Delay-Dependent Exponential Stability for Discrete-Time BAM Neural Networks with Time-Varying Delays

Abstract

This paper considers the delay-dependent exponential stability for discrete-time BAM neural networks with time-varying delays. By constructing the new Lyapunov functional, the improved delay-dependent exponential stability criterion is derived in terms of linear matrix inequality (LMI). Moreover, in order to reduce the conservativeness, some slack matrices are introduced in this paper. Two numerical examples are presented to show the effectiveness and less conservativeness of the proposed method.

1. Introduction

Bidirectional associative memory (BAM) neural networks were first introduced by Kosko [1, 2]. It generalizes the single-layer autoassociative Hebbian correlator to a two-layer pattern-matched heteroassociative circuits. This class of neural networks has been successfully applied to pattern recognition and artificial intelligence. In those applications, it is very important to guarantee the designed neural networks to be stable. On the other hand, time delays are unavoidably encountered in the implementation of neural networks, and its existence will lead to instability, oscillation, and poor performances of neural networks. Therefore, the the asymptotic or exponential stability analysis for BAM neural networks with time delays has received great attention during the past years, see, for example, [3–15]. The obtained results are classified into two categories: delay-independent results [3–5, 7, 10] and delay-dependent results [6, 9, 11, 12, 14–16]. Generally speaking, the delay-dependent results are less conservative than the delay-independent ones, especially when the size of time delay is small.

It should be pointed out that all of the above-mentioned references are concerned with continuous-time BAM neural networks. However, when implementing the continuous-time neural networks for computer simulation, for experimental or computational purposes, it is essential to formulate a discrete-time system that is an analogue of the continuous-time recurrent neural networks [17]. Generally speaking, the stability analysis of continuous-time neural networks is not applicable to the discrete version. Therefore, the stability analysis for various discrete-time neural networks with time delays is widely studied in recent years [17–28]. Using M-matrix method, Liang and Cao [25] studied the exponential stability of continuous-time BAM neural network with constant delays and its discrete analogue. Liang et al. [26] also studied the dynamics of discrete-time BAM neural networks with variable time delays based on unreasonably severe constraints on the delay functions. By using the Lyapunov functional method and linear matrix inequality technique, the exponential stability and robust exponential stability for discrete-time BAM neural networks with variable delays were considered in [27, 28], respectively. However, the results presented in [27, 28] are still conservative, since the the Lyapunov functionals constructed in [27, 28] are so simple. Therefore, there is much room to improve the results in [27, 28].

In this paper, we present a new exponential stability criterion for discrete-time BAM neural networks with time-varying delays. Compared with the existing methods, the main contributions of this paper are as follows. Firstly, more general Lyapunov functional is employed to obtain the improved exponential stability criterion; secondly, in order to reduce the conservativeness, some slack matrices, which bring much flexibility in solving LMI, are introduced in this paper. The proposed exponential stability criterion is expressed in LMI which can be efficiently solved by LMI toolbox in Matlab. Finally, two numerical examples are presented to show that our result is less conservative than some existing ones [27, 28].

The organization of this paper is as follows. Section 2 presents problem formulation of discrete-time BAM neural networks with time delay-varying. In Section 3, our main result of this paper is established and some remarks are given. In Section 4, numerical examples are given to demonstrate the proposed method. Finally, conclusion is drawn in Section 5.

Notation
Throughout this paper, the superscript “𝑇” stands for the transpose of a matrix. 𝑅^𝑛 and 𝑅^𝑛×𝑛 denote the 𝑛-dimensional Euclidean space and set of all 𝑛×𝑛 real matrices, respectively. A real symmetric matrix 𝑃>0(≥0) denotes 𝑃 being a positive definite (positive semidefinite) matrix. 𝐼 is used to denote an identity matrix with proper dimension. For integers 𝑎,𝑏, and 𝑎<𝑏,𝑁[𝑎,𝑏] denote the discrete interval 𝑁[𝑎,𝑏]={𝑎,𝑎+1,…,𝑏−1,𝑏}. 𝜆_𝑀(𝑋) and 𝜆_𝑚(𝑋) stand for the maximum and minimum eigenvalues of the symmetric matrix 𝑋, respectively. Matrices, if not explicitly stated, are assumed to have compatible dimensions. The symmetric terms in a symmetric matrix are denoted by ∗.

2. Problem Formulation

Consider the following discrete-time BAM neural network with time-varying delays:𝑢_𝑖(𝑘+1)=𝑎_𝑖𝑢_𝑖(𝑘)+^𝑚_𝑗=1𝑣𝑤_𝑖𝑗𝑓_𝑗(𝑣_𝑗(𝑘−𝜏(𝑘)))+𝐼_𝑖,𝑖=1,2,…,𝑛,_𝑗(𝑘+1)=𝑏_𝑗𝑣_𝑗(𝑘)+^𝑛_𝑖=1𝑣_𝑗𝑖𝑔_𝑖(𝑢_𝑖(𝑘−ℎ(𝑘)))+𝐽_𝑗,𝑗=1,2,…,𝑚,(2.1)where 𝑢_𝑖(𝑘) and 𝑣_𝑗(𝑘) are the states of the 𝑖th neuron from the neural field 𝐹_𝑋 and the 𝑗th neuron from the neural field 𝐹_𝑌 at time 𝑘, respectively; 𝑎_𝑖,𝑏_𝑗∈(0,1) denote the stability of internal neuron process on the 𝑋-layer and the 𝑌-layer, respectively; 𝑤_𝑖𝑗,𝑣_𝑗𝑖 are real constants, and denote the synaptic connection weights; 𝑓_𝑗(⋅)𝑔_𝑖(⋅) denote the activation functions of the 𝑗th neuron from the neural field 𝐹_𝑌 and the 𝑗th neuron from the neural field 𝐹_𝑋, respectively; 𝐼_𝑖 and 𝐽_𝑗 denote the external constant inputs from outside the network acting on the 𝑖th neuron from the neural field 𝐹_𝑋 and the 𝑗th neuron from the neural field 𝐹_𝑌, respectively. 𝜏(𝑘) and ℎ(𝑘) represent time-varying interval delays satisfying𝜏_𝑚≤𝜏(𝑘)≤𝜏_𝑀,ℎ_𝑚≤ℎ(𝑘)≤ℎ_𝑀,(2.2)where 𝜏_𝑚,𝜏_𝑀,ℎ_𝑚, and ℎ_𝑀 are positive integers.

Throughout this paper, we make the following assumption on the activation functions 𝑓_𝑗(⋅),𝑔_𝑖(⋅).

(A1) The activation functions 𝑓_𝑗(⋅),𝑔_𝑖(⋅)(𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚) are bounded on 𝑅.(A2) For any 𝜁₁,𝜁₂∈𝑅, there exist positive scalars 𝑙_1𝑗,𝑙_2𝑖 such that00≤≤||𝑓𝑔_𝑗𝑗(𝜁₁)−𝑓_𝑗(𝜁₂)|≤𝑙_1𝑗|𝜁₁−𝜁₂|,𝑗=1,2,…,𝑚,(𝜁₁)−𝑔_𝑗(𝜁₂)|≤𝑙_2𝑖|𝜁₁−𝜁₂|,𝑖=1,2,…,𝑛.(2.3)

It is clear that under Assumption (A1) and (A2), system (2.1) has at least one equilibrium. In order to simplify our proof, we shift the equilibrium point 𝑢^∗=[𝑢^∗1,𝑢^∗2,…,𝑢^∗𝑛]^𝑇,𝑣^∗=[𝑣^∗1,𝑣^∗2,…,𝑣^∗𝑚]^𝑇 of system (2.1) to the origin. Let 𝑥_𝑖(𝑘)=𝑢_𝑖(𝑘)−𝑢^∗𝑖,𝑦_𝑗(𝑘)=𝑣_𝑗(𝑘)−𝑣^∗𝑗,𝑓_𝑗(𝑦_𝑗(𝑘))=𝑓_𝑗(𝑣_𝑗(𝑘))−𝑓_𝑗(𝑣^∗𝑗),𝑔_𝑖(𝑥_𝑖(𝑘))=𝑔_𝑖(𝑢_𝑖(𝑘))−𝑔_𝑖(𝑢^∗𝑖), then system (2.1) can be transformed to𝑥_𝑖(𝑘+1)=𝑎_𝑖𝑥_𝑖(𝑘)+^𝑚_𝑗=1𝑦𝑤_𝑖𝑗𝑓_𝑗(𝑦_𝑗(𝑡−𝜏(𝑘))),𝑖=1,2,…,𝑛_𝑗(𝑘+1)=𝑏_𝑗𝑦_𝑗(𝑘)+^𝑛_𝑖=1𝑣_𝑗𝑖𝑔_𝑖(𝑥_𝑖(𝑡−ℎ(𝑘))),𝑗=1,2,…,𝑚.(2.4)Obviously, the the activation functions 𝑓_𝑗(⋅),𝑔_𝑖(⋅) satisfy the following conditions.

(A3) For any 𝜁∈𝑅, there exist positive scalars 𝑙_1𝑗,𝑙_2𝑖 such that0≤|𝑓_𝑗(𝜁)|≤𝑙_1𝑗|𝜁|,𝑗=1,2,…,𝑚,0≤|𝑔_𝑖(𝜁)|≤𝑙_2𝑖|𝜁|,𝑖=1,2,…,𝑛.(2.5)Also we assume that system (2.4) with initial value𝑥_𝑖(𝑠)=𝜙_𝑖(𝑠),𝑦_𝑗(𝑠)=𝜓_𝑗(𝑠),∀𝑠∈𝑁[−𝜏^∗,0],𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚,(2.6)where 𝜏^∗=max{𝜏_𝑀,ℎ_𝑀}.

For convenience, we rewrite system (2.4) in the form𝑥(𝑠)𝑥𝑦=((𝑘𝑘𝜙++(𝑠11)),)==𝑦𝐴𝐵(𝑠𝑥𝑦)((=𝑘𝑘))𝜓++(𝑠𝑊𝑉)𝑔,𝑓((𝑥𝑦∀((𝑘𝑠𝑘−∈−ℎ𝑁𝜏((𝑘[𝑘−))))𝜏)),_∗,,0],(2.7)where 𝑥(𝑘)=[𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘)]^𝑇,𝑦(𝑘)=[𝑦₁(𝑘),𝑦₂(𝑘),…,𝑦_𝑚(𝑘)]^𝑇,𝑓(𝑦(𝑘))=[𝑓₁(𝑦₁(𝑘)),𝑓₂(𝑦₂(𝑘)),…,𝑓_𝑚(𝑦_𝑚(𝑘))]^𝑇,(𝑤𝑔(𝑥_𝑖𝑗(𝑘))=[𝑔₁(𝑥₁(𝑘)),𝑔₂(𝑥₂(𝑘)),…,𝑔_𝑛(𝑥_𝑛(𝑘))]^𝑇,𝐴=diag{𝑎₁,𝑎₂,…,𝑎_𝑛},𝐵=diag{𝑏₁,𝑏₂,…,𝑏_𝑚},𝑊=)_𝑚×𝑛,𝑉=(𝑣_𝑖𝑗)_𝑛×𝑚,𝜙(𝑠)=[𝜙₁(𝑠),𝜙₂(𝑠),…,𝜙_𝑛(𝑠)]^𝑇,𝜓(𝑠)=[𝜓₁(𝑠),𝜓₂(𝑠),…,𝜓_𝑚(𝑠)]^𝑇.

From above analysis, we can see that the exponential stability problem of system (2.1) on equilibrium 𝑥^∗ is changed into the zero stability problem of system (2.7). Therefore, in the following part, we will devote into the exponential stability analysis problem of system (2.7).

Before giving the main result, we will firstly introduce the following definition and lemmas.

Definition 2.1. The trivial solution of BAM neural network (2.7) is said to be globally exponentially stable, if there exist scalars 𝑟>1 and 𝑀>1 such that‖𝑥(𝑘)‖²+‖𝑦(𝑘)‖²≤𝑀(sup_{𝑠∈𝑁[−ℎ𝑀,0]}‖𝜙(𝑠)‖²+sup_{𝑠∈𝑁[−𝜏𝑀,0]}‖𝜓(𝑠)‖²)𝑟^−𝑘.(2.8)

Lemma 2.2 (see [29]). For any real vectors 𝑎,𝑏, and any matrix 𝑄>0 with appropriate dimensions, it follows that 2𝑎^𝑇𝑏≤𝑎^𝑇𝑄𝑎+𝑏^𝑇𝑄⁻¹𝑏.(2.9)

3. Main Result

In this section, we are in a position to present the global exponential stability criterion of system (2.7).

Theorem 3.1. For given diagonal matrices 𝐿₁=diag{𝑙₁₁,𝑙₁₂,…,𝑙_1𝑚}and𝐿₂=diag{𝑙₂₁,𝑙₂₂,…,𝑙_2𝑛}. Under Assumptions (A1) and (A2), the system (2.7) is globally exponentially stable, if there exist matrices 𝑃_𝑖>0,𝑄_𝑖>0,𝑅_𝑖>0,𝑆_𝑖>0,𝑍_𝑖>0,(𝑖=1,2),𝐿_𝑗,𝑀_𝑗,𝑁_𝑗,𝑇_𝑗,(𝑗=1,2,…,8), and diagonal matrices 𝐷₁=diag{𝑑₁₁,𝑑₁₂,…,𝑑_1𝑚}>0,𝐷₂=diag{𝑑₂₁,𝑑₂₂,…,𝑑_2𝑛}>0, such that the following LMI holds: ⎡⎢⎢⎢⎢⎢⎣Ω∗√−ℎ𝑍_𝑀1𝐿√ℎ_𝑀−ℎ_𝑚𝑀√𝜏_𝑀𝑁√𝜏_𝑀−𝜏_𝑚𝑇∗∗−0𝑍₁00∗∗∗−0𝑍₂0∗∗∗∗−0𝑍₂⎤⎥⎥⎥⎥⎥⎦<0,(3.1)where 𝜔=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜔¹¹𝜔₁₂𝜔₁₃𝜔₁₄𝜔₁₅𝜔₁₆−𝑚₁+𝑙^𝑡7𝑙^𝑡8∗𝜔₂₂𝜔₂₃𝜔₂₄𝜔₂₅𝜔₂₆𝜔₂₇𝑡^𝑡8−𝑛^𝑡8∗∗𝜔₃₃−𝑡^𝑡4𝜔₃₅𝜔₃₆−𝑚₃−𝑡^𝑡7−𝑡^𝑡8∗∗∗∗∗∗𝜔∗₄₄𝜔𝑛₄₅₅𝑚₄−𝑙₄−𝑚₄0𝜔₅₆−𝑚₅+𝑛^𝑡7𝜔₅₈∗∗∗∗∗𝜔₆₆𝜔₆₇−𝑙^𝑡8+𝑚^𝑡8∗∗∗∗∗∗𝜔₇₇−𝑚^𝑡8∗∗∗∗∗∗∗𝜔₈₈𝑙=𝑙_𝑡1⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝑚=𝑚_𝑡1𝑙^𝑡2𝑙^𝑡3𝑙^𝑡4𝑙^𝑡5𝑙^𝑡6𝑙^𝑡7𝑙^𝑡8^𝑡,𝑛=𝑛_𝑡1𝑚^𝑡2𝑚^𝑡3𝑚^𝑡4𝑚^𝑡5𝑚^𝑡6𝑚^𝑡7𝑚^𝑡8^𝑡,𝑡=𝑡_𝑡1𝑛^𝑡2𝑛^𝑡3𝑛^𝑡4𝑛^𝑡5𝑛^𝑡6𝑛^𝑡7𝑛^𝑡8^𝑡,𝜔₁₁𝑡^𝑡2𝑡^𝑡3𝑡^𝑡4𝑡^𝑡5𝑡^𝑡6𝑡^𝑡7𝑡^𝑡8^𝑡,=𝑟𝑎^𝑡𝑝₁𝑎+𝑟^ℎ𝑚ℎ_𝑚(𝑎−𝑖)^𝑡𝜔𝑧₁(₁₂𝑎−𝑖)−𝑝₁+(ℎ_𝑚−ℎ_𝑚+1)𝑞₁+𝑟₁+𝑙₁+𝑙^𝑡1,𝜔=₁₃𝑙^𝑡2+𝑡₁−𝑛₁,𝜔₁₄=𝑙^𝑡3−𝑡₁,=𝑟𝑎^𝑡𝑝₁𝑤𝜔+₁₅𝑟^ℎ𝑚ℎ_𝑚(𝑎−𝑖)^𝑡𝑧₁𝑤+𝑙^𝑡4,𝜔₁₆=𝑙^𝑡5+𝑛₁,𝜔₂₂=−𝑙₁+𝑙^𝑡6+𝑚₁,=−𝑟𝜔^−𝜏𝑚₂₃𝑞₂+𝑡₂+𝑡^𝑡2+𝑑₁−𝑛₂−𝑛^𝑡2,𝜔=₂₄−𝑡₂+𝑡^𝑡3−𝑛^𝑡3,𝜔₂₅=𝑡^𝑡4−𝑛^𝑡4,𝜔₂₆=𝑡^𝑡5+𝑛₂−𝑛^𝑡5,𝜔₂₇=𝑡^𝑡6+𝑚₂−𝑙₂−𝑛^𝑡6,𝜔₃₃=−𝑚₂+𝑡^𝑡7−𝑛^𝑡7,=𝜔−₃₅𝑟^−𝜏𝑚𝑟₂−𝑡₃−𝑡^𝑡3,=𝑛₃−𝑡^𝑡5,(1)𝜔₄₄𝜔₃₆=𝑚₃−𝑙₃−𝑡^𝑡6,𝜔₅₅=𝑟𝑤^𝑡𝑝₁𝑤+𝑟^ℎ𝑚ℎ_𝑚𝑤^𝑡𝑧₁𝑤−𝑙¹⁻¹𝑑₁𝑙¹⁻¹,=𝑟𝑏^𝑡𝑝₂𝑏+𝑟^𝜏𝑚𝜏_𝑚(𝑏−𝑖)^𝑡𝜔𝑧₂(₅₆𝑏−𝑖)−𝑝₂+(𝜏_𝑚−𝜏_𝑚+1)𝑞₂+𝑟₂+𝑛₅+𝑛^𝑡5,𝜔₅₈=𝑚₅−𝑙₅+𝑛^𝑡6,𝜔₆₆=𝑟𝑏^𝑡𝑝₂𝑣+𝑟^𝜏𝑚𝜏_𝑚(𝑏−𝑖)^𝑡𝑧₂𝑣+𝑛^𝑡8,=−𝑟𝜔^−ℎ𝑚₆₇𝑞₁−𝑙₆−𝑙^𝑡6+𝑑₂+𝑚₆+𝑚^𝑡6,𝜔₇₇=−𝑙^𝑡7−𝑚₆+𝑚^𝑡7,𝜔₈₈=−𝑟^−ℎ𝑚𝑟₁−𝑚₇−𝑚^𝑡7,=𝑟𝑣^𝑡𝑝₂𝑣+𝑟^𝜏𝑚𝜏_𝑚𝑣^𝑡𝑧₂𝑣−𝑙²⁻¹𝑑₂𝑙²⁻¹.(3.2)

Proof. Choose the following Lyapunov-Krasovskii candidate function of system (2.7) as follows:𝑉(𝑘)=𝑉₁(𝑘)+𝑉₂(𝑘)+𝑉₃(𝑘)+𝑉₄(𝑘)+𝑉₅(𝑘),(3.3)where𝑣𝑣₁(𝑘)=𝑟^𝑘𝑥^𝑡(𝑘)𝑝₁𝑥(𝑘)+𝑟^𝑘𝑦^𝑡(𝑘)𝑝₂𝑦(𝑘),₂(𝑘)=^𝑘−1_𝑙=𝑘_−ℎ(𝑘)𝑟^𝑙𝑥^𝑡(𝑙)𝑞₁𝑥(𝑙)+^𝑘−1_𝑙=𝑘_{−𝜏(𝑘)}𝑣𝑟^𝑙𝑦^𝑡(𝑙)𝑞₂𝑦(𝑙),₃(𝑘)=^−ℎ𝑚_{𝜃=−ℎ𝑚+1𝑘−1𝑙=𝑘+𝜃}𝑟^𝑙𝑥^𝑡(𝑙)𝑞₁𝑥(𝑙)+^−𝜏𝑚_{𝜃=−𝜏𝑚+1𝑘−1𝑙=𝑘+𝜃}𝑣𝑟^𝑙𝑦^𝑡(𝑙)𝑞₂𝑦(𝑙),₄(𝑘)=^𝑘−1_𝑙=_{𝑘−ℎ𝑚}𝑟^𝑙𝑥^𝑡(𝑙)𝑟₁𝑥(𝑙)+^𝑘−1_𝑙=_{𝑘−𝜏𝑚}𝑣𝑟^𝑙𝑦^𝑡(𝑙)𝑟₂𝑦(𝑙),₅(𝑘)=𝑟^ℎ𝑚−1_{𝜃=−ℎ𝑚}^𝑘−1_{𝑙=𝑘+𝜃}𝑟^𝑙𝜂^𝑡1(𝑙)𝑧₁𝜂₁(𝑙)+𝑟^𝜏𝑚−1_{𝜃=−𝜏𝑚}^𝑘−1_{𝑙=𝑘+𝜃}𝑟^𝑙𝜂^𝑡2(𝑙)𝑧₂𝜂₂(𝑙),(3.4)and 𝜂₁(𝑙)=𝑥(𝑙+1)−𝑥(𝑙),𝜂₂(𝑙)=𝑦(𝑙+1)−𝑦(𝑙).
Calculating the difference of 𝑉(𝑘) along the trajectories of system (2.7), we can obtain 𝛿𝑣₁(𝑘)=+𝑟𝑟^𝑘+1[_𝑘+1𝑎𝑥(𝑘)+𝑤𝑓(𝑦(𝑘−𝜏(𝑘)))]^𝑡𝑝₁[𝑎𝑥(𝑘)+𝑤𝑓(𝑦(𝑘−𝜏(𝑘)))]−𝑟_𝑘[𝑏𝑦(𝑘)+𝑣𝑔(𝑥(𝑘−ℎ(𝑘)))]^𝑡𝑝₂[𝑏𝑦(𝑘)+𝑣𝑔(𝑥(𝑘−ℎ(𝑘)))]=𝑟_𝑘𝑥^𝑡(𝑘)𝑝₁𝑥(𝑘)−𝑟^𝑘𝑦^𝑡(𝑘)𝑝₂𝑦(𝑘){𝑥+^𝑡(𝑟𝑘𝑓)(_𝑡𝑟𝑎^𝑡𝑝₁𝑎−𝑝₁)𝑥(𝑘)+2𝑟𝑥^𝑡(𝑘)𝑎^𝑡𝑝₁𝑤𝑓(𝑦(𝑘−𝜏(𝑘)))+2𝑟𝑦(_𝑡𝑦(𝑘−𝜏(𝑘)))𝑤^𝑡𝑝₁𝑤𝑓(𝑦(𝑘−𝜏(𝑘)))+𝑦^𝑡(𝑘)(𝑟𝑏^𝑡𝑝₂𝑏−𝑝₂)𝑦(𝑘)𝛿𝑣₂(𝑘)𝑏^𝑡𝑝₂𝑣𝑔(𝑥(𝑘−ℎ(𝑘)))+𝑟𝑔^𝑡(𝑥(𝑘−ℎ(𝑘)))𝑣^𝑡𝑝₂𝑣𝑔(𝑥(𝑘−ℎ(𝑘)))},(3.5)(𝑘)≤+𝑟^𝑘𝑟𝑥^𝑡_𝑘(𝑘)𝑞₁𝑥(𝑘)−𝑟^{𝑘−ℎ(𝑘)}𝑥^𝑡(𝑘−ℎ(𝑘))𝑞₁𝑥(𝑘−ℎ(𝑘))+𝑦^𝑡(𝑘)𝑞₂𝑦(𝑘)−𝑟^{𝑘−𝜏(𝑘)}𝑦^𝑡(𝑘−𝜏(𝑘))𝑞₂𝑦(𝑘−𝜏(𝑘))_{𝑘−ℎ𝑚𝑙=𝑘+1}_{−ℎ(𝑘+1)}𝑟^𝑙𝑥^𝑡(𝑙)𝑞₁𝑥(𝑙)+^{𝑘−𝜏𝑚}_𝑙=𝑘+1_{−𝜏(𝑘+1)}≤𝑟_𝑘𝑟^𝑙𝑦^𝑡(𝑙)𝑞₂𝑦(𝑙)+𝑟𝑥^𝑡_𝑘(𝑘)𝑞₁𝑥(𝑘)−𝑟^{𝑘−ℎ𝑚}𝑥^𝑡(𝑘−ℎ(𝑘))𝑞₁𝑥(𝑘−ℎ(𝑘))+𝑦^𝑡(𝑘)𝑞₂𝑦(𝑘)−𝑟^{𝑘−𝜏𝑚}𝑦^𝑡(𝑘−𝜏(𝑘))𝑞₂𝑦(𝑘−𝜏(𝑘))_{𝑘−ℎ𝑚𝑙=𝑘}_+1−ℎ𝑚𝑟^𝑙𝑥^𝑡(𝑙)𝑞₁𝑥(𝑙)+^{𝑘−𝜏𝑚}_𝑙=𝑘_{+1−𝜏𝑚}𝛿𝑣₃𝑟^𝑙𝑦^𝑡(𝑙)𝑞₂𝑦(𝑙),(3.6)(𝑘)=−(ℎ_𝑚−ℎ_𝑚)𝑟^𝑘𝑥^𝑡(𝑘)𝑞₁𝑥(𝑘)+(𝜏_𝑚−𝜏_𝑚)𝑟^𝑘𝑦^𝑡(𝑘)𝑞₂𝑦(𝑘)_{𝑘−ℎ𝑚𝑙=𝑘}_+1−ℎ𝑚𝑟^𝑙𝑥^𝑡(𝑙)𝑞₁𝑥(𝑙)−^{𝑘−𝜏𝑚}_𝑙=𝑘_{+1−𝜏𝑚}𝛿𝑣₄𝑟^𝑙𝑦^𝑡(𝑙)𝑞₂𝑦(𝑙),(3.7)(𝑘)=+𝑟^𝑘𝑟𝑥^𝑡_𝑘(𝑘)𝑟₁𝑥(𝑘)−𝑟^{𝑘−ℎ𝑚}𝑥^𝑡(𝑘−ℎ_𝑚)𝑟₁𝑥(𝑘−ℎ_𝑚)𝛿𝑣₅𝑦^𝑡(𝑘)𝑟₂𝑦(𝑘)−𝑟^{𝑘−𝜏𝑚}𝑦^𝑡(𝑘−𝜏_𝑚)𝑟₂𝑦(𝑘−𝜏_𝑚),(3.8)(𝑘)=𝑟^𝑘𝑟^ℎ𝑚ℎ_𝑚𝜂^𝑡1(𝑘)𝑧₁𝜂₁(𝑘)−𝑟^{ℎ𝑚𝑘−1}_𝑙=_{𝑘−ℎ𝑚}+𝑟_𝑘𝑟^𝑙𝜂^𝑡1(𝑙)𝑧₁𝜂₁(𝑙)𝑟^𝜏𝑚𝜏_𝑚𝜂^𝑡2(𝑘)𝑧₂𝜂₂(𝑘)−𝑟^{𝜏𝑚𝑘−1}_𝑙=_{𝑘−𝜏𝑚}≤𝑟_𝑘𝑟^𝑙𝜂^𝑡2(𝑙)𝑧₂𝜂₂(𝑙)+𝑟𝑟^ℎ𝑚_𝑘ℎ_𝑚[(𝑎−𝑖)𝑥(𝑘)+𝑤𝑓(𝑦(𝑘−𝜏(𝑘)))]^𝑡𝑧₁[(𝑎−𝑖)𝑥(𝑘)+𝑤𝑓(𝑦(𝑘−𝜏(𝑘)))]−𝑟𝑟^𝜏𝑚𝜏_{𝑚𝑘𝑘−1}[(𝑏−𝑖)𝑦(𝑘)+𝑣𝑔(𝑥(𝑘−ℎ(𝑘)))]^𝑡𝑧₂[(𝑏−𝑖)𝑦(𝑘)+𝑣𝑔(𝑥(𝑘−ℎ(𝑘)))]_𝑙=𝑘_−ℎ(𝑘)𝜂^𝑡1(𝑙)𝑧₁𝜂₁(𝑙)−𝑟^{𝑘𝑘−ℎ(𝑘)}_𝑙=_{𝑘−ℎ𝑚}−𝑟_𝑘𝑘−1𝜂^𝑡1(𝑙)𝑧₁𝜂₁(𝑙)_𝑙=𝑘_{−𝜏(𝑘)}𝜂^𝑡2(𝑙)𝑧₂𝜂₂(𝑙)−𝑟^{𝑘𝑘−𝜏(𝑘)}_𝑙=_{𝑘−𝜏𝑚}𝜂^𝑡2(𝑙)𝑧₂𝜂₂(𝑙).(3.9)By Lemma 2.2, we have −^𝑘−1_𝑙=𝑘_−ℎ(𝑘)𝜂^𝑡1(𝑙)𝑧₁𝜂₁(𝑙)≤2^𝑘−1_𝑙=𝑘_−ℎ(𝑘)𝜂^𝑡1(𝑙)𝑙^𝑡𝜉(𝑘)+^𝑘−1_𝑙=𝑘_−ℎ(𝑘)=2[𝑥_𝑡𝜉^𝑡(𝑘)𝑙𝑧¹⁻¹𝑙^𝑡𝜉(𝑘)≤𝜉_𝑡(𝑘)−𝑥^𝑡(𝑘−ℎ(𝑘))]𝑙^𝑡𝜉(𝑘)+ℎ(𝑘)𝜉^𝑡(𝑘)𝑙𝑧¹⁻¹𝑙^𝑡𝜉(𝑘)−(𝑘)(𝜓₁𝑙^𝑡+𝑙𝜓₁)𝜉(𝑘)+ℎ_𝑚𝜉^𝑡(𝑘)𝑙𝑧¹⁻¹𝑙^𝑡𝜉(𝑘),(3.10)_{𝑘−ℎ(𝑘)𝑙=}_{𝑘−ℎ𝑚}𝜂^𝑡1(𝑙)𝑧₁𝜂₁(𝑙)≤2^{𝑘−ℎ(𝑘)}_𝑙=_{𝑘−ℎ𝑚}𝜂^𝑡1(𝑙)𝑚^𝑡𝜉(𝑘)+^{𝑘−ℎ(𝑘)}_𝑙=_{𝑘−ℎ𝑚}=2[𝑥_𝑡𝜉^𝑡(𝑘)𝑚𝑧¹⁻¹𝑚^𝑡𝜉(𝑘)≤𝜉_𝑡(𝑘−ℎ(𝑘))−𝑥^𝑡(𝑘−ℎ_𝑚)]𝑚^𝑡𝜉(𝑘)+(ℎ_𝑚−ℎ(𝑘))𝜉^𝑡(𝑘)𝑚𝑧¹⁻¹𝑚^𝑡𝜉(𝑘)−(𝑘)(𝜓₂𝑚^𝑡+𝑚𝜓₂)𝜉(𝑘)+(ℎ_𝑚−ℎ_𝑚)𝜉^𝑡(𝑘)𝑚𝑧¹⁻¹𝑚^𝑡𝜉(𝑘),(3.11)_{𝑘−1𝑙=𝑘}_{−𝜏(𝑘)}𝜂^𝑡2(𝑙)𝑧₂𝜂₂(𝑙)≤2^𝑘−1_𝑙=𝑘_{−𝜏(𝑘)}𝜂^𝑡2(𝑙)𝑛^𝑡𝜉(𝑘)+^𝑘−1_𝑙=𝑘_{−𝜏(𝑘)}=2[𝑦_𝑡𝜉^𝑡(𝑘)𝑛𝑧²⁻¹𝑛^𝑡𝜉(𝑘)≤𝜉_𝑡(𝑘)−𝑦^𝑡(𝑘−𝜏(𝑘))]𝑛^𝑡𝜉(𝑘)+𝜏(𝑘)𝜉^𝑡(𝑘)𝑛𝑧²⁻¹𝑛^𝑡𝜉(𝑘)−(𝑘)(𝜓₃𝑛^𝑡+𝑛𝜓₃)𝜉(𝑘)+𝜏_𝑚𝜉^𝑡(𝑘)𝑛𝑧²⁻¹𝑛^𝑡𝜉(𝑘),(3.12)_{𝑘−𝜏(𝑘)𝑙=}_{𝑘−𝜏𝑚}𝜂^𝑡2(𝑙)𝑧₂𝜂₂(𝑙)≤2^{𝑘−𝜏(𝑘)}_𝑙=_{𝑘−𝜏𝑚}𝜂^𝑡2(𝑙)𝑡^𝑡𝜉(𝑘)+^{𝑘−𝜏(𝑘)}_𝑙=_{𝑘−𝜏𝑚}=2[𝑦_𝑡𝜉^𝑡(𝑘)𝑡𝑧²⁻¹𝑡^𝑡𝜉(𝑘)≤𝜉_𝑡(𝑘−𝜏(𝑘))−𝑦^𝑡(𝑘−𝜏_𝑚)]𝑡^𝑡𝜉(𝑘)+(𝜏_𝑚−𝜏(𝑘))𝜉^𝑡(𝑘)𝑡𝑧²⁻¹𝑡^𝑡𝜉(𝑘)(𝑘)(𝜓₄𝑡^𝑡+𝑡𝜓₄)𝜉(𝑘)+(𝜏_𝑚−𝜏_𝑚)𝜉^𝑡(𝑘)𝑡𝑧²⁻¹𝑡^𝑡𝜉(𝑘),(3.13)where𝑚=𝑙𝑚=_𝑡1𝑙^𝑡1𝑙^𝑡2𝑙^𝑡3𝑙^𝑡4𝑙^𝑡5𝑙^𝑡6𝑙^𝑡7𝑙^𝑡8^𝑡,𝑛=𝑛_𝑡1𝑚^𝑡2𝑚^𝑡3𝑚^𝑡4𝑚^𝑡5𝑚^𝑡6𝑚^𝑡7𝑚^𝑡8^𝑡,𝑡=𝑡_𝑡1𝑛^𝑡2𝑛^𝑡3𝑛^𝑡4𝑛^𝑡5𝑛^𝑡6𝑛^𝑡7𝑛^𝑡8^𝑡,𝜓₁𝑡^𝑡2𝑡^𝑡3𝑡^𝑡4𝑡^𝑡5𝑡^𝑡6𝑡^𝑡7𝑡^𝑡8^𝑡,𝜓₂=𝑖0000−𝑖00^𝑡,𝜓₃=00000𝑖−𝑖0^𝑡,𝜓₄=0−𝑖00𝑖000^𝑡,=0𝑖−𝑖00000^𝑡,(3.14)and 𝜉(𝑘)=[𝑥^𝑇(𝑘)𝑦^𝑇(𝑘−𝜏(𝑘))𝑦^𝑇(𝑘−𝜏_𝑀)𝑓^𝑇(𝑦(𝑘−𝜏(𝑘)))𝑦^𝑇(𝑘)𝑥^𝑇(𝑘−ℎ(𝑘))𝑥^𝑇(𝑘−ℎ_𝑀)𝑔^𝑇(𝑥(𝑘−ℎ(𝑘)))]. By inequalities (2.5), it is well known that there exist positive diagonally matrices 𝐷_𝑖≥0(𝑖=1,2) such that the following inequalities hold:𝑟𝑟^𝑘[𝑦^𝑡(𝑡−𝜏(𝑡))𝑑₁𝑦(𝑡−𝜏(𝑡))−𝑓^𝑡(𝑦(𝑡−𝜏(𝑡)))𝑙¹⁻¹𝑑₁𝑙¹⁻¹𝑓(𝑦(𝑡−𝜏(𝑡)))]≥0,_𝑘[𝑥^𝑡(𝑡−ℎ(𝑡))𝑑₂𝑥(𝑡−ℎ(𝑡))−𝑔^𝑡(𝑥(𝑡−ℎ(𝑡)))𝑙²⁻¹𝑑₂𝑙²⁻¹𝑔(𝑥(𝑡−ℎ(𝑡)))]≥0,(3.15)where 𝐿₁=diag{𝑙₁₁,𝑙₁₂,…,𝑙_1𝑚},𝐿₂=diag{𝑙₂₁,𝑙₂₂,…,𝑙_2𝑛}.
Substituting (3.10)–(3.13) into (3.9), we have𝛿𝑣(𝑘)≤+𝛿𝑣𝑟_1𝑘(𝑘)+𝛿𝑣₂(𝑘)+𝛿𝑣₃(𝑘)+𝛿𝑣₄(𝑘)+𝛿𝑣₅(𝑘)+𝑟_𝑘[𝑦^𝑡(𝑡−𝜏(𝑡))𝑑₁𝑦(𝑡−𝜏(𝑡)))−𝑓^𝑡(𝑦(𝑡−𝜏(𝑡)))𝑙¹⁻¹𝑑₁𝑙¹⁻¹𝑓(𝑦(𝑡−𝜏(𝑡)))]≤𝑟_𝑘[𝑥^𝑡(𝑡−ℎ(𝑡))𝑑₂𝑥(𝑡−ℎ(𝑡)))−𝑔^𝑡(𝑥(𝑡−ℎ(𝑡)))𝑙²⁻¹𝑑₂𝑙²⁻¹𝑔(𝑥(𝑡−ℎ(𝑡)))]𝜉^𝑡(𝑘)[𝜔+ℎ_𝑚𝑙𝑧¹⁻¹𝑙^𝑡+(ℎ_𝑚−ℎ_𝑚)𝑚𝑧¹⁻¹𝑚^𝑡+𝜏_𝑚𝑛𝑧²⁻¹𝑛^𝑡+(𝜏_𝑚−𝜏_𝑚)𝑡𝑧²⁻¹𝑡^𝑡]𝜉(𝑘),(3.16)where Ω,𝐿,𝑀,𝑁, and 𝑇 are defined in Theorem 3.1. Thus, if LMI (3.1) holds, it can be concluded that Δ𝑉(𝑘)<0 according to Schur complement, which implies that 𝑉(𝑘)<𝑉(0). Note that𝑣₁(0)=≤𝜆𝑥^𝑡_𝑚(0)𝑝₁𝑥(0)+𝑦^𝑡(0)𝑝₂𝑦(0)𝑣(𝑝₁)sup_{𝑠∈𝑛[−ℎ𝑚,0]}‖𝜙(𝑠)‖²+𝜆_𝑚(𝑝₂)sup_{𝑠∈𝑛[−𝜏𝑚,0]}‖𝜓(𝑠)‖²,(3.17)₂(0)=⁻¹_𝑙=_−ℎ(0)𝑟^𝑙𝑥^𝑡(𝑙)𝑞₁𝑥(𝑙)+⁻¹_𝑙=_−𝜏(0)≤𝜆_𝑚𝑟^𝑙𝑦^𝑡(𝑙)𝑞₂𝑦(𝑙)(𝑞₁)1−𝑟^−ℎ𝑚𝑟−1sup_{𝑠∈𝑛[−ℎ𝑚,0]}‖𝜙(𝑠)‖²+𝜆_𝑚(𝑞₂)1−𝑟^−𝜏𝑚𝑟−1sup_{𝑠∈𝑛[−𝜏𝑚,0]}𝑣‖𝜓(𝑠)‖²,(3.18)₃(0)=^−ℎ𝑚_{𝜃=−ℎ𝑚+1−1𝑙=𝜃}𝑟^𝑙𝑥^𝑡(𝑙)𝑞₁𝑥(𝑙)+^−𝜏𝑚_{𝜃=−𝜏𝑚+1−1𝑙=𝜃}≤(ℎ_𝑚𝑟^𝑙𝑦^𝑡(𝑙)𝑞₂𝑦(𝑙)−ℎ_𝑚)⁻¹_𝑙_=ℎ𝑚𝑟^𝑙𝑥^𝑡(𝑙)𝑞₁𝑥(𝑙)+(𝜏_𝑚−𝜏_𝑚)⁻¹_𝑙_=𝜏𝑚≤(ℎ_𝑚𝑟^𝑙𝑦^𝑡(𝑙)𝑞₂𝑦(𝑙)−ℎ_𝑚)𝜆_𝑚(𝑞₁)1−𝑟^−ℎ𝑚𝑟−1sup_{𝑠∈𝑛[−ℎ𝑚,0]}‖𝜓(𝑠)‖²+(𝜏_𝑚−𝜏_𝑚)𝜆_𝑚(𝑞₂)1−𝑟^−𝜏𝑚𝑟−1sup_{𝑠∈𝑛[−𝜏𝑚,0]}𝑣‖𝜓(𝑠)‖²,(3.19)₄(0)=⁻¹_𝑙=_−ℎ𝑚𝑟^𝑙𝑥^𝑡(𝑙)𝑟₁𝑥(𝑙)+⁻¹_𝑙=_−𝜏𝑚≤𝜆_𝑚𝑟^𝑙𝑦^𝑡(𝑙)𝑟₂𝑦(𝑙)(𝑟₁)1−𝑟^−ℎ𝑚𝑟−1sup_{𝑠∈𝑛[−ℎ𝑚,0]}‖𝜙(𝑠)‖²+𝜆_𝑚(𝑟₂)1−𝑟^−𝜏𝑚𝑟−1sup_{𝑠∈𝑛[−𝜏𝑚,0]}𝑣‖𝜓(𝑠)‖²,(3.20)₅(0)=𝑟^ℎ𝑚−1_{𝜃=−ℎ𝑚}⁻¹_𝑙=𝜃𝑟^𝑙𝜂^𝑡1(𝑙)𝑧₁𝜂₁(𝑙)+𝑟^𝜏𝑚−1_{𝜃=−𝜏𝑚}¹_𝑙=𝜃≤𝑟_ℎ𝑚𝑟^𝑙𝜂^𝑡2(𝑙)𝑧₂𝜂₂(𝑙)ℎ_𝑚𝜆_𝑚(𝑧₁)⁻¹_𝑙=_−ℎ𝑚𝑟^𝑙𝜂^𝑡1(𝑙)𝜂₁(𝑙)+𝑟^𝜏𝑚𝜏_𝑚𝜆_𝑚(𝑧₂)⁻¹_𝑙=_−𝜏𝑚≤𝑟_ℎ𝑚𝑟^𝑙𝜂^𝑡2(𝑙)𝜂₂(𝑙)ℎ_𝑚𝜆_𝑚(𝑧₁)1−𝑟^−ℎ𝑚𝑟−1sup_{𝑠∈𝑛[−ℎ𝑚,−1]}‖𝜂₁(𝑠)‖²+𝑟^𝜏𝑚𝜏_𝑚𝜆_𝑚(𝑧₂)1−𝑟^−𝜏𝑚𝑟−1sup_{𝑠∈𝑛[−𝜏𝑚,−1]}‖𝜂₂(𝑠)‖²,(3.21)Substituting (3.22), (3.23) into (3.21), and combining (3.17)–(3.21), then we have𝑉(0)=⁵_𝑖=1𝑉_𝑖(0)≤𝜌₁sup_{𝑠∈𝑁[−ℎ𝑀,0]}‖𝜙(𝑠)‖²+𝜌₂sup_{𝑠∈𝑁[−𝜏𝑀,0]}‖𝜓(𝑠)‖²,(3.24)where𝜌₁=𝜆_𝑚(𝑝₁)+[(ℎ_𝑚−ℎ_𝑚+1)𝜆_𝑚(𝑞₁)+𝜆_𝑚(𝑟₁)+4𝑟^ℎ𝑚ℎ_𝑚𝜆_𝑚(𝑧₁)]1−𝑟^−ℎ𝑚𝜌𝑟−1₂,=𝜆_𝑚(𝑝₂)+[(𝜏_𝑚−𝜏_𝑚+1)𝜆_𝑚(𝑞₂)+𝜆_𝑚(𝑟₂)+4𝑟^𝜏𝑚𝜏_𝑚𝜆_𝑚(𝑧₂)]1−𝑟^−𝜏𝑚𝑟−1.(3.25)On the other hand,𝑉(𝑘)≥𝑟^𝑘𝑥^𝑇(𝑘)𝑃₁𝑥(𝑘)+𝑟^𝑘𝑦^𝑇(𝑘)𝑃₂𝑦(𝑘)≥𝑟^𝑘{𝜆_𝑚(𝑃₁)‖𝑥(𝑘)‖²+𝜆_𝑚(𝑃₂)‖𝑦(𝑘)‖²}.(3.26)Therefore, we can obtain‖𝑥(𝑘)‖²+‖𝑦(𝑘)‖²≤𝛼𝛽(sup_{𝑠∈𝑁[−ℎ𝑀,0]}‖𝜙(𝑠)‖²+sup_{𝑠∈𝑁[−𝜏𝑀,0]}‖𝜓(𝑠)‖²)𝑟^−𝑘,(3.27)where 𝛼=max{𝜌₁,𝜌₂},𝛽=min{𝜆_𝑚(𝑃₁),𝜆_𝑚(𝑃₂)}. Obviously, 𝛼/𝛽>1, by Definition 2.1, the system (2.7) or (2.1) is globally exponentially stable.

Remark 3.2.   Based on the linear matrix inequality (LMI) technique and Lyapunov stability theory, the exponential stability for discrete-time delayed BAM neural network (2.1) was investigated in [26–28]. However, it should be noted that the results in [26] are based on the unreasonably severe constraints on the delay functions 1<𝜏(𝑘+1)<1+𝜏(𝑘),1<ℎ(𝑘+1)<1+ℎ(𝑘), and the results in [27, 28] are based on the simple Lyapunov functionals. In this paper, in order to obtain the less conservative stability criterion, the novel Lyapunov functional 𝑉(𝑘) is employed, which contains 𝑉₄(𝑘),𝑉₅(𝑘), and more general than the traditional ones [26–28]. Moreover, some slack matrices, which bring much flexibility in solving LMI, are introduced in this paper.

Remark 3.3. By setting 𝑟=1 in Theorem 3.1, we can obtain the global asymptotic stability criterion of discrete-time BAM neural network (2.7).

Remark 3.4. Theorem 3.1 in this paper depends on both the delay upper bounds 𝜏_𝑀,ℎ_𝑀 and the delay intervals 𝜏_𝑀−𝜏_𝑚 and ℎ_𝑀−ℎ_𝑚.

Remark 3.5.  The proposed method in this paper can be generalized to more complex neural networks, such as delayed discrete-time BAM neural networks with parameter uncertainties [30, 31] and stochastic perturbations [30–32], delayed interval discrete-time BAM neural networks [28], and discrete-time analogues of BAM neural networks with mixed delays [32, 33].

4. Numerical Examples

Example 4.1. Consider the delayed discrete-time BAM neural network (2.7) with the following parameters:𝐴=00.800.9,𝐵=00.500.4,𝑊=−00.1.2−−00..011,𝑉=−0.01.5200.1.(4.1)The activation functions satisfy Assumptions (A1) and (A2) with𝐿₁=1001,𝐿₂=1001.(4.2)

For this example, by Theorem 3.1 in this paper with 𝑟=1, we can obtain some maximum allowable delay bounds for guaranteeing the asymptotic stability of this system. For a comparison with [27, Theorem 1] and in [28, Corollary 1], we made Table 1. For this example, it is obvious that our result is less conservative than those in [27, 28].

Table 1 

Maximum allowable delay bounds for Example 4.1.

Example 4.2 (see [27]). Consider the delayed discrete-time BAM NN (2.7) with the following parameters:𝐴=⎡⎢⎢⎣150015⎤⎥⎥⎦,𝐵=⎡⎢⎢⎣11000110⎤⎥⎥⎦,𝑊=⎡⎢⎢⎣011880⎤⎥⎥⎦,𝑉=⎡⎢⎢⎣−1020−0120⎤⎥⎥⎦.(4.3)The activation functions satisfy Assumptions (A1) and (A2) with𝐿₁=1001,𝐿₂=1001.(4.4)
Now, we assume 𝜏(𝑘)=ℎ(𝑘)=4−2sin((𝜋/2)𝑘),𝑟=1.55. It is obvious that 2≤𝜏(𝑘)=ℎ(𝑘)≤6, that is, ℎ_𝑚=𝜏_𝑚=2,ℎ_𝑀=𝜏_𝑚=6. In this case, it is found that the exponential conditions proposed in [27, 28] is not satisfied. However, it can be concluded that this system is globally exponentially stable by using Theorem 3.1 in this paper. If we assume ℎ_𝑚=𝜏_𝑚=𝜏_𝑀=2,𝑟=2.7. Using [27, Theorem 1] and [28, Corollary 1], the achieved maximum allowable delays for guaranteeing the globally exponentially stable of this system are ℎ_𝑀=3,ℎ_𝑀=3, respectively. However, we can obtain the delay bound ℎ_𝑀=4 by using Theorem 3.1 in this paper. For this example, it is seen that our result improve some existing results [27, 28].

5. Conclusion

In this paper, we consider the delay-dependent exponential stability for discrete-time BAM neural networks with time-varying delays. Based on the Lyapunov stability theory and linear matrix inequality technique, the novel delay-dependent stability criterion is obtained. Two numerical examples show that the proposed stability condition in this paper is less conservative than previously established ones.

Acknowledgments

The authors would like to thank the Associate Editor and the anonymous reviewers for their constructive comments and suggestions to improve the quality of the paper. This work was supported by the National Natural Science Foundation of China (no. 60643003) and the Natural Science Foundation of Henan Education Department (no. 2007120005).