Delay-Dependent Robust Exponential Stability for Uncertain Neutral Stochastic Systems with Interval Time-Varying Delay

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.


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 delaydependent 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 τ t < 1 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 delaydependent mean-square exponential stability criterion is formulated in terms of LMI, and the usual restriction of τ t < 1 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 A is a vector or matrix, its transpose is denoted by A T .P > 0 means that P is a symmetric positive definite matrix.
* denotes the symmetry part of a symmetry matrix.indicates terms that can be induced by symmetry, for example, A A T A and MP M T MP .Denote by λ M {•} and λ m {•} the maximum and minimum eigenvalue of a matrix, respectively.Let ρ • denote the spectral norm of a matrix, while | • | refers to the Euclidean vector norm.Let Ω, F, P be a complete probability space relative to an increasing family {F} t≥0 of σ algebras F t ⊂ F and E{•} the mathematical expectation operator with respect to the probability measure P. L 2 0, ∞ denotes the space of square integrable vector functions over 0, ∞ .Let τ > 0 and C −τ, 0 , R n denote the family of all continuous R n -valued functions φ on −τ, 0 with the norm φ sup{|φ θ | : −τ θ ≤ 0}.Let L 2 F 0 −τ, 0 ; R n be the family of all F 0 -measurable bounded C −τ, 0 , R n -valued random variables ϕ {ϕ θ : −τ θ 0}.

Preliminaries
Consider the following uncertain It ô-type neutral stochastic system with both discrete and distributed interval time-varying delays x s ds x s ds dB t , where x t ∈ R n is the state vector; B t is a standard scalar Brownian motion defined on a complete probability space Ω, F, P , we assume E{dB t } 0,

2.3
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 τ t < 1 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 τ t τ, 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.
A i , B j , i, j 0, 1, 2, are known real constant matrices of appropriate dimensions; ΔA i , ΔB j , i, j 0, 1, 2, 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 S i , T i , i 0, 1, 2, are known real constant matrices, F t is an unknown real timevarying function with appropriate dimension satisfying It is assumed that the elements of F t are Lebesgue measurable.Throughout this paper, the following assumption, definitions, and lemmas will be used to develop our results.

Main Results
In this section, a robustly stochastically exponentially stable criterion for the uncertain linear neutral stochastic distributed delayed system Σ will be established by applying the Lyapunov-Krasovskii theory and free-weighting matrices method.
For convenience, define a new state variable x s ds 3.1 and a new perturbation variable x s ds.

3.3
Applying Leibniz-Newton formula to 2.1 , it yields the following zero equations which will be used in our main result: where N, H, and M are any matrices with appropriate dimensions, and

3.5
With zero equations 3.4 above, we obtain the following theorem.

3.8
Proof.Choose a Lyapunov-Krasovskii functional candidate as where x s ds ; by Lemma 2.5, tr g T s T R 10 g s ds.

3.14
and by 2 of Lemma 2.6, for any 2 > 0 satisfying

3.16
In addition, it follows from 3.4 , 3 of Lemma 2.6 and Lemma 2.5 that 3.17 Note that g T s R 9 R 10 g s ds.

3.19
Applying inequalities from 3.13 to 3.19 to 3.12 , it yields where where ν λ m {−Λ}.Now, integrating both sides of 3.23 from 0 to t > 0, and then taking the expectation, we obtain On the other hand, it follows from 3.9 that Therefore, by 3.24 and 3.25 Let σ be any nonnegative real number.For all 0 t σ, it follows from 1 of Lemma 2.6 that

3.29
However, this holds for all −τ t 0 as well.Therefore, sup

3.36
Note that there exists a scalar α > 0 such that
Remark 3.2.Theorem 3.1 provides a delay-dependent condition with a new Lyapunov-Krasovskii functional, which makes full use of the relationship among the time-varying delay, its upper and lower bounds, and their difference.It is noted that this condition is obtained by free-weighting matrices techniques; model transformation is not resorted to in the derivation of Theorem 3.1.Thus, the conservatism caused by model transformation will be reduced.
If τ 1 τ 2 τ 0 , then τ 0, τ t τ 0 .Applying Leibniz-Newton formula to 2.1 , it yields the following zero equations: where L is any matrices with appropriate dimension, and x T s ds, x T t − 2τ 0 .

3.40
With zero equations 3.39 above, we obtain the following corollary.

Journal of Applied Mathematics
Applying Leibniz-Newton formula to 3.44 , it yields the following zero equations which will be used in our main result where N, H, and M are any matrices with appropriate dimensions, and

3.47
With zero equation 3.46 above, we obtain the following corollary.

3.52
Applying Leibniz-Newton formula to 3.51 , it yields the following zero equations where L is any matrices with appropriate dimension, and

3.54
With zero equations 3.53 above, we obtain the following corollary.

3.57
Deterministic systems may be regarded as special class of stochastic systems, let B i 0, ΔB i 0, i 0, 1, 2, then system 2.1 -2.2 becomes the following uncertain neutral system with both discrete and distributed interval time-varying delays x s ds,

3.59
Applying Leibniz-Newton formula to 3.58 , it yields the following zero equations which will be used in our main result: f s ds 0,

3.60
where N, H, and M are any matrices with appropriate dimensions, and ξ t is defined in 3.5 .With zero equations 3.60 above, we obtain Corollary 3.6.For given scalars 0 τ 1 τ 2 and τ, system Σ 0 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 P > 0, R i > 0, i 0, 1, 2, . . ., 8, scalars j , j 1, 2, 3 and any matrices N, H, M with appropriate dimensions, such that the following LMI 3.61 holds: where

3.63
Proof.Choose a Lyapunov-Krasovskii functional candidate as where V i t, ξ t , i 0, 1, 2, 3 are defined in 3.9 .It can be concluded from Theorem 3.1.

3.65
Applying Leibniz-Newton formula to 3.65 , it yields the following zero equations: where L is any matrices with appropriate dimension, f t A 0 x t A 1 x t − τ 0 , and

3.67
With zero equations 3.66 above, we obtain the following corollary.
Corollary 3.7.For given constant delay τ 0 , system 3.65 is robustly stochastically exponentially stable, if there exist symmetric positive definite matrices P 0 > 0, Z 1 > 0, Z 2 > 0, Z 3 > 0, Z 4 > 0, and any matrices L with appropriate dimensions, such that the following LMI 3.68 holds:  values with zero mean and variance 1.0, and F t sin t , the trajectories of x 1 t and x 2 t are shown in Figure 1.According to Theorem 3.1, the lower bounds and the upper bounds on the time delay to guarantee the system is robustly stochastically mean-square exponentially stable are listed in Table 1.The maximum upper bound τ M for this system in case of different cs is listed in Table 3, which shows that the results obtained by the methods proposed in this paper are less conservative than that in 19 .

Conclusion
The mean-square exponential stability for uncertain neutral stochastic system with both discrete and distributed time-varying delays has been investigated in this paper.Sufficient conditions have been established without ignoring any terms in the weak infinitesimal operator of Lyapunov-Krasovskii functional by considering the relationship among the timevarying delay, its upper bounds, and their difference.The usual restriction that τ t < 1 has been removed by free-weight matrices method.According to the numerical examples, it has been shown that the proposed results improve some existing ones.

Figure 1 :
Figure 1: Mean-square exponential stability of the neutral stochastic system with interval time-varying delay.

Example 4 . 3 .IfExample 4 . 4 .
Consider the uncertain neutral linear stochastic system 3.51 with Corollary 3.5 in this paper is applied, the maximal admissible delay τ M of this example is τ M 0.6001.Consider the uncertain neutral linear stochastic system 3.65 with R 10 g s ds dθ.
3.10 According to It ô's differential formula 1 , the stochastic differential is dV t, x t LV t, x t dt 2 x t − Dx t τ T Pg t dB t , 3.11 where LV t, x t dt 2 x t − Dx t τ T Pf t g t T Pg t

Table 1 :
Allowable lower bounds and upper bounds for different τ.
T, where x 1 θ , x 2 θ are random initial

Table 3 :
Allowable upper bounds of τ 2 for different c.According to Corollary 3.4, for τ 1 0, the upper bounds on the time delay to guarantee the system is robustly stochastically mean-square exponentially stable are listed in Table2.At the same time, Table2also lists the upper bounds obtained from the criterion in 15 .