Basically, sufficient conditions on stability of time delay systems are divided into two categories: delay-dependent and delay-independent cases. It is well known that the latter is less conservative than the former, especially when the delay is very small. Therefore, increasing attention has been focused on delay-dependent stability analysis of delay systems in recent years, see [1–25].

There are unstable systems without delay that can be stabilized with proper nonzero delay, see [26]. Therefore, it is of great significance to consider the stability of systems with interval time-varying delay, it is a time delay that varies in an interval, in which the lower bound is not restricted to 0. Recently, interval time-varying delay was introduced and investigated, see [11–17].

In the past years, as an effective approach of improving the performance of delay-dependent stability criteria, free-weighting matrices method has attracted much attention, see [2–5, 10–12]. The mean-square exponential stability was studied for neutral stochastic systems with fixed delays in [2, 3]. The global asymptotic stability for a class of neutral stochastic neural networks was studied in [4]. Improved delay-dependent robust stability criteria of uncertain stochastic systems with interval time-varying delay were proposed in [12].

The usual restriction on derivative of the time-varying delay that in [5–7, 18] brings conservatism, which shows that the resulting conditions can only bear slow-varying delay. In order to remove this limitation, fast-varying rate condition was studied in [4, 12, 23].

Stability of stochastic differential delay systems with nonlinear impulsive effects is studied in [24, 25], the equivalent relations are established between the stability of the stochastic differential delay equations with impulsive effects and that of a corresponding stochastic differential delay equations without impulses.

To our best knowledge, few works on the robust delay-dependent exponential stability analysis have been reported for uncertain neutral stochastic distributed system with fast-varying interval time-varying delay. This paper focuses on the stability analysis of uncertain neutral stochastic distributed system with interval time-varying delay. By Lyapunov-Krasovskii functional theory and free-weighting matrices method, a new delay-dependent mean-square exponential stability criterion is formulated in terms of LMI, and the usual restriction of is removed. Finally, three numerical example are given to illustrate the effectiveness of the proposed method.

Notation 1. Throughout this paper, the notations are standard. If is a vector or matrix, its transpose is denoted by . means that is a symmetric positive definite matrix. * denotes the symmetry part of a symmetry matrix. indicates terms that can be induced by symmetry, for example, and . Denote by and the maximum and minimum eigenvalue of a matrix, respectively. Let denote the spectral norm of a matrix, while refers to the Euclidean vector norm. Let be a complete probability space relative to an increasing family of algebras and the mathematical expectation operator with respect to the probability measure . denotes the space of square integrable vector functions over . Let and denote the family of all continuous valued functions on with the norm . Let be the family of all -measurable bounded -valued random variables .

2. Preliminaries

Consider the following uncertain Itô-type neutral stochastic system with both discrete and distributed interval time-varying delays where is the state vector; is a standard scalar Brownian motion defined on a complete probability space , we assume . is any given initial data in . Denote , is time-varying delays, satisfying

Remark 2.1. It should be noted that system (2.1)-(2.2) encompasses many state space models of neutral delay systems, which can be used to represent many important physical systems such as a large class of distributed networks containing lossless transmission lines, population ecology, heat exchangers, wind tunnel, and water resources systems (see [27–29] and the references therein). For example, signal transformation cannot be finished in time due to the finite signal propagation speed in distributed networks containing lossless transmission lines, or to the finite switching speed of amplifiers in electronic networks, so the models only depending on the discrete time delays are not complete, the more exact models should include the distributed time delays, and the delays are found both in the states and in the derivatives of the states.

Remark 2.2. It is worth pointing out that, when a time-varying delay appears, it is usually assumed that is satisfied and the lower bound of the delay is restricted to be zero in the literature [5, 7, 8] and so forth. However, in this paper, we only require , in addition, the range of delay may vary in a range for which the lower bound is not restricted to be zero. Therefore, the time-varying delay in this paper is more general.
, are known real constant matrices of appropriate dimensions; , are unknown matrices representing time-varying parameter uncertainties. For the sake of convenience, we assume that the uncertainties are norm-bounded and can be described as where , are known real constant matrices, is an unknown real time-varying function with appropriate dimension satisfying It is assumed that the elements of are Lebesgue measurable. Throughout this paper, the following assumption, definitions, and lemmas will be used to develop our results.

Assumption 2.3. The difference operator matrix satisfies .

Definition 2.4 (see [1]). The neutral stochastic delay system (2.1)-(2.2) is said to be mean-square exponentially stable if there is a pair of positive constants such that

Lemma 2.5 (see [30]). If there exists a vector function such that and are well defined, then the following inequality: holds for any pair of symmetric positive definite matrix and .

Lemma 2.6 (see [31]). For any vectors , matrices , and are real matrices of appropriate dimensions with and , the following inequalities hold:(1); (2)for any scalar such that , then ;(3).

3. Main Results

In this section, a robustly stochastically exponentially stable criterion for the uncertain linear neutral stochastic distributed delayed system will be established by applying the Lyapunov-Krasovskii theory and free-weighting matrices method.

For convenience, define a new state variable and a new perturbation variable Then, system (2.1) becomes Applying Leibniz-Newton formula to (2.1), it yields the following zero equations which will be used in our main result: where , , and are any matrices with appropriate dimensions, and With zero equations (3.4) above, we obtain the following theorem.

Theorem 3.1. For given scalars and , system is robustly stochastically exponentially stable for all time-varying delays satisfying (2.3) and for all admissible uncertainties satisfying (2.4) and (2.5), if there exist symmetric positive definite matrices , scalars and any matrices with appropriate dimensions, such that the following LMI (3.6) holds:
where with

Proof. Choose a Lyapunov-Krasovskii functional candidate as where According to Itô's differential formula [1], the stochastic differential is where Direct computations give By of Lemma 2.6, for any scalar , we have and by of Lemma 2.6, for any satisfying , we have For any scalar satisfying , the following inequality holds:
In addition, it follows from (3.4), (3)  of Lemma 2.6 and Lemma 2.5 that Note that Applying inequalities from (3.13) to (3.19) to (3.12), it yields where with being defined in Theorem 3.1.
By applying the Schur complement to (3.6) results in , which implies therefore, where . Now, integrating both sides of (3.23) from to , and then taking the expectation, we obtain On the other hand, it follows from (3.9) that Therefore, by (3.24) and (3.25) Since , choose such that , which implies , denote . Let be any nonnegative real number. For all , it follows from (1) of Lemma 2.6 that Hence By (3.25), (3.27), and (3.28), we then derive that for all , However, this holds for all as well. Therefore, Since and , we obtain that By supremum property, By (3.31) and (3.32), For any , it follows from (3.26), (3.33) and being an arbitrary nonnegative real number that Since is an arbitrary nonnegative number, we have Then, applying Gronwall-Bellman lemma to (3.35), it yields Note that there exists a scalar such that Denote , then we have By Definition 2.4, system () is robustly stochastically mean-square exponentially stable.