Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 748279, 23 pages

http://dx.doi.org/10.1155/2012/748279

## Delay-Dependent Exponential Stability for Uncertain Neutral Stochastic Systems with Mixed Delays and Markovian Jumping Parameters

Department of Mathematics, Nanchang University, Nanchang 330031, China

Received 9 December 2011; Accepted 5 March 2012

Academic Editor: Vimal Singh

Copyright © 2012 Huabin 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.

#### Abstract

This paper is mainly concerned with the globally exponential stability in mean square of uncertain neutral stochastic systems with mixed delays and Markovian jumping parameters. The mixed delays are comprised of the discrete interval time-varying delays and the distributed time delays. Taking the stochastic perturbation and Markovian jumping parameters into account, some delay-dependent sufficient conditions for the globally exponential stability in mean square of such systems can be obtained by constructing an appropriate Lyapunov-Krasovskii functional, which are given in the form of linear matrix inequalities (LMIs). The derived criteria are dependent on the upper bound and the lower bound of the time-varying delay and the distributed delay and are therefore less conservative. Two numerical examples are given to illustrate the effectiveness and applicability of our obtained results.

#### 1. Introduction

It is well known that many dynamical systems not only depend on the present and past states but also involve the derivative with delays as well as the functional of the past history. Neutral delay differential equations are often used to describe the following systems [1]: Many authors have considered the dynamical analysis of the neutral delay differential equations (see [2–7] and references therein). For example, Chen et al. in [3] and Wu et al. in [4, 5] have given some LMI-based conditions ensuring the stability analysis and the stabilization of neutral delay systems. Taking the environmental disturbances into account, the neutral stochastic delay differential equations can be given as follows: Some fundamental theories of neutral stochastic delay differential equations are introduced in [1, 8]. Since they can be extensively applied into many branches for the control field, the problem about the exponential stability and the asymptotical stability of the neutral stochastic delay systems has attracted many authors’ attention over the past few years, and many less conservative results of delay-dependent conditions ensuring the stabilization analysis and filtering design for such systems have been reported in many works, see, for example, [9–14] and references therein. The methods used include the Razumikhin-type theorems [10], the Lyapunov functional [13], the fixed point theorem [14], and the linear matrix inequality [9, 11, 12]. For example, Huang and Mao in [9] and Chen et al. in [11] have given the exponential stability criteria of neutral stochastic delay systems. Some LMI-based sufficient conditions for the mean-square exponential stability analysis of stochastic systems of neutral type have been obtained by introducing an auxiliary vector in [12]. In practice, the parameter uncertainty is considered as one of the main sources leading to undesirable behavior (e.g., instability) of dynamical systems, especially when implementing neural networks in applications. The stability analysis of the uncertain neutral stochastic system has received considerable research attention, see, for example, [15–18], and the problem of the filter design of the uncertain neutral stochastic delay systems has been discussed in [19, 20].

On the other hand, Markovian jump systems introduced by [21] are the hybrid systems with two components in the state. The first one refers to the mode that is described by a continuous-time finite-state Markovian process, and the second one refers to the state that is represented by a system of differential equations. The jump systems have the advantage of modeling the dynamic systems subject to abrupt variation in their structures, such as component failures or repairs, sudden environmental disturbance, changing subsystem interconnections, and operating in different points of a nonlinear plant [22]. The stability analysis and filter design of stochastic delay systems with Markovian jumping parameters and delay systems with Markovian jumping parameters have been widely studied, see, for example, [23–38]. For example, in [24], Liu et al. have discussed the exponential stability of delayed recurrent neural networks with Markovian jumping parameters; Liu et al. in [24] and Wang et al. in [25] have considered some sufficient conditions for the exponential stability of stochastic neural networks with mixed time delays and Markovian switching; Mao in [13] has also given some sufficient conditions for the exponential stability of stochastic delay interval systems with Markovian switching. More recently, He and Liu in [39] and Balasubramaniam et al. in [40] have presented some LMI-based sufficient conditions for the exponential stability of uncertain neutral systems with Markovian jumping parameters. Although Kolmanovskii et al. in [17], and Mao et al. in [18] have derived the exponential stability of the neutral stochastic delay systems with Markovian jumping parameters, some sufficient conditions obtained by using the estimate method are not easily checked. Thus, the problem of the stability analysis of the uncertain neutral stochastic systems with mixed delays and Markovian jumping parameters has not been fully investigated and there is still much room left for further consideration, which constitutes the motivation for the present research.

In this paper, the global exponential stability of a class of the uncertain neutral stochastic systems with mixed delays and Markovian jumping parameters is discussed. The delays include the discrete and distributed time delays, and the jumping parameters are generated from a finite state Markov chain. By constructing an appropriate Lyapunov functional, some LMIs-based sufficient conditions ensuring the exponential stability in mean square of the uncertain neutral stochastic systems with mixed delays and Markovian jumping parameters are obtained by using the stochastic analysis and some bounding technique. It is worth pointing out that compared with the earlier works in [17, 18], the obtained results given in the form of the linear matrix inequalities (LMIs) can be easily be solved by using the standard software packages. Two illustrative examples are exploited to demonstrate the effectiveness and applicability of the obtained results.

The content of the paper is arranged as follows. In Section 2, some necessary notations, definitions, and lemmas will be introduced. In Section 3, we mainly study the exponential stability in mean square of the uncertain neutral stochastic systems with mixed delays and Markovian jumping parameters. Two illustrative numerical examples are given to show the power of our obtained results in Section 4.

*Notations. *Unless otherwise specified, for a real square matrix , the matrix (, , ) means that is a positive definite (positive semidefinite, negative definite, and negative semidefinite, resp.); and denote the maximum and minimum eigenvalues of the square matrix , respectively. Let be a probability space with a natural and let stand for the mathematical expectation operator with respect to this probability measure. If is a vector or matrix, its transpose is denoted by . denotes the Euclidean norm of a vector and its induced norm of a matrix . Unless explicitly stated, matrices are assumed to have real entries and compatible dimensions. Let and be the family of all continuous -valued functions on the interval with the norm . Denote by the family of all -measurable -valued random variables such that . () is an -dimensional standard Brownian motion defined on the completed probability space .

#### 2. Problem Formulation

Let be a right-continuous Markov chain on the probability space taking values in a finite state space with generator given by where . Here, is the transition rate from to if while Now, we assume that the Markov chain is independent of the Brownian motion . It is well known that almost every sample path of is a right-continuous step function with a finite number of simple jumps in any finite subinterval of .

In this paper, we will consider the following uncertain neutral stochastic systems with mixed delays and Markovian switching: with the initial value (), where is the system state vector associated with the neurons and and are the time-varying delays. Here, we assume that the Markov chain is independent of the Brownian motion . is neuron activation function, and the noise perturbation is the noise intensity matrix. When , , , , , and are, respectively, denoted as , , , , and , and () are known matrices with . , , , and are matrix functions with time-varying uncertainties, that is, where , , , and () are known real constant matrices and , , , and () are unknown matrices representing time-varying parameter uncertainties in system model. We assume that the uncertainties are norm-bounded and can be described as where are known real matrices and is unknown real and possibly time-varying matrix for any given . It is assumed that the elements of are Lebesgue measurable. When , systems (2.3) have the following nominal case:

In order to obtain our results, we need some assumptions as follows.

*Assumption 2.1. *The neuron activation functions in (2.3) (or (2.6)) are bounded and satisfy the following Lipschitz condition:
where is known constant matrix and .

*Assumption 2.2. *The noise perturbation satisfies the following condition:
where , , and are known constant matrices with appropriate dimensions and ().

*Remark 2.3. *Under Assumptions 2.1 and 2.2, it is easily shown that the system (2.3) with uncertainties (2.4) admits a unique trivial solution when the initial data . The readers can refer to [41].

*Assumption 2.4. * and are two time-varying continuous functions that satisfy
where and are the lower and upper bounds of the time delay , respectively.

We present the definitions and three useful lemmas as follows.

*Definition 2.5. *The neutral stochastic systems with mixed delays and Markovian jump parameters (2.6) is said to be exponentially stable in mean square if there exist a pair of positive scalar and such that every solution of systems (2.6) satisfies
for any .

*Definition 2.6. *The uncertain neutral stochastic systems with mixed delays and Markovian jump parameters (2.3) are said to be exponentially stable in mean square if (2.10) holds for all admissible uncertainties (2.5).

Lemma 2.7 (see [8]). *For any vectors , the inequality
**
holds, in which is any matrix with .*

Lemma 2.8 (see [8]). *Let ; then
**
for any if is a symmetric matrix.*

Lemma 2.9 (see [42] Schur complement). *For a given matrix
**
with , , the following conditions are equivalent:*(1)*,*(2)*, ,*(3)*, .*

Lemma 2.10 (see [42]). *Let , , , and be real matrices of appropriate dimensions with satisfying ; then
**
if and only if there exist a scalar such that
*

Lemma 2.11 (see [43]). * For any positive symmetric constant matrix and a scalar , a vector function such that the integrations concerned are well defined, and then the following inequality holds:
*

#### 3. Main Results

Theorem 3.1. *Suppose that Assumptions 2.1–2.4 hold and for any given positive scalar , the neutral stochastic systems with mixed delays and Markovian switching (2.6) are exponentially stable in mean square if there exist () and some positive definite matrices () and () such that the following linear matrix inequalities (LMIs) are satisfied: for ,
**
where denotes the entries that are readily inferred bysymmetry of a symmetric matrix and
*

*Proof. *Denote by the family of all nonnegative functions on that are once differentiable with respect to the first variable and twice differentiable with respect to the second variable . To obtain the stability conditions, we consider the following Lyapunov functional:
where

The weak infinitesimal operator [17] along (2.6) from to is given by
where
By Lemma 2.7, we have
From Lemmas 2.7 and 2.11, it implies that
Substituting (3.8)–(3.13) into (3.7) and from Assumption 2.2, it follows that
On the other hand, we can obtain
Substituting (3.14)–(3.16) into (3.6), we have
where and, for ,
In view of (3.2), we have , for . By the Lyapunov functional ,
where

Letting , for system (2.6), we can define another operator as follows:

Now, we can choose sufficiently small such that
for .

By the weak infinitesimal operator along (2.6), it is obtained from (3.21) and (3.22) that

Using the definition of the Lyapunov functional (3.4) again, we have
for .

Thus, from (3.23) and (3.24), it follows that
where and .

From (), we obtain that there exist a positive scalar such that . So, for all and any , by using the elementary inequality, we derive
From for all , we have . Thus, we can choose sufficiently large such that

So,

For all , from (3.28), it follows that

that is,

When , it follows from (3.30) that
So, we can obtain
The proof of this theorem is completed.

Theorem 3.2. *Suppose that Assumptions 2.1–2.4 hold and for any given positive scalar , the uncertain neutral stochastic systems with mixed delays and Markovian switching (2.3) are robustly exponentially stable in mean square if there exist (), , (), and some positive definite matrices () and () such that the following linear matrix inequalities (LMIs) are satisfied:
**
where denotes the entries that are readily inferred by symmetry of a symmetric matrix and
*

*Proof. *Replacing , , , and in (3.2) with , , , and , , , , are described in (2.4) and (2.5), in view of Lemma 2.9 and Lemma 2.10, we obtain
where
where and , for any . The proof of the remainder can be easily finished by following a similar line as in the proof of Theorem 3.1. The proof is completed.

*Remark 3.3. *The delay-dependent sufficient conditions ensuring the robustly exponential stability in mean square of the uncertain neutral stochastic systems with mixed delays and Markovian switching (2.3) are provided in Theorem 3.2. It should be pointed out that such conditions are given in the form of LMIs, which could be easily solved by using the standard software packages. Besides, the criteria derived are dependent upon both the upper and lower bound of the time-varying delay and the distributed delay, which are less conservative.

*Remark 3.4. *Besides, by the Borel-Cantelli Lemma, we can also obtain the almost surely exponential stability of systems (2.3). Here, for the sake of brevity, we omit it and the readers can refer to [17]. Besides, we can easily come to a conclusion that the uncertain neutral stochastic delay systems with mixed delays and Markovian jumping parameters (2.3) are asymptotically stable in mean square from the conditions (). Thus, the results given in [15] are generalized.

*Case 1. *Consider the problem of delay-dependent robust exponential stability for a special case of the uncertain neutral stochastic systems with mixed delays and Markovian switching in (2.6), that is,
where , , , and are given in (2.3), while , , and are of the following form:
where , , , , , , and () are known constant matrices. For systems (3.37), we can also obtain the robust exponential stability. The proof can be easily established by following a similar line as in the proof of Theorem 3.1 and then is omitted here.

Theorem 3.5. *Suppose that Assumptions 2.1 and 2.2 hold and for any given positive scalar , the uncertain neutral stochastic systems with mixed delays and Markovian switching (3.37) are robustly exponentially stable in mean square if there exist , , , () and some positive definite matrices () and () such that the following linear matrix inequalities (LMIs) are satisfied:
**
where denotes the entries that are readily inferred by symmetry of a symmetric matrix and
*

*Case 2. *When , the uncertain neutral stochastic systems (2.3) are described as

Theorem 3.6. *Suppose that Assumptions 2.1 and 2.2 hold and for any given positive scalar , the uncertain neutral stochastic systems with mixed delays and Markovian switching (3.41) are robustly exponentially stable in mean square if there exist (), () and some positive definite matrices () and () such that the following linear matrix inequalities (LMIs) are satisfied: for ,
*