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Journal of Control Science and Engineering
Volume 2012 (2012), Article ID 517157, 12 pages
http://dx.doi.org/10.1155/2012/517157
Research Article

State Feedback Stabilization for Neutral-Type Neural Networks with Time-Varying Discrete and Unbounded Distributed Delays

School of Mathematical Science, Heilongjiang University, Harbin 150080, China

Received 19 December 2011; Revised 27 March 2012; Accepted 21 April 2012

Academic Editor: Onur Toker

Copyright © 2012 Yantao Wang 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

The problem of stabilization for a class of neutral-type neural networks with discrete and unbounded distributed delays is investigated. By introducing an appropriate Lyapunov-Krasovskii functional and using Jensen inequality technique to deal with its derivative, delay-range-dependent and rate-dependent stabilization criteria are presented in the form of LMIs with nonlinear constraints. In order to solve the nonlinear problem, a cone complementarity linearization (CCL) algorithm is offered. In addition, several numerical examples are provided to illustrate the applicability of the proposed approach.

1. Introduction

In 1943, McCulloch and Pitts [1] proposed the concept of artificial neurons and proved that a single neuron can perform logic functions. This is usually viewed as the beginning of era of artificial neural networks, which is called neural networks for short. Recently, neural networks have received considerable attention due to their wide applications in solving some optimization problems, pattern recognition, image processing, and signal processing. Since time delay is frequently a source of instability and/or oscillation of many practical systems, several approaches have been proposed for analysis and synthesis of delayed neural networks (see [28] and the references therein).

A neutral-type time-delay system contains delays in the state and its derivatives. Such system can be found in population ecology, lossless transmission lines, heat exchangers, and so forth. Because of its wider applications, the stability for the class of neutral-type systems has received considerable attention in the last several decades (see, e.g., [913]). Corresponding to the class of neutral-type time-delay systems, we will also get a class of neutral-type neural networks. Stability criteria for neutral-type neural networks have been proposed in [3, 1421] by constructing appropriate Lyapunov-Krasovskii functional and applying the linear matrix inequality (LMI) approach. Recently, the stability conditions for the neutral-type neural networks with discrete and unbounded distributed delays were provided in [1721]. Specifically, Rakkiyappan and Balasubramaniam [21] studied the global asymptotic stability for a class of neutral-type neural networks with two unbounded distributed delays by the so-called LMI technique. Lu [22] designed a state feedback controller stabilizing the neutral-type neural networks with discrete and bounded distributed delays. However, there are no results about the stabilization for neutral-type neural networks with unbounded distributed delay(s), this motivates our research.

This paper deals with the stabilization problem of a class of neutral-type neural networks with time-varying discrete and unbounded distributed delays. New sufficient conditions for the existence of the state feedback controller are proposed by introducing a Lyapunov-Krasovskii functional and using Jensen inequality technique to deal with its derivative. Since these sufficient conditions are presented in the form of LMIs with nonlinear constraints, the so-called CCL technique is employed to deal with the nonlinear constraints. This allows us to obtain a state feedback gain by LMI Control Toolbox of MATLAB. Several numerical examples are given to illustrate the effectiveness of the proposed approach.

2. Problem Formulation

Consider the class of neutral-type neural networks with time-varying discrete and unbounded distributed delays described by the integro differential equation: ̇𝑥𝑖(𝑡)=𝑎𝑖𝑥𝑖(𝑡)+𝑛𝑗=1𝑏𝑖𝑗𝑓𝑗𝑥𝑗+(𝑡)𝑛𝑗=1𝑐𝑖𝑗𝑓𝑗𝑥𝑗+(𝑡𝜏(𝑡))𝑛𝑗=1𝑒𝑖𝑗𝑡𝑧𝑗(𝑡𝑠)𝑓𝑗𝑥𝑗(𝑠)d𝑠+𝑛𝑗=1𝑑𝑖𝑗̇𝑥𝑗+(𝑡(𝑡))𝑛𝑗=1𝑓𝑖𝑗𝑡𝑧𝑗(𝑡𝑠)̇𝑥𝑗(𝑠)d𝑠+𝑖+𝐼𝑖𝑥(𝑡),(1)𝑖(𝜃)=𝜙𝑖(],𝜃),𝜃(,0(2) where 𝑖=1,2,,𝑛,𝑥𝑖(𝑡) is the state variable of the 𝑖th neuron, 𝑎𝑖>0 is a constant, 𝑏𝑖𝑗, 𝑐𝑖𝑗, 𝑑𝑖𝑗, 𝑒𝑖𝑗, and 𝑓𝑖𝑗 are connection weight coefficients of the neurons, 𝑓𝑗 is the neuron activation function, 𝑖 is the external bias vector element, 𝐼𝑖(𝑡) is the external control input vector, 𝜙𝑖(𝜃) denotes the initial condition, 𝜏(𝑡) and (𝑡) are time-varying delays satisfying 0𝜏1𝜏(𝑡)𝜏2,̇𝜏(𝑡)𝜏3,01(𝑡)2,̇(𝑡)3,(3) and 𝑧𝑗(𝑡) is a real value nonnegative continuous function on [0,+) and satisfies 0𝑧𝑗(𝑠)d𝑠=1.(4) When 𝐼𝑖(𝑡)0 for all 𝑖, the global asymptotic stability of the system (1) has been considered in [21]. Here, the control input item 𝐼𝑖(𝑡) in (1) is borrowed from [22].

Throughout this paper, we assume that the neuron activation functions 𝑓𝑗 (𝑗=1,2,,𝑛) satisfy the following condition: ||𝑓𝑗𝜁1𝑓𝑗𝜁2||𝐿𝑗||𝜁1𝜁2||𝜁1,𝜁2,𝜁1𝜁2,(5) where 𝐿𝑗 (𝑗=1,2,,𝑛) are known positive constants.

Let 𝑥=(𝑥1,𝑥2,,𝑥𝑛) be an equilibrium point of the system (1). Set 𝑦𝑖=𝑥𝑖𝑥𝑖,   𝑖=1,,𝑛. Then the system (1) can be written as ̇𝑦(𝑡)=𝐴𝑦(𝑡)+𝐵𝑔(𝑦(𝑡))+𝐶𝑔(𝑦(𝑡𝜏(𝑡)))+𝐷̇𝑦(𝑡(𝑡))+𝐸𝑡𝑍(𝑡𝑠)𝑔(𝑦(𝑠))d𝑠+𝐹𝑡],𝑍(𝑡𝑠)̇𝑦(𝑠)d𝑠+𝐼(𝑡),𝑦(𝜃)=𝜙(𝜃),𝜃(,0(6) where 𝑦(𝑡)=[𝑦1(𝑡)𝑦2(𝑡)𝑦𝑛(𝑡)]𝑇, 𝐴=diag(𝑎1,𝑎2,,𝑎𝑛), 𝐵=[𝑏𝑖𝑗], 𝐶=[𝑐𝑖𝑗], 𝐷=[𝑑𝑖𝑗], 𝐸=[𝑒𝑖𝑗], 𝐹=[𝑓𝑖𝑗], 𝑍(𝑡)=diag(𝑧1(𝑡),𝑧2(𝑡),,𝑧𝑛(𝑡)), 𝐼(𝑡)=[𝐼1(𝑡)𝐼𝑛(𝑡)]𝑇, 𝑔(𝑦(𝑡))=[𝑔1(𝑦1(𝑡))g2(𝑦2(𝑡))𝑔𝑛(𝑦𝑛(𝑡))]𝑇 with 𝑔𝑗(𝑦𝑗(𝑡))=𝑓𝑗(𝑦𝑗(𝑡)+𝑥𝑗)𝑓𝑗(𝑥𝑗), and 𝜙(𝜃)=[𝜙1(𝜃)𝑥1𝜙𝑛(𝜃)𝑥𝑛]. From (5), we have 𝑔2𝑗(𝜁)𝐿2𝑗𝜁2, 𝑗=1,2,,𝑛, for all 𝜁.

The main purpose of this paper is to develop a delay-range-dependent and rate-dependent condition for the existence of a state feedback controller 𝐼(𝑡)=𝐾𝑦(𝑡),(7) which stabilizes the neutral-type neural network (6) with time-varying discrete and unbounded distributed delays.

A delay-range-dependent and rate-dependent criterion stabilizing (6) is obtained based on a Lyapunov-Krasovskii functional approach accompanied with a CCL technique, a delay-range-dependent and rate-dependent criterion stabilizing (6) is obtained. Set 𝑦𝑡(𝜃)=𝑦(𝑡+𝜃), 𝜃(,0]. Assume that there exists a scalar 𝑉>0 such that 0𝑦𝑡(𝑠)2d𝑠𝑉 and 0̇𝑦𝑡(𝑠)2d𝑠𝑉. This guarantees that the so-called Lyapunov-Krasovskii functional theory can be applied to the considered stabilization problem.

3. Main Results

In this section, we will investigate a delay-range-dependent and rate-dependent stabilization criterion for the neutral-type neural network (6). This requires the following several lemmas.

Lemma 1 (Jensen inequality [23]). Given a real symmetry positive-definite matrix 𝑃𝑛×𝑛, a pair of scalars 𝑎 and 𝑏 satisfying 𝑏𝑎0. If a vector-valued function 𝑥[𝑎,𝑏]𝑛 is derivable on [𝑎,𝑏], then (𝑏𝑎)𝑏𝑎̇𝑥𝑇[](𝑤)𝑃̇𝑥(𝑤)d𝑤𝑥(𝑏)𝑥(𝑎)𝑇𝑃[].𝑥(𝑏)𝑥(𝑎)(8)

Lemma 2 (Schur complementary lemma [24]). For a given matrix 𝑆=𝑆11𝑆12𝑆𝑇12𝑆22 with 𝑆11=𝑆𝑇11 and 𝑆22=𝑆𝑇22, the following conditions are equivalent.(i)𝑆<0. (ii)𝑆22<0 and 𝑆11𝑆12𝑆122𝑆𝑇12<0.(iii)𝑆11<0 and 𝑆22𝑆𝑇12𝑆111𝑆12<0.

Lemma 3 (Cauchy-Schwarz inequality [25]). If the functions 𝑓 and 𝑔 are integral on [𝑎,𝑏], then 𝑏𝑎𝑓(𝑠)𝑔(𝑠)d𝑠2𝑏𝑎𝑓2(𝑠)d𝑠𝑏𝑎𝑔2.(𝑠)d𝑠(9)

Based on the above three lemmas, the following delay-range-dependent and rate-dependent stabilization criterion for the neutral-type neural network (6) can be investigated by constructing an appropriate Lyapunov-Krasovskii functional and applying the CCL technique.

Theorem 4. For given scalars 1, 2, 3, 𝜏1, 𝜏2, and 𝜏3 such that 2>1>0 and 𝜏2>𝜏1>0, the system (6) subject to (3) is asymptotically stabilizable via the controller (7) if there exist real symmetry positive-definite matrices 𝑃,𝑃,𝑇𝑖,𝑇𝑖,𝑄𝑖𝑄(𝑖=1,,6),4,𝑄5,𝑄6, 𝑅𝑗𝑅(𝑗=1,,5),4,𝑅5, 𝑀=diag(𝑚1,,𝑚𝑛), 𝑁=diag(𝑛1,,𝑛𝑛), 𝑁=diag(̃𝑛1,,̃𝑛𝑛), a matrix 𝐾,  and positive scalars 𝜀1 and 𝜀2,  such that Π𝐴𝑇𝐴𝑇𝐴𝑇𝐴𝑇𝐴𝑇𝐴𝑇𝐴𝑇100000𝐴0𝑇20000𝐴00𝑇3000𝐴000𝑇400𝐴0000𝑇50𝐴00000𝑇6𝑇<0,(10)1𝑃𝑄𝑃4𝑇0,2𝑃𝑃𝜏112𝑅4𝑇0,3𝑃𝑃𝜏21𝑅5𝑇0,(11)4𝑃𝑃112𝑄5𝑇0,5𝑃𝑃21𝑄6𝑇0,6𝑃𝑁𝑃𝑃0,(12)𝑃=𝐼,𝑇𝑖𝑇𝑖𝑅=𝐼(𝑖=1,,6),4𝑅4=𝐼,𝑅5𝑅5𝑄=𝐼,4𝑄4=𝐼,𝑄5𝑄5𝑄=𝐼,6𝑄6=𝐼,𝑁𝑁=𝐼,(13)

where Π=𝑒𝑇13𝑖=1𝑅𝑖+3𝑖=1𝑄𝑖+𝜀1𝐿𝑇𝐿𝑒1𝑒𝑇21𝜏3𝑅3𝜀2𝐿𝑇𝐿𝑒2+𝐴𝑇𝑒1+𝑒𝑇1𝐴13𝑒𝑇5𝑄3𝑒5112𝑒5𝑒6𝑇𝑄6𝑒5𝑒6𝜏112𝑒2𝑒4𝑇𝑅5𝑒2𝑒4+𝑒𝑇9𝑀𝜀1𝐼𝑒9𝑒𝑇11𝑀𝑒11𝑒𝑇12𝑁𝑒12𝜀2𝑒𝑇10𝑒1013𝑒𝑇8𝑄4𝑒8𝜏112𝑒2𝑒4𝑇𝑅4𝑒2𝑒4𝜏112𝑒2𝑒3𝑇𝑅4𝑒2𝑒3𝑒𝑇3𝑅1𝑒3𝑒𝑇4𝑅2𝑒4𝜏112𝑒2𝑒3𝑇𝑅5𝑒2𝑒3𝜏11𝑒1𝑒3𝑇𝑅5𝑒1𝑒3𝑒𝑇6𝑄1𝑒6𝑒𝑇7𝑄2𝑒7112𝑒5𝑒6𝑇𝑄5𝑒5𝑒6112𝑒5𝑒7𝑇𝑄6𝑒5𝑒7112𝑒5𝑒7𝑇𝑄5𝑒5𝑒711𝑒1𝑒6𝑇𝑄6𝑒1𝑒6,,𝐿𝐴=𝑃𝐴+𝐾000000𝑃𝐷𝑃𝐵𝑃𝐶𝑃𝐸𝑃𝐹𝐿=diag1,,𝐿𝑛,𝑒𝑖=0𝑛×(𝑖1)𝐼𝑛0𝑛×(12𝑖)(𝑖=1,,12),12=21,𝜏12=𝜏2𝜏1.(14)

Furthermore, when the LMIs (10)–(12) with the constraint (13) are feasible, a desired state feedback gain is given by 𝐾=𝑃1𝐾.

Proof. Choose a Lyapunov-Krasovskii functional as follows: 𝑉𝑦𝑡=6𝑖=1𝑉𝑖𝑦𝑡,(15) with 𝑉1𝑦𝑡=𝑦𝑇𝑉(𝑡)𝑃𝑦(𝑡),(16)2𝑦𝑡=𝑡𝑡(𝑡)𝑦𝑇(𝑠)𝑄3𝑦(𝑠)+̇𝑦𝑇(𝑠)𝑄4+̇𝑦(𝑠)d𝑠2𝑖=1𝑡𝑡𝑖𝑦𝑇(𝑠)𝑄𝑖+𝑦(𝑠)d𝑠𝑡𝑡𝜏(𝑡)𝑦𝑇(𝑠)𝑅3𝑦(𝑠)d𝑠+2𝑖=1𝑡𝑡𝜏𝑖𝑦𝑇(𝑠)𝑅𝑖𝑉𝑦(𝑠)d𝑠,3𝑦𝑡=𝜏1𝜏2𝑡𝑡+𝜃̇𝑦𝑇(𝑠)𝑅4+̇𝑦(𝑠)d𝑠d𝜃0𝜏2𝑡𝑡+𝜃̇𝑦𝑇(𝑠)𝑅5𝑉̇𝑦(𝑠)d𝑠d𝜃,4𝑦𝑡=12𝑡𝑡+𝜃̇𝑦𝑇(𝑠)𝑄5+̇𝑦(𝑠)d𝑠d𝜃02𝑡𝑡+𝜃̇𝑦𝑇(𝑠)𝑄6𝑉̇𝑦(𝑠)d𝑠d𝜃,5𝑦𝑡=𝑛𝑗=1𝑚𝑗0𝑧𝑗(𝛿)𝑡𝑡𝛿𝑔2𝑗𝑦𝑗𝑉(𝜉)d𝜉d𝛿,6𝑦𝑡=𝑛𝑗=1𝑛𝑗0𝑧𝑗(𝛿)𝑡𝑡𝛿̇𝑦2𝑗(𝜉)d𝜉d𝛿.(17)
Set 𝑦𝑡2𝑤=𝑦(𝑡)2+0𝑦𝑡(𝜃)2𝑑𝜃+0̇𝑦𝑡(𝜃)2d𝜃. By some derivation, we have 𝜆min(𝑃)𝑦(𝑡)2𝑦𝑉𝑡𝑎1+𝑎2+𝑎3𝑦𝑡2𝑤,(18) where 𝑎1=𝜆max(𝑃)+𝜆max𝑄1+𝜆max𝑄2+𝜆max𝑄3+𝜆max𝑄4,𝑎2=𝜆max𝑅1+𝜆max𝑅2+𝜆max𝑅3,𝑎3=𝜏12𝜆max𝑅4+𝜏2𝜆max𝑅5+12𝜆max𝑄5+2𝜆max𝑄6+𝜆max𝑀𝐿𝑇𝐿+𝜆max(𝑁),(19) which implies that the functional 𝑉(𝑦𝑡) is a legitimate due to [26]. Next we deal with the derivatives of 𝑉𝑖(𝑦𝑡)(𝑖=1,,6).
Set 𝐴1=[],𝑦𝐴+𝐾000000𝐷𝐵𝐶𝐸𝐹𝜉(𝑡)=𝑇(𝑡),𝑦𝑇(𝑡𝜏(𝑡)),𝑦𝑇𝑡𝜏1,𝑦𝑇𝑡𝜏2,𝑦𝑇𝑦(𝑡(𝑡)),𝑇𝑡1,𝑦𝑇𝑡2,̇𝑦𝑇𝑔(𝑡(𝑡)),𝑇(𝑦(𝑡)),𝑔𝑇(𝑦(𝑡𝜏(𝑡))),𝑡𝑍(𝑡𝑠)𝑔(𝑦(𝑠))d𝑠𝑇,𝑡𝑍(𝑡𝑠)̇𝑦(𝑠)d𝑠𝑇𝑇.(20)
When the controller (7) is applied to the system (6), the resultant closed-loop system is obtained as follows: 𝐴̇𝑦(𝑡)=1𝜉(𝑡).(21) Due to Lemma 1, (3), the derivatives of 𝑉𝑖(𝑦𝑡) (𝑖=1,,4) along the trajectory of (21) are ̇𝑉1𝑦𝑡=2𝑦𝑇(𝑡)𝑃̇𝑦(𝑡)=2𝑦𝑇(𝐴𝑡)𝑃1𝜉(𝑡)=𝜉𝑇(𝑒𝑡)𝑇1𝑃𝐴1+𝐴𝑇1𝑃𝑒1̇𝑉𝜉(𝑡),(22)2𝑦𝑡=𝑦𝑇(𝑡)3𝑖=1𝑄𝑖+3𝑖=1𝑅𝑖̇𝑦𝑦(𝑡)1(𝑡)𝑇(𝑡(𝑡))𝑄3̇𝑦(𝑡(𝑡))1(𝑡)̇𝑦𝑇(𝑡(𝑡))𝑄4̇𝑦(𝑡(𝑡))𝑦𝑇𝑡1𝑄1𝑦𝑡1𝑦𝑇𝑡2𝑄2𝑦𝑡2(1̇𝜏(𝑡))𝑦𝑇(𝑡𝜏(𝑡))𝑅3𝑦(𝑡𝜏(𝑡))𝑦𝑇𝑡𝜏1𝑅1𝑦𝑡𝜏1𝑦𝑇𝑡𝜏2𝑅2𝑦𝑡𝜏2+̇𝑦𝑇(𝑡)𝑄4̇𝑦(𝑡)𝜉𝑇(𝑡)𝑒𝑇13𝑖=1𝑄𝑖+3𝑖=1𝑅𝑖𝑒1𝜉(𝑡)+𝜉𝑇𝐴(𝑡)𝑇1𝑄4𝐴1𝜉(𝑡)𝜉𝑇(𝑡)𝑒𝑇7𝑄2𝑒7𝜉(𝑡)13𝜉𝑇(𝑡)𝑒𝑇8𝑄4𝑒8𝜉(𝑡)𝜉𝑇(𝑡)𝑒𝑇6𝑄1𝑒6𝜉(𝑡)13𝜉𝑇(𝑡)𝑒𝑇5𝑄3𝑒5𝜉(𝑡)1𝜏3𝜉𝑇(𝑡)𝑒𝑇2𝑅3𝑒2𝜉(𝑡)𝜉𝑇(𝑡)𝑒𝑇3𝑅1𝑒3𝜉(𝑡)𝜉𝑇(𝑡)𝑒𝑇4𝑅2𝑒4̇𝑉𝜉(𝑡),(23)3𝑦𝑡=̇𝑦𝑇𝜏(𝑡)12𝑅4+𝜏2𝑅5̇𝑦(𝑡)𝑡𝜏1𝑡𝜏2̇𝑦𝑇(𝑠)𝑅4̇𝑦(𝑠)d𝑠𝑡𝑡𝜏2̇𝑦𝑇(𝑠)𝑅5̇𝑦(𝑠)d𝑠=̇𝑦𝑇𝜏(𝑡)12𝑅4+𝜏2𝑅5̇𝑦(𝑡)𝑡𝜏(𝑡)𝑡𝜏2̇𝑦𝑇(𝑠)𝑅4̇𝑦(𝑠)d𝑠𝑡𝜏1𝑡𝜏(𝑡)̇𝑦𝑇(𝑠)𝑅4̇𝑦(𝑠)d𝑠𝑡𝜏(𝑡)𝑡𝜏2̇𝑦𝑇(𝑠)𝑅5̇𝑦(𝑠)d𝑠𝑡𝜏1𝑡𝜏(𝑡)̇𝑦𝑇(𝑠)𝑅5̇𝑦(𝑠)d𝑠𝑡𝑡𝜏1̇𝑦𝑇(𝑠)𝑅5̇𝑦(𝑠)d𝑠𝜉𝑇𝐴(𝑡)𝑇1𝜏12𝑅4+𝜏2𝑅5𝐴1𝜉(𝑡)𝜏112𝜉𝑇𝑒(𝑡)2𝑒4𝑇𝑅4𝑒2𝑒4𝜉(𝑡)𝜏112𝜉𝑇𝑒(𝑡)2𝑒3𝑇𝑅4𝑒2𝑒3𝜉(𝑡)𝜏112𝜉𝑇𝑒(𝑡)2𝑒4𝑇𝑅5𝑒2𝑒4𝜉(𝑡)𝜏112𝜉𝑇(𝑒𝑡)2𝑒3𝑇𝑅5𝑒2𝑒3𝜉(𝑡)𝜏11𝜉𝑇𝑒(𝑡)1𝑒3𝑇𝑅5𝑒1𝑒3̇𝑉𝜉(𝑡),(24)4𝑦𝑡=̇𝑦𝑇(𝑡)12𝑄5+2𝑄6̇𝑦(𝑡)𝑡1𝑡2̇𝑦𝑇(𝑠)𝑄5̇𝑦(𝑠)d𝑠𝑡𝑡2̇𝑦𝑇(𝑠)𝑄6̇𝑦(𝑠)d𝑠=̇𝑦𝑇(𝑡)12𝑄5+2𝑄6̇𝑦(𝑡)𝑡(𝑡)𝑡2̇𝑦𝑇(𝑠)𝑄5̇𝑦(𝑠)d𝑠𝑡1𝑡(𝑡)̇𝑦𝑇(𝑠)𝑄5̇𝑦(𝑠)d𝑠𝑡(𝑡)𝑡2̇𝑦𝑇(𝑠)𝑄6̇𝑦(𝑠)d𝑠𝑡1𝑡(𝑡)̇𝑦𝑇(𝑠)𝑄6̇𝑦(𝑠)d𝑠𝑡𝑡1̇𝑦𝑇(𝑠)𝑄6̇𝑦(𝑠)d𝑠𝜉𝑇𝐴(𝑡)𝑇112𝑄5+2𝑄6𝐴1𝜉(𝑡)112𝜉𝑇𝑒(𝑡)5𝑒7𝑇𝑄5𝑒5𝑒7𝜉(𝑡)112𝜉𝑇𝑒(𝑡)5𝑒6𝑇𝑄5𝑒5𝑒6𝜉(𝑡)112𝜉𝑇𝑒(𝑡)5𝑒7𝑇𝑄6𝑒5𝑒7𝜉(𝑡)112𝜉𝑇𝑒(𝑡)5𝑒6𝑇𝑄6𝑒5𝑒6𝜉(𝑡)11𝜉𝑇𝑒(𝑡)1𝑒6𝑇𝑄6𝑒1𝑒6𝜉(𝑡).(25)
By Lemma 3 and (4), we can obtain the derivatives of 𝑉5(𝑦𝑡) and 𝑉6(𝑦𝑡) as follows: ̇𝑉5𝑦𝑡=𝑛𝑗=1𝑚𝑗0𝑧𝑗(𝛿)𝑔2𝑗𝑦𝑗(𝑡)d𝛿𝑛𝑗=1𝑚𝑗0𝑧𝑗(𝛿)𝑔2𝑗𝑦𝑗=(𝑡𝛿)d𝛿𝑛𝑗=1𝑚𝑗𝑔2𝑗𝑦𝑗(𝑡)0𝑧𝑗(𝛿)d𝛿𝑛𝑗=1𝑚𝑗0𝑧𝑗×(𝛿)d𝛿0𝑧𝑗(𝛿)𝑔2𝑗𝑦𝑗(𝑡𝛿)d𝛿𝑔𝑇(𝑦(𝑡))𝑀𝑔(𝑦(𝑡))𝑛𝑗=1𝑚𝑗0𝑧𝑗(𝛿)𝑔𝑗𝑦𝑗(𝑡𝛿)d𝛿2=𝑔𝑇(𝑦(𝑡))𝑀𝑔(𝑦(𝑡))0𝑍(𝑡𝑠)𝑔(𝑦(𝑠))d𝑠𝑇×𝑀0𝑍(𝑡𝑠)𝑔(𝑦(𝑠))d𝑠=𝜉𝑇(𝑡)𝑒𝑇9𝑀𝑒9𝜉(𝑡)𝜉𝑇(𝑡)𝑒𝑇11𝑀𝑒11̇𝑉𝜉(𝑡),(26)6𝑦𝑡̇𝑦𝑇(𝑡)𝑁̇𝑦(𝑡)𝑛𝑗=1𝑛𝑗0𝑧𝑗(𝛿)̇𝑦𝑗(𝑡𝛿)d𝛿2=𝜉𝑇𝐴(𝑡)𝑇1𝑁𝐴1𝜉(𝑡)𝜉𝑇(𝑡)𝑒𝑇12𝑁𝑒12𝜉(𝑡).(27) For the functions 𝑔𝑗, 𝑗=1,2,,𝑛, using 𝑔2𝑗(𝜁𝑗)𝐿2𝑗𝜁2𝑗, we get that 0𝜀1𝜉𝑇(𝑡)𝑒𝑇9𝑒9𝜉(𝑡)+𝜀1𝜉𝑇(𝑡)𝑒𝑇1𝐿𝑇𝐿𝑒1𝜉(𝑡),0𝜀2𝜉𝑇(𝑡)𝑒𝑇10𝑒10𝜉(𝑡)+𝜀2𝜉𝑇(𝑡)𝑒𝑇2𝐿𝑇𝐿𝑒2𝜉(𝑡),(28) where 𝐿=diag(𝐿1,,𝐿𝑛). Noting that 𝐾=𝑃1𝐾, we have 𝐴𝐴=𝑃1. The combination of (22)–(28) gives ̇𝑉𝑦𝑡=6𝑖=1̇𝑉𝑖𝑦𝑡𝜉𝑇(𝑡)Π𝜉(𝑡),(29) where 𝐴Π=Π+𝑇1𝑄4+𝜏12𝑅4+𝜏2𝑅5+12𝑄5+2𝑄6𝐴+𝑁1.(30)
By Lemma 2 and (10)–(13), we can get that Π𝐴𝑇1𝑃𝐴𝑇1𝑃𝐴𝑇1𝑃𝐴𝑇1𝑃𝐴𝑇1𝑃𝐴𝑇1𝑃𝑃𝐴1Π22𝑃𝐴0000010Π33𝑃𝐴0000100Π44𝑃𝐴0001000Π55𝑃𝐴0010000Π660𝑃𝐴100000Π77<0,(31) where Π22=𝑃𝑄41𝑃,Π33=𝜏112𝑃𝑅41Π𝑃,44=𝜏21𝑃𝑅51𝑃,Π55=112𝑃𝑄51Π𝑃,66=21𝑃𝑄61𝑃,Π77=𝑃𝑁1𝑃.(32)
Thus it is easy to see that ̇𝑉(𝑦𝑡)<0, which, together with (7) and [26, Theorem  3.1.6], implies that the system (6) is asymptotically stabilizable. The proof is completed.

Remark 5. A delay-range-dependent and rate-dependent condition under which the system (6) is asymptotically stabilizable is investigated in Theorem 4. In [7, 8], the stability of a class of the neural networks with unbounded distributed delays has been studied; however, the neutral term was not considered in their models. In [18], only a unbounded distributed delay is taken into account. But in this paper the neutral-type neural networks with two unbounded distributed delays are studied. Therefore, Theorem 4 in this paper can be viewed as an extension of the corresponding results in [7, 8, 18].

Remark 6. In order to apply [26, Theorem 3.1.6] to conclude that the system (6) is asymptotically stabilizable, we denote a new norm 2𝑤 on a space of functions.
Noting that the stabilization criterion proposed in Theorem 4 includes the nonlinear constraint (13), we cannot obtain the desired gain 𝐾 by using the LMI Controller Toolbox of MATLAB. In order to deal with the nonlinear constraint (13), we offer the following CCL algorithm. The algorithm is available to determine the maximum value of 2 for given 1, 3, 𝜏1, 𝜏2, and 𝜏3 and the corresponding gain 𝐾.

Algorithm 7 (CCL algorithm). Step 1. Choose a sufficiently small initial 2>1, such that there exists a feasible solution to (10)–(12) and 𝐼𝑃𝐼𝑁𝑅𝑃𝐼0,𝑁𝐼0,4𝐼𝐼𝑅4𝑅0,5𝐼𝐼𝑅5𝑇0,𝑖𝐼𝐼𝑇𝑖𝑄0(𝑖=1,,6),𝑗𝐼𝐼𝑄𝑗0(𝑗=4,5,6).(33) Set max=2.

Step 2. Find a feasible set of 𝑃0, 𝑃0, 𝑁0, 𝑁0, 𝑇𝑖0, 𝑇𝑖0(𝑖=1,,6), 𝑅𝑙0, 𝑅𝑙0(𝑙=4,5), 𝑄𝑗0, and 𝑄𝑗0(𝑗=4,5,6) satisfying (10)–(12), (33). Set 𝑘=0.

Step 3. Solve the following LMI problem for the variables 𝐾, 𝑃, 𝑃, 𝑁, 𝑁, 𝑇𝑖, 𝑇𝑖(𝑖=1,,6), 𝑅𝑙, 𝑅𝑙(𝑙=4,5), 𝑄𝑗, and 𝑄𝑗(𝑗=4,5,6): minsubjectto(8)(10),(29)trΘ𝑘,(34) where Θ𝑘=𝑃𝑘𝑃𝑃+𝑃𝑘+𝑁𝑘𝑁𝑁+𝑁𝑘+6𝑖=1𝑇𝑖𝑇𝑖𝑘+𝑇𝑖𝑘𝑇𝑖+5𝑙=4𝑅𝑙𝑅𝑙𝑘+𝑅𝑙𝑘𝑅𝑙+6𝑗=4𝑄𝑗𝑄𝑗𝑘+𝑄𝑗𝑘𝑄𝑗.(35)

Set 𝑃𝑘+1=𝑃, 𝑃𝑘+1=𝑃, 𝑁𝑘+1=𝑁, 𝑁𝑘+1=𝑁, 𝑇𝑖𝑘+1=𝑇𝑖, 𝑇𝑖𝑘+1=𝑇𝑖(𝑖=1,,6), 𝑅𝑙𝑘+1=𝑅𝑙, 𝑅𝑙𝑘+1=𝑅𝑙(𝑙=4,5), 𝑄𝑗𝑘+1=𝑄𝑗 and 𝑄𝑗𝑘+1=𝑄𝑗(𝑗=4,5,6).

Step 4. If the LMI. Π𝐴Π=𝑇1𝑄4𝐴𝑇1𝑅4𝐴𝑇1𝑅5𝐴𝑇1𝑄5𝐴𝑇1𝑄6𝐴𝑇1𝑁𝑄4𝐴1𝑄4𝑅000004𝐴10𝜏112𝑅4𝑅00005𝐴100𝜏21𝑅5𝑄0005𝐴1000112𝑄5𝑄006𝐴1000021𝑄60𝑁𝐴100000𝑁<0(36) is feasible for the variables 𝑁, 𝑇𝑖(𝑖=1,,6), 𝑅𝑙(𝑙=4,5), and 𝑄𝑗(𝑗=4,5,6) and the matrices 𝐾 and 𝑃 obtained in Step 3, then set 𝐾=𝑃1𝐾 and stop. If the LMI (36) is infeasible within a specified number of iterations, then stop; otherwise, set 𝑘=𝑘+1 and go to Step 3.

Remark 8. The idea of the above algorithm is taken from [27]. Compared with the CCL algorithms proposed in [28, 29], the merit of the above algorithm is that more freedoms are provided to test the iteration stop conditions (see [27, Remark  4] for details).

Next, we consider the following neutral-type neural network (37), which is a special case of (6). A stability criterion for the neural network (37) has been presented in [21]. The following corollary can be immediately obtained from Theorem 4, which gives a stabilization criterion for the neutral-type neural network (37): ̇𝑦(𝑡)=𝐴𝑦(𝑡)+𝐵𝑔(𝑦(𝑡))+𝐶𝑔(𝑦(𝑡(𝑡)))+𝐷̇𝑦(𝑡(𝑡))+𝐸𝑡𝑍(𝑡𝑠)𝑔(𝑦(𝑠))d𝑠+𝐹𝑡𝑍(𝑡𝑠)̇𝑦(𝑠)d𝑠+𝐼(𝑡),(37) where 0(𝑡)2,̇(𝑡)3.(38)

Corollary 9. For given scalars 2 and 3 such that 2>0, the neutral-type neural network (37) subject to (38) is asymptotically stabilizable via the controller (7) if there exist real symmetry positive-definite matrices 𝑃,𝑃,𝑄𝑖𝑄(𝑖=1,,4),2,𝑄4,𝑇𝑗,𝑇𝑗(𝑗=1,2,3),𝑀=diag(𝑚1,,𝑚𝑛),𝑁=diag(𝑛1,,𝑛𝑛), and 𝑁=diag(̃𝑛1,,̃𝑛𝑛), a matrix 𝐾 and positive scalars 𝜀1 and 𝜀2, such that Σ=Σ𝒜𝑇𝒜𝑇𝒜𝑇𝒜𝑇100𝒜0𝑇20𝒜00𝑇3𝑇<0,1𝑃𝑄𝑃2𝑇0,2𝑃𝑃21𝑄4𝑇0,3𝑃𝑁𝑃𝑃0,(39)𝑃=𝐼,𝑇𝑗𝑇𝑗𝑄=𝐼(𝑗=1,2,3),2𝑄2=𝐼,𝑄4𝑄4=𝐼,𝑁𝑁=𝐼,(40) where Σ=̂𝑒𝑇1𝑄1+𝑄3+𝜀1𝐿𝑇𝐿̂𝑒1̂𝑒213𝑄1𝜀2𝐿𝑇𝐿̂𝑒2+𝒜𝑇̂𝑒1+̂𝑒𝑇1𝒜13̂𝑒𝑇4𝑄2̂𝑒4̂𝑒𝑇3𝑄3̂𝑒3+̂𝑒𝑇5𝑀𝜀1𝐼̂𝑒5̂𝑒𝑇7𝑀̂𝑒721̂𝑒2̂𝑒3𝑇𝑄4̂𝑒2̂𝑒321̂𝑒1̂𝑒2𝑇𝑄4̂𝑒1̂𝑒2̂𝑒𝑇8𝑁̂𝑒8𝜀2̂𝑒𝑇6̂𝑒6,,𝒜=𝑃𝐴+𝐾00𝑃𝐷𝑃𝐵𝑃𝐶𝑃𝐸𝑃𝐹̂𝑒𝑖=0𝑛×(𝑖1)𝐼𝑛0𝑛×(8𝑖),𝑖=1,,8.(41)
Furthermore, when the LMIs (39) with the constraint (40) are feasible, a desired state feedback gain is given by 𝐾=𝑃1𝐾.

Proof. If we use the following ̂𝜉(𝑡) and 𝑉(𝑦𝑡) instead of 𝜉(𝑡) and 𝑉(𝑦𝑡) in the proof of Theorem 4, respectively, then the proof can be easily completed: 𝑉𝑦𝑡=5𝑖=1𝑉𝑖𝑦𝑡,(42) with 𝑉1𝑦𝑡=𝑦𝑇𝑉(𝑡)𝑃𝑦(𝑡),2𝑦𝑡=𝑡𝑡(𝑡)𝑦𝑇(𝑠)𝑄1𝑦(𝑠)+̇𝑦𝑇(𝑠)𝑄2+̇𝑦(𝑠)d𝑠𝑡𝑡2𝑦𝑇(𝑠)𝑄3𝑉𝑦(𝑠)d𝑠,3𝑦𝑡=02𝑡𝑡+𝜃̇𝑦𝑇(𝑠)𝑄4𝑉̇𝑦(𝑠)d𝑠d𝜃,4𝑦𝑡=𝑛𝑗=1𝑚𝑗0𝑍𝑗(𝛿)𝑡𝑡𝛿𝑔2𝑗𝑦𝑗𝑉(𝜉)d𝜉d𝛿,5𝑦𝑡=𝑛𝑗=1𝑛𝑗0𝑍𝑗(𝛿)𝑡𝑡𝛿̇𝑦2𝑗̂𝑦(𝜉)d𝜉d𝛿.𝜉(𝑡)=𝑇(𝑡),𝑦𝑇(𝑡(𝑡)),𝑦𝑇𝑡2,̇𝑦𝑇(𝑡(𝑡)),𝑔𝑇(𝑦(𝑡)),𝑔𝑇(𝑦(𝑡(𝑡))),𝑡𝑍(𝑡𝑠)𝑔(𝑦(𝑠))d𝑠𝑇,𝑡𝑍(𝑡𝑠)̇𝑦(𝑠)d𝑠𝑇𝑇.(43)

The following system model is a practical partial element equivalent circuit (PEEC) that is described in [30, Figure 1] by Bellen et al.: [],̇𝑦(𝑡)=𝐴𝑦(𝑡)+𝐵𝑦(t𝜏)+𝐶̇𝑦(𝑡𝜏)+𝐼(𝑡),(44)𝑦(𝜃)=𝜙(𝜃),𝜃𝜏,0(45) where 𝜏 is a positive scalar representing the system delay.

It should be emphasized that the matrix 𝐴 in (44) is not necessarily diagonal, which is different from the one in (6). However, one can easily find that the proof of Theorem 4 is always available whether the matrix 𝐴 is diagonal or not. For this reason, we can derive the following Corollaries 10 and 12 by a method similar to obtaining Theorem 4.

Corollary 10. For a given scalar 𝜏>0, the neutral-type system (44) is asymptotically stabilizable via the controller (7) if there exist real symmetry positive-definite matrices 𝑃, 𝑃,  𝑄𝑖(𝑖=1,,3), 𝑄2, 𝑄3, 𝑇𝑗, and 𝑇𝑗(𝑗=1,2) and a matrix 𝐾, such that Ξ𝐴Ξ=𝑇𝐴𝑇𝐴𝑇10𝐴0𝑇2𝑇<0,(46)1𝑃𝑄𝑃2𝑇0,2𝑃𝑃𝜏1𝑄3𝑃0,(47)𝑃=𝐼,𝑇𝑗𝑇𝑗𝑄=𝐼(𝑗=1,2),2𝑄2=𝐼,𝑄3𝑄3=𝐼,(48) where Ξ=𝑒𝑇1𝑄1𝑒1𝑒𝑇2𝑄1𝑒2+𝐴𝑇𝑒1+𝑒𝑇1𝐴𝑒𝑇3𝑄2𝑒3𝜏1𝑒1𝑒2𝑇𝑄3𝑒1𝑒2,,𝐴=𝑃𝐴+𝐾𝑃𝐵𝑃𝐶𝑒𝑖=0𝑛×(𝑖1)𝐼𝑛0𝑛×(3𝑖),𝑖=1,2,3.(49)
Furthermore, when the LMIs (46) and (47) with the nonlinear constraint (48) are feasible, a desired state feedback gain is given by 𝐾=𝑃1𝐾.

Remark 11. Similar to Algorithm 7, one can solve the stabilization criteria proposed in Corollaries 9 and 10.

Corollary 12. For a given scalar 𝜏>0, the unforced system of (44) is asymptotically stable if there exist real symmetry positive-definite matrices 𝑃 and 𝑄𝑖(𝑖=1,2,3) such that Θ=𝑒𝑇1𝑄1𝑒1𝑒𝑇2𝑄1𝑒2+𝐴𝑇𝑃𝑒1+𝑒𝑇1𝑃𝐴𝑒𝑇3𝑄2𝑒3𝜏1𝑒1𝑒2𝑇𝑄3𝑒1𝑒2+𝐴𝑇𝑄2+𝜏𝑄3𝐴<0,(50) where ,𝐴=𝐴𝐵𝐶(51) and 𝑒𝑖 is defined as in Corollary 10.

4. Numerical Examples

In this section, we will illustrate our approach by several numerical examples.

Example 13. Consider the neutral-type neural network (6), where ,,,𝐿𝐴=2003,𝐵=0.3400.10.1𝐶=0.10.20.150.18,𝐷=0.1000.1𝐸=0.410.50.690.31,𝐹=0.30.150.50.2𝑗𝜏=1,𝑗=1,2,3=0.5,𝜏2=4.8,𝜏1=4.1,3=0.5,1=4.5,2=7.5,𝑔(𝑦(𝑠))=tanh𝑦(𝑠)2.(52) Solving the LMIs (10)–(12) with nonlinear constraint (13) by Algorithm 7 and the LMI Control Toolbox of MATLAB, we can get a desired state feedback gain stabilizing the system as follows: 𝐾=0.77630.10700.10701.7242.(53)
The state response curves of the closed-loop system are given in Figures 1 and 2, where the initial functions in Figures 1 and 2 are, respectively, chosen as 𝜙(𝜃)=0.120, 𝜃(,0] and 𝜙(𝜃)=e𝜃e2𝜃, 𝜃(,0].

fig1
Figure 1: State responses of the closed-loop system for 𝑧1(𝑠)=𝑧2(𝑠)=1|𝑠1|, 0𝑠2.
fig2
Figure 2: State responses of the closed-loop system for 𝑧1(𝑠)=𝑧2(𝑠)=(2/𝜋)𝑒(1/2)𝑠2, 𝑠(,0].

Example 14. Consider the system (37) with the following parameter matrices: ,,,𝐿𝐴=3.6003.6,𝐵=1.1980.10.11.198𝐶=0.10.160.050.1,𝐷=0.2000.2𝐸=0.40.20.30.2,𝐹=0.30.150.50.2𝑗,]=1,𝑗=1,2,𝜙(𝜃)=0.120𝜃(,0,𝑔(𝑦(𝑠))=tanh𝑦(𝑠)2,||||𝑧(𝑠)=1𝑠1(0𝑠2),2=8.(54) When 3=0.15 by the LMI Control Toolbox of MATLAB, it can be verified that the LMI in [21, Theorem 3.1] is not feasible, so it cannot guarantee that the unforced system is stable. Now we check the stabilization condition proposed in Corollary 9 of this paper and find that it is feasible. So, by Corollary 9 the system is asymptotically stabilizable by state feedback controller, and a desired state feedback gain is given by 𝐾=1.66550.31680.31681.8069,(55) and, further, the state response curves of the closed-loop system are given in Figure 3.
When 3=0, by the LMI Control Toolbox of MATLAB, it can be verified that the LMI in [21, Theorem 3.2] is not feasible, which cannot guarantee the unforced system is stable. However, the stabilization condition proposed in Corollary 9 can be satisfied. Hence, by Corollary 9, the neutral-type neural network is asymptotically stabilizable via state feedback controller, and a desired state feedback controller gain is given by 𝐾=1.68470.31100.31101.8198,(56) and, further, the state response curves of the closed-loop system are given in Figure 4.

fig3
Figure 3: State responses of the closed-loop system when 3=0.15.
fig4
Figure 4: State responses of the closed-loop system when 3=0.

Example 15. Consider the PEEC system (44) with the following parameter matrices: ,1𝐴=100×𝛽12390126,𝐵=100×1030.50.510.51.50𝐶=.72152403241(57)
Applying Corollaries 12 and 10, the maximum delay bounds 𝜏𝑀 for the various 𝛽 are presented in Table 1, which clearly shows the effectiveness of our approach. Through Table 1, we can get that the stability criterion proposed in this paper is less conservative than the ones in [3133].
The following example is provided for applying the approach proposed in this paper to a realistic problem which is motivated by the small PEEC model in [30, Figure 1].

tab1
Table 1: Allowable upper bound of 𝜏𝑀 for different 𝛽.

Example 16. Consider the PEEC system (44) with ,1𝐴=100×712390126,𝐵=100×1030.50.510.51.50𝐶=72152403241,𝜏=1.(58) When 𝐼(𝑡)0, Bellen et al. [30] gave sufficient conditions for the asymptotic stability of the zero solution to (44) by utilizing a suitable reformulation of the system.
By applying Algorithm 7 and the LMI Control Toolbox of MATLAB to solve LMIs (46) and (47) with the nonlinear constraint (48), a required state feedback gain is obtained as follows: 𝐾=66.1662283.98480.7157487.718545.601631.171574.6213146.45001.7714.(59) The simulation results for the PEEC system are shown by Figures 5 and 6 with the initial value 𝜙(𝜃)=[sin(𝑡)sin(2𝑡)sin(3𝑡)]𝑇. Figure 5 represents the state responses of the PEEC system when 𝐼(𝑡)0, and Figure 6 represents the state responses of the resultant closed-loop system. From Figures 5 and 6, we can see that both the open-loop and closed-loop systems are stable, however, the speed of convergence towards the null point in the closed-loop system is faster than one in the open-loop system.

517157.fig.005
Figure 5: State responses of the open-loop system.
517157.fig.006
Figure 6: State responses of the closed-loop system.

5. Conclusions

In this paper, the delay-range-dependent and rate-dependent stabilization criteria for a class of neutral-type neural networks with time-varying discrete and unbounded distributed delays are established. The criteria are derived by constructing an appropriate Lyapunov-Krasovskii functional and using certain matrix technique. A CCL algorithm is developed to obtain the state feedback gain 𝐾. Numerical examples are provided to show that our results are more suitable than some existing ones, which illustrate the merits of the proposed approach.

Acknowledgments

This work is supported by the fund of Heilongjiang Education Committee under Grant no. 12521429 and the fund of Heilongjiang University Innovation Team Support Plan under Grant no. Hdtd2010-03. The authors thank the anonymous referees for their helpful comments and suggestions that improved greatly this paper.

References

  1. W. S. McCulloch and W. Pitts, “A logical calculus of the ideas immanent in nervous activity,” The Bulletin of Mathematical Biophysics, vol. 5, no. 4, pp. 115–133, 1943. View at Publisher · View at Google Scholar · View at Scopus
  2. J. Li, “New delay-dependent robust stability condition for neutral-type neural networks with mixed time delays,” in Procedings of the 2nd International Conference on Industrial Mechatronics and Automation (ICIMA '10), vol. 1, pp. 486–489, May 2010. View at Publisher · View at Google Scholar · View at Scopus
  3. M. S. Mahmoud and A. Ismail, “Improved results on robust exponential stability criteria for neutral-type delayed neural networks,” Applied Mathematics and Computation, vol. 217, no. 7, pp. 3011–3019, 2010. View at Publisher · View at Google Scholar · View at Scopus
  4. Y. Sun and J. Cao, “Stabilization of stochastic delayed neural networks with markovian switching,” Asian Journal of Control, vol. 10, no. 3, pp. 327–340, 2008. View at Publisher · View at Google Scholar · View at Scopus
  5. J. F. Wang, J. G. Jian, and P. Yan, “Finite-time boundedness analysis of a class of neutral type neural networks with time delays,” in Proceedings of the 6th International Symposium on Neural Networks on Advances in Neural Networks, vol. 1, pp. 395–404, 2009.
  6. J. Liu and G. Zong, “New delay-dependent asymptotic stability conditions concerning BAM neural networks of neutral type,” Neurocomputing, vol. 72, no. 10–12, pp. 2549–2555, 2009. View at Publisher · View at Google Scholar · View at Scopus
  7. Q. Zhang, X. Wei, and J. Xu, “Global exponential stability for nonautonomous cellular neural networks with unbounded delays,” Chaos, Solitons and Fractals, vol. 39, no. 3, pp. 1144–1151, 2009. View at Publisher · View at Google Scholar · View at Scopus
  8. H. Yang and T. Chu, “LMI conditions for stability of neural networks with distributed delays,” Chaos, Solitons and Fractals, vol. 34, no. 2, pp. 557–563, 2007. View at Publisher · View at Google Scholar · View at Scopus
  9. B. Song, S. Xu, and Y. Zou, “Delay-dependent robust H filtering for uncertain neutral stochastic time-delay systems,” Circuits, Systems, and Signal Processing, vol. 28, no. 2, pp. 241–256, 2009. View at Publisher · View at Google Scholar · View at Scopus
  10. Y. Li, S. Xu, B. Zhang, and Y. Chu, “Delay-dependent guaranteed cost control for uncertain neutral systems with distributed delays,” International Journal of Control, Automation and Systems, vol. 6, no. 1, pp. 15–23, 2008. View at Scopus
  11. Y. Chen, W. X. Zheng, and A. Xue, “A new result on stability analysis for stochastic neutral systems,” Automatica, vol. 46, no. 12, pp. 2100–2104, 2010. View at Publisher · View at Google Scholar · View at Scopus
  12. P. Balasubramaniam, A. Manivannan, and R. Rakkiyappan, “Exponential stability results for uncertain neutral systems with interval time-varying delays and Markovian jumping parameters,” Applied Mathematics and Computation, vol. 216, no. 11, pp. 3396–3407, 2010. View at Publisher · View at Google Scholar · View at Scopus
  13. Y. Wang, Y. He, and X. Zhang, “Refined delay-dependent robust stability criteria of a class of uncertain mixed neutral and lur'e dynamical systems with interval time-varying,” in Proceedings of the Chinese Control and Decision Conference (CCDC '11), pp. 4179–4184, Sichuan, China, May 2011. View at Publisher · View at Google Scholar
  14. T. Yi, G. Liu, Y. Liu, and R. Wang, “Global robust stability analysis for uncertain stochastic neural networks of neutral-type with time-varying delays,” in Proceedings of the International Conference on Electric Information and Control Engineering (ICEICE '11), pp. 1399–1403, April 2011. View at Publisher · View at Google Scholar · View at Scopus
  15. H. Chen, Y. Zhang, and P. Hu, “Novel delay-dependent robust stability criteria for neutral stochastic delayed neural networks,” Neurocomputing, vol. 73, no. 13–15, pp. 2554–2561, 2010. View at Publisher · View at Google Scholar · View at Scopus
  16. W. Su and Y. Chen, “Global asymptotic stability analysis for neutral stochastic neural networks with time-varying delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 1576–1581, 2009. View at Publisher · View at Google Scholar · View at Scopus
  17. L. Jin, “New LMI-based stability condition for neutral-type neural networks with variable and distributed delays,” in Proceedings of the International Conference on Intelligent System Design and Engineering Application (ISDEA '10), pp. 1002–1005, October 2010. View at Publisher · View at Google Scholar · View at Scopus
  18. J. E. Feng, S. Xu, and Y. Zou, “Delay-dependent stability of neutral type neural networks with distributed delays,” Neurocomputing, vol. 72, no. 10-12, pp. 2576–2580, 2009. View at Publisher · View at Google Scholar · View at Scopus
  19. X. Li, “Global robust stability for stochastic interval neural networks with continuously distributed delays of neutral type,” Applied Mathematics and Computation, vol. 215, no. 12, pp. 4370–4384, 2010. View at Publisher · View at Google Scholar · View at Scopus
  20. R. Rakkiyappan and P. Balasubramaniam, “New global exponential stability results for neutral type neural networks with distributed time delays,” Neurocomputing, vol. 71, no. 4–6, pp. 1039–1045, 2008. View at Publisher · View at Google Scholar · View at Scopus
  21. R. Rakkiyappan and P. Balasubramaniam, “LMI conditions for global asymptotic stability results for neutral-type neural networks with distributed time delays,” Applied Mathematics and Computation, vol. 204, no. 1, pp. 317–324, 2008. View at Publisher · View at Google Scholar · View at Scopus
  22. C. Y. Lu, “A delay-dependent approach to robust control for neutral uncertain neural networks with mixed interval time-varying delays,” Nonlinearity, vol. 24, no. 4, pp. 1121–1136, 2011. View at Publisher · View at Google Scholar · View at Scopus
  23. K. Gu, “An integral inequality in the stability problem of time-delay systems,” in Proceedings of the 39th IEEE Confernce on Decision and Control, pp. 2805–2810, December 2000. View at Scopus
  24. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15, SIAM Studies in Applied Mathematics, Philadelphia, Pa, USA, 1994.
  25. G. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, UK, 1988.
  26. V. B. Kolmanovski and A. D. Myshkis, Applied Theory of Functional Differential Equations, vol. 85, Springer, 1992.
  27. Y. He, M. Wu, G. P. Liu, and J. H. She, “Output feedback stabilization for a discrete-time system with a time-varying delay,” IEEE Transactions on Automatic Control, vol. 53, no. 10, pp. 2372–2377, 2008. View at Publisher · View at Google Scholar · View at Scopus
  28. H. Gao, J. Lam, C. Wang, and Y. Wang, “Delay-dependent output-feedback stabilisation of discrete-time systems with time-varying state delay,” IEE Proceedings Control Theory & Applications, vol. 151, no. 6, pp. 691–698, 2004.
  29. L. El Ghaoui, F. Oustry, and M. AitRami, “A cone complementarity linearization algorithm for static output-feedback and related problems,” IEEE Transactions on Automatic Control, vol. 42, no. 8, pp. 1171–1176, 1997. View at Scopus
  30. A. Bellen, N. Guglielmi, and A. E. Ruehli, “Methods for linear systems of circuit delay differential equations of neutral type,” IEEE Transactions on Circuits and Systems I, vol. 46, no. 1, pp. 212–215, 1999. View at Scopus
  31. R. Rakkiyappan, P. Balasubramaniam, and R. Krishnasamy, “Delay-dependent stability of neutral systems with time-varying delays using delay-decomposition approach,” Applied Mathematical Modelling, vol. 36, no. 5, pp. 2253–2261, 2012. View at Publisher · View at Google Scholar
  32. M. N. A. Parlakçi, “Extensively augmented Lyapunov functional approach for the stability of neutral time-delay systems,” IET Control Theory and Applications, vol. 2, no. 5, pp. 431–436, 2008. View at Publisher · View at Google Scholar · View at Scopus
  33. M. N. A. Parlakçi, “Robust stability of uncertain neutral systems: a novel augmented Lyapunov functional approach,” IET Control Theory and Applications, vol. 1, no. 3, pp. 802–809, 2007. View at Publisher · View at Google Scholar · View at Scopus