Robust Stability Criteria for Uncertain Neutral Systems with Interval Nondifferentiable Time-Varying Delay and Nonlinear Perturbations

Abstract

We study the robust stability criteria for uncertain neutral systems with interval time-varying delays and time-varying nonlinear perturbations simultaneously. The constraint on the derivative of the time-varying delay is not required, which allows the time-delay to be a fast time-varying function. Based on the Lyapunov-Krasovskii theory, we derive new delay-dependent stability conditions in terms of linear matrix inequalities (LMIs) which can be solved by various available algorithms. Numerical examples are given to demonstrate that the derived conditions are much less conservative than those given in the literature.

1. Introduction

It is well known that the existence of time delay in a system may cause instability and oscillations. Example, of time-delay systems are chemical engineering systems, biological modeling, electrical networks, physical networks, and many others, [7–16]. The stability criteria for system with time delays can be classified into two categories: delay-independent and delay-dependent. Delay-independent criteria do not employ any information on the size of the delay; while delay-dependent criteria make use of such information at different levels. Delay-dependent stability conditions are generally less conservative than delay-independent ones especially when the delay is small.

In many practical systems, models of system are described by neutral differential equations, in which the models depend on the delays of state and state derivatives. Heat exchanges, distributed networks containing lossless transmission lines and population ecology are examples of neutral systems because of its wider application. Therefore, several researchers have studied neutral systems and provided sufficient conditions to guarantee the asymptotic stability of neutral time delay systems, see [5, 9, 11–14, 16, 17] and references cited therein.

Well-known nonlinearities, as the delays, may cause instability and poor performance of practical systems, which have driven many researchers to study the problem of nonlinear perturbed systems with state delays during the recent years [5, 7, 9, 18]. In [18], the delay-dependent robust stability for linear time-varying systems with nonlinear perturbations is given, by using the Newton-Leibniz formula which has been taken into account instead of applying an integral inequality. In [7], a model transformation technique is used to deal with the stability of system with time varying for delays and nonlinear perturbations. In [9], based on a descriptor model transformation combined with a matrix decomposition approach, the robust stability of uncertain systems with time varying discrete delay is studied by applying an integral inequality. However, these model transformations often introduce additional dynamics which leads to relatively conservative results. In [5], the neutral delay and the discrete delay are all time-varying, while the derivative of discrete delay is less than 1 which limits its bigger application. In most studies the time-varying delays are required to be differentiable [1–5, 7, 9, 11–14, 16, 18]. Therefore their methods have a conservatism which can be improved upon. However, in most cases, these conditions are difficult to satisfy. From these reasons, the conditions are interesting to study, but there are fewer results for removing restriction to the derivative of interval time-varying delays. Therefore, in this paper we will employ some new techniques so that the above conditions can be removed.

In this paper, the problem of delay-dependent criterion for asymptotic stability for uncertain neutral system is studied with interval time-varying delay and time-varying nonlinear perturbations simultaneously. The restriction to the derivative of the interval time-varying delays is removed, which means that a fast interval time-varying delay is allowed. Based on the Lyapunov-Krasovskii theory, we derive new delay-dependent stability conditions in terms of linear matrix inequalities (LMIs) which can be solved by various available algorithms. The new stability condition is much less conservative and is more general than some existing results. Numerical examples are given to illustrate the effectiveness of our theoretical results.

2. Problem Formulation and Preliminaries

The following notations will be used in this paper: ℝ⁺ denotes the set of all real nonnegative numbers; ℝ^𝑛 denotes the 𝑛-dimensional space and the vector norm ‖⋅‖; 𝑀^𝑛×𝑟 denotes the space of all matrices of (𝑛×𝑟)-dimensions. 𝐴^𝑇 denotes the transpose of 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)}.𝑥_𝑡∶={𝑥(𝑡+𝑠)∶𝑠∈[−ℎ,0]}, ‖𝑥_𝑡‖=sup_{𝑠∈[−ℎ,0]}‖𝑥(𝑡+𝑠)‖; 𝐶([0,𝑡],ℝ^𝑛) denotes the set of all ℝ^𝑛-valued continuous functions on [0,𝑡]; 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; 𝐴>𝐵 means 𝐴−𝐵>0. The symmetric term in a matrix is denoted by ∗.

Consider the following neutral system with time-varying delay:𝑥̇(𝑡)−𝐶𝑥̇(𝑡−𝑑(𝑡))=𝐴(𝑡)𝑥(𝑡)𝑥+𝑡𝐵(𝑡)𝑥(𝑡−𝜏(𝑡))+𝐷₁(𝑡)𝑓₁(𝑡,𝑥(𝑡))+𝐷₂(𝑡)𝑓₂(𝑡,𝑥(𝑡−𝜏(𝑡))),₀+𝜃=𝜙(𝜃),𝜃∈[−ℎ,0],(2.1) where 𝑥(𝑡)∈𝑅^𝑛 is the state vector, 𝑑(𝑡) is a neutral delay, 𝜏(𝑡) is a time-varying continuous function which satisfies0≤𝜏_𝑚≤𝜏(𝑡)≤𝜏_𝑀,0≤𝑑(𝑡)≤𝑑,𝑑̇(𝑡)≤𝛿,(2.2) where 𝜏_𝑚, 𝜏_𝑀, 𝑑, 𝛿 are constants and ℎ=max{𝑑,𝜏_𝑀}; the initial condition function 𝜙(𝑡) denotes a continuous vector-valued initial function of 𝑡∈[−ℎ,0], 𝑓₁(𝑡,𝑥(𝑡)) and 𝑓₂(𝑡,𝑥(𝑡−𝜏(𝑡))) are unknown nonlinear perturbations satisfying 𝑓₁(𝑡,0)=0, 𝑓₂(𝑡,0)=0 and𝑓𝑓^𝑇1(𝑡,𝑥(𝑡))𝑓₁(𝑡,𝑥(𝑡))≤𝛼²𝑥^𝑇(𝑡)𝑥(𝑡),_𝑇2(𝑡,𝑥(𝑡−𝜏(𝑡)))𝑓₂(𝑡,𝑥(𝑡−𝜏(𝑡)))≤𝛽²𝑥^𝑇(𝑡−𝜏(𝑡))𝑥(𝑡−𝜏(𝑡)),(2.3) where 𝛼 and 𝛽 are positive real numbers.

The uncertain matrices 𝐴(𝑡), 𝐵(𝑡), 𝐷₁(𝑡), and 𝐷₂(𝑡) satisfy 𝐷𝐴(𝑡)=𝐴+Δ𝐴(𝑡),𝐵(𝑡)=𝐵+Δ𝐵(𝑡),₁(𝑡)=𝐷₁+Δ𝐷₁(𝑡),𝐷₂(𝑡)=𝐷₂+Δ𝐷₂(𝑡),(2.4) where 𝐴, 𝐵, 𝐷₁, 𝐷₂∈ℝ^𝑛×𝑛 are constant matrices with appropriate dimension, and Δ𝐴(𝑡), Δ𝐵(𝑡), Δ𝐷₁(𝑡), and Δ𝐷₂(𝑡) are unknown real matrices of appropriate dimension representing the systems time-varying parameter uncertainties which satisfyΔ𝐷Δ𝐴₁(𝑡)=𝐺₁𝐹(𝑡)𝐸_𝐴,Δ𝐵(𝑡)=𝐺₂𝐹(𝑡)𝐸_𝐵,(𝑡)=𝐺₃𝐹(𝑡)𝐸_𝐷1,Δ𝐷₂(𝑡)=𝐺₄𝐹(𝑡)𝐸_𝐷2,(2.5) where 𝐺₁, 𝐺₂, 𝐺₃, 𝐺₄, 𝐸_𝐴, 𝐸_𝐵, 𝐸_𝐷1, and 𝐸_𝐷2 are known real constant matrices of appropriate dimension. 𝐹(𝑡) is unknown time-varying matrix satisfying𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼.(2.6) For simplicity, we denote 𝑓₁(𝑡,𝑥(𝑡)), 𝑓₂(𝑡,𝑥(𝑡−𝜏(𝑡))), by 𝑓₁, 𝑓₂, respectively.

Let 𝜏_𝑒=(1/2)(𝜏_𝑀+𝜏_𝑚) and 𝜌=(1/2)(𝜏_𝑀−𝜏_𝑚). Then 𝜏(𝑡) can be expressed as𝜏(𝑡)=𝜏_𝑒+𝜌𝜉(𝑡),(2.7) where 𝜉(𝑡)=⎧⎪⎨⎪⎩2𝜏(𝑡)−𝜏^𝑀+𝜏_𝑚𝜏_𝑀−𝜏_𝑚0,,𝜏𝜏_𝑀𝑀>𝜏_𝑚,=𝜏_𝑚.(2.8) Obviously, |𝜉(𝑡)|≤1. For this case, 𝜏(𝑡) is a function belonging to the interval [𝜏_𝑒−𝜌,𝜏_𝑒+𝜌], where 𝜌 can be taken as the range of variation of the time-varying delay 𝜏(𝑡). Using the fact that 𝑥𝑡−𝜏_𝑒−𝑥(𝑡−𝜏(𝑡))=^{𝑡−𝜏𝑒}_{𝑡−𝜏(𝑡)}𝑥̇(𝑠)𝑑𝑠(2.9) system (2.1) can be rewritten as𝑥̇(𝑡)−𝐶𝑥̇(𝑡−𝑑(𝑡))=+𝐴𝐷(𝑡₁)𝑥(𝑡)+𝐵(𝑡)𝑥𝑡−𝜏_𝑒−𝐵(𝑡)^{𝑡−𝜏𝑒}_{𝑡−𝜏(𝑡)}𝑥̇(𝑠)𝑑𝑠(𝑡)𝑓₁+𝐷₂(𝑡)𝑓₂.(2.10)

Lemma 2.1 (see [17]). There exists a symmetric matrix 𝑋 such that ⎡⎢⎢⎣𝑃₁−𝐿𝑋𝐿^𝑇𝑄₁𝑄^𝑇1𝑅₁⎤⎥⎥⎦<0,⎡⎢⎢⎣𝑃₂+𝑋𝑄₂𝑄^𝑇2𝑅₂⎤⎥⎥⎦<0(2.11) if and only if ⎡⎢⎢⎢⎢⎣𝑃₁+𝐿𝑃₂𝐿^𝑇𝑄₁𝐿𝑄₂𝑄𝑄^𝑇1_𝑇2𝑅₁0𝐿^𝑇0𝑅₂⎤⎥⎥⎥⎥⎦<0.(2.12)

Lemma 2.2 (see [3]). For any constant symmetric matrix 𝑀∈𝑅^𝑛×𝑛, 𝑀=𝑀^𝑇>0, 0≤ℎ_𝑚≤ℎ(𝑡)≤ℎ_𝑀, 𝑡≥0, and any differentiable vector function 𝑥(𝑡)∈𝑅^𝑛, we have (a)^{𝑡𝑡−ℎ}_𝑚𝑥̇(𝑠)𝑑𝑠^𝑇𝑀^{𝑡𝑡−ℎ}_𝑚𝑥̇(𝑠)𝑑𝑠≤ℎ_𝑚^{𝑡𝑡−ℎ}_𝑚(b)_{𝑡−ℎ𝑚𝑡−ℎ(𝑡)}𝑥̇^𝑇(𝑠)𝑀𝑥̇(𝑠)𝑑𝑠,𝑥̇(𝑠)𝑑𝑠^𝑇𝑀^{𝑡−ℎ𝑚}_{𝑡−ℎ(𝑡)}𝑥̇(𝑠)𝑑𝑠≤≤ℎℎ(𝑡)−ℎ_𝑚^{𝑡−ℎ𝑚}_{𝑡−ℎ(𝑡)}𝑥̇^𝑇(𝑠)𝑀𝑥̇(𝑠)𝑑𝑠_𝑀−ℎ_𝑚^{𝑡−ℎ𝑚}_{𝑡−ℎ(𝑡)}𝑥̇^𝑇(𝑠)𝑀𝑥̇(𝑠)𝑑𝑠.(2.13)

Lemma 2.3 (see [19]). Given matrices 𝑄=𝑄^𝑇, 𝐻, 𝐸, and 𝑅=𝑅^𝑇>0 with appropriate dimensions. Then 𝑄+𝐻𝐹𝐸+𝐸^𝑇𝐹^𝑇𝐻^𝑇<0(2.14) for all 𝐹 satisfying 𝐹^𝑇𝐹≤𝑅, if and only if there exists an 𝜖>0 such that 𝑄+𝜖𝐻𝐻^𝑇+𝜖⁻¹𝐸^𝑇𝑅𝐸<0.(2.15)

Proposition 2.4 (Cauchy inequality). For any symmetric positive definite matrix 𝑁∈𝑀^𝑛×𝑛 and 𝑥,𝑦∈ℝ^𝑛, we have ±2𝑥^𝑇𝑦≤𝑥^𝑇𝑁𝑥+𝑦^𝑇𝑁⁻¹𝑦.(2.16)

3. Main Results

Now we present a new delay-dependent condition for the asymptotic stability of system (2.1).

Assumption 3.1. All the eigenvalues of matrix C are inside the unit circle.
First, we study the problem of stability for nominal system of (2.10) with Δ𝐴(𝑡)=0, Δ𝐵(𝑡)=0, Δ𝐷₁(𝑡)=0, and Δ𝐷₂(𝑡)=0.

Theorem 3.2. Under Assumption 3.1, nominal system of (2.10) with time-varying delay satisfying (2.2) is asymptotically stable if there exist positive definite matrices 𝑃, 𝑄, 𝑄₁, 𝑅, 𝑆, 𝑊, matrices 𝐾₁, 𝐾₂, 𝐿_𝑖, 𝑀_𝑖,𝑖=1,2,…,7 of appropriate dimension and 𝛿₁,𝛿₂>0 such that Σ₁=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜙¹¹𝜙₁₂𝜙₁₃𝜙₁₄𝜙₁₅𝜙₁₆𝜙₁₇𝜏_𝑒𝐿^𝑇1𝜌𝑀₁∗𝜙₂₂𝜙₂₃𝜙₂₄𝜙₂₅𝜙₂₆𝜙₂₇𝜏_𝑒𝐿^𝑇2𝜌𝑀₂∗∗𝜙₃₃𝜙₃₄𝜙₃₅𝜙₃₆𝜙₃₇𝜏_𝑒𝐿^𝑇3𝜌𝑀₃∗∗∗∗∗∗𝜙∗₄₄𝜙𝜙₄₅₅₅𝜙₄₆𝜙₄₇𝜏_𝑒𝐿^𝑇4𝜌𝐾^𝑇1𝐵+𝑀₄∗∗∗∗∗𝜙𝜙₅₆₆₆𝜙₅₇𝜏_𝑒𝐿^𝑇5𝜌𝐾^𝑇2𝐵+𝑀₅0𝜏_𝑒𝐿^𝑇6𝜌𝑀₆∗∗∗∗∗∗𝜙₇₇𝜏_𝑒𝐿^𝑇7𝜌𝑀₇∗∗∗∗∗∗∗∗∗∗∗∗∗∗−𝜏∗_𝑒𝑅−𝜌0𝑆⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.1) where 𝜙₁₁=𝜙𝑄+𝐿₁+𝐿^𝑇1+𝜀₁𝛼²𝐼,₁₂𝜙₁₃=𝑀^𝑇1+𝐿₂,𝜙₁₄=−𝐿^𝑇1+𝐿₃+𝑀₁,𝜙=₁₅𝑃+𝐴^𝑇𝐾₁+𝐿₄,=𝜙𝐴^𝑇₁₆𝐾₂+𝐿₅,𝜙₁₇=𝐿₆,𝜙₂₂=𝐿₇,𝜙=𝑀^𝑇2+𝑀₂+𝜀₂𝛽²𝐼−𝑊,₂₃=−𝐿𝜙^𝑇2−𝑀^𝑇2+𝑀₃+𝑊,₂₄𝜙₂₅=𝑀₄,𝜙₂₆=𝑀₅,𝜙₂₇=𝑀₆,𝜙₃₃=𝑀₇,=−𝜙𝑄−𝐿^𝑇3−𝐿₃−𝑀^𝑇3−𝑀₃−𝑊,₃₄𝜙₃₅=𝐵^𝑇𝐾₁−𝐿₄−𝑀₄,𝜙=₃₆𝐵^𝑇𝐾₂−𝐿₅−𝑀₅,𝜙₃₇=−𝐿₆−𝑀₆,𝜙₄₄=−𝐿₇−𝑀₇,=𝑄₁+𝜙𝜏_𝑒𝑅+𝜌𝑆−𝐾^𝑇1−𝐾₁+𝜌²𝑊,₄₅𝜙=₄₆𝐾^𝑇1𝐶−𝐾₂,𝜙₄₇=𝐾^𝑇1𝐷₁,𝜙₅₅=𝐾^𝑇1𝐷₂,=−(1−𝜙𝛿₅₆)𝑄₁+𝐾^𝑇2𝐶+𝐶^𝑇𝐾₂,𝜙₅₇=𝐾^𝑇2𝐷₁,𝜙₆₆=𝐾^𝑇2𝐷₂,𝜙=−𝛿₁𝐼,₇₇=−𝛿₂𝐼.(3.2)

Proof. We prove that Theorem 3.2 is true for three cases, namely, 𝜏_𝑚≤𝜏(𝑡)<𝜏_𝑒; 𝜏(𝑡)=𝜏_𝑒; 𝜏_𝑒<𝜏(𝑡)≤𝜏_𝑀.
Case 1 (𝜏_𝑚≤𝜏(𝑡)<𝜏_𝑒). Choose a Lyapunov-Krasovskii functional candidate as 𝑉₁𝑥_𝑡=+𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+^{𝑡𝑡−𝜏}_𝑒𝑥^𝑇(𝑠)𝑄𝑥(𝑠)𝑑𝑠_{𝑡𝑡−𝑑(𝑡)}𝑥̇^𝑇(𝑠)𝑄₁𝑥̇(𝑠)𝑑𝑠^0−𝜏_𝑒^{𝑡𝑡+𝑠}+𝑥̇^𝑇(𝜃)𝑅𝑥̇(𝜃)𝑑𝜃𝑑𝑠_{−𝜏𝑚−𝜏𝑒}^{𝑡𝑡+𝑠}𝑥̇^𝑇(𝜃)𝑆𝑥̇(𝜃)𝑑𝜃𝑑𝑠+𝜌^−𝜏𝑚_−𝜏𝑒^{𝑡𝑡+𝑠}𝑥̇^𝑇(𝜃)𝑊𝑥̇(𝜃)𝑑𝜃𝑑𝑠,(3.3) where 𝑃, 𝑄, 𝑄₁, 𝑅, 𝑆, and 𝑊 are positive definite matrices. Taking the derivative of 𝑉₁(𝑥_𝑡) with respect to 𝑡 along the trajectory of (2.10) yields 𝑉₁̇𝑥_𝑡=+2𝑥𝑥^𝑇̇_𝑇(𝑡)𝑃𝑥̇(𝑡)+𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−𝑥^𝑇𝑡−𝜏_𝑒𝑄𝑥𝑡−𝜏_𝑒+𝑥̇_𝑇(𝑡)𝑄₁𝑥̇(𝑡)−1−𝑑̇(𝑡)𝑥̇^𝑇(𝑡−𝑑(𝑡))𝑄₁𝑥̇^𝑇(𝑡−𝑑(𝑡))−(𝑡)𝜏_𝑒𝑅+𝜌²𝑊+𝜌𝑆𝑥̇(𝑡)_{𝑡𝑡−𝜏𝑒}−𝜌𝑥_{𝑡−𝜏𝑚𝑡−𝜏𝑒}̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠−^{𝑡−𝜏𝑚}_{𝑡−𝜏𝑒}𝑥̇^𝑇(𝑠)𝑆𝑥̇(𝑠)𝑑𝑠≤2𝑥_𝑇𝑥̇^𝑇(𝑠)𝑊𝑥̇(𝑠)𝑑𝑠+𝑥̇_𝑇(𝑡)𝑃𝑥̇(𝑡)+𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−𝑥^𝑇𝑡−𝜏_𝑒𝑄𝑥𝑡−𝜏_𝑒+𝑥̇_𝑇(𝑡)𝑄₁𝑥̇(𝑡)−(1−𝛿)𝑥̇^𝑇(𝑡−𝑑(𝑡))𝑄₁𝑥̇^𝑇(𝑡−𝑑(𝑡))−(𝑡)𝜏_𝑒𝑅+𝜌²𝑊+𝜌𝑆𝑥̇(𝑡)_{𝑡𝑡−𝜏𝑒}−𝜌𝑥_{𝑡−𝜏𝑚𝑡−𝜏𝑒}̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠−^{𝑡−𝜏𝑚}_{𝑡−𝜏𝑒}𝑥̇^𝑇(𝑠)𝑆𝑥̇(𝑠)𝑑𝑠𝑥̇^𝑇(𝑠)𝑊𝑥̇(𝑠)𝑑𝑠,(3.4) since −𝜌−^{𝑡−𝜏𝑚}_{𝑡−𝜏𝑒𝑡−𝜏𝑚𝑡−𝜏𝑒}𝑥̇^𝑇(𝑠)𝑆𝑥̇(𝑠)𝑑𝑠≤−^{𝑡−𝜏(𝑡)𝑡−𝜏}_𝑒𝑥̇^𝑇(𝑠)𝑆𝑥̇(𝑠)𝑑𝑠,𝑥̇^𝑇(𝑠)𝑊𝑥̇(𝑠)𝑑𝑠≤−𝜌^{𝑡−𝜏(𝑡)𝑡−𝜏}_𝑒𝑥̇^𝑇(𝑠)𝑊𝑥̇(𝑠)𝑑𝑠.(3.5) Based on Lemma 2.2, we obtain −𝜌^{𝑡−𝜏(𝑡)𝑡−𝜏}_𝑒𝑥̇^𝑇(𝑠)𝑊𝑥̇(𝑠)𝑑𝑠≤≤−−𝑥𝜏_𝑒𝑇−𝜏(𝑡)^{𝑡−𝜏(𝑡)𝑡−𝜏}_𝑒𝑥̇^𝑇(𝑠)𝑊𝑥̇(𝑠)𝑑𝑠−𝑥_𝑇𝑡−𝜏_𝑒𝑊𝑥𝑡−𝜏_𝑒+2𝑥^𝑇𝑡−𝜏_𝑒𝑊𝑥(𝑡−𝜏(𝑡))(𝑡−𝜏(𝑡))𝑊𝑥(𝑡−𝜏(𝑡)),(3.6) and from the following equalities: 2𝑥̇^𝑇(𝑡)𝐾^𝑇1+𝑥̇^𝑇(𝑡−𝑑(𝑡))𝐾^𝑇2+𝐷×₁𝐴𝑥(𝑡)+𝐵𝑥𝑡−𝜏_𝑒−𝐵^{𝑡−𝜏𝑒}_{𝑡−𝜏(𝑡)}𝑥̇(𝑠)𝑑𝑠2𝑥𝑓₁+𝐷₂𝑓₂+𝐶𝑥̇(𝑡−𝑑(𝑡))−𝑥̇(𝑡)=0,(3.7)_𝑇(𝑡)𝐿^𝑇1+𝑥^𝑇(𝑡−𝜏(𝑡))𝐿^𝑇2+𝑥^𝑇𝑡−𝜏_𝑒𝐿^𝑇3+𝑥̇^𝑇(𝑡)𝐿^𝑇4+𝑥̇^𝑇(𝑡−𝑑(𝑡))𝐿^𝑇52𝑥+𝑓^𝑇1𝐿^𝑇6+𝑓^𝑇2𝐿^𝑇7×𝑥(𝑡)−𝑥𝑡−𝜏_𝑒−^{𝑡𝑡−𝜏}_𝑒𝑥̇(𝑠)𝑑𝑠=0,(3.8)_𝑇(𝑡)𝑀^𝑇1+𝑥^𝑇(𝑡−𝜏(𝑡))𝑀^𝑇2+𝑥^𝑇𝑡−𝜏_𝑒𝑀^𝑇3+𝑥̇^𝑇(𝑡)𝑀^𝑇4+𝑥̇^𝑇(𝑡−𝑑(𝑡))𝑀^𝑇5+𝑓^𝑇1𝑀^𝑇6+𝑓^𝑇2𝑀^𝑇7×𝑥(𝑡−𝜏(𝑡))−𝑥𝑡−𝜏_𝑒−^{𝑡−𝜏(𝑡)𝑡−𝜏}_𝑒𝑥̇(𝑠)𝑑𝑠=0,(3.9) where 𝐾₁, 𝐾₂, and 𝐿_𝑖, 𝑀_𝑖, 𝑖=1,2,…,7 are some matrices of appropriate dimension. Next, from (4.5), for any scalars 𝛿₁>0 and 𝛿₂>0, we obtain 𝛿𝛿₁𝛼²𝑥^𝑇(𝑡)𝑥(𝑡)−𝑓^𝑇1𝑓₁≥0,₂𝛽²𝑥^𝑇(𝑡−𝜏(𝑡))𝑥(𝑡−𝜏(𝑡))−𝑓^𝑇2𝑓₂≥0.(3.10) By adding the terms on left of (3.7)–(3.10) to 𝑉₁̇(𝑥_𝑡), we may express 𝑉₁̇(𝑥_𝑡) as 𝑉₁̇𝑥_𝑡≤−2𝑥(1^𝑇−(𝑡)𝛿𝑃)𝑥𝑥̇̇_𝑇(𝑡)+𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−𝑥^𝑇𝑡−𝜏_𝑒𝑄𝑥𝑡−𝜏_𝑒+𝑥̇^𝑇(𝑡)𝑄₁𝑥̇(𝑡)−(𝑡−𝑑(𝑡))𝑄₁𝑥̇(𝑡−𝑑(𝑡))+𝑥̇^𝑇(𝑡)𝜏_𝑒𝑅+𝜌𝑆+𝜌²𝑊𝑥̇(𝑡)_{𝑡𝑡−𝜏𝑒}×𝐴𝑥𝑥(𝑡̇^𝑇)(+𝑠)𝐵𝑅𝑥𝑥̇(𝑡𝑠−)𝑑𝜏𝑠_𝑒−^{𝑡−𝜏(𝑡)𝑡−𝜏}_𝑒𝑥̇^𝑇(𝑠)𝑆𝑥̇(𝑠)𝑑𝑠+2𝑥̇^𝑇(𝑡)𝐾^𝑇1+𝑥̇^𝑇(𝑡−𝑑(𝑡))𝐾^𝑇2+2𝑥_𝑇+𝐵^{𝑡−𝜏(𝑡)𝑡−𝜏}_𝑒𝑥̇(𝑠)𝑑𝑠+𝐷₁𝑓₁+𝐷₂𝑓₂+𝐶𝑥̇(𝑡−𝑑(𝑡))−𝑥̇(𝑡)(𝑡)𝐿^𝑇1+𝑥^𝑇(𝑡−𝜏(𝑡))𝐿^𝑇2+𝑥^𝑇𝑡−𝜏_𝑒𝐿^𝑇3+𝑥̇^𝑇(𝑡)𝐿^𝑇4+𝑥̇^𝑇(𝑡−𝑑(𝑡))𝐿^𝑇5+2+𝑥𝑓^𝑇1_𝑇𝐿^𝑇6+𝑓^𝑇2𝐿^𝑇7×𝑥(𝑡)−𝑥𝑡−𝜏_𝑒−^{𝑡𝑡−𝜏}_𝑒𝑥̇(𝑠)𝑑𝑠(𝑡)𝑀^𝑇1+𝑥^𝑇(𝑡−𝜏(𝑡))𝑀^𝑇2×𝑥+(𝑥𝑡^𝑇−𝜏𝑡−(𝑡)𝜏)_𝑒−𝑀𝑥^𝑇3𝑡+−𝑥𝜏̇^𝑇_𝑒(𝑡)𝑀^𝑇4+𝑥̇^𝑇(𝑡−𝑑(𝑡))𝑀^𝑇5+𝑓^𝑇1𝑀^𝑇6+𝑓^𝑇2𝑀^𝑇7−𝑥_𝑇−^{𝑡−𝜏(𝑡)𝑡−𝜏}_𝑒𝑥̇(𝑠)𝑑𝑠+𝛿₁𝛼²𝑥^𝑇(𝑡)𝑥(𝑡)−𝑓^𝑇1𝑓₁−𝑥_𝑇𝑡−𝜏_𝑒𝑊𝑥𝑡−𝜏_𝑒+2𝑥^𝑇𝑡−𝜏_𝑒𝑊𝑥(𝑡−𝜏(𝑡))=1(𝑡−𝜏(𝑡))𝑊𝑥(𝑡−𝜏(𝑡))+𝛿₂𝛽²𝑥^𝑇(𝑡−𝜏(𝑡))𝑥(𝑡−𝜏(𝑡))−𝑓^𝑇2𝑓₂𝜏_𝑒^{𝑡𝑡−𝜏}_𝑒𝜔^𝑇(𝑡,𝑠)𝜙₁𝜔(𝑡,𝑠)𝑑𝑠+1𝜏_𝑒−𝜏(𝑡)^{𝑡−𝜏(𝑡)𝑡−𝜏}_𝑒𝜔^𝑇(𝑡,𝑠)𝜙₂𝜔(𝑡,𝑠)𝑑𝑠,(3.11) where 𝜔𝜙^𝑇(𝑡,𝑠)=𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−𝜏(𝑡))𝑥^𝑇𝑡−𝜏_𝑒𝑥̇^𝑇(𝑡)𝑥̇^𝑇(𝑡−𝑑(𝑡))𝑓^𝑇1𝑓^𝑇2−𝑥̇^𝑇(𝑠)₁,=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜙¹¹𝜙₁₂𝜙₁₃𝜙₁₄𝜙₁₅𝜙₁₆𝜙₁₇𝜏_𝑒𝐿^𝑇1∗𝜙₂₂𝜙₂₃𝜙₂₄𝜙₂₅𝜙₂₆𝜙₂₇𝜏_𝑒𝐿^𝑇2∗∗𝜙₃₃𝜙₃₄𝜙₃₅𝜙₃₆𝜙₃₇𝜏_𝑒𝐿^𝑇3∗∗∗𝜙₄₄𝜙₄₅𝜙₄₆𝜙₄₇𝜏_𝑒𝐿^𝑇4∗∗∗∗𝜙₅₅𝜙₅₆𝜙₅₇𝜏_𝑒𝐿^𝑇5∗∗∗∗∗𝜙₆₆0𝜏_𝑒𝐿^𝑇6∗∗∗∗∗∗𝜙₇₇𝜏_𝑒𝐿^𝑇7𝜙∗∗∗∗∗∗∗−𝜏_𝑒𝑅⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+𝑍,₂=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝑍¹¹−𝑍₁₂−𝑍₁₃−𝑍₁₄−𝑍₁₅−𝑍₁₆−𝑍₁₇𝜏_𝑒−𝜏(𝑡)𝑀₁∗−𝑍₂₂−𝑍₂₃−𝑍₂₄−𝑍₂₅−𝑍₂₆−𝑍₂₇𝜏_𝑒−𝜏(𝑡)𝑀₂∗∗−𝑍₃₃−𝑍₃₄−𝑍₃₅−𝑍₃₆−𝑍₃₇𝜏_𝑒−𝜏(𝑡)𝑀₃∗∗∗−𝑍₄₄−𝑍₄₅−𝑍₄₆−𝑍₄₇𝜑₁∗∗∗∗−𝑍₅₅−𝑍₅₆−𝑍₅₇𝜑₂∗∗∗∗∗−𝑍₆₆−𝑍₆₇𝜏_𝑒−𝜏(𝑡)𝑀₆∗∗∗∗∗∗−𝑍₇₇𝜏_𝑒−𝜏(𝑡)𝑀₇𝑍=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑍∗₁₁∗∗∗∗∗∗−||𝜏_𝑒−𝜏(𝑡)||𝑆⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,∗𝑍𝑍₁₂₂₂𝑍₁₃𝑍₁₄𝑍₁₅𝑍₁₆𝑍₁₇0∗∗𝑍𝑍₂₃₃₃𝑍₂₄𝑍₂₅𝑍₂₆𝑍₂₇0∗∗∗𝑍𝑍₃₄₄₄𝑍₃₅𝑍₃₆𝑍₃₇0∗∗∗∗𝑍𝑍₄₅₅₅𝑍₄₆𝑍₄₇0∗∗∗∗∗𝑍𝑍₅₆₆₆𝑍₅₇0∗∗∗∗∗∗𝑍𝑍₆₇₇₇0∗∗∗∗∗∗∗00⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(3.12)𝜑_𝑖=(𝜏_𝑒−𝜏(𝑡))(−𝐾^𝑇𝑖𝐵+𝑀_𝑖), 𝑖=1,2, 𝑍_𝑖𝑖>0, 𝑖=1,2,…,7, 𝑍_𝑖𝑗, 𝑖=1,2,…,7, 𝑗=𝑖+1,…,7 are some parameter matrices of appropriate dimensions. From (3.11) if 𝜙₁<0 and 𝜙₂<0, then 𝑉₁̇(𝑥_𝑡)≤−𝜆₁‖𝑥(𝑡)‖² for some 𝜆₁>0. Pre and postmultiplying both sides of 𝜙₂<0 by diag{𝐼,𝐼,𝐼,𝐼,𝐼,𝐼,𝐼,sgn(𝜏(𝑡)−𝜏_𝑒)}, we get that 𝜙₂≤⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝑍¹¹−𝑍₁₂−𝑍₁₃−𝑍₁₄−𝑍₁₅−𝑍₁₆−𝑍₁₇𝜌𝑀₁∗−𝑍₂₂−𝑍₂₃−𝑍₂₄−𝑍₂₅−𝑍₂₆−𝑍₂₇𝜌𝑀₂∗∗−𝑍₃₃−𝑍₃₄−𝑍₃₅−𝑍₃₆−𝑍₃₇𝜌𝑀₃∗∗∗∗∗∗−𝑍∗₄₄−−𝑍𝑍₄₅₅₅−𝑍₄₆−𝑍₄₇𝜌−𝐾^𝑇1𝐵+𝑀₄∗∗∗∗∗−−𝑍𝑍₅₆₆₆−𝑍₅₇𝜌−𝐾^𝑇2𝐵+𝑀₅−𝑍₆₇𝜌𝑀₆∗∗∗∗∗∗−𝑍₇₇𝜌𝑀₇∗∗∗∗∗∗∗−𝜌𝑆⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0.(3.13) By Schur complement lemma, this implies 𝜙₂<0. In light of Lemma 2.1, (3.1) holds if and only if 𝜙₁<0 and (3.13) simultaneously hold. Then (3.1) holds if and only if there exists a symmetric matrix 𝑍,𝜙₁<0 and (3.13) simultaneously hold. Therefore, nominal system of (2.10) is asymptotically stable.
Case 2 (𝜏(𝑡)=𝜏_𝑒). For this case, we choose a Lyapunov-Krasovskii functional candidate as 𝑉₂𝑥_𝑡=+𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+^{𝑡𝑡−𝜏}_𝑒𝑥^𝑇(𝑠)𝑄𝑥(𝑠)𝑑𝑠+^{𝑡𝑡−𝑑(𝑡)}𝑥̇^𝑇(𝑠)𝑄₁𝑥̇(𝑠)𝑑𝑠_0−𝜏𝑒𝑑𝑠^{𝑡𝑡+𝑠}𝑥̇^𝑇(𝜃)𝑅𝑥̇(𝜃)𝑑𝜃,(3.14) where 𝑃, 𝑄, 𝑄₁, and 𝑅 positive definite matrices are the same as those in 𝑉₁(𝑥_𝑡).
Case 3 (𝜏_𝑒<𝜏(𝑡)<𝜏_𝑀). For this case, we choose the Lyapunov-Krasovskii functional candidate as 𝑉₃𝑥_𝑡=+𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+^{𝑡𝑡−𝜏}_𝑒𝑥^𝑇(𝑠)𝑄𝑥(𝑠)𝑑𝑠_{𝑡𝑡−𝑑(𝑡)}𝑥̇^𝑇(𝑠)𝑄₁𝑥̇(𝑠)𝑑𝑠^0−𝜏_𝑒^{𝑡𝑡+𝑠}+𝑥̇^𝑇(𝜃)𝑅𝑥̇(𝜃)𝑑𝜃𝑑𝑠_{−𝜏𝑒−𝜏𝑀}^{𝑡𝑡+𝑠}𝑥̇^𝑇(𝜃)𝑆𝑥̇(𝜃)𝑑𝜃𝑑𝑠+𝜌^−𝜏𝑒_−𝜏𝑀^{𝑡𝑡+𝑠}𝑥̇^𝑇(𝜃)𝑊𝑥̇(𝜃)𝑑𝜃𝑑𝑠,(3.15) where 𝑃, 𝑄, 𝑄₁, 𝑅, 𝑆, and 𝑊 are positive definite matrices and are the same as those in 𝑉₁(𝑥_𝑡).
By similar arguments used in proof of Theorem 3.2, we conclude that the nominal system of (2.10) is robustly asymptotically stable. The proof is complete.

Based on Theorem 3.2, we can perform the robust stability analysis for system (2.10) with uncertainties (2.5) and (2.6).

Theorem 3.3. Under Assumption 3.1, system (2.10) with time-varying delay satisfying (2.2) and uncertainties (2.5) and (2.6) is asymptotically stable if there exist positive definite matrices 𝑃, 𝑄, 𝑄₁, 𝑅, 𝑆, 𝑊 and 𝐾₁, 𝐾₂, 𝐿_𝑖, 𝑀_𝑖, 𝑖=1,2,…,7 of appropriate dimension and scalars 𝜖_𝑖>0, 𝑖=1,2,…,10 such that ℳ=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑀¹¹𝜙₁₂𝜙₁₃𝜙₁₄𝜙₁₅𝜙₁₆𝜙₁₇𝜏_𝑒𝐿^𝑇1𝜌𝑀₁∗𝜙₂₂𝜙₂₃𝜙₂₄𝜙₂₅𝜙₂₆𝜙₂₇𝜏_𝑒𝐿^𝑇2𝜌𝑀₂∗∗𝑀₃₃𝜙₃₄𝜙₃₅𝜙₃₆𝜙₃₇𝜏_𝑒𝐿^𝑇3𝜌𝑀₃∗∗∗∗∗∗𝑀∗₄₄𝑀𝜙₄₅₅₅𝜙₄₆𝜙₄₇𝜏_𝑒𝐿^𝑇4𝜌𝐾^𝑇1𝐵+𝑀₄∗∗∗∗∗𝑀𝜙₅₆₆₆𝜙₅₇𝜏_𝑒𝐿^𝑇5𝜌𝐾^𝑇2𝐵+𝑀₅0𝜏_𝑒𝐿^𝑇6𝜌𝑀₆∗∗∗∗∗∗𝑀₇₇𝜏_𝑒𝐿^𝑇7𝜌𝑀₇∗∗∗∗∗∗∗∗∗∗∗∗∗∗−𝜏∗_𝑒𝑅𝑀0₉₉ℳ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,₁=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−0.1𝐾^𝑇1−0.1𝐾₁𝐾^𝑇1𝐺₁𝐾^𝑇1𝐺₂𝐾^𝑇1𝐺₂𝐾^𝑇1𝐺₃𝐾^𝑇1𝐺₄𝐺𝐺^𝑇1𝐾₁−𝜖₁𝐼0000_𝑇2𝐺𝐾₁0−𝜖₂𝐼000_𝑇2𝐺𝐾₁00−𝜖₃𝐼00_𝑇3𝐺𝐾₁000−𝜖₄𝐼0_𝑇4ℳ𝐾₁0000−𝜖₅𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,₂=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−0.1𝑄¹+0.1𝐾^𝑇2𝐶+0.1𝐶^𝑇𝐾₂𝐾^𝑇2𝐺₁𝐾^𝑇2𝐺₂𝐾^𝑇2𝐺₂𝐾^𝑇2𝐺₃𝐾^𝑇2𝐺₄𝐺𝐺^𝑇1𝐾₂−𝜖₆𝐼0000_𝑇2𝐺𝐾₂0−𝜖₇𝐼000_𝑇2𝐺𝐾₂00−𝜖₈𝐼00_𝑇3𝐺𝐾₂000−𝜖₉𝐼0_𝑇4𝐾₂0000−𝜖₁₀𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.16) where 𝑀𝑀₁₁₃₃=𝜙₁₁+𝜖₁𝐸^𝑇𝐴𝐸_𝐴+𝜖₆𝐸^𝑇𝐴𝐸_𝐴,𝑀₄₄=𝜙₃₃+𝜖₂𝐸^𝑇𝐵𝐸_𝐵+𝜖₇𝐸^𝑇𝐵𝐸_𝐵,𝑀=𝑄₁+𝜏_𝑒𝑅+𝜌𝑆−0.9𝐾^𝑇1−0.9𝐾₁+𝜌²𝑊,₅₅𝑀₆₆=−0.9𝑄₁+𝛿𝑄₁+0.9𝐾^𝑇2𝐶+0.9𝐶^𝑇𝐾₂,𝑀₇₇=𝜙₆₆+𝜖₄𝐸^𝑇𝐷₁𝐸_𝐷1+𝜖₉𝜌²𝐸^𝑇𝐵𝐸_𝐵,𝑀₉₉=𝜙₇₇+𝜖₅𝐸^𝑇𝐷₂𝐸_𝐷2+𝜖₁₀𝜌²𝐸^𝑇𝐵𝐸_𝐵,=−𝜌𝑆+𝜖₃𝜌²𝐸^𝑇𝐵𝐸_𝐵+𝜖₈𝜌²𝐸^𝑇𝐵𝐸_𝐵.(3.17)

Proof. We choose Lyapunov-Krasovskii functional as in Theorem 3.2, we may proof this Theorem by using a similar arguments as in the proof of Theorem 3.2. By replacing 𝐴, 𝐵, 𝐷₁, and 𝐷₂ in (3.11) with 𝐴+𝐺𝐹(𝑡)𝐸_𝐴, 𝐵+𝐺𝐹(𝑡)𝐸_𝐵, 𝐷₁+𝐺𝐹(𝑡)𝐸_𝐷1 and 𝐷₂+𝐺𝐹(𝑡)𝐸_D2, respectively. For Case 1𝑉₁̇𝑥_𝑡≤−2𝑥(1^𝑇−(𝑡)𝛿𝑃)𝑥𝑥̇̇_𝑇(𝑡)+𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−𝑥^𝑇𝑡−𝜏_𝑒𝑄𝑥𝑡−𝜏_𝑒+𝑥̇^𝑇(𝑡)𝑄₁𝑥̇(𝑡)−(𝑡−𝑑(𝑡))𝑄₁𝑥̇(𝑡−𝑑(𝑡))+𝑥̇^𝑇(𝑡)𝜏_𝑒𝑅+𝜌𝑆+𝜌²𝑊𝑥̇(𝑡)_{𝑡𝑡−𝜏𝑒}×𝐴𝑥+̇^𝑇𝐺(𝑠₁)𝑅𝑥̇(𝑠)𝑑𝑠−^{𝑡−𝜏(𝑡)𝑡−𝜏}_𝑒𝑥̇^𝑇(𝑠)𝑆𝑥̇(𝑠)𝑑𝑠+2𝑥̇^𝑇(𝑡)𝐾^𝑇1+𝑥̇^𝑇(𝑡−𝑑(𝑡))𝐾^𝑇2×_{𝑡−𝜏(𝑡)𝑡−𝜏𝑒}𝐹(𝑡)𝐸_𝐴𝑥(𝑡)+𝐵+𝐺₂𝐹(𝑡)𝐸_𝐵𝑥𝑡−𝜏_𝑒+𝐵+𝐺₂𝐹(𝑡)𝐸_𝐵+2𝑥_𝑇𝑥̇(𝑠)𝑑𝑠+𝐷₁+𝐺₃𝐹(𝑡)𝐸_𝐷1𝑓₁+𝐷₂+𝐺₄𝐹(𝑡)𝐸_𝐷2𝑓₂+𝐶𝑥̇(𝑡−𝑑(𝑡))−𝑥̇(𝑡)(𝑡)𝐿^𝑇1+𝑥^𝑇(𝑡−𝜏(𝑡))𝐿^𝑇2+𝑥^𝑇𝑡−𝜏_𝑒𝐿^𝑇3+𝑥̇^𝑇(𝑡)𝐿^𝑇4+𝑥̇^𝑇(𝑡−𝑑(𝑡))𝐿^𝑇5+2+𝑥𝑓^𝑇1_𝑇𝐿^𝑇6+𝑓^𝑇2𝐿^𝑇7×𝑥(𝑡)−𝑥𝑡−𝜏_𝑒−^{𝑡𝑡−𝜏}_𝑒𝑥̇(𝑠)𝑑𝑠(𝑡)𝑀^𝑇1+𝑥^𝑇(𝑡−𝜏(𝑡))𝑀^𝑇2+𝑥^𝑇𝑡−𝜏_𝑒𝑀^𝑇3+𝑥̇^𝑇(𝑡)𝑀^𝑇4+𝑥̇^𝑇(𝑡−𝑑(𝑡))𝑀^𝑇5−𝑥_𝑇+𝑓^𝑇1𝑀^𝑇6+𝑓^𝑇2𝑀^𝑇7𝑥(𝑡−𝜏(𝑡))−𝑥𝑡−𝜏_𝑒−^{𝑡−𝜏(𝑡)𝑡−𝜏}_𝑒𝑥̇(𝑠)𝑑𝑠+𝛿₁𝛼²𝑥^𝑇(𝑡)𝑥(𝑡)−𝑓^𝑇1𝑓₁−𝑥_𝑇𝑡−𝜏_𝑒𝑊𝑥𝑡−𝜏_𝑒+2𝑥^𝑇𝑡−𝜏_𝑒𝑊𝑥(𝑡−𝜏(𝑡))(𝑡−𝜏(𝑡))𝑊𝑥(𝑡−𝜏(𝑡))+𝛿₂𝛽²𝑥^𝑇(𝑡−𝜏(𝑡))𝑥(𝑡−𝜏(𝑡))−𝑓^𝑇2𝑓₂.(3.18) Applying Lemmas 2.3. and 2.4., the following estimations hold: 2𝑥̇^𝑇(𝑡)𝐾^𝑇1𝐴+𝐺₁𝐹(𝑡)𝐸_𝐴𝑥(𝑡)≤+2𝑥𝜖̇^𝑇₁₋₁(𝑡)𝐾^𝑇1𝐴𝑥(𝑡)+𝜖₁𝑥̇^𝑇(𝑡)𝐾^𝑇1𝐺₁𝐺^𝑇1𝐾₁𝑥̇(𝑡)𝑥^𝑇(𝑡)𝐸^𝑇𝐴𝐸_𝐴𝑥(𝑡),(3.19)2𝑥̇^𝑇(𝑡)𝐾^𝑇1𝐵+𝐺₂𝐹(𝑡)𝐸_𝐵𝑥𝑡−𝜏_𝑒≤+2𝑥𝜖̇^𝑇₂₋₁(𝑡)𝐾^𝑇1𝐵𝑥𝑡−𝜏_𝑒+𝜖₂𝑥̇^𝑇(𝑡)𝐾^𝑇1𝐺₂𝐺^𝑇2𝐾₁𝑥̇(𝑡)𝑥^𝑇𝑡−𝜏_𝑒𝐸^𝑇𝐵𝐸_𝐵𝑥𝑡−𝜏_𝑒,(3.20)2𝑥̇^𝑇(𝑡)𝐾^𝑇1𝐵+𝐺₂𝐹(𝑡)𝐸_𝐵^{𝑡−𝜏(𝑡)𝑡−𝜏}_𝑒𝑥̇(𝑠)𝑑𝑠≤+2𝑥𝜖̇^𝑇₃₋₁(𝑡)𝐾^𝑇1𝐵^{𝑡−𝜏(𝑡)𝑡−𝜏}_𝑒𝑥̇(𝑠)𝑑𝑠+𝜖₃𝑥̇^𝑇(𝑡)𝐾^𝑇1𝐺₂𝐺^𝑇2𝐾₄𝑥̇^𝑇(𝑡)𝜌^{𝑡−𝜏(𝑡)𝑡−𝜏}_𝑒𝑥̇(𝑠)𝐸^𝑇𝐵𝐸_𝐵𝑥̇(𝑠)𝑑𝑠,(3.21)2𝑥̇^𝑇(𝑡)𝐾^𝑇1𝐷₁+𝐺₃𝐹(𝑡)𝐸_𝐷1𝑓₁≤2𝑥̇^𝑇(𝑡)𝐾^𝑇1𝐷₁𝑓₁++𝜖𝜖⁴⁻¹₄𝑥̇^𝑇(𝑡)𝐾^𝑇1𝐺₃𝐺^𝑇3𝐾₁𝑥̇(𝑡)𝑓^𝑇1𝐸^𝑇𝐷₁𝐸_𝐷1𝑓₁,(3.22)2𝑥̇^𝑇(𝑡)𝐾^𝑇1𝐷₂+𝐺₄𝐹(𝑡)𝐸_𝐷2𝑓₂≤2𝑥̇^𝑇(𝑡)𝐾^𝑇1𝐷₂𝑓₂++𝜖𝜖⁵⁻¹₅𝑥̇^𝑇(𝑡)𝐾^𝑇1𝐺₄𝐺^𝑇4𝐾₁𝑥̇(𝑡)𝑓^𝑇2𝐸^𝑇𝐷₂𝐸_𝐷2𝑓₂,(3.23)2𝑥̇^𝑇(𝑡−𝑑(𝑡))𝐾^𝑇2𝐴+𝐺₁𝐹(𝑡)𝐸_𝐴𝑥(𝑡)≤+2𝑥𝜖̇^𝑇₆₋₁(𝑡−𝑑(𝑡))𝐾^𝑇2𝐴𝑥(𝑡)+𝜖₆𝑥̇^𝑇(𝑡−𝑑(𝑡))𝐾^𝑇2𝐺₁𝐺^𝑇1𝐾₂𝑥̇(𝑡−𝑑(𝑡))𝑥^𝑇(𝑡−𝑑(𝑡))𝐸^𝑇𝐴𝐸_𝐴𝑥(𝑡),(3.24)2𝑥̇^𝑇(𝑡−𝑑(𝑡))𝐾^𝑇2𝐵+𝐺₂𝐹(𝑡)𝐸_𝐵𝑥𝑡−𝜏_𝑒≤+2𝑥𝜖̇^𝑇₇₋₁(𝑡−𝑑(𝑡))𝐾^𝑇1𝐵𝑥𝑡−𝜏_𝑒+𝜖₇𝑥̇^𝑇(𝑡−𝑑(𝑡))𝐾^𝑇1𝐺₂𝐺^𝑇2𝐾₁𝑥̇(𝑡−𝑑(𝑡))𝑥^𝑇𝑡−𝜏_𝑒𝐸^𝑇𝐵𝐸_𝐵𝑥𝑡−𝜏_𝑒,(3.25)2𝑥̇^𝑇(𝑡−𝑑(𝑡))𝐾^𝑇2𝐵+𝐺₂𝐹(𝑡)𝐸_𝐵^{𝑡−𝜏(𝑡)𝑡−𝜏}_𝑒𝑥̇(𝑠)𝑑𝑠≤+2𝑥𝜖̇^𝑇₈₋₁(𝑡−𝑑(𝑡))𝐾^𝑇2𝐵^{𝑡−𝜏(𝑡)𝑡−𝜏}_𝑒𝑥̇(𝑠)𝑑𝑠+𝜖₈𝑥̇^𝑇(𝑡−𝑑(𝑡))𝐾^𝑇2𝐺₂𝐺^𝑇2𝐾₂𝑥̇^𝑇(𝑡−𝑑(𝑡))𝜌^{𝑡−𝜏(𝑡)𝑡−𝜏}_𝑒𝑥̇(𝑠)𝐸^𝑇𝐵𝐸_𝐵𝑥̇(𝑠)𝑑𝑠,(3.26)2𝑥̇^𝑇(𝑡−𝑑(𝑡))𝐾^𝑇2𝐷₁+𝐺₃𝐹(𝑡)𝐸_𝐷1𝑓₁≤2𝑥̇^𝑇(𝑡−𝑑(𝑡))𝐾^𝑇2𝐷₁𝑓₁++𝜖𝜖⁹⁻¹₉𝑥̇^𝑇(𝑡−𝑑(𝑡))𝐾^𝑇2𝐺₃𝐺^𝑇3𝐾₂𝑥̇(𝑡−𝑑(𝑡))𝑓^𝑇1𝐸^𝑇𝐷₁𝐸_𝐷1𝑓₁,(3.27)2𝑥̇^𝑇(𝑡−𝑑(𝑡))𝐾^𝑇2𝐷₂+𝐺₄𝐹(𝑡)𝐸_𝐷2𝑓₂≤2𝑥̇^𝑇(𝑡−𝑑(𝑡))𝐾^𝑇2𝐷₂𝑓₂++𝜖𝜖⁻¹¹⁰₁₀𝑥̇^𝑇(𝑡−𝑑(𝑡))𝐾^𝑇2𝐺₄𝐺^𝑇4𝐾₂𝑥̇(𝑡−𝑑(𝑡))𝑓^𝑇2𝐸^𝑇𝐷₂𝐸_𝐷2𝑓₂.(3.28) Therefore, from (3.18)–(3.28), it follows that 𝑉₁̇𝑥_𝑡≤𝜔^𝑇(𝑡,𝑠)ℳ𝜔(𝑡,𝑠)+𝑥̇^𝑇(𝑡)Ω₁𝑥̇(𝑡)+𝑥̇^𝑇(𝑡−𝑑(𝑡))Ω₂𝑥̇(𝑡−𝑑(𝑡)),(3.29) where Ω₁=−0.1𝐾^𝑇1−0.1𝐾₁+𝜖¹⁻¹𝐾^𝑇1𝐺₁𝐺^𝑇1𝐾₁+𝜖²⁻¹𝐾^𝑇1𝐺₂𝐺^𝑇2𝐾₁Ω₂+𝜖³⁻¹𝐾^𝑇1𝐺₂𝐺^𝑇2𝐾₁+𝜖⁴⁻¹𝐾^𝑇1𝐺₃𝐺^𝑇3𝐾₁+𝜖⁵⁻¹𝐾^𝑇1𝐺₄𝐺^𝑇4𝐾₁,=−0.1𝑄^𝑇1+0.1𝐾^𝑇2𝐶+0.1𝐶^𝑇𝐾₂+𝜖⁶⁻¹𝐾^𝑇2𝐺₁𝐺^𝑇1𝐾₂+𝜖⁷⁻¹𝐾^𝑇2𝐺₂𝐺^𝑇2𝐾₂+𝜖⁸⁻¹𝐾^𝑇2𝐺₂𝐺^𝑇2𝐾₂+𝜖⁹⁻¹𝐾^𝑇2𝐺₃𝐺^𝑇3𝐾₂+𝜖⁻¹¹⁰𝐾^𝑇2𝐺₄𝐺^𝑇4𝐾₂.(3.30) Applying Schur complement lemma, the inequalities Ω₁<0 and Ω₂<0 are equivalent to ℳ₁<0 and ℳ₂<0, respectively. Therefore, system (2.10) is robust asymptotically stable if the condition (3.16) holds.
By using arguments similar to the proof of Case 1 for Case 2 and Case 3, we may conclude that the close-loop system (2.10) is robust asymptotically stable.

Remark 3.4. In this paper, the restriction that the state delay is differentiable is not required, which allows state delay to be fast time varying. Meanwhile, this restriction is required in some existing results, see [1–5, 7, 9, 11–14, 16, 18].

Remark 3.5. In the proof of Theorem 3.3, we need negative definiteness of matrices ℳ, Ω₁ and Ω₂ simultaneously. In order to do so, we need to have certain diagonal terms of matrices ℳ, Ω₁ and Ω₂ being negative. This leads to the splitting of the term 𝐾₁ as (0.1+0.9)𝐾₁ which is one possibility to achieve such goal.

4. Numerical Examples

In this section, we provide numerical examples to show the effectiveness of our theoretical results.

Example 4.1. Consider the following uncertain neutral system with time-varying delay and nonlinear uncertainties which is studied in [1, 2]: 𝑥̇(𝑡)−𝐶𝑥̇(𝑡−𝑑)=+(𝐴𝐷+Δ𝐴(𝑡))𝑥(𝑡)+(𝐵+Δ𝐵(𝑡))𝑥(𝑡−𝜏(𝑡))+𝐷₁+Δ𝐷₁(𝑡)𝑓₁(𝑡,𝑥(𝑡))₂+Δ𝐷₂(𝑡)𝑓₂(𝑡,𝑥(𝑡−𝜏(𝑡))),(4.1) where 𝐴=⎡⎢⎢⎣−−10..21𝐷−−01.1⎤⎥⎥⎦,𝐵=⎡⎢⎢⎣−−01.6−00.7.8⎤⎥⎥⎦,𝐶=⎡⎢⎢⎣−00.2.2−00.1₁⎤⎥⎥⎦,Δ𝐴(𝑡)=𝐺𝐹(𝑡)𝐸=_𝐴⎡⎢⎢⎣−00.1−00.1⎤⎥⎥⎦,𝐷₂=⎡⎢⎢⎣−00.1−00.1⎤⎥⎥⎦,Δ𝐷₂,Δ𝐵(𝑡)=𝐺𝐹(𝑡)𝐸_𝐵,Δ𝐷₁(𝑡)=𝐺𝐹(𝑡)𝐸_𝐷1,(𝑡)=𝐺𝐹(𝑡)𝐸𝐸_𝐷2,𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼,𝐺=𝛾𝐼,𝐸_𝐴=𝐼,_𝐵=𝐼,𝐸_𝐷1=𝐼,𝐸_𝐷2=𝐼.(4.2) It is assumed that the nonlinear uncertainties satisfy ‖‖𝑓₁(𝑡,𝑥(𝑡))‖‖≤𝛼‖𝑥(𝑡)‖,‖‖𝑓₂(𝑡,𝑥(𝑡−𝜏(𝑡)))‖‖≤𝛽‖𝑥(𝑡−𝜏(𝑡))‖,𝛼>0,𝛽>0.(4.3) Applying Theorem 3.3, the maximum allowable value of 𝜏_𝑀 is given in Table 1 when 𝛾=0.1 and in Table 2 for 𝛾=0.5. The results obtained in [1, 2] may not be used for the case when 𝜏_𝑚≠0. Moreover, the differentiability of the time delay 𝜏(𝑡) is not required in Theorem 3.3. Tables 1 and 2 show that our results significantly improve the results of [1, 2].
Moreover, it should be pointed out that if we let 𝜏_𝑚=0.1 and 𝜏_𝑀=0.85, then from Theorem 3.3, the solutions of LMI (3.16) are given as follows: 𝑄𝑃=⎡⎢⎢⎣20..4355907901..39580790⎤⎥⎥⎦,𝑄=⎡⎢⎢⎣00..9199565701..10926376₁⎤⎥⎥⎦,𝐿𝑅==⎡⎢⎢⎣⎡⎢⎢⎣−1000...825.017656114485501−..0270.762.19028959⎤⎥⎥⎦425,⎤⎥⎥⎦,𝑊𝑆==⎡⎢⎢⎣00⎡⎢⎢⎣..401015..41322210121400..030157..11262308₁29⎤⎥⎥⎦43,⎤⎥⎥⎦,𝐿=⎡⎢⎢⎣−−10..81329195−−01..09443000⎤⎥⎥⎦,𝐿₂=⎡⎢⎢⎣−11.5.130461111..2027296₃7⎤⎥⎥⎦,𝐿=⎡⎢⎢⎣−−00..11645367−−00..03210527⎤⎥⎥⎦,𝐿₄=⎡⎢⎢⎣−01.1.3810464−−01..43556870₅⎤⎥⎥⎦,𝐿=⎡⎢⎢⎣−−00..10541140−00.0.0452932⎤⎥⎥⎦,𝐿₆=⎡⎢⎢⎣−−00..00004376−00.0.0006445₇⎤⎥⎥⎦,𝐾=⎡⎢⎢⎣−−00..00004377−00.0.0006446⎤⎥⎥⎦,𝐾₁=⎡⎢⎢⎣10..7069522801..24290570₂⎤⎥⎥⎦,𝑀=⎡⎢⎢⎣−00.0.2480186−00.0.1230436⎤⎥⎥⎦,𝑀₁=⎡⎢⎢⎣−−10..40452385−−01..03806976₂⎤⎥⎥⎦,𝑀=⎡⎢⎢⎣−11.1.2276860−−10..09285489⎤⎥⎥⎦,𝑀₃=⎡⎢⎢⎣00..20361620−00.4.137811₄7⎤⎥⎥⎦,𝑀=⎡⎢⎢⎣00..39912068−01.5.0860857⎤⎥⎥⎦,𝑀₅=⎡⎢⎢⎣−00.3.169021800..11606945₆⎤⎥⎥⎦,=𝛿⎡⎢⎢⎣00..00013120−00.0.0008416⎤⎥⎥⎦,𝑀₇=⎡⎢⎢⎣00..00013121−00.0.000841₁7⎤⎥⎥⎦,𝜖=2.0156,𝛿₂=1.9922,𝜖₁=0.4383,₂𝜖=0.4197,𝜖₃=0.5513,𝜖₄=0.9140,₅𝜖=0.9042,𝜖₆=0.3288,𝜖₇=0.3143,₈=0.4376,𝜖₉=0.9021,𝜖₁₀=0.9003.(4.4)
Figure 1 shows the trajectories of solutions 𝑥₁(𝑡) and 𝑥₂(𝑡) of the system (4.1) with time-varying delay 𝜏(𝑡)=0.1+0.75|sin10𝑡|, 𝑑=1, 𝜙(𝑡)=[sin𝑡,cos𝑡)], for all 𝑡∈[−1,0], 𝑓₁(𝑡,𝑥(𝑡))=[0.1sin|𝑥₁(𝑡)|,0.1cos|𝑥₂(𝑡)|]^𝑇, 𝑓₂(𝑡,𝑥(𝑡−𝜏(𝑡)))=[0.1𝑒^{−sin2𝑥1(𝑡−𝜏(𝑡))},0.1𝑒^{−cos2𝑥2(𝑡−𝜏(𝑡))}]^𝑇 and 𝐹(𝑡)=diag{sin²(𝑡),sin²(𝑡)}. Since the time-delay 𝜏(𝑡) is not differentiable, the stability criterion in [1, 2] cannot be applied to this case because it is only applicable to the system with the differentiable delay.

Table 1 

Comparison of the maximum value 𝜏_𝑀 for 𝛾=0.1.

Table 2 

Comparison of the maximum value 𝜏_𝑀 for 𝛾=0.5.

Figure 1 

The trajectories of 𝑥1(𝑡) and 𝑥2(𝑡) of the system (4.1) with time-varying delay 𝜏(𝑡)=0.1+0.75|sin10𝑡|.

Example 4.2. Consider the following uncertain neutral system with time-varying delay in [3, 4]: 𝑥̇(𝑡)−𝐶𝑥̇(𝑡−𝑑(𝑡))=(𝐴+Δ𝐴(𝑡))𝑥(𝑡)+(𝐵+Δ𝐵(𝑡))𝑥(𝑡−𝜏(𝑡)),(4.5) where 𝐴=Δ⎡⎢⎢⎣−𝐴02(𝑡−)01=⎤⎥⎥⎦⎡⎢⎢⎣,𝛾₁𝐵=⎡⎢⎢⎣−−11−01⎤⎥⎥⎦,𝐶=⎡⎢⎢⎣0𝑐0𝑐⎤⎥⎥⎦,0𝛾0₂⎤⎥⎥⎦,Δ𝐵(𝑡)=⎡⎢⎢⎣𝛾0₃𝛾0₄⎤⎥⎥⎦,(4.6) where 0≤|𝑐|<1, and 𝛾_𝑖, 𝑖=1,2,…,4 are unknown parameter satisfying |𝛾₁|≤1.6, |𝛾₂|≤0.05, |𝛾₃|<0.1, and |𝛾₄|<0.3.

Case 1. For 𝑐=0.1, 𝛿=0, the maximum values of 𝜏_𝑀 are listed in Table 3 for 𝑐=0.1 by applying criteria in [3, 4] and in this paper. We see that the maximum allowable bounds for 𝜏_𝑀 obtained from Theorem 3.3 are much better than that obtained in [3, 4].

Table 3 

Maximum allowable upper bounds 𝜏_𝑀 of the time-varying delay for different values of the lower bounds 𝜏_𝑚 and 𝑐=0.1.

Case 2. For 𝑐=0.1, 𝛿=0.1, the maximum value 𝜏_𝑀 obtained form Theorem 3.3 is listed in Table 4. In [3, 4] the neutral delay is constant, then its stability criterion cannot be applied to systems with time-varying neutral delay. Furthermore, the stability criterion in [5, 6] cannot be applied to this case because Theorem 3.3 does not have restriction on the derivative of time-varying delay. It is obvious that the obtained results are significantly better than those in [3–6].

Table 4 

Maximum allowable upper bounds 𝜏_𝑀 of the time-varying delay for different values of the lower bounds 𝜏_𝑚, 𝑐=0.1 and 𝛿=0.1.

Figure 2 shows the trajectories of solutions 𝑥₁(𝑡) and 𝑥₂(𝑡) of the system (4.5) with time-varying delay 𝜏(𝑡)=0.3+0.5|cos10𝑡|, 𝑑(𝑡)=0.1sin²𝑡, 𝜙(𝑡)=[sin𝑡,cos𝑡)], for all 𝑡∈[−0.8,0].

Figure 2 

The trajectories of 𝑥1(𝑡) and 𝑥2(𝑡) of the system (4.5) with time-varying delay 𝜏(𝑡)=0.3+0.5|cos10𝑡|.

5. Conclusions

In this paper, we have investigated the delay-dependent robust stability criteria for uncertain neutral systems with interval time-varying delays and time-varying nonlinear perturbations simultaneously. Based on Lyapunov-krasovskii theory, new delay-dependent sufficient conditions for robust stability have been derived in terms of LMIs. The interval time-varying delay function is not required to be differentiable, which allows time-delay function to be a fast time-varying function. Numerical examples are given to illustrate the effectiveness of the theoretic results which show that our results are much less conservative than some existing results in the literature.

Acknowledgments

Financial support from the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant no. PHD/0355/2552) to W. Weera and P. Niamsup is acknowledged. The first author is also supported by the Graduate School, Chiang Mai University, and the Center of Excellence in Mathematics, Thailand. The second author is also supported by the Center of Excellence in Mathematics, Thailand and Commission for Higher Education, Thailand.