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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 950269, 19 pages
http://dx.doi.org/10.1155/2012/950269
Research Article

Improved Stability Analysis for Neural Networks with Time-Varying Delay

State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, China

Received 20 February 2012; Accepted 30 March 2012

Academic Editor: Cristian Toma

Copyright © 2012 Yongming Li et al. 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

This paper concerned the problem of delay-dependent asymptotic stability for neural networks with time-varying delay. A new class of Lyapunov functional dividing the interval delay is constructed to derive some new delay-dependent stability criteria. The obtained criteria are less conservative because free-weighting matrices method, a convex optimization approach, and a mixed dividing delay interval approach are considered. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.

1. Introduction

In the past few decades, delayed neural networks have been investigated extensively because of their successful applications in various scientific areas, such as pattern recognition, image processing, associative memories, and parallel computation. It is well known that time delay is frequently encountered in neural networks, and it is often a major cause of instability and oscillation. Thus, the stability analysis of delayed neural networks has been widely considered by many research results, delay-independent ones [13], and delay-dependent ones [437]. Generally speaking, the delay-dependent stability criteria are less conservative than delay-independent ones when the time delay is small. Therefore, much attention has been paid to develop delay-dependent stability conditions. Some less conservative stability criteria were derived in [6] by considering some useful terms and using the free-weighting matrices method. By the fact that the neuron activation functions are sector bounded and nondecreasing, [7] presents an improved method, named the delay-slope-dependent method, for stability analysis of neural networks with time-varying delays. The method includes more information on the slope of neuron activation functions and fewer matrix variables in the constructing Lyapunov functionals. Then some new delay-dependent stability criteria with less conservatism are obtained. Recently, some new Lyapunov functionals based on the idea of decomposing the delay were introduced to investigate the stability of neural networks with time-invariant delay [1012] and time-varying delay [1316], which significantly reduced the conservativeness of the derived stability criteria. In [13], different from some previous results, the delay interval [0,𝑑(𝑡)] is divided into some variable subintervals by employing weighting delays. Thus, some new delay-dependent stability criteria for neural networks with time varying delay are derived by applying the weighting-delay method, which are less conservative than the existing results. However, when the delay is time-varying, the information of subinterval is not considered sufficiently. For example, the time-varying delay 𝜏(𝑡) satisfies 0𝜏(𝑡). When the delay interval [0,𝜏(𝑡)] is divided into some subintervals, the delay interval [0,] is also divided into some subintervals, in essence. But in the construction of Lyapunov functional in [15], this important information is ignored, which is a major source of conservativeness. Furthermore, the purpose of reducing conservatism is still limited due to the existence of multiple coefficients and the impact of subintervals with uniform size. Thus, it is still a quite difficult task to divide interval [0,𝜏(𝑡)] in a more reasonable manner, so that the functional with the augmented matrix can easily be constructed to obtain less conservative stability results, which motivates our present study.

In this paper, the problem of delay-dependent asymptotic stability criterion for neural networks with time-varying delay has been considered. A new class of Lyapunov functional is constructed to derive some new delay-dependent stability criteria. The obtained criteria are less conservative because a mixed dividing delay interval approach is considered. Finally, the numerical examples are given to indicate significant improvements over some existing results.

2. Problem Formulation

Consider the following neural networks with time-varying delay:̇𝑥(𝑡)=𝐶𝑥(𝑡)+𝐴𝑔(𝑥(𝑡))+𝐵𝑔(𝑥(𝑡𝜏(𝑡)))+𝜇,(2.1) where 𝑥(𝑡)=[𝑥1(𝑡),𝑥2(𝑡),,𝑥𝑛(𝑡)]𝑇𝑛 is the neuron state vector, 𝑔(𝑥())=[𝑔1(𝑥1()),𝑔2(𝑥2()),,𝑔𝑛(𝑥𝑛())]𝑇𝑛 denotes the neuron activation function, and 𝜇=(𝜇1,𝜇2,,𝜇𝑛)𝑇𝑛 is a constant input vector. 𝐴,𝐵𝑛×𝑛 are the connection weight matrix and the delayed connection weight matrix, respectively. 𝐶=diag(𝐶1,𝐶2,,𝐶𝑛) with 𝐶𝑖>0,𝑖=1,2,,𝑛. 𝜏(𝑡) is a time-varying continuous function that satisfies 0𝜏(𝑡), .𝜏(𝑡)𝑢, where and 𝑢 are constants. In addition, it is assumed that each neuron activation function in (2.1), 𝑔𝑖(),𝑖=1,2,,𝑛, is bounded and satisfies the following condition:𝑔0𝑖(𝑥)𝑔𝑖(𝑦)𝑥𝑦𝑘𝑖,𝑥,𝑦𝑅,𝑥𝑦,𝑖=1,2,,𝑛,(2.2) where 𝑘𝑖,𝑖=1,2,,𝑛 are positive constants.

Assuming that 𝑥=[𝑥1,𝑥2,,𝑥𝑛]𝑇 is the equilibrium point of (2.1) whose uniqueness has been given in [31] and using the transformation 𝑧()=𝑥()𝑥, system (2.1) can be converted to the following system:̇𝑧(𝑡)=𝐶𝑧(𝑡)+𝐴𝑓(𝑧(𝑡))+𝐵𝑓(𝑧(𝑡𝜏(𝑡))),(2.3) where 𝑧(𝑡)=[𝑧1(𝑡),𝑧2(𝑡),,𝑧𝑛(𝑡)]𝑇, 𝑓(𝑧())=[𝑓1(𝑧1()),𝑓2(𝑧2()),,𝑓𝑛(𝑧𝑛())]𝑇, and 𝑓𝑖(𝑧𝑖())=𝑔𝑖(𝑧𝑖()+𝑥𝑖)𝑔𝑖(𝑥𝑖),𝑖=1,2,,𝑛. According to the inequality (2.2), one can obtain that𝑓0𝑖𝑧𝑖(𝑡)𝑧𝑖(𝑡)𝑘𝑖,𝑓𝑖(0)=0,𝑖=1,2,,𝑛.(2.4) Thus, under this assumption, the following inequality holds for any diagonal matrix 𝑄>0,𝑧𝑇(𝑡)𝐾𝑄𝐾𝑧(𝑡)𝑓𝑇(𝑧(𝑡))𝑄𝑓(𝑧(𝑡))0,(2.5) where 𝐾=diag(𝑘1,𝑘2,,𝑘𝑛).

Lemma 2.1 (see [38]). For any constant matrix 𝑍𝑛×𝑛,𝑍=𝑍𝑇>0, scalars 2>1>0, such that the following integrations are well defined, then 21𝑡1𝑡2𝑥𝑇(𝑠)𝑍𝑥(𝑠)𝑑𝑠𝑡1𝑡2𝑥𝑇(𝑠)𝑑𝑠𝑍𝑡1𝑡2𝑥(𝑠)𝑑𝑠.(2.6)

3. Main Results

In this section, a new Lyapunov functional is constructed, and a new delay-dependent stability criterion is obtained.

Theorem 3.1. For given scalars 𝐾=diag(𝑘1,𝑘2,,𝑘𝑛), 0,𝑢,and0<𝛼<1, the system (2.3) is globally asymptotically stable if there exist symmetric positive matrices [𝑃=𝑃𝑖𝑗]3×3,𝑄𝑖(𝑖=1,2,,6), 𝑅𝑖(𝑖=1,2,,6), positive diagonal matrices 𝑇1,𝑇2,𝑄,Δ=diag(𝛿1,𝛿2,,𝛿𝑛),Λ=diag(𝜆1,𝜆2,,𝜆𝑛), and any matrices 𝑃1,𝑃2,𝑁𝑖,𝑀𝑖,𝐿𝑖,𝐻𝑖,𝑍𝑖,𝑆𝑖,𝑈𝑖,𝑉𝑖,𝑊𝑖(𝑖=1,2,3) with appropriate dimensions, such that the following LMIs hold: 𝐸1=𝛼𝐸𝛼𝑁222𝐻1𝛼222𝑀𝛼𝑅3𝛼00222𝑅501𝛼222𝑅6𝐸<0,(3.1)2=𝐸(1𝛼)𝛼𝐿𝛼2𝛼𝑆222𝐻1𝛼222𝑀(1𝛼)𝛼𝑅3000𝛼2𝑅3𝛼00222𝑅501𝛼222𝑅6Φ<0,(3.2)1=𝛼Φ(1𝛼)𝛼𝑈(1𝛼)𝑊222𝐻1𝛼222𝑀(1𝛼)𝛼𝑅3000(1𝛼)𝑅4𝛼00222𝑅501𝛼222𝑅6Φ<0,(3.3)2=𝛼Φ(1𝛼)𝛼𝑉(1𝛼)𝑍222𝐻1𝛼222𝑀(1𝛼)𝛼𝑅3000(1𝛼)𝑅4𝛼00222𝑅501𝛼222𝑅6<0,(3.4) where 𝐸𝐸=11𝐸12𝐸13𝐸140𝑃13𝐸17𝑃1𝐴+𝐾𝑇1𝑃1𝐵𝑃22𝐻1𝑃23𝑀1𝐸22𝐸23𝑁200000𝐻2𝑀2𝐸33𝑁30000𝐾𝑇2𝐻3𝑀3𝐸440𝑅4(1𝛼)000𝑃22+𝑃𝑇23𝑃23+𝑃33𝑄5000000𝐸66000𝑃𝑇23𝑃33𝐸77𝑃2𝐴+ΛΔ𝑃2𝐵𝑃12𝑃13𝑄62𝑇1𝑄000𝐸9900𝑅10𝑅2,ΦΦ=11Φ12Φ13Φ14𝑉1+𝑅3𝛼2𝑃13𝑊1Φ17𝑃1𝐴+𝐾𝑇1𝑃1𝐵𝑃22𝐻1𝑃23𝑀1Φ22Φ23𝑈2+𝑍2𝑉2𝑊2000𝐻2𝑀2Φ33𝑈3+𝑍3𝑉3𝑊300𝐾𝑇2𝐻3𝑀3𝑄300000𝑃22+𝑃𝑇23𝑃23+𝑃33𝑄5𝑅3𝛼2000000𝑄4000𝑃𝑇23𝑃33Φ77𝑃2𝐴+ΛΔ𝑃2𝐵𝑃12𝑃13𝑄62𝑇1𝑄000Φ9900𝑅10𝑅2𝐸11=𝑃12+𝑃𝑇12+𝑄1+𝑄2+𝑄3+𝑄4+𝑄5+𝛼22𝑅1+(1𝛼)22𝑅2+𝑆1+𝑆𝑇1𝑃1𝐶𝐶𝑃𝑇1𝐻+𝛼1+𝐻𝑇1𝑀+(1𝛼)1+𝑀𝑇1𝐸+𝐾𝑄𝐾,12=𝐿1+𝑆𝑇2𝑆1+𝛼𝐻𝑇2+(1𝛼)𝑀𝑇2,𝐸13=𝑁1𝐿1+𝑆𝑇3+𝛼𝐻𝑇3+(1𝛼)𝑀𝑇3,𝐸14=𝑃12+𝑃13𝑁1,𝐸17=𝑃11+𝐾Δ𝑃1𝐶𝑃𝑇2,𝐸22=(1𝛼𝑢)𝑄1+𝐿2+𝐿𝑇2𝑆2𝑆𝑇2,𝐸23=𝐿2+𝐿𝑇3𝑆𝑇3+𝑁2,𝐸33=(1𝑢)𝑄2+𝑁3+𝑁𝑇3𝐿3𝐿𝑇3𝐸(1𝑢)𝐾𝑄𝐾,44=𝑄3𝑅4,𝐸(1𝛼)66=𝑄4𝑅4,𝐸(1𝛼)77=𝛼𝑅3+(1𝛼)𝑅4+𝛼222𝑅5+1𝛼222𝑅6𝑃2𝑃𝑇2,𝐸99=(1𝑢)𝑄62𝑇2Φ+(1𝑢)𝑄,11=𝑃12+𝑃𝑇12+𝑄1+𝑄2+𝑄3+𝑄4+𝑄5+𝛼22𝑅1+(1𝛼)22𝑅2𝑃1𝐶𝐶𝑃𝑇1𝐻+𝛼1+𝐻𝑇1+𝑀(1𝛼)1+𝑀𝑇1𝑅3𝛼2Φ+𝐾𝑄𝐾,12=𝑈1𝑉1+𝛼𝐻𝑇2+(1𝛼)𝑀𝑇2,Φ13=𝑊1𝑍1+𝛼𝐻𝑇3+(1𝛼)𝑀𝑇3,Φ14=𝑃12+𝑃13𝑈1+𝑍1,Φ17=𝑃11+𝐾Δ𝑃1𝐶𝑃𝑇2,Φ22=(1𝛼𝑢)𝑄1+𝑈2+𝑈𝑇2𝑉2𝑉𝑇2,Φ23=𝑈𝑇3𝑉𝑇3+𝑊2𝑍2,Φ33=(1𝑢)𝑄2+𝑊3+𝑊𝑇3𝑍3𝑍𝑇3Φ(1𝑢)𝐾𝑄𝐾,77=𝛼𝑅3+(1𝛼)𝑅4+𝛼222𝑅5+1𝛼222𝑅6𝑃2𝑃𝑇2,Φ99=(1𝑢)𝑄62𝑇2+𝑁(1𝑢)𝑄,𝑁=𝑇1𝑁𝑇2𝑁𝑇300000000𝑇,𝐿𝐿=𝑇1𝐿𝑇2𝐿𝑇300000000𝑇,𝑆𝑆=𝑇1𝑆𝑇2𝑆𝑇300000000𝑇,𝐻𝐻=𝑇1𝐻𝑇2𝐻𝑇300000000𝑇,𝑀𝑀=𝑇1𝑀𝑇2𝑀𝑇300000000𝑇,𝑈𝑈=𝑇1𝑈𝑇2𝑉𝑇300000000𝑇,𝑉𝑉=𝑇1𝑉𝑇2𝑉𝑇300000000𝑇,𝑊𝑊=𝑇1𝑊𝑇2𝑊𝑇300000000𝑇,𝑍𝑍=𝑇1𝑍𝑇2𝑍𝑇300000000𝑇.(3.5)

Proof. Construct a new class of Lyapunov functional candidate as follows: 𝑉(𝑧(𝑡))=6𝑖=1𝑉𝑖(𝑧(𝑡)),(3.6) where 𝑉1(𝑧(𝑡))=𝜉𝑇(𝑡)𝑃𝜉(𝑡)+2𝑛𝑖=1𝜆𝑖𝑧𝑖0(𝑡)𝑓𝑖(𝑠)𝑑𝑠+2𝑛𝑖=1𝛿𝑖𝑧𝑖0(𝑡)𝑘𝑖𝑠𝑓𝑖𝑉(𝑠)𝑑𝑠,2(𝑧(𝑡))=𝑡𝑡𝛼𝜏(𝑡)𝑧𝑇(𝑠)𝑄1𝑧(𝑠)𝑑𝑠+𝑡𝑡𝜏(𝑡)𝑧𝑇(𝑠)𝑄2𝑧(𝑠)𝑑𝑠+𝑡𝑡𝛼𝑧𝑇(𝑠)𝑄3+𝑧(𝑠)𝑑𝑠𝑡𝑡𝑧𝑇(𝑠)𝑄4𝑧(𝑠)𝑑𝑠+𝑡𝑡𝛼2𝑧𝑇(𝑠)𝑄5𝑧(𝑠)𝑑𝑠+𝑡𝑡𝜏(𝑡)𝑓𝑇(𝑧(𝑠))𝑄6𝑓𝑉(𝑧(𝑠))𝑑𝑠,3(𝑧(𝑡))=𝛼0𝛼𝑡𝑡+𝜃𝑧𝑇(𝑠)𝑅1𝑧(𝑠)𝑑𝑠d𝜃+(1𝛼)𝛼𝑡𝑡+𝜃𝑧𝑇(𝑠)𝑅2𝑉𝑧(𝑠)𝑑𝑠𝑑𝜃,4(𝑧(𝑡))=0𝛼𝑡𝑡+𝜃̇𝑧𝑇(𝑠)𝑅3̇𝑧(𝑠)𝑑𝑠𝑑𝜃+𝛼𝑡𝑡+𝜃̇𝑧𝑇(𝑠)𝑅4𝑉̇𝑧(𝑠)𝑑𝑠𝑑𝜃,5(𝑧(𝑡))=0𝛼0𝜃𝑡𝑡+𝜆̇𝑧𝑇(𝑠)𝑅5̇𝑧(𝑠)𝑑𝑠𝑑𝜆𝑑𝜃+𝛼0𝜃𝑡𝑡+𝜆̇𝑧𝑇(𝑠)𝑅6𝑉̇𝑧(𝑠)𝑑𝑠𝑑𝜆𝑑𝜃,6(𝑧(𝑡))=𝑡𝑡𝜏(𝑡)𝑧𝑇(𝑠)𝐾𝑄𝐾𝑧(𝑠)𝑓𝑇(𝑧(𝑠))𝑄𝑓(𝑧(𝑠))𝑑𝑠,(3.7) where 𝜉𝑇𝑧(𝑡)=𝑇(𝑡)𝑡𝑡𝛼𝑧𝑇(𝑠)𝑑𝑠𝑡𝛼𝑡𝑧𝑇(𝑠)𝑑𝑠,0<𝛼<1.(3.8)

Remark 3.2. Since the term 2𝑛𝑖=1𝛿𝑖𝑧𝑖0(𝑡)(𝑘𝑖𝑠𝑓𝑖(𝑠))𝑑𝑠 in 𝑉1(𝑧(𝑡)) and 𝑉6(𝑧(𝑡))=𝑡𝑡𝜏(𝑡)[𝑧𝑇(𝑠)𝐾𝑄𝐾𝑧(𝑠)𝑓𝑇(𝑧(𝑠))𝑄𝑓(𝑧(𝑠))]𝑑𝑠 is taken into account, it is clear that the Lyapunov functional candidate in this paper is more general than that in [5, 6, 8, 9]. So the stability criteria in this paper may be more applicable.

The time derivative of 𝑉(𝑧(𝑡)) along the trajectory of system (2.3) is given bẏ𝑉(𝑧(𝑡))=6𝑖=1̇𝑉𝑖(𝑧(𝑡)),(3.9) where ̇𝑉1(𝑧(𝑡))=2𝜉𝑇(𝑡)𝑃̇𝑧(𝑡)𝑧(𝑡)𝑧(𝑡𝛼)𝑧(𝑡𝛼)𝑧(𝑡)+2𝑓𝑇(𝑧(𝑡))Λ̇𝑧(𝑡)+2(𝐾𝑧(𝑡))𝑓(𝑧(𝑡))𝑇̇𝑉Δ̇𝑧(𝑡),2(𝑧(𝑡))=𝑧𝑇(𝑄𝑡)1+𝑄2+𝑄3+𝑄4+𝑄5𝑧(𝑡)+𝑓𝑇(𝑧(𝑡))𝑄6𝑓(𝑧(𝑡))(1𝛼𝑢)𝑧𝑇(𝑡𝛼𝜏(𝑡))𝑄1𝑧(𝑡𝛼𝜏(𝑡))(1𝑢)𝑧𝑇(𝑡𝜏(𝑡))𝑄2𝑧(𝑡𝜏(𝑡))𝑧𝑇(𝑡𝛼)𝑄3𝑧(𝑡𝛼)𝑧𝑇(𝑡)𝑄4𝑧(𝑡)𝑧𝑇𝑡𝛼2𝑄5𝑧𝑡𝛼2(1𝑢)𝑓𝑇(𝑧(𝑡𝜏(𝑡)))𝑄6𝑓(𝑧(𝑡𝜏(𝑡))).(3.10) Using Lemma 2.1, one can obtain that ̇𝑉3(𝑧(𝑡))=𝛼22𝑧𝑇(𝑡)𝑅1𝑧(𝑡)+(1𝛼)22𝑧𝑇(𝑡)𝑅2𝑧(𝑡)𝛼𝑡𝑡𝛼𝑧𝑇(𝑠)𝑅1𝑧(𝑠)𝑑𝑠(1𝛼)𝑡𝛼𝑡𝑧𝑇(𝑠)𝑅2𝑧(𝑠)𝑑𝑠𝛼22𝑧𝑇(𝑡)𝑅1𝑧(𝑡)+(1𝛼)22𝑧𝑇(𝑡)𝑅2𝑧(𝑡)𝑡𝑡𝛼𝑧(𝑠)𝑑𝑠𝑇𝑅1𝑡𝑡𝛼𝑧(𝑠)𝑑𝑠𝑡𝛼𝑡𝑧(𝑠)𝑑𝑠𝑇𝑅2𝑡𝛼𝑡,̇𝑉𝑧(𝑠)𝑑𝑠4(𝑧(𝑡))=̇𝑧𝑇(𝑡)𝛼𝑅3+(1𝛼)𝑅4̇𝑧(𝑡)𝑡𝑡𝛼̇𝑧𝑇(𝑠)𝑅3̇𝑧(𝑠)𝑑𝑠𝑡𝛼𝑡̇𝑧𝑇(𝑠)𝑅4̇𝑉̇𝑧(𝑠)𝑑𝑠,5(𝑧(𝑡))=̇𝑧𝑇𝛼(𝑡)222𝑅5+1𝛼222𝑅6̇𝑧(𝑡)0𝛼𝑡𝑡+𝜃̇𝑧𝑇(𝑠)𝑅5̇𝑧(𝑠)𝑑𝑠𝑑𝜃𝛼𝑡𝑡+𝜃̇𝑧𝑇(𝑠)𝑅6̇𝑉̇𝑧(𝑠)𝑑𝑠𝑑𝜃,6(𝑧(𝑡))𝑧𝑇(𝑡)𝐾𝑄𝐾𝑧(𝑡)𝑓𝑇(𝑧(𝑡))𝑄𝑓(𝑧(𝑡))(1𝑢)𝑧𝑇(𝑡𝜏(𝑡))𝐾𝑄𝐾𝑧(𝑡𝜏(𝑡))+(1𝑢)𝑓𝑇(𝑧(𝑡𝜏(𝑡)))𝑄𝑓(𝑧(𝑡𝜏(𝑡))).(3.11)(1) For the case of 0𝜏(𝑡)𝛼, then it gets 𝑡𝑡𝛼̇𝑧𝑇(𝑠)𝑅3̇𝑧(𝑠)𝑑𝑠=𝑡𝜏(𝑡)𝑡𝛼̇𝑧𝑇(𝑠)𝑅3̇𝑧(𝑠)𝑑𝑠𝑡𝛼𝜏(𝑡)𝑡𝜏(𝑡)̇𝑧𝑇(𝑠)𝑅3̇𝑧(𝑠)𝑑𝑠𝑡𝑡𝛼𝜏(𝑡)̇𝑧𝑇(𝑠)𝑅3̇𝑧(𝑠)𝑑𝑠.(3.12) Similar to [8], the following equalities hold: 2𝜁𝑇(𝑡)𝑁𝑧(𝑡𝜏(𝑡))𝑧(𝑡𝛼)𝑡𝜏(𝑡)𝑡𝛼̇𝑧(𝑠)𝑑𝑠=0,(3.13)2𝜁𝑇(𝑡)𝐿𝑧(𝑡𝛼𝜏(𝑡))𝑧(𝑡𝜏(𝑡))𝑡𝛼𝜏(𝑡)𝑡𝜏(𝑡)̇𝑧(𝑠)𝑑𝑠=0,(3.14)2𝜁𝑇(𝑡)𝑆𝑧(𝑡)𝑧(𝑡𝛼𝜏(𝑡))𝑡𝑡𝛼𝜏(𝑡)2𝑧̇𝑧(𝑠)𝑑𝑠=0,(3.15)𝑇(𝑡)𝑃1+̇𝑧𝑇(𝑡)𝑃2[]̇𝑧(𝑡)𝐶𝑧(𝑡)+𝐴𝑓(𝑧(𝑡))+𝐵𝑓(𝑧(𝑡𝜏(𝑡)))=0,(3.16)2𝜁𝑇(𝑡)𝐻𝛼𝑧(𝑡)𝑡𝑡𝛼𝑧(𝑠)𝑑𝑠0𝛼𝑡𝑡+𝜃̇𝑧(𝑠)𝑑𝑠𝑑𝜃=0,(3.17)2𝜁𝑇((𝑡)𝑀1𝛼)𝑧(𝑡)𝑡𝛼𝑡𝑧(𝑠)𝑑𝑠𝛼𝑡𝑡+𝜃𝜁̇𝑧(𝑠)𝑑𝑠𝑑𝜃=0,(3.18)𝑇𝑧(𝑡)=𝑇(𝑡)𝑧𝑇(𝑡𝛼𝜏(𝑡))𝑧𝑇(𝑡𝜏(𝑡))𝑧𝑇(𝑡𝛼)𝑧𝑇𝑡𝛼2𝑧𝑇(𝑡)̇𝑧𝑇𝑓(𝑡)𝑇(𝑧(𝑡))𝑓𝑇(𝑧(𝑡𝜏(𝑡)))𝑡𝑡𝛼𝑧𝑇(𝑠)𝑑𝑠𝑡𝛼𝑡𝑧𝑇.(𝑠)𝑑𝑠(3.19) It is easy to see that 2𝜁𝑇(𝑡)𝑁𝑡𝜏(𝑡)𝑡𝛼̇𝑧(𝑠)𝑑𝑠(𝛼𝜏(𝑡))𝜁𝑇(𝑡)𝑁𝑅31𝑁𝑇𝜁(𝑡)+𝑡𝜏(𝑡)𝑡𝛼̇𝑧𝑇(𝑠)𝑅3̇𝑧(𝑠)𝑑𝑠,(3.20)2𝜁𝑇(𝑡)𝐿𝑡𝛼𝜏(𝑡)𝑡𝜏(𝑡)̇𝑧(𝑠)𝑑𝑠(1𝛼)𝜏(𝑡)𝜁𝑇(𝑡)𝐿𝑅31𝐿𝑇𝜁(𝑡)+𝑡𝛼𝜏(𝑡)𝑡𝜏(𝑡)̇𝑧𝑇(𝑠)𝑅3̇𝑧(𝑠)𝑑𝑠,(3.21)2𝜁𝑇𝑆𝑡𝑡𝛼𝜏(𝑡)̇𝑧(𝑠)𝑑𝑠𝛼𝜏(𝑡)𝜁𝑇(𝑡)𝑆𝑅31𝑆𝑇𝜁(𝑡)+𝑡𝑡𝛼𝜏(𝑡)̇𝑧𝑇(𝑠)𝑅3̇𝑧(𝑠)𝑑𝑠,(3.22)2𝜁𝑇𝐻0𝛼𝑡𝑡+𝜃𝛼̇𝑧(𝑠)𝑑𝑠𝑑𝜃222𝜁𝑇(𝑡)𝐻𝑅51𝐻𝑇𝜁(𝑡)+0𝛼𝑡𝑡+𝜃̇𝑧𝑇(𝑠)𝑅5̇𝑧(𝑠)𝑑𝑠𝑑𝜃,(3.23)2𝜁𝑇𝑀𝛼𝑡𝑡+𝜃̇𝑧(𝑠)𝑑𝑠𝑑𝜃1𝛼222𝜁𝑇(𝑡)𝑀𝑅61𝑀𝑇𝜁(𝑡)+𝛼𝑡𝑡+𝜃̇𝑧𝑇(𝑠)𝑅6̇𝑧(𝑠)𝑑𝑠𝑑𝜃.(3.24) Using Lemma 2.1, it is easy to obtain that 𝑡𝛼𝑡̇𝑧𝑇(𝑠)𝑅4[𝑧]̇𝑧(𝑠)𝑑𝑠(𝑡𝛼)𝑧(𝑡)𝑇𝑅4[𝑧].(1𝛼)(𝑡𝛼)𝑧(𝑡)(3.25) Furthermore, there exist positive diagonal matrices 𝑇1,𝑇2, such that the following inequalities hold based on (2.4): 2𝑓𝑇(𝑧(𝑡))𝑇1𝑓(𝑧(𝑡))+2𝑧𝑇(𝑡)𝐾𝑇1𝑓(𝑧(𝑡))0,(3.26)2𝑓𝑇(𝑧(𝑡𝜏(𝑡)))𝑇2𝑓(𝑧(𝑡𝜏(𝑡)))+2𝑧𝑇(𝑡𝜏(𝑡))𝐾𝑇2𝑓(𝑧(𝑡𝜏(𝑡)))0.(3.27) From (3.10)–(3.27), one can obtain that ̇𝑉(𝑧(𝑡))𝜁𝑇(𝑡)Σ1𝜁(𝑡),(3.28) where Σ1=𝐸+(𝛼𝜏(𝑡))𝑁𝑅31𝑁𝑇+(1𝛼)𝜏(𝑡)𝐿𝑅31𝐿𝑇+𝛼𝜏(𝑡)𝑆𝑅31𝑆𝑇+𝛼222𝐻𝑅51𝐻𝑇+1𝛼222𝑀𝑅61𝑀𝑇.(3.29) Note that 0𝜏(𝑡)𝛼, (𝛼𝜏(𝑡))𝑁𝑅31𝑁𝑇+(1𝛼)𝜏(𝑡)𝐿𝑅31𝐿𝑇+𝛼𝜏(𝑡)𝑆𝑅31𝑆𝑇 can be seen as the convex combination of 𝑁𝑅31𝑁𝑇,𝐿𝑅31𝐿𝑇, and 𝑆𝑅31𝑆𝑇 on 𝜏(𝑡). Therefore, Σ1<0 holds if and only if 𝐸+𝛼𝑁𝑅31𝑁𝑇+𝛼222𝐻𝑅51𝐻𝑇+1𝛼222𝑀𝑅61𝑀𝑇<0,(3.30)𝐸+(1𝛼)𝛼𝐿𝑅31𝐿𝑇+𝛼2𝑆𝑅31𝑆𝑇+𝛼222𝐻𝑅51𝐻𝑇+1𝛼222𝑀𝑅61𝑀𝑇<0.(3.31) Applying the Schur complement, the inequalities (3.30) and (3.31) are equivalent to the LMI (3.1) and (3.2), respectively. (2) When 𝛼𝜏(𝑡), then it gets 𝑡𝑡𝛼̇𝑧𝑇(𝑠)𝑅3̇𝑧(𝑠)𝑑𝑠=𝑡𝛼𝜏(𝑡)𝑡𝛼̇𝑧𝑇(𝑠)𝑅3̇𝑧(𝑠)𝑑𝑠𝑡𝛼2𝑡𝛼𝜏(𝑡)̇𝑧𝑇(𝑠)𝑅3̇𝑧(𝑠)𝑑𝑠𝑡𝑡𝛼2̇𝑧𝑇(𝑠)𝑅3̇𝑧(𝑠)𝑑𝑠,𝑡𝛼𝑡̇𝑧𝑇(𝑠)𝑅4̇𝑧(𝑠)𝑑𝑠=𝑡𝜏(𝑡)𝑡̇𝑧𝑇(𝑠)𝑅4̇𝑧(𝑠)𝑑𝑠𝑡𝛼𝑡𝜏(𝑡)̇𝑧𝑇(𝑠)𝑅4̇𝑧(𝑠)𝑑𝑠.(3.32) Similar to [8], the following equalities hold: 2𝜁𝑇(𝑡)𝑈𝑧(𝑡𝛼𝜏(𝑡))𝑧(𝑡𝛼)𝑡𝛼𝜏(𝑡)𝑡𝛼̇𝑧(𝑠)𝑑𝑠=0,2𝜁𝑇𝑧(𝑡)𝑉𝑡𝛼2𝑧(𝑡𝛼𝜏(𝑡))𝑡𝛼2𝑡𝛼𝜏(𝑡)̇𝑧(𝑠)𝑑𝑠=0,2𝜁𝑇(𝑡)𝑊𝑧(𝑡𝜏(𝑡))𝑧(𝑡)𝑡𝜏(𝑡)𝑡̇𝑧(𝑠)𝑑𝑠=0,2𝜁𝑇(𝑡)𝑍𝑧(𝑡𝛼)𝑧(𝑡𝜏(𝑡))𝑡𝛼𝑡𝜏(𝑡)̇𝑧(𝑠)𝑑𝑠=0.(3.33) It is easy to obtain that 2𝜁𝑇(𝑡)𝑈𝑡𝛼𝜏(𝑡)𝑡𝛼̇𝑧(𝑠)𝑑𝑠𝛼(𝜏(𝑡))𝜁𝑇(𝑡)𝑈𝑅31𝑈𝑇𝜁(𝑡)+𝑡𝛼𝜏(𝑡)𝑡𝛼̇𝑧𝑇(𝑠)𝑅3̇𝑧(𝑠)𝑑𝑠,2𝜁𝑇(𝑡)𝑉𝑡𝛼2𝑡𝛼𝜏(𝑡)̇𝑧(𝑠)𝑑𝑠𝛼(𝜏(𝑡)𝛼)𝜁𝑇(𝑡)𝑉𝑅31𝑉𝑇𝜁(𝑡)+𝑡𝛼2𝑡𝛼𝜏(𝑡)̇𝑧𝑇(𝑠)𝑅3̇𝑧(𝑠)𝑑𝑠,2𝜁𝑇(𝑡)𝑊𝑡𝜏(𝑡)𝑡̇𝑧(𝑠)𝑑𝑠(𝜏(𝑡))𝜁𝑇(𝑡)𝑊𝑅41𝑊𝑇𝜁(𝑡)+𝑡𝜏(𝑡)𝑡̇𝑧𝑇(𝑠)𝑅4̇𝑧(𝑠)𝑑𝑠,2𝜁𝑇(𝑡)𝑍𝑡𝛼𝑡𝜏(𝑡)̇𝑧(𝑠)𝑑𝑠(𝜏(𝑡)𝛼)𝜁𝑇(𝑡)𝑍𝑅41𝑍𝑇𝜁(𝑡)+𝑡𝛼𝑡𝜏(𝑡)̇𝑧𝑇(𝑠)𝑅4̇𝑧(𝑠)𝑑𝑠.(3.34) Using Lemma 2.1, one can obtain that 𝑡𝑡𝛼2̇𝑧𝑇(𝑠)𝑅41̇𝑧(𝑠)𝑑𝑠𝛼2𝑧(𝑡)𝑧𝑡𝛼2𝑇𝑅3𝑧(𝑡)𝑧𝑡𝛼2.(3.35) From (3.10)–(3.11), (3.16)–(3.18), (3.23), (3.24), (3.26), (3.27) and (3.33)–(3.35), one can obtain that ̇𝑉(𝑧(𝑡))𝜁𝑇(𝑡)Σ2𝜁(𝑡),(3.36) where Σ2=Φ+𝛼(𝜏(𝑡))𝑈𝑅31𝑈𝑇+𝛼(𝜏(𝑡)𝛼)𝑉𝑅31𝑉𝑇+(𝜏(𝑡))𝑊𝑅41𝑊𝑇+(𝜏(𝑡)𝛼)𝑍𝑅41𝑍𝑇+𝛼222𝐻𝑅51𝐻𝑇+1𝛼222𝑀𝑅61𝑀𝑇.(3.37) Note that 𝛼𝜏(𝑡), 𝛼(𝜏(𝑡))𝑈𝑅31𝑈𝑇+𝛼(𝜏(𝑡)𝛼)𝑉𝑅31𝑉𝑇+(𝜏(𝑡))𝑊𝑅41𝑊𝑇+(𝜏(𝑡)𝛼)𝑍𝑅41𝑍𝑇 can be seen as the convex combination of 𝑈𝑅31𝑈𝑇,𝑉𝑅31𝑉𝑇,𝑊𝑅41𝑊𝑇, and 𝑍𝑅41𝑍𝑇 on 𝜏(𝑡). Therefore, Σ2<0 holds if and only if Φ+𝛼(1𝛼)𝑈𝑅31𝑈𝑇+(1𝛼)𝑊𝑅41𝑊𝑇+𝛼222𝐻𝑅51𝐻𝑇+1𝛼222𝑀𝑅61𝑀𝑇<0,(3.38)Φ+𝛼(1𝛼)𝑉𝑅31𝑉𝑇+(1𝛼)𝑍𝑅41𝑍𝑇+𝛼222𝐻𝑅51𝐻𝑇+1𝛼222𝑀𝑅61𝑀𝑇<0.(3.39) Applying the Schur complement, the inequalities (3.38) and (3.39) are equivalent to the LMI (3.3) and (3.4), respectively. Therefore, if the LMIs (3.1)–(3.4) are satisfied, then the system (2.3) is guaranteed to be asymptotically stable for 0𝜏(𝑡).

Remark 3.3. It is well known that the delay-dividing approach can reduce the conservatism notably. But some previous literature only uses single method to divide the delay interval [0,]. Unlike [10, 25], the new Lyapunov functional in our paper which not only divides the delay interval [0,] into two ones [0,𝛼] and [𝛼,] but also divides the delay interval [0,] into three ones [0,𝛼𝜏(𝑡)], [𝛼𝜏(𝑡),𝜏(𝑡)], and [𝜏(𝑡),] is proposed. Each segment has a different positive matrix, which has the potential to yield less conservative results.

Remark 3.4. In this paper, by taking the states 𝑧(𝑡𝛼),𝑧(𝑡𝜏(𝑡)),𝑧(𝑡𝛼2),𝑧(𝑡), and 𝑧(𝑡𝛼𝜏(𝑡)) as augmented variables, the stability in Theorem 3.1 utilizes more information on state variables. And in deriving upper bounds of integral terms in ̇𝑉4(𝑧(𝑡)), different free-weighting matrices are introduced in two different intervals 0𝜏(𝑡)𝛼 and 𝛼𝜏(𝑡). These methods mentioned above may lead to obtain an improved feasible region for delay-dependent stability criteria.

Remark 3.5. In (3.28), 𝐸+(𝛼𝜏(𝑡))𝑁𝑅31𝑁𝑇+(1𝛼)𝜏(𝑡)𝐿𝑅31𝐿𝑇+𝛼𝜏(𝑡)𝑆𝑅31𝑆𝑇+(𝛼22/2)𝐻𝑅51𝐻𝑇+((1𝛼2)2/2)𝑀𝑅61𝑀𝑇 is not simply guaranteed by 𝐸+𝛼𝑁𝑅31𝑁𝑇+(1𝛼)𝛼𝐿𝑅31𝐿𝑇+𝛼2𝑆𝑅31𝑆𝑇+(𝛼22/2)𝐻𝑅51𝐻𝑇+((1𝛼2)2/2)𝑀𝑅61𝑀𝑇 but is evaluated by the LMIs (3.1) and (3.2), which can help reduce much more conservatism than the results in [8].

Remark 3.6. In many cases, 𝑢 is unknown. For this situation, a rate-independent criterion for a delay satisfying 0𝜏(𝑡) is derived as follows by setting 𝑄1=0,𝑄2=0,𝑄6=0, and 𝑄=0 in the proof of Theorem 3.1.

Corollary 3.7. For given scalars 𝐾=diag(𝑘1,𝑘2,,𝑘𝑛), 0, and 0<𝛼<1, the system (2.3) is globally asymptotically stable if there exist symmetric positive matrices [𝑃=𝑃𝑖𝑗]3×3,𝑄3,𝑄4,𝑄5, 𝑅𝑖(𝑖=1,2,,6), positive diagonal matrices 𝑇1,𝑇2,Λ=diag(𝜆1,𝜆2,,𝜆𝑛),Δ=diag(𝛿1,𝛿2,,𝛿𝑛), and any matrices  𝑃1,𝑃2,𝑁𝑖,𝑀𝑖,𝐿𝑖,𝐻𝑖,𝑍𝑖,𝑆𝑖,𝑈𝑖,𝑉𝑖,𝑊𝑖(𝑖=1,2,3) with appropriate dimensions, such that the following LMIs hold: 𝐸1=𝛼𝐸𝛼𝑁222𝐻1𝛼222𝑀𝛼𝑅3𝛼00222𝑅501𝛼222𝑅6<0,𝐸2=𝐸(1𝛼)𝛼𝐿𝛼2𝛼𝑆222𝐻1𝛼222𝑀(1𝛼)𝛼𝑅3000𝛼2𝑅3𝛼00222𝑅501𝛼222𝑅6<0,Φ1=𝛼Φ(1𝛼)𝛼𝑈(1𝛼)𝑊222𝐻1𝛼222𝑀(1𝛼)𝛼𝑅3000(1𝛼)𝑅4𝛼00222𝑅501𝛼222𝑅6<0,Φ2=𝛼Φ(1𝛼)𝛼𝑉(1𝛼)𝑍222𝐻1𝛼222𝑀(1𝛼)𝛼𝑅3000(1𝛼)𝑅4𝛼00222𝑅501𝛼222𝑅6<0,(3.40) where 𝐸=𝐸11𝐸12𝐸13𝐸140𝑃13𝐸17𝑃1𝐴+𝐾𝑇1𝑃1𝐵𝑃22𝐻1𝑃23𝑀1𝐸22𝐸23𝑁200000𝐻2𝑀2𝐸33𝑁30000𝐾𝑇2𝐻3𝑀3𝐸440𝑅4(1𝛼)000𝑃22+𝑃𝑇23𝑃23+𝑃33𝑄5000000𝐸66000𝑃𝑇23𝑃33𝐸77𝑃2𝐴+ΛΔ𝑃2𝐵𝑃12𝑃132𝑇10002𝑇200𝑅10𝑅2,Φ=Φ11Φ12Φ13Φ14𝑉1+𝑅3𝛼2𝑃13𝑊1Φ17𝑃1𝐴+𝐾𝑇1𝑃1𝐵𝑃22𝐻1𝑃23𝑀1Φ22Φ23𝑈2+𝑍2𝑉2𝑊2000𝐻2𝑀2Φ33𝑈3+𝑍3𝑉3𝑊300𝐾𝑇2𝐻3𝑀3𝑄300000𝑃22+𝑃𝑇23𝑃23+𝑃33𝑄5𝑅3𝛼2000000𝑄4000𝑃𝑇23𝑃33Φ77𝑃2𝐴+ΛΔ𝑃2𝐵𝑃12𝑃132𝑇10002𝑇200𝑅10𝑅2𝐸11=𝑃12+𝑃𝑇12+𝑄3+𝑄4+𝑄5+𝛼22𝑅1+(1𝛼)22𝑅2+𝑆1+𝑆𝑇1𝑃1𝐶𝐶𝑃𝑇1𝐻+𝛼1+𝐻𝑇1𝑀+(1𝛼)1+𝑀𝑇1,𝐸22=𝐿2+𝐿𝑇2𝑆2𝑆𝑇2,𝐸33=𝑁3+𝑁𝑇3𝐿3𝐿𝑇3,Φ11=𝑃12+𝑃𝑇12+𝑄3+𝑄4+𝑄5+𝛼22𝑅1+(1𝛼)22𝑅2𝑃1𝐶𝐶𝑃𝑇1𝐻+𝛼1+𝐻𝑇1+𝑀(1𝛼)1+𝑀𝑇1𝑅3𝛼2,Φ22=𝑈2+𝑈𝑇2𝑉2𝑉𝑇2,Φ33=𝑊3+𝑊𝑇3𝑍3𝑍𝑇3.(3.41) The other 𝐸𝑖𝑗,Φ𝑖𝑗 are defined in Theorem 3.1.

4. Numerical Examples

Example 4.1. Consider the stability of system (2.3) with time-varying delay and 𝐶=diag(2,2),𝐴=1111,𝐵=0.88111,𝜎1𝜎=0.4,2=0.8,𝛾1=0,𝛾2=0.(4.1) Our purpose is to estimate the allowable upper bounds delay under different 𝑢 such that the system (2.3) is globally asymptotically stable. According to Table 1, this example shows that the stability criterion in this paper gives much less conservative results than those in the literature. By using the Matlab LMI toolbox, we solve LMIs (3.1)–(3.4) for the case 𝛼=0.4,𝑢=0.8, and =2.9144 and obtain 𝑃11=0.00050.00020.00020.0018,𝑃12=1.0𝑒003×,𝑃0.03630.00810.16630.194513=1.0𝑒006×0.10020.02520.39930.2498,𝑃22=1.0𝑒004×,𝑃0.16780.21520.21520.504723=1.0𝑒006×0.02210.08360.04680.3688,𝑃33=1.0𝑒006×,𝑄0.26140.21460.21460.54181=1.0𝑒006×0.23680.24540.24540.4008,𝑄2=1.0𝑒005×,𝑄0.01740.00870.00870.19763=1.0𝑒003×0.05260.02960.02960.2541,𝑄4=1.0𝑒004×,𝑄0.99710.51100.51100.46395=1.0𝑒005×0.02320.05610.05610.1584,𝑄6=,𝑅0.00210.00350.00350.00571=1.0𝑒004×0.45230.60200.60200.8760,𝑅2=1.0𝑒006×,𝑅0.10490.21590.21590.52993=1.0𝑒004×0.79150.36500.36500.4110,𝑅4=1.0𝑒004×,𝑅0.97090.50710.50710.42125=1.0𝑒003×0.10830.18720.18720.3776,𝑅6=1.0𝑒006×,0.06980.12660.12660.2910Λ=0.0000000.0014,Δ=1.0𝑒003×,𝑇0.1686000.00271=0.0037000.0039,𝑇2=.0.0009000.0010,𝑄=0.0000000.0038(4.2) Therefore, it follows from Theorem 3.1 that the system (2.3) with given parameters is globally asymptotically stable.

tab1
Table 1: Allowable upper bound of for different 𝑢 (Example 4.1).

Example 4.2. Consider the stability of system (2.3) with time-varying delay and ,𝜎𝐶=diag(1.5,0.7),𝐴=0.05030.04540.09870.2075,𝐵=0.23810.93200.03880.50621=0.3,𝜎2=0.8,𝛾1=0,𝛾2=0.(4.3)

Table 2 gives the comparison results on the maximum delay bound allowed via the methods in recent paper and our new study. According to Table 2, this example shows that the stability criterion in this paper can lead to less conservative results. By using the Matlab LMI toolbox, we solve LMIs (3.1)–(3.4) for the case 𝛼=0.4,𝑢=0.4, and =4.7444 and obtain𝑃11=1.02060.75280.75281.2378,𝑃12=1.0𝑒003×,𝑃0.00450.04170.06520.013713=1.0𝑒003×0.24670.64040.18390.3921,𝑃22=,𝑃0.00730.00200.00200.002223=1.0𝑒004×0.00860.46160.41080.1085,𝑃33=1.0𝑒003×,𝑄0.19040.12920.12920.10591=0.00130.00120.00120.0012,𝑄2=0.02580.05660.05660.1268,𝑄3=,𝑄0.06530.00510.00510.00504=0.01140.03000.03000.0839,𝑄5=0.00140.00090.00090.0006,𝑄6=,𝑅0.03400.03000.03000.03101=0.00900.00660.00660.0062,𝑅2=1.0𝑒003×,𝑅0.19430.13610.13610.10363=0.24850.38460.38460.7348,𝑅4=,𝑅0.08020.21970.21970.61225=0.15530.08210.08210.0709,𝑅6=1.0𝑒003×,0.76510.49060.49060.5825Λ=0.0318000.0344,Δ=1.0𝑒003×,𝑇0.0797000.35931=0.0136000.1968,𝑇2=.0.3056000.3423,𝑄=0.4567000.0141(4.4) Therefore, it follows from Theorem 3.1 that the system (2.3) with given parameters is globally asymptotically stable.

tab2
Table 2: Allowable upper bound of for different 𝑢 (Example 4.2).

5. Conclusions

In this paper, a new delay-dependent asymptotic stability criterion for neural networks with time-varying delay has been proposed. A new class of Lyapunov functional has been introduced to derive some less conservative delay-dependent stability criteria by using the free-weighting matrices method and the technique of dealing with some integral terms. Finally, numerical examples have been given to illustrate the effectiveness of the proposed method.

Acknowledgments

The authors would like to thank the editors and the reviewers for their valuable suggestions and comments which have led to a much improved paper. This work was supported by Sichuan Provincial Youth Science and Technology Fund (Grant no. 2012JQ0010), Program for New Century Excellent Talents in University (Grant no. NCET-11-1062), National Science Fund for Distinguished Young Scholars of China (Grant no. 51125019), PetroChina Innovation Foundation (Grant no. 2011D-5006-0201), and research on the model and method of parameter identification in reservoir simulation under Grant PLN1121.

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