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

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. 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; 𝜆_max(𝐴)=max{Re𝜆∶𝜆∈𝜆(𝐴)}; 𝜆_min(𝐴)=min{Re𝜆∶𝜆∈𝜆(𝐴)}; matrix 𝐴 is called semipositive definite (𝐴≥0) if 𝑥^𝑇𝐴𝑥≥0, for all 𝑥∈ℝ^𝑛;𝐴 is positive definite (𝐴>0) if 𝑥^𝑇𝐴𝑥>0 for all 𝑥≠0; matrix 𝐵 is called seminegative definite (𝐵≤0) if 𝑥^𝑇𝐵𝑥≤0, for all 𝑥∈ℝ^𝑛;𝐵 is negative definite (𝐵<0) if 𝑥^𝑇𝐵𝑥<0 for all 𝑥≠0; 𝐴>𝐵 means 𝐴−𝐵>0; 𝐴≥𝐵 means 𝐴−𝐵≥0; 𝐶([−ℎ,0],ℝ^𝑛) 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 𝑥̇(𝑡)−𝐶𝑥̇(𝑡−𝑟(𝑡))=𝐴(𝑡)𝑥(𝑡)+𝐵(𝑡)𝑥(𝑡−ℎ(𝑡))+𝑓₁(𝑡,𝑥(𝑡))+𝑓₂(𝑡,𝑥(𝑡−ℎ(𝑡)))+𝐷(𝑡)^𝑡_{(𝑡)𝑡−𝑔}𝑥(𝑠)𝑥𝑡𝑑𝑠,𝑡≥0,₀+𝜃=𝜙(𝜃),𝑥̇𝑡₀(𝜃),𝐴(𝑡)=[(𝑡(𝑡)=[(𝑡(𝑡)=[(𝑡,+𝜃=𝜑,∀𝜃∈−max𝑟,ℎ,𝑔,0𝐴+Δ𝐴)],𝐵𝐵+Δ𝐵)],𝐷𝐷+Δ𝐷)] (2.1) 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, (𝑡)𝑟̇(𝑡)(𝑡)ℎ̇(𝑡)(𝑡)0≤𝑟≤𝑟,≤𝛿<1,0≤ℎ≤ℎ,≤𝜇<+∞,0≤𝑔≤𝑔, (2.2) where ℎ,𝑟,𝑔,𝜇,and𝛿 are given positive constants. The initial condition functions 𝜙(𝑡),𝜑(𝑡)∈𝐶([−max{𝑟,ℎ,𝑔},0],ℝ^𝑛) denote continuous vector-valued initial function of 𝑡∈[−max{𝑟,ℎ,𝑔},0] with the norm ‖𝜙‖=sup_{𝑠∈[−max{𝑟,ℎ,𝑔},0]}‖𝜙(𝑠)‖, ‖𝜑‖=sup_{𝑠∈[−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,𝑁_𝑖,𝑖=1,2,3, are known constant matrices with appropriate dimensions. The uncertain matrix 𝐹(𝑡) satisfies 𝐹(𝑡)^𝑇𝐹(𝑡)≤𝐼. (2.4) The uncertainties 𝑓₁(⋅) and 𝑓₂(⋅) 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: 𝑓^𝑇₁(𝑡,𝑥(𝑡))𝑓₁(𝑡,𝑥(𝑡))≤𝛽₁𝑥^𝑇(𝑡)𝑥(𝑡),𝑓^𝑇₂(𝑡,𝑥(𝑡−ℎ(𝑡)))𝑓₂(𝑡,𝑥(𝑡−ℎ(𝑡)))≤𝛽₂𝑥^𝑇(𝑡−ℎ(𝑡))𝑥(𝑡−ℎ(𝑡)), (2.5) where 𝜂,𝜌 are given positive constants. First, we consider nominal system (2.1) which is defined to be 𝑥̇(𝑡)−𝐶𝑥̇(𝑡−𝑟(𝑡))=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−ℎ(𝑡))+𝑓₁(𝑡,𝑥(𝑡))+𝑓₂(𝑡,𝑥(𝑡−ℎ(𝑡)))+𝐷^𝑡_{(𝑡)𝑡−𝑔}𝑥(𝑠)𝑥𝑡𝑑𝑠,𝑡≥0,₀+𝜃=𝜙(𝜃),𝑥̇𝑡₀+𝜃=𝜑(𝜃),∀𝜃∈−max𝑟,ℎ,𝑔,0. (2.6) Rewrite the system (2.6) in the following descriptor system: 𝑥̇(𝑡)(𝑡),𝑦(𝑡)(𝑡)((𝑡=𝑦=𝐴𝑥+𝐵𝑥𝑡−ℎ))+𝑓₁(𝑡,𝑥(𝑡))+𝑓₂(((𝑡((𝑡𝑡,𝑥𝑡−ℎ)))+𝐶𝑦𝑡−𝑟))+𝐷^𝑡_{(𝑡)𝑡−𝑔}𝑥(𝑠)𝑑𝑠. (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 𝐵=𝐵₁+𝐵₂, (2.8) where 𝐵₁,𝐵₂∈ℝ^𝑛×𝑛 are constant matrices. Utilize the following zero equation: 0=𝐺𝑥(𝑡)−𝐺𝑥(𝑡−ℎ(𝑡))−𝐺^𝑡_{(𝑡)𝑡−ℎ}𝑥̇(𝑠)𝑑𝑠, (2.9) 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: 𝑥̇(𝑡)=𝑦(𝑡)+𝐺𝑥(𝑡)−𝐺𝑥(𝑡−ℎ(𝑡))−𝐺^𝑡_{(𝑡)𝑡−ℎ}𝑦(𝑠)𝑦(𝑡)𝑑𝑠,=𝐴₁𝑥(𝑡)+𝐵₂𝑥(𝑡−ℎ(𝑡))+𝑓₁(𝑡,𝑥(𝑡))+𝑓₂(((𝑡((𝑡𝑡,𝑥𝑡−ℎ)))+𝐶𝑦𝑡−𝑟))−𝐵₁^𝑡_{(𝑡)𝑡−ℎ}𝑦(𝑠)𝑑𝑠+𝐷^𝑡_{(𝑡)𝑡−𝑔}𝑥(𝑠)𝑑𝑠, (2.10) where 𝐴₁=𝐴+𝐵₁.

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 𝑥𝑡,𝜙,𝜑≤𝑀max𝜙,𝜑𝑒^−𝛼𝑡,∀𝑡∈ℝ⁺. (2.11)

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 −ℎ^𝑡_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑥(𝑡)𝑥()𝑑𝑠≤𝑡−ℎ^𝑇𝑥(𝑡)𝑥()−𝑅𝑅𝑅−𝑅𝑡−ℎ. (2.12)

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 𝐶 are inside the unit circle.

Theorem 3.1. Under Assumption 1 and given positive constants ℎ,𝑟,𝑔,𝜇,𝛿,𝛼,𝛽₁, and 𝛽₂, the nominal system (2.1) is exponentially stable if there exist positive definite matrices 𝑃_𝑖, 𝑖=1,2,…,6, any appropriate dimensional matrices 𝑊₁,𝑄₁,𝑄₂,𝑀_𝑖, 𝑖=1,2,…,8, and positive constants 𝜖₁,𝜖₂ satisfying the following LMI: ΩΩ=¹¹Ω₁₂Ω₁₃Ω₁₄Ω₁₅Ω₁₆Ω₁₇Ω₁₈∗Ω₂₂Ω₂₃Ω₂₄𝑄^𝑇₂𝑄^𝑇₂𝑄^𝑇₂𝐶𝑄^𝑇₂𝐷∗∗Ω₃₃Ω₃₄−𝑀₅−𝑀₆−𝑀₇−𝑀₈∗∗∗Ω₄₄−𝑀₅−𝑀₆−𝑀₇−𝑀₈∗∗∗∗−𝜖₁𝐼000∗∗∗∗∗−𝜖₂𝐼00∗∗∗∗∗∗Ω₇₇0∗∗∗∗∗∗∗Ω₈₈<0, (3.1) where Ω₁₁=𝑊₁+𝑊^𝑇₁+𝑄^𝑇₁𝐴₁+𝐴^𝑇₁𝑄₁+𝑃₂−𝑒^−2𝛼ℎ𝑃₄+𝑔²𝑃₆+𝑀^𝑇₁+𝑀₁+𝜖₁𝛽²₁𝐼+2𝛼𝑃₁,Ω₂₂=−𝑄₂−𝑄^𝑇₂+ℎ²𝑃₃+ℎ²𝑃₄+𝑃₅,Ω₂₃=𝑄^𝑇₂𝐵₂−𝑀^𝑇₂,Ω₂₄=𝑄^𝑇₂𝐵₁−𝑀^𝑇₂,Ω₃₃=−𝑒^−2𝛼ℎ𝑃₂+𝜇𝑃₂−𝑒^−2𝛼ℎ𝑃₄−𝑀^𝑇₃−𝑀₃+𝜖₂𝛽²₂𝐼, Ω₃₄=−𝑀^𝑇₃−𝑀₄,Ω₄₄=−𝑒^−2𝛼ℎ𝑃₃−𝑀₄−𝑀^𝑇₄,Ω₇₇=−(1−𝛿)𝑒^−2𝛼𝑟𝑃₅, Ω₈₈=−𝑒^−2𝛼𝑔𝑃₆,Ω₁₂=𝑃₁−𝑄^𝑇₁+𝐴^𝑇₁𝑄₂+𝑀₂,Ω₁₃=−𝑊₁+𝑄^𝑇₁𝐵₂+𝑒^−2𝛼ℎ𝑃₄−𝑀^𝑇₁+𝑀₃,Ω₁₄=−𝑊₁−𝑄^𝑇₁𝐵₁−𝑀^𝑇₁+𝑀₄,Ω₁₅=𝑄^𝑇₁+𝑀₅,Ω₁₆=𝑄^𝑇₁+𝑀₆,Ω₁₇=𝑄^𝑇₁𝐶+𝑀₇, Ω₁₈=𝑄^𝑇₁𝐷+𝑀₈.Moreover, the solution 𝑥(𝑡,𝜙,𝜑) satisfies the inequality 𝑥𝑡,𝜙,𝜑≤𝑁𝜆_min𝑃₁max𝜙,𝜑𝑒^−𝛼𝑡,𝑡∈ℝ⁺, (3.2) where 𝑁=𝜆_max𝑃₁+ℎ𝜆_max𝑃₂+ℎ³𝜆_max𝑃₃+ℎ³𝜆_max𝑃₄+𝑟𝜆_max𝑃₅+𝑔³𝜆_max𝑃₆. (3.3)

Proof. Construct a Lyapunov functional candidate for the nominal system (2.1) of the form 𝑉(𝑡)=⁶_𝑖=1𝑉_𝑖(𝑡), (3.4) where 𝑉₁(𝑡)=𝑥^𝑇(𝑡)𝑃₁𝑥(𝑡)=𝑥(𝑡)𝑦(𝑡)^𝑇𝐼0𝑃00¹0𝑄₁𝑄₂𝑥(𝑡)𝑦(𝑡),𝑉₂(𝑡)=^𝑡_{(𝑡)𝑡−ℎ}𝑒^{2𝛼(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑃₂𝑥(𝑠)𝑉𝑑𝑠,₃(𝑡)=ℎ⁰_−ℎ^𝑡_𝑡+𝑠𝑒^{2𝛼(𝑠−𝑡)}𝑦^𝑇(𝜃)𝑃₃𝑦(𝜃)𝑉𝑑𝜃𝑑𝑠,₄(𝑡)=ℎ⁰_−ℎ^𝑡_𝑡+𝑠𝑒^{2𝛼(𝑠−𝑡)}𝑦^𝑇(𝜃)𝑃₄𝑦(𝜃)𝑉𝑑𝜃𝑑𝑠,₅(𝑡)=^𝑡_{(𝑡)𝑡−𝑟}𝑒^{2𝛼(𝑠−𝑡)}𝑦^𝑇(𝑠)𝑃₅𝑦(𝑠)𝑉𝑑𝑠,₆(𝑡)=𝑔⁰_−𝑔^𝑡_𝑡+𝑠𝑒^{2𝛼(𝑠−𝑡)}𝑥^𝑇(𝜃)𝑃₆𝑥(𝜃)𝑑𝜃𝑑𝑠. (3.5) The derivative of 𝑉(⋅) along the trajectories of nominal system (2.1) is given by 𝑉̇(𝑡)=⁶_𝑖=1𝑉̇_𝑖(𝑡), (3.6) where 𝑉̇₁(𝑡)𝑥(𝑡)𝑦(𝑡)=2^𝑇𝑃₁𝑄^𝑇₁0𝑄^𝑇₂𝑥̇(𝑡)0=2𝑥^𝑇(𝑡)𝑃₁𝑦(𝑡)+𝐺𝑥(𝑡)−𝐺𝑥(𝑡−ℎ(𝑡))−𝐺^𝑡_{(𝑡)𝑡−ℎ}𝑦(𝑠)𝑑𝑠+2𝑥^𝑇(𝑡)𝑄^𝑇₁−𝑦(𝑡)+𝐴₁𝑥(𝑡)+𝐵₂𝑥(𝑡−ℎ(𝑡))+𝑓₁(𝑡,𝑥(𝑡))+𝑓₂(𝑡,𝑥(𝑡−ℎ(𝑡)))+𝐶𝑦(𝑡−𝑟(𝑡))−𝐵₁^𝑡_{(𝑡)𝑡−ℎ}𝑦(𝑠)𝑑𝑠+𝐷^𝑡_{(𝑡)𝑡−𝑔}𝑥(𝑠)𝑑𝑠+2𝑦^𝑇(𝑡)𝑄^𝑇₂−𝑦(𝑡)+𝐴₁𝑥(𝑡)+𝐵₂𝑥(𝑡−ℎ(𝑡))+𝑓₁(𝑡,𝑥(𝑡))+𝑓₂(((𝑡((𝑡𝑡,𝑥𝑡−ℎ)))+𝐶𝑦𝑡−𝑟))−𝐵₁^𝑡_{(𝑡)𝑡−ℎ}𝑦(𝑠)𝑑𝑠+𝐷^𝑡_{(𝑡)𝑡−𝑔}𝑥(𝑠)𝑑𝑠+2𝛼𝑥^𝑇(𝑡)𝑃₁𝑥(𝑡)−2𝛼𝑉₁(𝑡),𝑉̇₂(𝑡)=𝑥^𝑇(𝑡)𝑃₂𝑥(𝑡)−1−ℎ̇(𝑡)𝑒^{−2𝛼ℎ(𝑡)}𝑥^𝑇(𝑡−ℎ(𝑡))𝑃₂𝑥(𝑡−ℎ(𝑡))−2𝛼𝑉₂(𝑡)≤𝑥^𝑇(𝑡)𝑃₂𝑥(𝑡)−𝑒^−2𝛼ℎ𝑥^𝑇(𝑡−ℎ(𝑡))𝑃₂𝑥(𝑡−ℎ(𝑡))+𝜇𝑥^𝑇(𝑡−ℎ(𝑡))𝑃₂𝑥(𝑡−ℎ(𝑡))−2𝛼𝑉₂(𝑡). (3.7) Obviously, for any scalar 𝑠∈[𝑡−ℎ,𝑡], we get 𝑒^−2𝛼ℎ≤𝑒^{−2𝛼(𝑠−𝑡)}≤1. Together with Lemma 2.2 (Jensen's inequality), we obtain 𝑉̇₃(𝑡)=ℎ²𝑦^𝑇(𝑡)𝑃₃𝑦(𝑡)−ℎ⁰_−ℎ𝑒^2𝛼𝑠𝑦^𝑇(𝑡+𝑠)𝑃₃𝑦(𝑡+𝑠)𝑑𝑠−2𝛼𝑉₃(𝑡)≤ℎ²𝑦^𝑇(𝑡)𝑃₃𝑦(𝑡)−ℎ^𝑡_𝑡−ℎ𝑒^{2𝛼(𝑠−𝑡)}𝑦^𝑇(𝑠)𝑃₃𝑦(𝑠)𝑑𝑠−2𝛼𝑉₃(𝑡)≤ℎ²𝑦^𝑇(𝑡)𝑃₃𝑦(𝑡)−ℎ𝑒^{−2𝛼ℎ𝑡}_𝑡−ℎ𝑦^𝑇(𝑠)𝑃₃𝑦(𝑠)𝑑𝑠−2𝛼𝑉₃(𝑡)≤ℎ²𝑦^𝑇(𝑡)𝑃₃𝑦(𝑡)−𝑒^{−2𝛼ℎ𝑡}_𝑡−ℎ𝑦^𝑇(𝑠)𝑑𝑠𝑃₃^𝑡_𝑡−ℎ𝑦(𝑠)𝑑𝑠−2𝛼𝑉₃(𝑡)≤ℎ²𝑦^𝑇(𝑡)𝑃₃𝑦(𝑡)−𝑒^{−2𝛼ℎ𝑡}_{(𝑡)𝑡−ℎ}𝑦^𝑇(𝑠)𝑑𝑠𝑃₃^𝑡_{(𝑡)𝑡−ℎ}𝑦(𝑠)𝑑𝑠−2𝛼𝑉₃(𝑡),𝑉̇₄(𝑡)≤ℎ²𝑦^𝑇(𝑡)𝑃₄𝑦(𝑡)−𝑒^{−2𝛼ℎ𝑡}_{(𝑡)𝑡−ℎ}𝑦^𝑇(𝑠)𝑑𝑠𝑃₄^𝑡_{(𝑡)𝑡−ℎ}𝑦(𝑠)𝑑𝑠−2𝛼𝑉₄(𝑡)=ℎ²𝑦^𝑇(𝑡)𝑃₄𝑦(𝑡)−𝑒^−2𝛼ℎ𝑥^𝑇(𝑡)−𝑥^𝑇(𝑡−ℎ(𝑡))𝑃₄[𝑥(𝑡)−𝑥(𝑡−ℎ(𝑡))]−2𝛼𝑉₄(𝑡),𝑉̇₅(𝑡)=𝑥^𝑇(𝑡)𝑃₅𝑥(𝑡)−(1−𝑟̇(𝑡))𝑒^{−2𝛼𝑟(𝑡)}𝑥^𝑇(𝑡−𝑟(𝑡))𝑃₂𝑥(𝑡−𝑟(𝑡))−2𝛼𝑉₅(𝑡)≤𝑥^𝑇(𝑡)𝑃₅𝑥(𝑡)−(1−𝛿)𝑒^−2𝛼𝑟𝑥^𝑇(𝑡−𝑟(𝑡))𝑃₅𝑥(𝑡−𝑟(𝑡))−2𝛼𝑉₅(𝑡),𝑉̇₆(𝑡)≤𝑔²𝑥^𝑇(𝑡)𝑃₆𝑥(𝑡)−𝑒^{−2𝛼𝑔𝑡}_{(𝑡)𝑡−𝑔}𝑥^𝑇(𝑠)𝑑𝑠𝑃₆^𝑡_{(𝑡)𝑡−𝑔}𝑥(𝑠)𝑑𝑠−2𝛼𝑉₆(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 2𝑥^𝑇(𝑡)𝑀^𝑇₁+𝑦^𝑇(𝑡)𝑀^𝑇₂+𝑥^𝑇(𝑡−ℎ(𝑡))𝑀^𝑇₃+^𝑡_{(𝑡)𝑡−ℎ}𝑦^𝑇(𝑠)𝑑𝑠𝑀₄+𝑓^𝑇₁(𝑡,𝑥(𝑡))𝑀₅+𝑓^𝑇₂(𝑡,𝑥(𝑡−ℎ(𝑡)))𝑀^𝑇₆+𝑦^𝑇(𝑡−𝑟(𝑡))𝑀^𝑇₇+^𝑡_{(𝑡)𝑡−𝑔}𝑥^𝑇(𝑠)𝑑𝑠𝑀₈×𝑥(𝑡)−𝑥(𝑡−ℎ(𝑡))−^𝑡_{(𝑡)𝑡−ℎ}𝑦(𝑠)𝑑𝑠=0. (3.9) From (2.5), we obtain for any scalars 𝜖₁,𝜖₂>00≤𝜖₁𝛽₁𝑥^𝑇(𝑡)𝑥(𝑡)−𝜖₁𝑓^𝑇₁(𝑡,𝑥(𝑡))𝑓₁(𝑡,𝑥(𝑡)),0≤𝜖₂𝛽₂𝑥^𝑇(𝑡−ℎ(𝑡))𝑥(𝑡−ℎ(𝑡))−𝜖₂𝑓^𝑇₂(𝑡,𝑥(𝑡−ℎ(𝑡)))𝑓₂(𝑡,𝑥(𝑡−ℎ(𝑡))). (3.10) Let us define 𝑊₁=𝑃₁𝐺. From (3.6), (3.9), and (3.10), we obtain 𝑉̇(𝑡)+2𝛼𝑉(𝑡)≤𝜉^𝑇(𝑡)Ω𝜉(𝑡), (3.11) where 𝜉^𝑇(𝑡)=[𝑥^𝑇(𝑡),𝑦^𝑇(𝑡),𝑥^𝑇(𝑡−ℎ(𝑡)),∫^𝑡_{𝑡−ℎ(𝑡)}𝑦^𝑇(𝑠)𝑑𝑠,𝑓^𝑇(𝑡,𝑥(𝑡)),𝑔^𝑇(𝑡,𝑥(𝑡−ℎ(𝑡))),𝑦^𝑇(𝑡−𝑟(𝑡)),∫^𝑡_{𝑡−𝑔(𝑡)}𝑥^𝑇(𝑠)𝑑𝑠] and ΩΩ=¹¹Ω₁₂Ω₁₃Ω₁₄Ω₁₅Ω₁₆Ω₁₇Ω₁₈∗Ω₂₂Ω₂₃Ω₂₄𝑄^𝑇₂𝑄^𝑇₂𝑄^𝑇₂𝐶𝑄^𝑇₂𝐷∗∗Ω₃₃Ω₃₄−𝑀₅−𝑀₆−𝑀₇−𝑀₈∗∗∗Ω₄₄−𝑀₅−𝑀₆−𝑀₇−𝑀₈∗∗∗∗−𝜖₁𝐼000∗∗∗∗∗−𝜖₂𝐼00∗∗∗∗∗∗Ω₇₇0∗∗∗∗∗∗∗Ω₈₈. (3.12) It is a fact that, if Ω<0, then 𝑉̇(𝑡)+2𝛼𝑉(𝑡)≤0,∀𝑡∈ℝ⁺, (3.13) which gives 𝑉(𝑡)≤𝑉(0)𝑒^−2𝛼𝑡,∀𝑡∈ℝ⁺. (3.14) From (3.14), it is easy to see that 𝜆_min𝑃₁‖𝑥(𝑡)‖²≤𝑉(𝑡)≤𝑉(0)𝑒^−2𝛼𝑡,𝑉(0)=⁶_𝑖=1𝑉_𝑖(0), (3.15) where 𝑉₁(0)=𝑥^𝑇(0)𝑃₁𝑥(0),𝑉₂(0)=⁰_(0)−ℎ𝑒^2𝛼𝑠𝑥^𝑇(𝑠)𝑃₂𝑥(𝑠)𝑉𝑑𝑠,₃(0)=ℎ⁰_−ℎ⁰_𝑠𝑒^2𝛼𝑠𝑦^𝑇(𝜃)𝑃₃𝑦(𝜃)𝑉𝑑𝜃𝑑𝑠,₄(0)=ℎ⁰_−ℎ⁰_𝑠𝑒^2𝛼𝑠𝑦^𝑇(𝜃)𝑃₄𝑦(𝜃)𝑉𝑑𝜃𝑑𝑠,₅(0)=⁰_(0)−𝑟𝑒^2𝛼𝑠𝑦^𝑇(𝑠)𝑃₅𝑦(𝑠)𝑉𝑑𝑠,₆(0)=𝑔⁰_−𝑔⁰_𝑠𝑒^2𝛼𝑠𝑥^𝑇(𝜃)𝑃₆𝑥(𝜃)𝑑𝜃𝑑𝑠. (3.16) Therefore, we get 𝜆_min𝑃₁‖𝑥(𝑡)‖²≤𝑉(0)𝑒^−2𝛼𝑡≤𝑁max𝜙,𝜑²𝑒^−2𝛼𝑡, (3.17) where 𝑁=𝜆_max(𝑃₁)+ℎ𝜆_max(𝑃₂)+ℎ³𝜆_max(𝑃₃)+ℎ³𝜆_max(𝑃₄)+𝑟𝜆_max(𝑃₅)+𝑔³𝜆_max(𝑃₆). From (3.17), we get 𝑥𝑡,𝜙,𝜑≤𝑁𝜆_min𝑃₁max𝜙,𝜑𝑒^−𝛼𝑡,∀𝑡∈ℝ⁺. (3.18) This means that the system (2.1) is exponentially stable. The proof of the theorem is complete.

If 𝐶=Δ𝐴(𝑡)=Δ𝐵(𝑡)=𝑓₁(𝑡,𝑥(𝑡))=𝑓₂(𝑡,𝑥(𝑡−ℎ(𝑡)))=𝐷(𝑡)=0, then system (2.1) reduces to the following system: 𝑥̇(𝑡)(𝑡)((𝑡𝑥𝑡=𝐴𝑥+𝐵𝑥𝑡−ℎ)),𝑡≥0,₀+𝜃=𝜙(𝜃),𝑥̇𝑡₀+𝜃=𝜑(𝜃),∀𝜃∈[−ℎ,0]. (3.19) Take the Lyapunov functional as 𝑉̇(𝑡)=⁴_𝑖=1𝑉̇_𝑖(𝑡), (3.20) where 𝑉₁(𝑡),𝑉₂(𝑡),𝑉₃(𝑡), and 𝑉₄(𝑡) 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 ℎ,𝜇,and𝛼, the system (3.19) is exponentially stable if there exist positive definite matrices 𝑃_𝑖, 𝑖=1,2,…,4, and any appropriate dimensional matrices 𝑊₁,𝑄₁,𝑄₂,𝑀_𝑖, 𝑖=1,2,…,4, satisfying the following LMI: ΦΦ=¹¹Φ₁₂Φ₁₃Φ₁₄∗Φ₂₂Φ₂₃Φ₂₄∗∗Φ₃₃Φ₃₄∗∗∗Φ₄₄<0, (3.21) where Φ₁₁=𝑊₁+𝑊^𝑇₁+𝑄^𝑇₁𝐴₁+𝐴^𝑇₁𝑄₁+𝑃₂−𝑒^−2𝛼ℎ𝑃₄+𝑀^𝑇₁+𝑀₁+2𝛼𝑃₁,Φ₂₂=−𝑄₂−𝑄^𝑇₂+ℎ²𝑃₃+ℎ²𝑃₄,Φ₂₃=𝑄^𝑇₂𝐵₂−𝑀^𝑇₂,Φ₂₄=𝑄^𝑇₂𝐵₁−𝑀^𝑇₂,Φ₃₃=−𝑒^−2𝛼ℎ𝑃₂+𝜇𝑃₂−𝑒^−2𝛼ℎ𝑃₄−𝑀^𝑇₃−𝑀₃,Ω₃₄=−𝑀^𝑇₃−𝑀₄,Φ₄₄=−𝑒^−2𝛼ℎ𝑃₃−𝑀₄−𝑀^𝑇₄,Φ₁₂=𝑃₁−𝑄^𝑇₁+𝐴^𝑇₁𝑄₂+𝑀₂, Φ₁₃=−𝑊₁+𝑄^𝑇₁𝐵₂+𝑒^−2𝛼ℎ𝑃₄−𝑀^𝑇₁+𝑀₃, Φ₁₄=−𝑊₁−𝑄^𝑇₁𝐵₁−𝑀^𝑇₁+𝑀₄. Moreover, the solution 𝑥(𝑡,𝜙,𝜑) satisfies the inequality 𝑥𝑡,𝜙,𝜑≤𝐾𝜆_min𝑃₁max𝜙,𝜑𝑒^−𝛼𝑡,𝑡∈ℝ⁺, (3.22) where 𝐾=𝜆_max𝑃₁+ℎ𝜆_max𝑃₂+ℎ³𝜆_max𝑃₃+ℎ³𝜆_max𝑃₄. (3.23)

If Δ𝐴(𝑡)=Δ𝐵(𝑡)=𝐷(𝑡)=0, then system (2.1) reduces to the following system: 𝑥̇(𝑡)−𝐶𝑥̇(𝑡−𝑟(𝑡))=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−ℎ(𝑡))+𝑓₁(𝑡,𝑥(𝑡))+𝑓₂(((𝑡𝑥𝑡𝑡,𝑥𝑡−ℎ))),𝑡≥0,₀+𝜃=𝜙(𝜃),𝑥̇𝑡₀+𝜃=𝜑(𝜃),∀𝜃∈[−max{𝑟,ℎ},0]. (3.24) Take the Lyapunov functional as 𝑉̇(𝑡)=⁵_𝑖=1𝑉̇_𝑖(𝑡), (3.25) where 𝑉₁(𝑡),𝑉₂(𝑡),𝑉₃(𝑡),𝑉₄(𝑡), and 𝑉₅(𝑡) 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 ℎ,𝑟,𝜇,𝛿,𝛼,𝛽₁, and 𝛽₂, the system (3.24) is exponentially stable if there exist positive definite matrices 𝑃_𝑖, 𝑖=1,2,…,5, any appropriate dimensional matrices 𝑊₁,𝑄₁,𝑄₂,𝑀_𝑖, 𝑖=1,2,…,7, and positive constants 𝜖₁,𝜖₂ satisfying the following LMI: ΥΥ=¹¹Υ₁₂Υ₁₃Υ₁₄Υ₁₅Υ₁₆Υ₁₇∗Υ₂₂Υ₂₃Υ₂₄𝑄^𝑇₂𝑄^𝑇₂𝑄^𝑇₂𝐶∗∗Υ₃₃Υ₃₄−𝑀₅−𝑀₆−𝑀₇∗∗∗Υ₄₄−𝑀₅−𝑀₆−𝑀₇∗∗∗∗−𝜖₁𝐼00∗∗∗∗∗−𝜖₂𝐼0∗∗∗∗∗∗Υ₇₇<0, (3.26) where Υ₁₁=𝑊₁+𝑊^𝑇₁+𝑄^𝑇₁𝐴₁+𝐴^𝑇₁𝑄₁+𝑃₂−𝑒^−2𝛼ℎ𝑃₄+𝑀^𝑇₁+𝑀₁+𝜖₁𝛽²₁𝐼+2𝛼𝑃₁,Υ₂₂=−𝑄₂−𝑄^𝑇₂+ℎ²𝑃₃+ℎ²𝑃₄+𝑃₅, Υ₂₃=𝑄^𝑇₂𝐵₂−𝑀^𝑇₂, Υ₂₄=𝑄^𝑇₂𝐵₁−𝑀^𝑇₂, Υ₃₃=−𝑒^−2𝛼ℎ𝑃₂+𝜇𝑃₂−𝑒^−2𝛼ℎ𝑃₄−𝑀^𝑇₃−𝑀₃+𝜖₂𝛽²₂𝐼,Υ₃₄=−𝑀^𝑇₃−𝑀₄,Υ₄₄=−𝑒^−2𝛼ℎ𝑃₃−𝑀₄−𝑀^𝑇₄, Υ₁₂=𝑃₁−𝑄^𝑇₁+𝐴^𝑇₁𝑄₂+𝑀₂,Υ₁₃=−𝑊₁+𝑄^𝑇₁𝐵₂+𝑒^−2𝛼ℎ𝑃₄−𝑀^𝑇₁+𝑀₃,Υ₁₄=−𝑊₁−𝑄^𝑇₁𝐵₁−𝑀^𝑇₁+𝑀₄,Υ₁₅=𝑄^𝑇₁+𝑀₅,Υ₁₆=𝑄^𝑇₁+𝑀₆,Υ₁₇=𝑄^𝑇₁𝐶+𝑀₇,Υ₇₇=−(1−𝛿)𝑒^−2𝛼𝑟𝑃₅. Moreover, the solution 𝑥(𝑡,𝜙,𝜑) satisfies the inequality 𝑥𝑡,𝜙,𝜑≤𝐾𝜆_min𝑃₁max𝜙,𝜑𝑒^−𝛼𝑡,𝑡∈ℝ⁺, (3.27) where 𝐾=𝜆_max𝑃₁+ℎ𝜆_max𝑃₂+ℎ³𝜆_max𝑃₃+ℎ³𝜆_max𝑃₄. (3.28)

4. Robust Exponential Stability Conditions

Theorem 4.1. Under Assumption 1 and given positive constants ℎ,𝑟,𝑔,𝜇,𝛿,𝛼,𝛽₁, and 𝛽₂, the system (2.1) is exponentially stable if there exist positive definite matrices 𝑃_𝑖, 𝑖=1,2,…,6, any appropriate dimensional matrices 𝑊₁,𝑄₁,𝑄₂,𝑀_𝑖, 𝑖=1,2,…,8, and positive constants 𝜖₁,𝜖₂,and𝜀 satisfying the following LMI: ΩΩ=¹¹Ω₁₂Ω₁₃Ω₁₄Ω₁₅Ω₁₆Ω₁₇Ω₁₈𝑄^𝑇₁𝐸𝜀𝑁^𝑇₁∗Ω₂₂Ω₂₃Ω₂₄𝑄^𝑇₂𝑄^𝑇₂𝑄^𝑇₂𝐶𝑄^𝑇₂𝐷𝑄^𝑇₁𝐸0∗∗Ω₃₃Ω₃₄−𝑀₅−𝑀₆−𝑀₇−𝑀₈0𝜀𝑁^𝑇₂∗∗∗Ω₄₄−𝑀₅−𝑀₆−𝑀₇−𝑀₈00∗∗∗∗−𝜖₁𝐼00000∗∗∗∗∗−𝜖₂𝐼0000∗∗∗∗∗∗Ω₇₇000∗∗∗∗∗∗∗Ω₈₈0𝜀𝑁^𝑇₃∗∗∗∗∗∗∗∗−𝜀𝐼0∗∗∗∗∗∗∗∗∗−𝜀𝐼<0, (4.1) where Ω₁₁=𝑊₁+𝑊^𝑇₁+𝑄^𝑇₁𝐴₁+𝐴^𝑇₁𝑄₁+𝑃₂−𝑒^−2𝛼ℎ𝑃₄+𝑔²𝑃₆+𝑀^𝑇₁+𝑀₁+𝜖₁𝛽²₁𝐼+2𝛼𝑃₁,Ω₂₂=−𝑄₂−𝑄^𝑇₂+ℎ²𝑃₃+ℎ²𝑃₄+𝑃₅,Ω₂₃=𝑄^𝑇₂𝐵₂−𝑀^𝑇₂,Ω₂₄=𝑄^𝑇₂𝐵₁−𝑀^𝑇₂, Ω₃₃=−𝑒^−2𝛼ℎ𝑃₂+𝜇𝑃₂−𝑒^−2𝛼ℎ𝑃₄−𝑀^𝑇₃−𝑀₃+𝜖₂𝛽²₂𝐼,Ω₃₄=−𝑀^𝑇₃−𝑀₄,Ω₄₄=−𝑒^−2𝛼ℎ𝑃₃−𝑀₄−𝑀^𝑇₄,Ω₇₇=−(1−𝛿)𝑒^−2𝛼𝑟𝑃₅,Ω₈₈=−𝑒^−2𝛼𝑔𝑃₆,Ω₁₂=𝑃₁−𝑄^𝑇₁+𝐴^𝑇₁𝑄₂+𝑀₂,Ω₁₃=−𝑊₁+𝑄^𝑇₁𝐵₂+𝑒^−2𝛼ℎ𝑃₄−𝑀^𝑇₁+𝑀₃, Ω₁₄=−𝑊₁−𝑄^𝑇₁𝐵₁−𝑀^𝑇₁+𝑀₄,Ω₁₅=𝑄^𝑇₁+𝑀₅,Ω₁₆=𝑄^𝑇₁+𝑀₆,Ω₁₇=𝑄^𝑇₁𝐶+𝑀₇,Ω₁₈=𝑄^𝑇₁𝐷+𝑀₈. Moreover, the solution 𝑥(𝑡,𝜙,𝜑) satisfies the inequality 𝑥𝑡,𝜙,𝜑≤𝑁𝜆_min𝑃₁max𝜙,𝜑𝑒^−𝛼𝑡,𝑡∈ℝ⁺, (4.2) where 𝑁=𝜆_max𝑃₁+ℎ𝜆_max𝑃₂+ℎ³𝜆_max𝑃₃+ℎ³𝜆_max𝑃₄+𝑟𝜆_max𝑃₅+𝑔³𝜆_max𝑃₆. (4.3)

Proof. Replacing 𝐴₁,𝐵₂, and 𝐷 in (3.26) with 𝐴₁(𝑡)=𝐴+𝐵₁+𝐸𝐹(𝑡)𝑁₁, 𝐵₂(𝑡)=𝐵₂+𝐸𝐹(𝑡)𝑁₂, and 𝐷(𝑡)=𝐷+𝐸𝐹(𝑡)𝑁₃, respectively, we find that condition (3.26) for system (2.1) is equivalent to the following condition: Ω+Γ𝐹(𝑡)Π+Π^𝑇𝐹(𝑡)^𝑇Γ^𝑇<0, (4.4) where Γ^𝑇=[𝐸^𝑇𝑄₁,𝐸^𝑇𝑄₂,0,0,0,0,0,0] and Π=[𝑁₁,0,𝑁₂,0,0,0,0,𝑁₃]. 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: Ω+𝜀⁻¹ΓΓ^𝑇+𝜀Π^𝑇Π<0. (4.5) Applying Lemma  1 (Schur complement lemma) in [12], we obtain ΩΓ𝜀Π^𝑇Γ^𝑇−𝜀𝐼0𝜀Π0−𝜀𝐼<0. (4.6) Hence, we conclude that (4.6) is equivalent to (4.1). The proof of the theorem is complete.

If 𝑓₁(𝑡,𝑥(𝑡))=𝑓₂(𝑡,𝑥(𝑡−ℎ(𝑡)))=𝐷(𝑡)=0, then system (2.1) reduces to the following system: 𝑥̇(𝑡)−𝐶𝑥̇(𝑡−𝑟(𝑟))=𝐴+𝐸𝐹(𝑡)𝑁₁𝑥(𝑡)+𝐵+𝐸𝐹(𝑡)𝑁₂𝑥((𝑡𝑥𝑡𝑡−ℎ)),𝑡≥0,₀+𝜃=𝜙(𝜃),𝑥̇𝑡₀+𝜃=𝜑(𝜃),∀𝜃∈[−max{𝑟,ℎ},0]. (4.7) Take the Lyapunov functional as 𝑉̇(𝑡)=⁵_𝑖=1𝑉̇_𝑖(𝑡), (4.8) where 𝑉₁(𝑡),𝑉₂(𝑡),𝑉₃(𝑡),𝑉₄(𝑡), and 𝑉₅(𝑡) 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 ℎ,𝑟,𝜇,𝛿, and 𝛼, the system (4.7) is robustly exponentially stable if there exist positive definite matrices 𝑃_𝑖, 𝑖=1,2,…,5, any appropriate dimensional matrices 𝑊₁,𝑄₁,𝑄₂,𝑀₇,𝑀_𝑖, 𝑖=1,2,…,4, and positive constant 𝜀 satisfying the following LMI: ΩΩ=¹¹Ω₁₂Ω₁₃Ω₁₄Ω₁₇𝑄^𝑇₁𝐸𝜀𝑁^𝑇₁∗Ω₂₂Ω₂₃Ω₂₄𝑄^𝑇₂𝐶𝑄^𝑇₁𝐸0∗∗Ω₃₃Ω₃₄−𝑀₇0𝜀𝑁^𝑇₂∗∗∗Ω₄₄−𝑀₇00∗∗∗∗Ω₇₇00∗∗∗∗∗−𝜀𝐼0∗∗∗∗∗∗−𝜀𝐼<0, (4.9) where Ω₁₁=𝑊₁+𝑊^𝑇₁+𝑄^𝑇₁𝐴₁+𝐴^𝑇₁𝑄₁+𝑃₂−𝑒^−2𝛼ℎ𝑃₄+𝑀^𝑇₁+𝑀₁+2𝛼𝑃₁,Ω₂₂=−𝑄₂−𝑄^𝑇₂+ℎ²𝑃₃+ℎ²𝑃₄+𝑃₅,Ω₂₃=𝑄^𝑇₂𝐵₂−𝑀^𝑇₂,Ω₂₄=𝑄^𝑇₂𝐵₁−𝑀^𝑇₂,Ω₃₃=−𝑒^−2𝛼ℎ𝑃₂+𝜇𝑃₂−𝑒^−2𝛼ℎ𝑃₄−𝑀^𝑇₃−𝑀₃,Ω₃₄=−𝑀^𝑇₃−𝑀₄,Ω₄₄=−𝑒^−2𝛼ℎ𝑃₃−𝑀₄−𝑀^𝑇₄,Ω₇₇=−(1−𝛿)𝑒^−2𝛼𝑟𝑃₅,Ω₁₂=𝑃₁−𝑄^𝑇₁+𝐴^𝑇₁𝑄₂+𝑀₂,Ω₁₃=−𝑊₁+𝑄^𝑇₁𝐵₂+𝑒^−2𝛼ℎ𝑃₄−𝑀^𝑇₁+𝑀₃, Ω₁₄=−𝑊₁−𝑄^𝑇₁𝐵₁−𝑀^𝑇₁+𝑀₄,Ω₁₇=𝑄^𝑇₁𝐶+𝑀₇. Moreover, the solution 𝑥(𝑡,𝜙,𝜑) satisfies the inequality 𝑥𝑡,𝜙,𝜑≤𝑍𝜆_min𝑃₁max𝜙,𝜑𝑒^−𝛼𝑡,𝑡∈ℝ⁺, (4.10) where 𝑍=𝜆_max𝑃₁+ℎ𝜆_max𝑃₂+ℎ³𝜆_max𝑃₃+ℎ³𝜆_max𝑃₄+𝑟𝜆_max𝑃₅. (4.11)

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 (𝛼=0) of the linear system (3.19) with time-varying delays when 𝐴=−200−0.9,𝐵=−10−1−1. (5.1) Decompose matrix 𝐵 as follows: 𝐵=𝐵₁+𝐵₂, where 𝐵₁=−0.830−1−0.83,𝐵₂=−0.1700−0.17. (5.2)
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 Table 1 that our results (Corollary 3.2) are superior to those in Lemma  4 [16], Lemma  1 [4], and Theorem  1 [18].

Table 1 

Comparison of the maximum allowed time delay ℎ_𝑚𝑎𝑥.

Example 5.2. Consider the exponential stability of the neutral system (3.24) with time-varying delays and nonlinear perturbations with ,𝛽𝐴=−200−0.9,𝐵=−10−1−1,𝐶=0.1000.1₁=0.05,𝛽₂=0.1,ℎ(𝑡)=𝑟(𝑡). (5.3) Decompose matrix 𝐵 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].

Table 2 

Comparison of the maximum allowed time delay ℎ_𝑚𝑎𝑥.

Example 5.3. Consider the asymptotic stability (𝛼=0) of the uncertain neutral system (4.7) with time-varying delays with 𝐴=−200−0.9,𝐵=−10−1−1,𝐶=𝑐00𝑐,0≤𝑐<1,𝐸=𝐼,𝑁₁=𝑁₂=0.2𝐼,ℎ(𝑡)=𝑟(𝑡). (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].

Table 3 

Comparison of the maximum allowed time delay ℎ_𝑚𝑎𝑥.

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.