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Journal of Applied Mathematics
Volume 2011, Article ID 138912, 20 pages
http://dx.doi.org/10.1155/2011/138912
Research Article

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

W. Weera1,2 and P. Niamsup1,2

1Department of Mathematics, Chiang Mai University, Chiang Mai 50200, Thailand
2Center of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 7 July 2011; Accepted 13 September 2011

Academic Editor: James Buchanan

Copyright © 2011 W. Weera and P. Niamsup. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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, [716]. 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, 1114, 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 [15, 7, 9, 1114, 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: + denotes the set of all real nonnegative numbers; 𝑛 denotes the 𝑛-dimensional space and the vector norm ; 𝑀𝑛×𝑟 denotes the space of all matrices of (𝑛×𝑟)-dimensions. 𝐴𝑇 denotes the transpose of matrix 𝐴; 𝐴 is symmetric if 𝐴=𝐴𝑇; 𝐼 denotes the identity matrix; 𝜆(𝐴) denotes the set of all eigenvalues of 𝐴; 𝜆max(𝐴)=max{Re𝜆;𝜆𝜆(𝐴)}.𝑥𝑡={𝑥(𝑡+𝑠)𝑠[,0]}, 𝑥𝑡=sup𝑠[,0]𝑥(𝑡+𝑠); 𝐶([0,𝑡],𝑛) denotes the set of all 𝑛-valued continuous functions on [0,𝑡]; Matrix 𝐴 is called semipositive definite (𝐴0) if 𝑥𝑇𝐴𝑥0, for all 𝑥𝑛; 𝐴 is positive definite (𝐴>0) if 𝑥𝑇𝐴𝑥>0, for all 𝑥0; 𝐴>𝐵 means 𝐴𝐵>0. The symmetric term in a matrix is denoted by .

Consider the following neutral system with time-varying delay:̇𝑥(𝑡)𝐶̇𝑥(𝑡𝑑(𝑡))=𝐴(𝑡)𝑥(𝑡)+𝐵(𝑡)𝑥(𝑡𝜏(𝑡))+𝐷1(𝑡)𝑓1(𝑡,𝑥(𝑡))+𝐷2(𝑡)𝑓2𝑥𝑡(𝑡,𝑥(𝑡𝜏(𝑡))),0[],+𝜃=𝜙(𝜃),𝜃,0(2.1) where 𝑥(𝑡)𝑅𝑛 is the state vector, 𝑑(𝑡) is a neutral delay, 𝜏(𝑡) is a time-varying continuous function which satisfies0𝜏𝑚𝜏(𝑡)𝜏𝑀̇,0𝑑(𝑡)𝑑,𝑑(𝑡)𝛿,(2.2) where 𝜏𝑚, 𝜏𝑀, 𝑑, 𝛿 are constants and =max{𝑑,𝜏𝑀}; the initial condition function 𝜙(𝑡) denotes a continuous vector-valued initial function of 𝑡[,0], 𝑓1(𝑡,𝑥(𝑡)) and 𝑓2(𝑡,𝑥(𝑡𝜏(𝑡))) are unknown nonlinear perturbations satisfying 𝑓1(𝑡,0)=0, 𝑓2(𝑡,0)=0 and𝑓𝑇1(𝑡,𝑥(𝑡))𝑓1(𝑡,𝑥(𝑡))𝛼2𝑥𝑇𝑓(𝑡)𝑥(𝑡),𝑇2(𝑡,𝑥(𝑡𝜏(𝑡)))𝑓2(𝑡,𝑥(𝑡𝜏(𝑡)))𝛽2𝑥𝑇(𝑡𝜏(𝑡))𝑥(𝑡𝜏(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, 𝐷2𝑛×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 satisfyΔ𝐴(𝑡)=𝐺1𝐹(𝑡)𝐸𝐴,Δ𝐵(𝑡)=𝐺2𝐹(𝑡)𝐸𝐵,Δ𝐷1(𝑡)=𝐺3𝐹(𝑡)𝐸𝐷1,Δ𝐷2(𝑡)=𝐺4𝐹(𝑡)𝐸𝐷2,(2.5) where 𝐺1, 𝐺2, 𝐺3, 𝐺4, 𝐸𝐴, 𝐸𝐵, 𝐸𝐷1, and 𝐸𝐷2 are known real constant matrices of appropriate dimension. 𝐹(𝑡) is unknown time-varying matrix satisfying𝐹𝑇(𝑡)𝐹(𝑡)𝐼.(2.6) For simplicity, we denote 𝑓1(𝑡,𝑥(𝑡)), 𝑓2(𝑡,𝑥(𝑡𝜏(𝑡))), by 𝑓1, 𝑓2, respectively.

Let 𝜏𝑒=(1/2)(𝜏𝑀+𝜏𝑚) and 𝜌=(1/2)(𝜏𝑀𝜏𝑚). Then 𝜏(𝑡) can be expressed as𝜏(𝑡)=𝜏𝑒+𝜌𝜉(𝑡),(2.7) where 𝜏𝜉(𝑡)=2𝜏(𝑡)𝑀+𝜏𝑚𝜏𝑀𝜏𝑚,𝜏𝑀>𝜏𝑚,0,𝜏𝑀=𝜏𝑚.(2.8) Obviously, |𝜉(𝑡)|1. For this case, 𝜏(𝑡) is a function belonging to the interval [𝜏𝑒𝜌,𝜏𝑒+𝜌], where 𝜌 can be taken as the range of variation of the time-varying delay 𝜏(𝑡). Using the fact that 𝑥𝑡𝜏𝑒𝑥(𝑡𝜏(𝑡))=𝑡𝜏𝑒𝑡𝜏(𝑡)̇𝑥(𝑠)𝑑𝑠(2.9) system (2.1) can be rewritten aṡ𝑥(𝑡)𝐶̇𝑥(𝑡𝑑(𝑡))=𝐴(𝑡)𝑥(𝑡)+𝐵(𝑡)𝑥𝑡𝜏𝑒𝐵(𝑡)𝑡𝜏𝑒𝑡𝜏(𝑡)̇𝑥(𝑠)𝑑𝑠+𝐷1(𝑡)𝑓1+𝐷2(𝑡)𝑓2.(2.10)

Lemma 2.1 (see [17]). There exists a symmetric matrix 𝑋 such that 𝑃1𝐿𝑋𝐿𝑇𝑄1𝑄𝑇1𝑅1𝑃<0,2+𝑋𝑄2𝑄𝑇2𝑅2<0(2.11) if and only if 𝑃1+𝐿𝑃2𝐿𝑇𝑄1𝐿𝑄2𝑄𝑇1𝑅10𝑄𝑇2𝐿𝑇0𝑅2<0.(2.12)

Lemma 2.2 (see [3]). For any constant symmetric matrix 𝑀𝑅𝑛×𝑛, 𝑀=𝑀𝑇>0, 0𝑚(𝑡)𝑀, 𝑡0, and any differentiable vector function 𝑥(𝑡)𝑅𝑛, we have (a)𝑡𝑡𝑚̇𝑥(𝑠)𝑑𝑠𝑇𝑀𝑡𝑡𝑚̇𝑥(𝑠)𝑑𝑠𝑚𝑡𝑡𝑚̇𝑥𝑇(𝑠)𝑀̇𝑥(𝑠)𝑑𝑠,(b)𝑡𝑚𝑡(𝑡)̇𝑥(𝑠)𝑑𝑠𝑇𝑀𝑡𝑚𝑡(𝑡)̇𝑥(𝑠)𝑑𝑠(𝑡)𝑚𝑡𝑚𝑡(𝑡)̇𝑥𝑇(𝑠)𝑀̇𝑥(𝑠)𝑑𝑠𝑀𝑚𝑡𝑚𝑡(𝑡)̇𝑥𝑇(𝑠)𝑀̇𝑥(𝑠)𝑑𝑠.(2.13)

Lemma 2.3 (see [19]). Given matrices 𝑄=𝑄𝑇, 𝐻, 𝐸, and 𝑅=𝑅𝑇>0 with appropriate dimensions. Then 𝑄+𝐻𝐹𝐸+𝐸𝑇𝐹𝑇𝐻𝑇<0(2.14) for all 𝐹 satisfying 𝐹𝑇𝐹𝑅, if and only if there exists an 𝜖>0 such that 𝑄+𝜖𝐻𝐻𝑇+𝜖1𝐸𝑇𝑅𝐸<0.(2.15)

Proposition 2.4 (Cauchy inequality). For any symmetric positive definite matrix 𝑁𝑀𝑛×𝑛 and 𝑥,𝑦𝑛, we have ±2𝑥𝑇𝑦𝑥𝑇𝑁𝑥+𝑦𝑇𝑁1𝑦.(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, 𝐿𝑖, 𝑀𝑖,𝑖=1,2,,7 of appropriate dimension and 𝛿1,𝛿2>0 such that Σ1=𝜙11𝜙12𝜙13𝜙14𝜙15𝜙16𝜙17𝜏𝑒𝐿𝑇1𝜌𝑀1𝜙22𝜙23𝜙24𝜙25𝜙26𝜙27𝜏𝑒𝐿𝑇2𝜌𝑀2𝜙33𝜙34𝜙35𝜙36𝜙37𝜏𝑒𝐿𝑇3𝜌𝑀3𝜙44𝜙45𝜙46𝜙47𝜏𝑒𝐿𝑇4𝜌𝐾𝑇1𝐵+𝑀4𝜙55𝜙56𝜙57𝜏𝑒𝐿𝑇5𝜌𝐾𝑇2𝐵+𝑀5𝜙660𝜏𝑒𝐿𝑇6𝜌𝑀6𝜙77𝜏𝑒𝐿𝑇7𝜌𝑀7𝜏𝑒𝑅0𝜌𝑆<0,(3.1) where 𝜙11=𝑄+𝐿1+𝐿𝑇1+𝜀1𝛼2𝜙𝐼,12=𝑀𝑇1+𝐿2,𝜙13=𝐿𝑇1+𝐿3+𝑀1,𝜙14=𝑃+𝐴𝑇𝐾1+𝐿4,𝜙15=𝐴𝑇𝐾2+𝐿5,𝜙16=𝐿6,𝜙17=𝐿7,𝜙22=𝑀𝑇2+𝑀2+𝜀2𝛽2𝜙𝐼𝑊,23=𝐿𝑇2𝑀𝑇2+𝑀3𝜙+𝑊,24=𝑀4,𝜙25=𝑀5,𝜙26=𝑀6,𝜙27=𝑀7,𝜙33=𝑄𝐿𝑇3𝐿3𝑀𝑇3𝑀3𝜙𝑊,34=𝐵𝑇𝐾1𝐿4𝑀4,𝜙35=𝐵𝑇𝐾2𝐿5𝑀5,𝜙36=𝐿6𝑀6,𝜙37=𝐿7𝑀7,𝜙44=𝑄1+𝜏𝑒𝑅+𝜌𝑆𝐾𝑇1𝐾1+𝜌2𝜙𝑊,45=𝐾𝑇1𝐶𝐾2,𝜙46=𝐾𝑇1𝐷1,𝜙47=𝐾𝑇1𝐷2,𝜙55=(1𝛿)𝑄1+𝐾𝑇2𝐶+𝐶𝑇𝐾2,𝜙56=𝐾𝑇2𝐷1,𝜙57=𝐾𝑇2𝐷2,𝜙66=𝛿1𝜙𝐼,77=𝛿2𝐼.(3.2)

Proof. We prove that Theorem 3.2 is true for three cases, namely, 𝜏𝑚𝜏(𝑡)<𝜏𝑒; 𝜏(𝑡)=𝜏𝑒; 𝜏𝑒<𝜏(𝑡)𝜏𝑀.
Case 1 (𝜏𝑚𝜏(𝑡)<𝜏𝑒). Choose a Lyapunov-Krasovskii functional candidate as 𝑉1𝑥𝑡=𝑥𝑇(𝑡)𝑃𝑥(𝑡)+𝑡𝑡𝜏𝑒𝑥𝑇(+𝑠)𝑄𝑥(𝑠)𝑑𝑠𝑡𝑡𝑑(𝑡)̇𝑥𝑇(𝑠)𝑄1̇𝑥(𝑠)𝑑𝑠0𝜏𝑒𝑡𝑡+𝑠̇𝑥𝑇+(𝜃)𝑅̇𝑥(𝜃)𝑑𝜃𝑑𝑠𝜏𝑚𝜏𝑒𝑡𝑡+𝑠̇𝑥𝑇(𝜃)𝑆̇𝑥(𝜃)𝑑𝜃𝑑𝑠+𝜌𝜏𝑚𝜏𝑒𝑡𝑡+𝑠̇𝑥𝑇(𝜃)𝑊̇𝑥(𝜃)𝑑𝜃𝑑𝑠,(3.3) where 𝑃, 𝑄, 𝑄1, 𝑅, 𝑆, and 𝑊 are positive definite matrices. Taking the derivative of 𝑉1(𝑥𝑡) with respect to 𝑡 along the trajectory of (2.10) yields ̇𝑉1𝑥𝑡=2𝑥𝑇(𝑡)𝑃̇𝑥(𝑡)+𝑥𝑇(𝑡)𝑄𝑥(𝑡)𝑥𝑇𝑡𝜏𝑒𝑄𝑥𝑡𝜏𝑒+̇𝑥𝑇(𝑡)𝑄1̇̇𝑥(𝑡)1𝑑(𝑡)̇𝑥𝑇(𝑡𝑑(𝑡))𝑄1̇𝑥𝑇(𝑡𝑑(𝑡))+̇𝑥𝑇𝜏(𝑡)𝑒𝑅+𝜌2𝑊+𝜌𝑆̇𝑥(𝑡)𝑡𝑡𝜏𝑒̇𝑥𝑇(𝑠)𝑅̇𝑥(𝑠)𝑑𝑠𝑡𝜏𝑚𝑡𝜏𝑒̇𝑥𝑇(𝑠)𝑆̇𝑥(𝑠)𝑑𝑠𝜌𝑡𝜏𝑚𝑡𝜏𝑒̇𝑥𝑇(𝑠)𝑊̇𝑥(𝑠)𝑑𝑠2𝑥𝑇(𝑡)𝑃̇𝑥(𝑡)+𝑥𝑇(𝑡)𝑄𝑥(𝑡)𝑥𝑇𝑡𝜏𝑒𝑄𝑥𝑡𝜏𝑒+̇𝑥𝑇(𝑡)𝑄1̇𝑥(𝑡)(1𝛿)̇𝑥𝑇(𝑡𝑑(𝑡))𝑄1̇𝑥𝑇(𝑡𝑑(𝑡))+̇𝑥𝑇𝜏(𝑡)𝑒𝑅+𝜌2𝑊+𝜌𝑆̇𝑥(𝑡)𝑡𝑡𝜏𝑒̇𝑥𝑇(𝑠)𝑅̇𝑥(𝑠)𝑑𝑠𝑡𝜏𝑚𝑡𝜏𝑒̇𝑥𝑇(𝑠)𝑆̇𝑥(𝑠)𝑑𝑠𝜌𝑡𝜏𝑚𝑡𝜏𝑒̇𝑥𝑇(𝑠)𝑊̇𝑥(𝑠)𝑑𝑠,(3.4) since 𝑡𝜏𝑚𝑡𝜏𝑒̇𝑥𝑇(𝑠)𝑆̇𝑥(𝑠)𝑑𝑠𝑡𝜏(𝑡)𝑡𝜏𝑒̇𝑥𝑇(𝑠)𝑆̇𝑥(𝑠)𝑑𝑠,𝜌𝑡𝜏𝑚𝑡𝜏𝑒̇𝑥𝑇(𝑠)𝑊̇𝑥(𝑠)𝑑𝑠𝜌𝑡𝜏(𝑡)𝑡𝜏𝑒̇𝑥𝑇(𝑠)𝑊̇𝑥(𝑠)𝑑𝑠.(3.5) Based on Lemma 2.2, we obtain 𝜌𝑡𝜏(𝑡)𝑡𝜏𝑒̇𝑥𝑇𝜏(𝑠)𝑊̇𝑥(𝑠)𝑑𝑠𝑒𝜏(𝑡)𝑡𝜏(𝑡)𝑡𝜏𝑒̇𝑥𝑇(𝑠)𝑊̇𝑥(𝑠)𝑑𝑠𝑥𝑇𝑡𝜏𝑒𝑊𝑥𝑡𝜏𝑒+2𝑥𝑇𝑡𝜏𝑒𝑊𝑥(𝑡𝜏(𝑡))𝑥𝑇(𝑡𝜏(𝑡))𝑊𝑥(𝑡𝜏(𝑡)),(3.6) and from the following equalities: 2̇𝑥𝑇(𝑡)𝐾𝑇1+̇𝑥𝑇(𝑡𝑑(𝑡))𝐾𝑇2×𝐴𝑥(𝑡)+𝐵𝑥𝑡𝜏𝑒𝐵𝑡𝜏𝑒𝑡𝜏(𝑡)̇𝑥(𝑠)𝑑𝑠+𝐷1𝑓1+𝐷2𝑓22𝑥+𝐶̇𝑥(𝑡𝑑(𝑡))̇𝑥(𝑡)=0,(3.7)𝑇(𝑡)𝐿𝑇1+𝑥𝑇(𝑡𝜏(𝑡))𝐿𝑇2+𝑥𝑇𝑡𝜏𝑒𝐿𝑇3+̇𝑥𝑇(𝑡)𝐿𝑇4+̇𝑥𝑇(𝑡𝑑(𝑡))𝐿𝑇5+𝑓𝑇1𝐿𝑇6+𝑓𝑇2𝐿𝑇7×𝑥(𝑡)𝑥𝑡𝜏𝑒𝑡𝑡𝜏𝑒2𝑥̇𝑥(𝑠)𝑑𝑠=0,(3.8)𝑇(𝑡)𝑀𝑇1+𝑥𝑇(𝑡𝜏(𝑡))𝑀𝑇2+𝑥𝑇𝑡𝜏𝑒𝑀𝑇3+̇𝑥𝑇(𝑡)𝑀𝑇4+̇𝑥𝑇(𝑡𝑑(𝑡))𝑀𝑇5+𝑓𝑇1𝑀𝑇6+𝑓𝑇2𝑀𝑇7×𝑥(𝑡𝜏(𝑡))𝑥𝑡𝜏𝑒𝑡𝜏(𝑡)𝑡𝜏𝑒̇𝑥(𝑠)𝑑𝑠=0,(3.9) where 𝐾1, 𝐾2, and 𝐿𝑖, 𝑀𝑖, 𝑖=1,2,,7 are some matrices of appropriate dimension. Next, from (4.5), for any scalars 𝛿1>0 and 𝛿2>0, we obtain 𝛿1𝛼2𝑥𝑇(𝑡)𝑥(𝑡)𝑓𝑇1𝑓1𝛿0,2𝛽2𝑥𝑇(𝑡𝜏(𝑡))𝑥(𝑡𝜏(𝑡))𝑓𝑇2𝑓20.(3.10) By adding the terms on left of (3.7)–(3.10) to ̇𝑉1(𝑥𝑡), we may express ̇𝑉1(𝑥𝑡) as ̇𝑉1𝑥𝑡2𝑥𝑇(𝑡)𝑃̇𝑥(𝑡)+𝑥𝑇(𝑡)𝑄𝑥(𝑡)𝑥𝑇𝑡𝜏𝑒𝑄𝑥𝑡𝜏𝑒+̇𝑥𝑇(𝑡)𝑄1̇𝑥(𝑡)(1𝛿)̇𝑥𝑇(𝑡𝑑(𝑡))𝑄1̇𝑥(𝑡𝑑(𝑡))+̇𝑥𝑇𝜏(𝑡)𝑒𝑅+𝜌𝑆+𝜌2𝑊̇𝑥(𝑡)𝑡𝑡𝜏𝑒̇𝑥𝑇(𝑠)𝑅̇𝑥(𝑠)𝑑𝑠𝑡𝜏(𝑡)𝑡𝜏𝑒̇𝑥𝑇(𝑠)𝑆̇𝑥(𝑠)𝑑𝑠+2̇𝑥𝑇(𝑡)𝐾𝑇1+̇𝑥𝑇(𝑡𝑑(𝑡))𝐾𝑇2×𝐴𝑥(𝑡)+𝐵𝑥𝑡𝜏𝑒+𝐵𝑡𝜏(𝑡)𝑡𝜏𝑒̇𝑥(𝑠)𝑑𝑠+𝐷1𝑓1+𝐷2𝑓2𝑥+𝐶̇𝑥(𝑡𝑑(𝑡))̇𝑥(𝑡)+2𝑇(𝑡)𝐿𝑇1+𝑥𝑇(𝑡𝜏(𝑡))𝐿𝑇2+𝑥𝑇𝑡𝜏𝑒𝐿𝑇3+̇𝑥𝑇(𝑡)𝐿𝑇4+̇𝑥𝑇(𝑡𝑑(𝑡))𝐿𝑇5+𝑓𝑇1𝐿𝑇6+𝑓𝑇2𝐿𝑇7×𝑥(𝑡)𝑥𝑡𝜏𝑒𝑡𝑡𝜏𝑒𝑥̇𝑥(𝑠)𝑑𝑠+2𝑇(𝑡)𝑀𝑇1+𝑥𝑇(𝑡𝜏(𝑡))𝑀𝑇2+𝑥𝑇𝑡𝜏𝑒𝑀𝑇3+̇𝑥𝑇(𝑡)𝑀𝑇4+̇𝑥𝑇(𝑡𝑑(𝑡))𝑀𝑇5+𝑓𝑇1𝑀𝑇6+𝑓𝑇2𝑀𝑇7×𝑥(𝑡𝜏(𝑡))𝑥𝑡𝜏𝑒𝑡𝜏(𝑡)𝑡𝜏𝑒̇𝑥(𝑠)𝑑𝑠+𝛿1𝛼2𝑥𝑇(𝑡)𝑥(𝑡)𝑓𝑇1𝑓1𝑥𝑇𝑡𝜏𝑒𝑊𝑥𝑡𝜏𝑒+2𝑥𝑇𝑡𝜏𝑒𝑊𝑥(𝑡𝜏(𝑡))𝑥𝑇(𝑡𝜏(𝑡))𝑊𝑥(𝑡𝜏(𝑡))+𝛿2𝛽2𝑥𝑇(𝑡𝜏(𝑡))𝑥(𝑡𝜏(𝑡))𝑓𝑇2𝑓2=1𝜏𝑒𝑡𝑡𝜏𝑒𝜔𝑇(𝑡,𝑠)𝜙11𝜔(𝑡,𝑠)𝑑𝑠+𝜏𝑒𝜏(𝑡)𝑡𝜏(𝑡)𝑡𝜏𝑒𝜔𝑇(𝑡,𝑠)𝜙2𝜔(𝑡,𝑠)𝑑𝑠,(3.11) where 𝜔𝑇𝑥(𝑡,𝑠)=𝑇(𝑡)𝑥𝑇(𝑡𝜏(𝑡))𝑥𝑇𝑡𝜏𝑒̇𝑥𝑇(𝑡)̇𝑥𝑇(𝑡𝑑(𝑡))𝑓𝑇1𝑓𝑇2̇𝑥𝑇,𝜙(𝑠)1=𝜙11𝜙12𝜙13𝜙14𝜙15𝜙16𝜙17𝜏𝑒𝐿𝑇1𝜙22𝜙23𝜙24𝜙25𝜙26𝜙27𝜏𝑒𝐿𝑇2𝜙33𝜙34𝜙35𝜙36𝜙37𝜏𝑒𝐿𝑇3𝜙44𝜙45𝜙46𝜙47𝜏𝑒𝐿𝑇4𝜙55𝜙56𝜙57𝜏𝑒𝐿𝑇5𝜙660𝜏𝑒𝐿𝑇6𝜙77𝜏𝑒𝐿𝑇7𝜏𝑒𝑅𝜙+𝑍,2=𝑍11𝑍12𝑍13𝑍14𝑍15𝑍16𝑍17𝜏𝑒𝑀𝜏(𝑡)1𝑍22𝑍23𝑍24𝑍25𝑍26𝑍27𝜏𝑒𝑀𝜏(𝑡)2𝑍33𝑍34𝑍35𝑍36𝑍37𝜏𝑒𝑀𝜏(𝑡)3𝑍44𝑍45𝑍46𝑍47𝜑1𝑍55𝑍56𝑍57𝜑2𝑍66𝑍67𝜏𝑒𝑀𝜏(𝑡)6𝑍77𝜏𝑒𝑀𝜏(𝑡)7||𝜏𝑒||𝑆,𝑍𝜏(𝑡)𝑍=11𝑍12𝑍13𝑍14𝑍15𝑍16𝑍170𝑍22𝑍23𝑍24𝑍25𝑍26𝑍270𝑍33𝑍34𝑍35𝑍36𝑍370𝑍44𝑍45𝑍46𝑍470𝑍55𝑍56𝑍570𝑍66𝑍670𝑍770,0(3.12)𝜑𝑖=(𝜏𝑒𝜏(𝑡))(𝐾𝑇𝑖𝐵+𝑀𝑖), 𝑖=1,2, 𝑍𝑖𝑖>0, 𝑖=1,2,,7, 𝑍𝑖𝑗, 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 ̇𝑉1(𝑥𝑡)𝜆1𝑥(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)𝜏𝑒)}, we get that 𝜙2𝑍11𝑍12𝑍13𝑍14𝑍15𝑍16𝑍17𝜌𝑀1𝑍22𝑍23𝑍24𝑍25𝑍26𝑍27𝜌𝑀2𝑍33𝑍34𝑍35𝑍36𝑍37𝜌𝑀3𝑍44𝑍45𝑍46𝑍47𝜌𝐾𝑇1𝐵+𝑀4𝑍55𝑍56𝑍57𝜌𝐾𝑇2𝐵+𝑀5𝑍66𝑍67𝜌𝑀6𝑍77𝜌𝑀7𝜌𝑆<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)=𝜏𝑒). For this case, we choose a Lyapunov-Krasovskii functional candidate as 𝑉2𝑥𝑡=𝑥𝑇(𝑡)𝑃𝑥(𝑡)+𝑡𝑡𝜏𝑒𝑥𝑇(𝑠)𝑄𝑥(𝑠)𝑑𝑠+𝑡𝑡𝑑(𝑡)̇𝑥𝑇(𝑠)𝑄1+̇𝑥(𝑠)𝑑𝑠0𝜏𝑒𝑑𝑠𝑡𝑡+𝑠̇𝑥𝑇(𝜃)𝑅̇𝑥(𝜃)𝑑𝜃,(3.14) where 𝑃, 𝑄, 𝑄1, and 𝑅 positive definite matrices are the same as those in 𝑉1(𝑥𝑡).
Case 3 (𝜏𝑒<𝜏(𝑡)<𝜏𝑀). For this case, we choose the Lyapunov-Krasovskii functional candidate as 𝑉3𝑥𝑡=𝑥𝑇(𝑡)𝑃𝑥(𝑡)+𝑡𝑡𝜏𝑒𝑥𝑇(+𝑠)𝑄𝑥(𝑠)𝑑𝑠𝑡𝑡𝑑(𝑡)̇𝑥𝑇(𝑠)𝑄1̇𝑥(𝑠)𝑑𝑠0𝜏𝑒𝑡𝑡+𝑠̇𝑥𝑇+(𝜃)𝑅̇𝑥(𝜃)𝑑𝜃𝑑𝑠𝜏𝑒𝜏𝑀𝑡𝑡+𝑠̇𝑥𝑇(𝜃)𝑆̇𝑥(𝜃)𝑑𝜃𝑑𝑠+𝜌𝜏𝑒𝜏𝑀𝑡𝑡+𝑠̇𝑥𝑇(𝜃)𝑊̇𝑥(𝜃)𝑑𝜃𝑑𝑠,(3.15) where 𝑃, 𝑄, 𝑄1, 𝑅, 𝑆, and 𝑊 are positive definite matrices and are the same as those in 𝑉1(𝑥𝑡).
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, 𝐿𝑖, 𝑀𝑖, 𝑖=1,2,,7 of appropriate dimension and scalars 𝜖𝑖>0, 𝑖=1,2,,10 such that 𝑀=11𝜙12𝜙13𝜙14𝜙15𝜙16𝜙17𝜏𝑒𝐿𝑇1𝜌𝑀1𝜙22𝜙23𝜙24𝜙25𝜙26𝜙27𝜏𝑒𝐿𝑇2𝜌𝑀2𝑀33𝜙34𝜙35𝜙36𝜙37𝜏𝑒𝐿𝑇3𝜌𝑀3𝑀44𝜙45𝜙46𝜙47𝜏𝑒𝐿𝑇4𝜌𝐾𝑇1𝐵+𝑀4𝑀55𝜙56𝜙57𝜏𝑒𝐿𝑇5𝜌𝐾𝑇2𝐵+𝑀5𝑀660𝜏𝑒𝐿𝑇6𝜌𝑀6𝑀77𝜏𝑒𝐿𝑇7𝜌𝑀7𝜏𝑒𝑅0𝑀99<0,1=0.1𝐾𝑇10.1𝐾1𝐾𝑇1𝐺1𝐾𝑇1𝐺2𝐾𝑇1𝐺2𝐾𝑇1𝐺3𝐾𝑇1𝐺4𝐺𝑇1𝐾1𝜖1𝐺𝐼0000𝑇2𝐾10𝜖2𝐺𝐼000𝑇2𝐾100𝜖3𝐺𝐼00𝑇3𝐾1000𝜖4𝐺𝐼0𝑇4𝐾10000𝜖5𝐼<0,2=0.1𝑄1+0.1𝐾𝑇2𝐶+0.1𝐶𝑇𝐾2𝐾𝑇2𝐺1𝐾𝑇2𝐺2𝐾𝑇2𝐺2𝐾𝑇2𝐺3𝐾𝑇2𝐺4𝐺𝑇1𝐾2𝜖6𝐺𝐼0000𝑇2𝐾20𝜖7𝐺𝐼000𝑇2𝐾200𝜖8𝐺𝐼00𝑇3𝐾2000𝜖9𝐺𝐼0𝑇4𝐾20000𝜖10𝐼<0,(3.16) where 𝑀11=𝜙11+𝜖1𝐸𝑇𝐴𝐸𝐴+𝜖6𝐸𝑇𝐴𝐸𝐴,𝑀33=𝜙33+𝜖2𝐸𝑇𝐵𝐸𝐵+𝜖7𝐸𝑇𝐵𝐸𝐵,𝑀44=𝑄1+𝜏𝑒𝑅+𝜌𝑆0.9𝐾𝑇10.9𝐾1+𝜌2𝑀𝑊,55=0.9𝑄1+𝛿𝑄1+0.9𝐾𝑇2𝐶+0.9𝐶𝑇𝐾2,𝑀66=𝜙66+𝜖4𝐸𝑇𝐷1𝐸𝐷1+𝜖9𝜌2𝐸𝑇𝐵𝐸𝐵,𝑀77=𝜙77+𝜖5𝐸𝑇𝐷2𝐸𝐷2+𝜖10𝜌2𝐸𝑇𝐵𝐸𝐵,𝑀99=𝜌𝑆+𝜖3𝜌2𝐸𝑇𝐵𝐸𝐵+𝜖8𝜌2𝐸𝑇𝐵𝐸𝐵.(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)𝐸𝐴, 𝐵+𝐺𝐹(𝑡)𝐸𝐵, 𝐷1+𝐺𝐹(𝑡)𝐸𝐷1 and 𝐷2+𝐺𝐹(𝑡)𝐸D2, respectively. For Case 1̇𝑉1𝑥𝑡2𝑥𝑇(𝑡)𝑃̇𝑥(𝑡)+𝑥𝑇(𝑡)𝑄𝑥(𝑡)𝑥𝑇𝑡𝜏𝑒𝑄𝑥𝑡𝜏𝑒+̇𝑥𝑇(𝑡)𝑄1̇𝑥(𝑡)(1𝛿)̇𝑥𝑇(𝑡𝑑(𝑡))𝑄1̇𝑥(𝑡𝑑(𝑡))+̇𝑥𝑇𝜏(𝑡)𝑒𝑅+𝜌𝑆+𝜌2𝑊̇𝑥(𝑡)𝑡𝑡𝜏𝑒̇𝑥𝑇(𝑠)𝑅̇𝑥(𝑠)𝑑𝑠𝑡𝜏(𝑡)𝑡𝜏𝑒̇𝑥𝑇(𝑠)𝑆̇𝑥(𝑠)𝑑𝑠+2̇𝑥𝑇(𝑡)𝐾𝑇1+̇𝑥𝑇(𝑡𝑑(𝑡))𝐾𝑇2×𝐴+𝐺1𝐹(𝑡)𝐸𝐴𝑥(𝑡)+𝐵+𝐺2𝐹(𝑡)𝐸𝐵𝑥𝑡𝜏𝑒+𝐵+𝐺2𝐹(𝑡)𝐸𝐵×𝑡𝜏(𝑡)𝑡𝜏𝑒𝐷̇𝑥(𝑠)𝑑𝑠+1+𝐺3𝐹(𝑡)𝐸𝐷1𝑓1+𝐷2+𝐺4𝐹(𝑡)𝐸𝐷2𝑓2𝑥+𝐶̇𝑥(𝑡𝑑(𝑡))̇𝑥(𝑡)+2𝑇(𝑡)𝐿𝑇1+𝑥𝑇(𝑡𝜏(𝑡))𝐿𝑇2+𝑥𝑇𝑡𝜏𝑒𝐿𝑇3+̇𝑥𝑇(𝑡)𝐿𝑇4+̇𝑥𝑇(𝑡𝑑(𝑡))𝐿𝑇5+𝑓𝑇1𝐿𝑇6+𝑓𝑇2𝐿𝑇7×𝑥(𝑡)𝑥𝑡𝜏𝑒𝑡𝑡𝜏𝑒𝑥̇𝑥(𝑠)𝑑𝑠+2𝑇(𝑡)𝑀𝑇1+𝑥𝑇(𝑡𝜏(𝑡))𝑀𝑇2+𝑥𝑇𝑡𝜏𝑒𝑀𝑇3+̇𝑥𝑇(𝑡)𝑀𝑇4+̇𝑥𝑇(𝑡𝑑(𝑡))𝑀𝑇5+𝑓𝑇1𝑀𝑇6+𝑓𝑇2𝑀𝑇7𝑥(𝑡𝜏(𝑡))𝑥𝑡𝜏𝑒𝑡𝜏(𝑡)𝑡𝜏𝑒̇𝑥(𝑠)𝑑𝑠+𝛿1𝛼2𝑥𝑇(𝑡)𝑥(𝑡)𝑓𝑇1𝑓1𝑥𝑇𝑡𝜏𝑒𝑊𝑥𝑡𝜏𝑒+2𝑥𝑇𝑡𝜏𝑒𝑊𝑥(𝑡𝜏(𝑡))𝑥𝑇(𝑡𝜏(𝑡))𝑊𝑥(𝑡𝜏(𝑡))+𝛿2𝛽2𝑥𝑇(𝑡𝜏(𝑡))𝑥(𝑡𝜏(𝑡))𝑓𝑇2𝑓2.(3.18) Applying Lemmas 2.3. and 2.4., the following estimations hold: 2̇𝑥𝑇(𝑡)𝐾𝑇1𝐴+𝐺1𝐹(𝑡)𝐸𝐴𝑥(𝑡)2̇𝑥𝑇(𝑡)𝐾𝑇1𝐴𝑥(𝑡)+𝜖11̇𝑥𝑇(𝑡)𝐾𝑇1𝐺1𝐺𝑇1𝐾1̇𝑥(𝑡)+𝜖1𝑥𝑇(𝑡)𝐸𝑇𝐴𝐸𝐴𝑥(𝑡),(3.19)2̇𝑥𝑇(𝑡)𝐾𝑇1𝐵+𝐺2𝐹(𝑡)𝐸𝐵𝑥𝑡𝜏𝑒2̇𝑥𝑇(𝑡)𝐾𝑇1𝐵𝑥𝑡𝜏𝑒+𝜖21̇𝑥𝑇(𝑡)𝐾𝑇1𝐺2𝐺𝑇2𝐾1̇𝑥(𝑡)+𝜖2𝑥𝑇𝑡𝜏𝑒𝐸𝑇𝐵𝐸𝐵𝑥𝑡𝜏𝑒,(3.20)2̇𝑥𝑇(𝑡)𝐾𝑇1𝐵+𝐺2𝐹(𝑡)𝐸𝐵𝑡𝜏(𝑡)𝑡𝜏𝑒̇𝑥(𝑠)𝑑𝑠2̇𝑥𝑇(𝑡)𝐾𝑇1𝐵𝑡𝜏(𝑡)𝑡𝜏𝑒̇𝑥(𝑠)𝑑𝑠+𝜖31̇𝑥𝑇(𝑡)𝐾𝑇1𝐺2𝐺𝑇2𝐾4̇𝑥𝑇(𝑡)+𝜖3𝜌𝑡𝜏(𝑡)𝑡𝜏𝑒̇𝑥(𝑠)𝐸𝑇𝐵𝐸𝐵,̇𝑥(𝑠)𝑑𝑠(3.21)2̇𝑥𝑇(𝑡)𝐾𝑇1𝐷1+𝐺3𝐹(𝑡)𝐸𝐷1𝑓12̇𝑥𝑇(𝑡)𝐾𝑇1𝐷1𝑓1+𝜖41̇𝑥𝑇(𝑡)𝐾𝑇1𝐺3𝐺𝑇3𝐾1̇𝑥(𝑡)+𝜖4𝑓𝑇1𝐸𝑇𝐷1𝐸𝐷1𝑓1,(3.22)2̇𝑥𝑇(𝑡)𝐾𝑇1𝐷2+𝐺4𝐹(𝑡)𝐸𝐷2𝑓22̇𝑥𝑇(𝑡)𝐾𝑇1𝐷2𝑓2+𝜖51̇𝑥𝑇(𝑡)𝐾𝑇1𝐺4𝐺𝑇4𝐾1̇𝑥(𝑡)+𝜖5𝑓𝑇2𝐸𝑇𝐷2𝐸𝐷2𝑓2,(3.23)2̇𝑥𝑇(𝑡𝑑(𝑡))𝐾𝑇2𝐴+𝐺1𝐹(𝑡)𝐸𝐴𝑥(𝑡)2̇𝑥𝑇(𝑡𝑑(𝑡))𝐾𝑇2𝐴𝑥(𝑡)+𝜖61̇𝑥𝑇(𝑡𝑑(𝑡))𝐾𝑇2𝐺1𝐺𝑇1𝐾2̇𝑥(𝑡𝑑(𝑡))+𝜖6𝑥𝑇(𝑡𝑑(𝑡))𝐸𝑇𝐴𝐸𝐴𝑥(𝑡),(3.24)2̇𝑥𝑇(𝑡𝑑(𝑡))𝐾𝑇2𝐵+𝐺2𝐹(𝑡)𝐸𝐵𝑥𝑡𝜏𝑒2̇𝑥𝑇(𝑡𝑑(𝑡))𝐾𝑇1𝐵𝑥𝑡𝜏𝑒+𝜖71̇𝑥𝑇(𝑡𝑑(𝑡))𝐾𝑇1𝐺2𝐺𝑇2𝐾1̇𝑥(𝑡𝑑(𝑡))+𝜖7𝑥𝑇𝑡𝜏𝑒𝐸𝑇𝐵𝐸𝐵𝑥𝑡𝜏𝑒,(3.25)2̇𝑥𝑇(𝑡𝑑(𝑡))𝐾𝑇2𝐵+𝐺2𝐹(𝑡)𝐸𝐵𝑡𝜏(𝑡)𝑡𝜏𝑒̇𝑥(𝑠)𝑑𝑠2̇𝑥𝑇(𝑡𝑑(𝑡))𝐾𝑇2𝐵𝑡𝜏(𝑡)𝑡𝜏𝑒̇𝑥(𝑠)𝑑𝑠+𝜖81̇𝑥𝑇(𝑡𝑑(𝑡))𝐾𝑇2𝐺2𝐺𝑇2𝐾2̇𝑥𝑇(𝑡𝑑(𝑡))+𝜖8𝜌𝑡𝜏(𝑡)𝑡𝜏𝑒̇𝑥(𝑠)𝐸𝑇𝐵𝐸𝐵,̇𝑥(𝑠)𝑑𝑠(3.26)2̇𝑥𝑇(𝑡𝑑(𝑡))𝐾𝑇2𝐷1+𝐺3𝐹(𝑡)𝐸𝐷1𝑓12̇𝑥𝑇(𝑡𝑑(𝑡))𝐾𝑇2𝐷1𝑓1+𝜖91̇𝑥𝑇(𝑡𝑑(𝑡))𝐾𝑇2𝐺3𝐺𝑇3𝐾2̇𝑥(𝑡𝑑(𝑡))+𝜖9𝑓𝑇1𝐸𝑇𝐷1𝐸𝐷1𝑓1,(3.27)2̇𝑥𝑇(𝑡𝑑(𝑡))𝐾𝑇2𝐷2+𝐺4𝐹(𝑡)𝐸𝐷2𝑓22̇𝑥𝑇(𝑡𝑑(𝑡))𝐾𝑇2𝐷2𝑓2+𝜖110̇𝑥𝑇(𝑡𝑑(𝑡))𝐾𝑇2𝐺4𝐺𝑇4𝐾2̇𝑥(𝑡𝑑(𝑡))+𝜖10𝑓𝑇2𝐸𝑇𝐷2𝐸𝐷2𝑓2.(3.28) Therefore, from (3.18)–(3.28), it follows that ̇𝑉1𝑥𝑡𝜔𝑇(𝑡,𝑠)𝜔(𝑡,𝑠)+̇𝑥𝑇(𝑡)Ω1̇𝑥(𝑡)+̇𝑥𝑇(𝑡𝑑(𝑡))Ω2̇𝑥(𝑡𝑑(𝑡)),(3.29) where Ω1=0.1𝐾𝑇10.1𝐾1+𝜖11𝐾𝑇1𝐺1𝐺𝑇1𝐾1+𝜖21𝐾𝑇1𝐺2𝐺𝑇2𝐾1+𝜖31𝐾𝑇1𝐺2𝐺𝑇2𝐾1+𝜖41𝐾𝑇1𝐺3𝐺𝑇3𝐾1+𝜖51𝐾𝑇1𝐺4𝐺𝑇4𝐾1,Ω2=0.1𝑄𝑇1+0.1𝐾𝑇2𝐶+0.1𝐶𝑇𝐾2+𝜖61𝐾𝑇2𝐺1𝐺𝑇1𝐾2+𝜖71𝐾𝑇2𝐺2𝐺𝑇2𝐾2+𝜖81𝐾𝑇2𝐺2𝐺𝑇2𝐾2+𝜖91𝐾𝑇2𝐺3𝐺𝑇3𝐾2+𝜖110𝐾𝑇2𝐺4𝐺𝑇4𝐾2.(3.30) Applying Schur complement lemma, the inequalities Ω1<0 and Ω2<0 are equivalent to 1<0 and 2<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 [15, 7, 9, 1114, 16, 18].

Remark 3.5. In the proof of Theorem 3.3, we need negative definiteness of matrices , Ω1 and Ω2 simultaneously. In order to do so, we need to have certain diagonal terms of matrices , 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]: 𝐷̇𝑥(𝑡)𝐶̇𝑥(𝑡𝑑)=(𝐴+Δ𝐴(𝑡))𝑥(𝑡)+(𝐵+Δ𝐵(𝑡))𝑥(𝑡𝜏(𝑡))+1+Δ𝐷1𝑓(𝑡)1+𝐷(𝑡,𝑥(𝑡))2+Δ𝐷2(𝑓𝑡)2(𝑡,𝑥(𝑡𝜏(𝑡))),(4.1) where ,𝐷𝐴=1.20.10.11,𝐵=0.60.710.8,𝐶=0.200.20.11=0.1000.1,𝐷2=,0.1000.1Δ𝐴(𝑡)=𝐺𝐹(𝑡)𝐸𝐴,Δ𝐵(𝑡)=𝐺𝐹(𝑡)𝐸𝐵,Δ𝐷1(𝑡)=𝐺𝐹(𝑡)𝐸𝐷1,Δ𝐷2(𝑡)=𝐺𝐹(𝑡)𝐸𝐷2,𝐹𝑇(𝑡)𝐹(𝑡)𝐼,𝐺=𝛾𝐼,𝐸𝐴𝐸=𝐼,𝐵=𝐼,𝐸𝐷1=𝐼,𝐸𝐷2=𝐼.(4.2) It is assumed that the nonlinear uncertainties satisfy 𝑓1𝑓(𝑡,𝑥(𝑡))𝛼𝑥(𝑡),2(𝑡,𝑥(𝑡𝜏(𝑡)))𝛽𝑥(𝑡𝜏(𝑡)),𝛼>0,𝛽>0.(4.3) Applying Theorem 3.3, the maximum allowable value of 𝜏𝑀 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 𝜏𝑚0. 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 𝜏𝑚=0.1 and 𝜏𝑀=0.85, then from Theorem 3.3, the solutions of LMI (3.16) are given as follows: ,𝑄𝑃=2.45970.35090.35091.9870,𝑄=0.99550.19670.19671.02361=,,𝐿0.56150.05450.05450.2292,𝑆=1.42110.12240.12241.1393𝑅=1.81640.27180.27181.7699,𝑊=0.41310.05200.05200.37681=1.83990.04300.12151.9400,𝐿2=,𝐿1.53411.07971.10611.22263=0.16560.02020.14370.3157,𝐿4=,𝐿1.31440.45670.18061.35805=0.15140.05920.04100.0423,𝐿6=,𝐿0.00470.00640.00360.00457=0.00470.00640.00370.0046,𝐾1=,𝐾1.76520.22070.09281.49502=0.28160.02030.04080.1346,𝑀1=,𝑀1.44280.08670.05351.30962=1.27801.02581.12660.9849,𝑀3=,𝑀0.23120.17170.06600.43814=0.39261.06870.91080.5805,𝑀5=,𝑀0.36010.16640.19280.10956=0.00320.00460.01100.0081,𝑀7=,𝛿0.00320.00470.01110.00811=2.0156,𝛿2=1.9922,𝜖1𝜖=0.4383,2=0.4197,𝜖3=0.5513,𝜖4𝜖=0.9140,5=0.9042,𝜖6=0.3288,𝜖7𝜖=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(𝑡𝜏(𝑡)))=[0.1𝑒sin2𝑥1(𝑡𝜏(𝑡)),0.1𝑒cos2𝑥2(𝑡𝜏(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.

tab1
Table 1: Comparison of the maximum value 𝜏𝑀 for 𝛾=0.1.
tab2
Table 2: Comparison of the maximum value 𝜏𝑀 for 𝛾=0.5.
138912.fig.001
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]: ̇𝑥(𝑡)𝐶̇𝑥(𝑡𝑑(𝑡))=(𝐴+Δ𝐴(𝑡))𝑥(𝑡)+(𝐵+Δ𝐵(𝑡))𝑥(𝑡𝜏(𝑡)),(4.5) where ,𝛾𝐴=2001,𝐵=1011,𝐶=𝑐00𝑐Δ𝐴(𝑡)=100𝛾2𝛾,Δ𝐵(𝑡)=300𝛾4,(4.6) where 0|𝑐|<1, and 𝛾𝑖, 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 𝜏𝑀 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 𝜏𝑀 obtained from Theorem 3.3 are much better than that obtained in [3, 4].

tab3
Table 3: Maximum allowable upper bounds 𝜏𝑀 of the time-varying delay for different values of the lower bounds 𝜏𝑚 and 𝑐=0.1.

Case 2. For 𝑐=0.1, 𝛿=0.1, the maximum value 𝜏𝑀 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 [36].

tab4
Table 4: Maximum allowable upper bounds 𝜏𝑀 of the time-varying delay for different values of the lower bounds 𝜏𝑚, 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].

138912.fig.002
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.

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