New Robust Exponential Stability Criterion for Uncertain Neutral Systems with Discrete and Distributed Time-Varying Delays and Nonlinear Perturbations

and Applied Analysis 3 where h, r, g, μ, and δ are given positive constants. The initial condition functions φ t , φ t ∈ C −max{r, h, g}, 0 ,R denote continuous vector-valued initial function of t ∈ −max{r, h, g}, 0 with the norm ‖φ‖ sups∈ −max{r,h,g},0 ‖φ s ‖, ‖φ‖ sups∈ −max{r,h,g},0 ‖φ s ‖. The uncertain matrices ΔA t , ΔB t , and ΔD t are norm bounded and can be described as ΔA t EF t N1, ΔB t EF t N2, ΔD t EF t N3, 2.3 where E,Ni, i 1, 2, 3, are known constant matrices with appropriate dimensions. The uncertain matrix F t satisfies


Introduction
In the past decades, the problem of stability for neutral differential systems, which have delays in both its state and the derivatives of its states, has been widely investigated by many researchers.Such systems are often encountered in engineering, biology, and economics.The existence of time delay is frequently a source of instability or poor performances in the systems.Recently, some stability criteria for neutral system with time delay have been given 1-13 .The problem of delay-dependent stability for uncertain neutral systems with time-varying delays was considered 6 .Reference 13 investigated the problem of delaydependent robust stability for delay neutral type control system with time-varying structured uncertainties and time-varying delays.The problem of a new delay-dependent robust stability criterion for neutral systems was investigated.The considered system has timevarying structured uncertainties and interval time-varying delays in 7 .However, many researchers have studied the problem of stability for systems with discrete and distributed delays such as 12 which presented some stability conditions for uncertain neutral systems with discrete and distributed delays.The robust stability of uncertain linear neutral systems

Problem Formulation and Preliminaries
We introduce some notations and definitions that will be used throughout the paper.R denotes the set of all real nonnegative numbers; R n denotes the n-dimensional space with the vector norm • ; x denotes the Euclidean vector norm of x ∈ R n ; R n×r denotes the set of n × r real matrices; A T denotes the transpose of the matrix A; A is symmetric if A A T ; I denotes the identity matrix; λ A denotes the set of all eigenvalues of A; denotes the space of all continuous vector functions mapping −h, 0 into R n ; * represents the elements below the main diagonal of a symmetric matrix.
Consider the uncertain neutral system with discrete and distributed time-varying delays and nonlinear perturbations of the form where x t ∈ R n is the state, A,B,C, D ∈ R n×n are known real constant matrices.r t , h t , and g t are neutral, discrete, and distributed time-varying delays, respectively,

2.4
The uncertainties f 1 • and f 2 • represent the nonlinear parameter perturbations with respect to the current state x t and the delayed state x t − h t , respectively, and are bounded in magnitude: where η, ρ are given positive constants.First, we consider nominal system 2.1 which is defined to be

2.7
In order to improve the bound of the discrete delay h t in 2.6 , let us decompose the constant matrix B as where B 1 , B 2 ∈ R n×n are constant matrices.Utilize the following zero equation: where G ∈ R n×n will be chosen to guarantee the exponential stability of the system 2.6 .By 2.8 and 2.9 , the system 2.7 can be represented in the form of a descriptor system with discrete and distributed delays and nonlinear perturbations: x s ds,

2.10
where Definition 2.1.The system 2.1 is robustly exponentially stable if there exist real scalars α > 0, M > 0 such that, for each φ t ∈ C − max{r, h, g}, 0 , R n , the solution x t, φ, ϕ of the system 2.1 satisfies Lemma 2.2 Jensen's inequality 17 .For any constant symmetric matrix R > 0, scalar h > 0, and vector function ẋ t : −h, 0 → R n such that the following integral is well defined, one has

Exponential Stability Conditions
In this section, we first study the exponential stability criteria for the nominal system 2.1 by using the combination of linear matrix inequality LMI technique and Lyapunov method. where Moreover, the solution x t, φ, ϕ satisfies the inequality where Proof.Construct a Lyapunov functional candidate for the nominal system 2.1 of the form where e 2α s−t x T θ P 6 x θ dθ ds.

3.5
The derivative of V • along the trajectories of nominal system 2.1 is given by x s ds − 2αV 6 t .

3.8
From the Leibniz-Newton formula, the following equation is true for any real matrices M i , i 1, 2, . . ., 8, with appropriate dimensions y s ds 0. 3.9 From 2.5 , we obtain for any scalars 1 , 2 > 0

3.10
Let us define W 1 P 1 G. From 3.6 , 3.9 , and 3.10 , we obtain It is a fact that, if Ω < 0, then From 3.14 , it is easy to see that where

3.18
This means that the system 2.1 is exponentially stable.The proof of the theorem is complete. where where Γ T E T Q 1 , E T Q 2 , 0, 0, 0, 0, 0, 0 and Π N 1 , 0, N 2 , 0, 0, 0, 0, N 3 .Using Lemma 1 in 9 , we have that there exists a positive number ε > 0. We can find that 4.4 is equivalent to the LMI as follows: Abstract and Applied Analysis 13 Applying Lemma 1 Schur complement lemma in 12 , we obtain Hence, we conclude that 4.6 is equivalent to 4.1 .The proof of the theorem is complete. If where

5.3
Decompose matrix B as follows which is the same as the decomposition in Example 5.1.

5.4
Decompose matrix B as follows which is the same as the decomposition in Example 5.1.Table 3 shows the comparison of the upper bounds delay allowed obtained from Corollary 4.2 for asymptotic stability α 0 of system 4.7 with 5.4 by other methods.It can be found from Table 3 that our results are significantly better than those in 2, 5, 9 .

Conclusions
The problem of robust exponential stability for uncertain neutral systems with discrete and distributed time-varying delays and nonlinear perturbations was studied.Based on the combination of descriptor model transformation, decomposition technique of coefficient matrix, and utilization of zero equation and new Lyapunov functional, sufficient conditions for robust exponential stability were obtained and formulated in terms of linear matrix inequalities LMIs .Numerical examples have shown significant improvements over some existing results.

ee
2αs y T s P 5 y s ds, 2αs x T θ P 6 x θ dθ ds.

Example 5 . 3 .
Consider the asymptotic stability α 0 of the uncertain neutral system 4.7 with time-varying delays with A Assumption 1.All the eigenvalues of matrix C are inside the unit circle.

Table 1 :
Comparison of the maximum allowed time delay h max .In order to show the effectiveness of the approaches presented in Sections 3 and 4, three numerical examples are provided.

Table 1
lists the comparison of the upper bounds delays for asymptotic stability α 0 of system 5.1 by different methods.We can see from Table1that our results Corollary 3.2 are superior to those in Lemma 4 16 , Lemma 1 4 , and Theorem 1 18 .

Table 2 :
Comparison of the maximum allowed time delay h max .

Table 3 :
Comparison of the maximum allowed time delay h max .

Table 2
lists the comparison of the upper bounds delay for exponential stability of 5.3 by different methods.It is clear that our results Corollary 3.3 are superior to those in Theorem 1 1 .