#### Abstract

We investigate the problem of robust exponential stability for uncertain neutral systems with discrete and distributed time-varying delays and nonlinear perturbations. 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 are obtained and formulated in terms of linear matrix inequalities (LMIs). The new stability conditions are less conservative and more general than some existing results.

#### 1. 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 delay-dependent 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 time-varying 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 with discrete and distributed delays has been studied in [3]. In [14, 15], they studied the problem of stability for linear switching system with discrete and distributed delays. Moreover, a descriptor model transformation and a corresponding Lyapunov-Krasovskii functional have been introduced for stability analysis of systems with delays in [10, 16].

In this paper, we will study the problems of robust exponential stability for uncertain neutral systems with discrete and distributed time-varying delays and nonlinear perturbations. Based on a combination of descriptor model transformation, decomposition technique of coefficient matrix, and utilization of zero equation and new Lyapunov functional which improved delay-dependent stability criteria for the considered systems, sufficient conditions for the robust exponential stability are obtained and formulated in terms of linear matrix inequalities (LMIs). The new stability conditions will be less conservative and more general than some existing results.

#### 2. Problem Formulation and Preliminaries

We introduce some notations and definitions that will be used throughout the paper. denotes the set of all real nonnegative numbers; denotes the -dimensional space with the vector norm ; denotes the Euclidean vector norm of ; denotes the set of real matrices; denotes the transpose of the matrix ; is symmetric if ; denotes the identity matrix; denotes the set of all eigenvalues of ; ; ; matrix is called semipositive definite () if , for all is positive definite () if for all ; matrix is called seminegative definite () if , for all is negative definite () if for all ; means ; means ; denotes the space of all continuous vector functions mapping into ; 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 is the state, ,, are known real constant matrices. , and are neutral, discrete, and distributed time-varying delays, respectively, where are given positive constants. The initial condition functions denote continuous vector-valued initial function of with the norm , . The uncertain matrices , , and are norm bounded and can be described as where , are known constant matrices with appropriate dimensions. The uncertain matrix satisfies The uncertainties and represent the nonlinear parameter perturbations with respect to the current state and the delayed state , respectively, and are bounded in magnitude: where are given positive constants. First, we consider nominal system (2.1) which is defined to be Rewrite the system (2.6) in the following descriptor system: In order to improve the bound of the discrete delay in (2.6), let us decompose the constant matrix as where are constant matrices. Utilize the following zero equation: where 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: where .

*Definition 2.1. *The system (2.1) is robustly exponentially stable if there exist real scalars , such that, for each , the solution of the system (2.1) satisfies

Lemma 2.2 (Jensen's inequality [17]). *For any constant symmetric matrix , scalar , and vector function such that the following integral is well defined, one has
*

#### 3. 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.

*Assumption 1. *All the eigenvalues of matrix are inside the unit circle.

Theorem 3.1. *Under Assumption 1 and given positive constants , and , the nominal system (2.1) is exponentially stable if there exist positive definite matrices , , any appropriate dimensional matrices , , and positive constants satisfying the following LMI:
**
where ,,,,, ,,, ,,,,,,, Moreover, the solution satisfies the inequality
**
where
*

*Proof. *Construct a Lyapunov functional candidate for the nominal system (2.1) of the form
where
The derivative of along the trajectories of nominal system (2.1) is given by
where
Obviously, for any scalar , we get . Together with Lemma 2.2 (Jensen's inequality), we obtain
From the Leibniz-Newton formula, the following equation is true for any real matrices , with appropriate dimensions
From (2.5), we obtain for any scalars
Let us define . From (3.6), (3.9), and (3.10), we obtain
where and
It is a fact that, if , then
which gives
From (3.14), it is easy to see that
where
Therefore, we get
where . From (3.17), we get
This means that the system (2.1) is exponentially stable. The proof of the theorem is complete.

If , then system (2.1) reduces to the following system: Take the Lyapunov functional as where , and are defined in Theorem 3.1. According to Theorem 3.1, we have the following corollary for the delay-dependent exponential stability criterion of system (3.19).

Corollary 3.2. *For given positive constants , the system (3.19) is exponentially stable if there exist positive definite matrices , , and any appropriate dimensional matrices , , satisfying the following LMI:
**
where ,,,,,,,, , . Moreover, the solution satisfies the inequality
**
where
*

If , then system (2.1) reduces to the following system: Take the Lyapunov functional as where , and are defined in Theorem 3.1. According to Theorem 3.1, we have the following corollary for the delay-dependent exponential stability criterion of system (3.24).

Corollary 3.3. *For given positive constants , and , the system (3.24) is exponentially stable if there exist positive definite matrices , , any appropriate dimensional matrices , , and positive constants satisfying the following LMI:
**
where ,, , , ,,, ,,,,,,. Moreover, the solution satisfies the inequality
**
where
*

#### 4. Robust Exponential Stability Conditions

Theorem 4.1. *Under Assumption 1 and given positive constants , and , the system (2.1) is exponentially stable if there exist positive definite matrices , , any appropriate dimensional matrices , , and positive constants satisfying the following LMI:
**
where ,,,, ,,,,,,, ,,,,. Moreover, the solution satisfies the inequality
**
where
*

*Proof. *Replacing , and in (3.26) with , , and , respectively, we find that condition (3.26) for system (2.1) is equivalent to the following condition:
where and . Using Lemma 1 in [9], we have that there exists a positive number . We can find that (4.4) is equivalent to the LMI as follows:
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 , then system (2.1) reduces to the following system: Take the Lyapunov functional as where , and are defined in Theorem 3.1. According to Theorems 3.1 and 4.1, we have the following corollary for the delay-dependent robust exponential stability of system (4.7).

Corollary 4.2. *Under Assumption 1 and given positive constants , and , the system (4.7) is robustly exponentially stable if there exist positive definite matrices , , any appropriate dimensional matrices , , and positive constant satisfying the following LMI:
**
where ,,,,,,,,,, ,. Moreover, the solution satisfies the inequality
**
where
*

#### 5. Numerical Examples

In order to show the effectiveness of the approaches presented in Sections 3 and 4, three numerical examples are provided.

*Example 5.1. *Consider the asymptotic stability of the linear system (3.19) with time-varying delays when
Decompose matrix as follows: , where

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

*Example 5.2. *Consider the exponential stability of the neutral system (3.24) with time-varying delays and nonlinear perturbations with
Decompose matrix as follows which is the same as the decomposition in Example 5.1.

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].

*Example 5.3. *Consider the asymptotic stability of the uncertain neutral system (4.7) with time-varying delays with
Decompose matrix 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 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].

#### 6. 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.

#### Acknowledgment

This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission, through the Cluster of Research to Enhance the Quality of Basic Education.