Novel Global Exponential Stability Criterion for Recurrent Neural Networks with Time-Varying Delay

Abstract

The problem of global exponential stability for recurrent neural networks with time-varying delay is investigated. By dividing the time delay interval [0,𝜏(𝑡)] into 𝐾+1 dynamical subintervals, a new Lyapunov-Krasovskii functional is introduced; then, a novel linear-matrix-inequality (LMI-) based delay-dependent exponential stability criterion is derived, which is less conservative than some previous literatures (Zhang et al., 2005; He et al., 2006; and Wu et al., 2008). An illustrate example is finally provided to show the effectiveness and the advantage of the proposed result.

1. Introduction

In the past decades, recurrent neural networks (RNNs) have been extensively investigated because of their applications, such as combinatorial optimization [1, 2], associative memories [3–5], signal processing [6], image processing [7], pattern recognition [8, 9], and so forth. Some of these applications often require that equilibrium points of the designed networks be stable. Meanwhile, in the hard implementation of RNNs, time delay commonly occurrs due to the finite switching speed of amplifiers or finite speed of signal processing, and its existence is always an origin of oscillation, divergence, and instability in neural networks. Therefore, the stability of RNNs with time delay has received much attention, and a large amount of results have been proposed to ensure the asymptotic or exponential stability of delayed neural networks [10–21].

So far, there is a main means handling the stability of delayed neural networks: free-weighting matrix approach [22–26]. Recently, a novel method was proposed for Hopfield neural networks with constant delay in [27], which brings more free-weighting matrices by dividing equally the constant time delay interval [0,𝜏] into 𝑚 subintervals. Further more, by dividing the time delay interval [0,𝜏(𝑡)] into 𝐾+1 dynamical subintervals, Zhang et al. [28] generalize this method to study global asymptotic stability of RNNs with time-varying delay. This method mainly utilizes the information in the time delay interval [0,𝜏(𝑡)], which brings more freedom degrees and can reduce conservativeness.

Motivated by above-mentioned discussions, in this paper, we consider the global exponential stability of RNNs with time-varying delay. By dividing the time delay interval [0,𝜏(𝑡)] into 𝐾+1 dynamical subintervals, we construct a new Lyapunov-Krasovskii functional (LKF) and derive a novel sufficient condition, which is presented in term of linear matrix inequality (LMI). The obtained stability result is less conservative than some existing results [22, 23, 29]. Finally, an illustrating example is given to verify the effectiveness and the advantage of the proposed result.

The rest of this paper is organized as follows. In Section 2, the problem of exponential stability analysis for RNNs with time-varying delay is formulated. Section 3 presents our main results. An illustrating example is provided in Section 4. The conclusion is stated in Section 5.

Throughout this paper, 𝐶=[𝐶_𝑖𝑗]_𝑛×𝑛 denotes an 𝑛×𝑛 real matrix. 𝐶^𝑇,  ‖𝐶‖,  𝜆_𝑚(𝐶), and  𝜆_𝑀(𝐶) represent the transpose, the Euclidean norm, the minimum eigenvalue, and the maximum eigenvalue of matrix 𝐶, respectively. 𝐶>0(𝐶<0) denotes that 𝐶 is a positive (negative) definite matrix. 𝐼 denotes an identify matrix with compatible dimensions, and * denotes the symmetric terms in a symmetric matrix.

2. Problem Formulation

Consider the following RNNs with time-varying delay: 𝑧̇(𝑡)(𝑡)(𝑧(𝑡(𝑧((𝑡𝑧(𝑡)(𝑡)[],=−𝐷𝑧+𝐴𝑓))+𝐵𝑓𝑡−𝜏)))+𝐽,=𝜓∀𝑡∈−𝜏,0 (1) where 𝑧(⋅)=[𝑧₁(⋅),𝑧₂(⋅),…,𝑧_𝑛(⋅)]^𝑇 is the state vector, 𝑓(𝑧(⋅))=[𝑓₁(𝑧₁(⋅)),𝑓₂(𝑧₂(⋅)),…,𝑓_𝑛(𝑧_𝑛(⋅))]^𝑇 denotes the neuron activation function, and 𝐽 is a bias value vector. 𝐷=diag(𝑑_𝑖) is diagonal matrix with 𝑑_𝑖>0, 𝑖=1,2,…,𝑛. 𝐴 and 𝐵 are connection weight matrix and the delay connection weight matrix, respectively. The initial condition 𝜓(𝑡) is a continuous and differentiable vector-valued function, where 𝑡∈[−𝜏,0]. The time delay 𝜏(𝑡) is a differentiable function that satisfies: 0≤𝜏(𝑡)≤𝜏,  𝜏̇(𝑡)≤𝜇, where 𝜏>0 and 𝜇≥0.

To obtain the proposal result, we assume that each 𝑓_𝑖 is bounded and satisfies 𝑙⁻_𝑖≤𝑓_𝑖(𝑢)−𝑓_𝑖(𝑣)𝑢−𝑣≤𝑙⁺_𝑖, (2) where ∀𝑢,𝑣∈𝑅,  𝑢≠𝑣. 𝑙⁻_𝑖 and 𝑙⁺_𝑖 are some constants, 𝑖=1,2,…,𝑛.

Let 𝑔_𝑖(𝑧_𝑖(𝑡))=𝑓_𝑖(𝑧_𝑖(𝑡))−𝑙⁻_𝑖𝑧_𝑖(𝑡),  𝑙_𝑖=𝑙⁺_𝑖−𝑙⁻_𝑖, system (1) is equivalent to the following form: 𝑧̇(𝑡)=−𝐷−𝐴𝐿₀𝑧(𝑡)+𝐴𝑔(𝑧(𝑡))+𝐵𝑔(𝑧(𝑡−𝜏(𝑡)))+𝐵𝐿₀𝑧(𝑡−𝜏(𝑡))+𝐽, (3) where 𝐿₀=diag(𝑙⁻₁,𝑙⁻₂,…,𝑙⁻_𝑛),  𝐿₁=diag(𝑙⁺₁,𝑙⁺₂,…,𝑙⁺_𝑛),  𝐿=diag(𝑙₁,𝑙₂,…,𝑙_𝑛)=𝐿₁−𝐿₀.

Noting assumption (2), we have 𝑔0≤^𝑖(𝑢)−𝑔_𝑖(𝑣)𝑢−𝑣≤𝑙_𝑖∀𝑢,𝑣∈𝑅,𝑢≠𝑣,𝑙_𝑖=𝑙⁺_𝑖−𝑙⁻_𝑖(𝑖=1,2,…,𝑛). (4) Assuming system (3) has an equilibrium point 𝑧^∗=(𝑧^∗₁,𝑧^∗₂,…,𝑧^∗_𝑛)^𝑇. Then, let 𝑥=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇, and we define 𝑥_𝑖(𝑡)=𝑧_𝑖(𝑡)−𝑧^∗_𝑖, system (3) is transformed into the following form: 𝑥̇(𝑡)=−𝐷−𝐴𝐿₀𝑥(𝑡)+𝐴𝑔(𝑥(𝑡))+𝐵𝑔(𝑥(𝑡−𝜏(𝑡)))+𝐵𝐿₀𝑥(𝑡−𝜏(𝑡)), (5) where 𝑔_𝑖(𝑥_𝑖(𝑡))=𝑔_𝑖(𝑥_𝑖(𝑡)+𝑧^∗_𝑖)−𝑔_𝑖(𝑧^∗_𝑖) with 𝑔_𝑖(0)=0,  𝑔_𝑖(𝑥_𝑖(𝑡−𝜏(𝑡)))=𝑔_𝑖(𝑥_𝑖(𝑡−𝜏(𝑡))+𝑧^∗_𝑖)−𝑔_𝑖(𝑧^∗_𝑖).

In the derivation of the main results, we need the following lemmas and definitions.

Definition 1 (global exponential stability). System (5) is said to be globally exponentially stable with convergence rate 𝑘, if there exist constants 𝑘>0 and 𝑀≥1, such that ‖𝑥(𝑡)‖≤𝑀𝜙𝑒^−𝑘𝑡,∀𝑡≥0, (6) where 𝜙=sup_{−𝜏≤𝜃≤0}‖𝑥(𝜃)‖.

Lemma 2. Let 𝑥(𝑡)∈𝑅^𝑛 be a vector-valued function with the first-order continuous-derivative entries. Then, the following integral inequality holds for matrix 𝑋=𝑋^𝑇>0 and any matrices 𝑀₁,  𝑀₂, and two scalar functions ℎ₁(𝑡) and ℎ₂(𝑡), where ℎ₂(𝑡)≥ℎ₁(𝑡)≥0−^{𝑡−ℎ1(𝑡)}_{𝑡−ℎ2(𝑡)}𝑥̇^𝑇(𝑠)𝑋𝑥̇(𝑠)𝑑𝑠≤𝜁^𝑇(𝑡)𝑀^𝑇₁+𝑀₁−𝑀^𝑇₁+𝑀₂∗−𝑀^𝑇₂−𝑀₂𝜁(𝑡)+ℎ₁(𝑡)−ℎ₂(𝑡)𝜁^𝑇(𝑡)𝐻^𝑇𝑅⁻¹𝐻𝜁(𝑡), (7) where 𝐻=[𝑀₁,𝑀₂]∈𝑅^𝑛×2𝑛 and 𝜁(𝑡)=[𝑥^𝑇(𝑡−ℎ₁(𝑡))𝑥^𝑇(𝑡−ℎ₂(𝑡))]^𝑇.

Proof. This proof can be completed in a manner similar to [30].

Lemma 3 (see [31]). For any two vectors 𝑎,  𝑏∈𝑅^𝑛, any matrix A, any positive definite symmetric matrix 𝐵 with the same dimensions, and any two positive constants m, n, the following inequality holds: −𝑚𝑎^𝑇𝐵𝑎+2𝑛𝑎^𝑇𝐴𝑏≤𝑛²𝑏^𝑇𝐴^𝑇(𝑚𝐵)⁻¹𝐴𝑏. (8)

3. Main Results

In this section, we will consider the delay interval [0,𝜏(𝑡)], which is divided into 𝐾+1 dynamical subintervals, namely, [0,𝜌₁𝜏(𝑡)],…,[𝜌_𝐾𝜏(𝑡),𝜏(𝑡)], where 𝜌₁<⋯<𝜌_𝐾. This is to say, there is a parameter sequence (𝜌₁,…,𝜌_𝐾), which satisfies the following conditions: 0<𝜌₁𝜏(𝑡)<𝜌₂𝜏(𝑡)<⋯<𝜌_𝐾𝜏(𝑡)<𝜏(𝑡),0≤𝜌_𝑖𝜏̇(𝑡)≤𝜌_𝑖𝜇, (9) where 𝜌_𝑖∈(0,1),𝑖=1,2,…,𝐾,𝐾 is positive integer.

Utilizing the useful information of 𝐾+1 dynamical subintervals, a novel LKF is constructed, and then a newly LMI-based delay-dependent sufficient condition can be proposed to guarantee the global exponential stability of RNNs with time-varying delay.

Theorem 4. The equilibrium point of system (5) with 𝜇<1 is globally exponentially stable with convergence rate 𝑘>0, if there exists parameter 𝜌_𝑖 satisfying 0<𝜌₁<⋯<𝜌_𝐾<1, some positive definite symmetric matrices 𝑃,  𝑅₁,  𝑅₂,  𝑅₃,  𝑄_𝑖,  𝑍, some positive definite diagonal matrices Λ,  𝑋₁,  𝑋₂,  𝑌₁ and 𝑌₂, and any matrices 𝑀_𝑗, where 𝑖=1,2,…,𝐾,  𝑗=1,2,…,2𝐾+4, and 𝐾 is a positive integer, such that the following LMI has feasible solution: Σ𝑊₀∗000−𝑍𝜌₁𝑊₁00∗∗−𝑍𝜌₂−𝜌₁⋯0∗∗∗⋯𝑊_𝐾∗∗∗∗−𝑍1−𝜌_𝐾<0, (10) where 𝑊_𝑖=𝜏𝑒^−𝑘𝜏𝐻^𝑇_𝑖,𝐻_𝑖=[𝑀_2𝑖+1𝑀_2𝑖+2]∈𝑅^𝑛×2𝑛,𝑖=0,1,2,…,𝐾.

ΣΣ=^1,1Σ_1,20000Σ_1,𝐾+20Σ_1,𝐾+4Σ_1,𝐾+5∗Σ_2,2Σ_2,30000000∗∗Σ_3,3Σ_3,4000000∗∗∗⋯000000∗∗∗∗Σ_𝐾,𝐾Σ_𝐾,𝐾+10000∗∗∗∗∗Σ_{𝐾+1,𝐾+1}Σ_{𝐾+1,𝐾+2}000∗∗∗∗∗∗Σ_{𝐾+2,𝐾+2}Σ_{𝐾+2,𝐾+3}Σ_{𝐾+2,𝐾+4}Σ_{𝐾+2,𝐾+5}∗∗∗∗∗∗∗Σ_{𝐾+3,𝐾+3}00∗∗∗∗∗∗∗∗Σ_{𝐾+4,𝐾+4}Σ_{𝐾+4,𝐾+5}∗∗∗∗∗∗∗∗∗Σ_{𝐾+5,𝐾+5}, (11)Σ_1,1=2𝑘𝑃−𝑃𝐷−𝐴𝐿₀−𝐷−𝐴𝐿₀^𝑇𝑃+𝑅₁+𝑅₃+𝜏²𝐷−𝐴𝐿₀^𝑇𝑍𝐷−𝐴𝐿₀+^𝐾_𝑖=1𝑄_𝑖+𝐿𝑋₁𝐿+𝜏𝑒^−2𝑘𝜏𝑀^𝑇₁+𝑀₁,Σ_1,2=𝜏𝑒^−2𝑘𝜏−𝑀^𝑇₁+𝑀₂,Σ_1,𝐾+2=𝑃𝐵𝐿₀−𝜏²𝐷−𝐴𝐿₀𝑍𝐵𝐿₀,Σ_1,𝐾+4=𝑃𝐴+2𝑘Λ−𝐷−𝐴𝐿₀^𝑇Λ−𝜏²𝐷−𝐴𝐿₀^𝑇𝑍𝐴+𝐿𝑋₂,Σ_1,𝐾+5=𝑃𝐵−𝜏²𝐷−𝐴𝐿₀^𝑇Σ𝑍𝐵,_2,2=−𝑒^{−2𝑘𝜌1𝜏}𝑄₁1−𝜌₁𝜇+𝜏𝑒^−2𝑘𝜏−𝑀^𝑇₂−𝑀₂+𝜏𝑒^−2𝑘𝜏𝑀^𝑇₃+𝑀₃,Σ_2,3=𝜏𝑒^−2𝑘𝜏−𝑀^𝑇₃+𝑀₄,Σ_3,3=−𝑒^{−2𝑘𝜌2𝜏}𝑄₂1−𝜌₂𝜇+𝜏𝑒^−2𝑘𝜏−𝑀^𝑇₄−𝑀₄+𝜏𝑒^−2𝑘𝜏𝑀^𝑇₅+𝑀₅,Σ_3,4=𝜏𝑒^−2𝑘𝜏−𝑀^𝑇₅+𝑀₆Σ,…,_{𝐾+1,𝐾+1}=−𝑒^{−2𝑘𝜌𝐾𝜏}𝑄_𝐾1−𝜌_𝐾𝜇+𝜏𝑒^−2𝑘𝜏−𝑀^𝑇_2𝐾−𝑀_2𝐾+𝜏𝑒^−2𝑘𝜏𝑀^𝑇_2𝐾+1+𝑀_2𝐾+1,Σ_{𝐾+1,𝐾+2}=𝜏𝑒^−2𝑘𝜏−𝑀^𝑇_2𝐾+1+𝑀_2𝐾+2,Σ_{𝐾+2,𝐾+2}=−𝑒^−2𝑘𝜏𝑅₁1−𝜇+𝜏²𝐿₀𝐵^𝑇𝑍𝐵𝐿₀+𝐿𝑌₁𝐿+𝜏𝑒^−2𝑘𝜏−𝑀^𝑇_2𝐾+2−𝑀_2𝐾+2+𝜏𝑒^−2𝑘𝜏𝑀^𝑇_2𝐾+3+𝑀_2𝐾+3Σ_{𝐾+2,𝐾+3}=𝜏𝑒^−2𝑘𝜏−𝑀^𝑇_2𝐾+3+𝑀_2𝐾+4,Σ_{𝐾+2,𝐾+4}=𝐿₀𝐵^𝑇Λ+𝜏²𝐿₀𝐵^𝑇Σ𝑍𝐴,_{𝐾+2,𝐾+5}=𝜏²𝐿₀𝐵^𝑇𝑍𝐵+𝐿𝑌₂,Σ_{𝐾+3,𝐾+3}=𝑒^−2𝑘𝜏𝑅₃+𝜏𝑒^−2𝑘𝜏−𝑀^𝑇_2𝐾+4−𝑀_2𝐾+4,Σ_{𝐾+4,𝐾+4}=Λ𝐴+𝐴^𝑇Λ+𝑅₂+𝜏²𝐴^𝑇𝑍𝐴−𝑋₁−2𝑋₂,Σ_{𝐾+4,𝐾+5}=Λ𝐵+𝜏²𝐴^𝑇Σ𝑍𝐵,_{𝐾+2,𝐾+5}=−𝑒^−2𝑘𝜏𝑅₂1−𝜇+𝜏²𝐵^𝑇𝑍𝐵−𝑌₁−2𝑌₂,𝐿₀=diag𝑙⁻_𝑖,𝐿₁=diag𝑙⁺_𝑖,𝐿₀=diag𝑙_𝑖,𝐷=diag𝑑_𝑖,𝐴=𝑎_{𝑖𝑗𝑛×𝑛},𝐵=𝑏_{𝑖𝑗𝑛×𝑛}. (12)

Proof. Construct the following Lyapunov-Krasovskii functional candidate: 𝑉(𝑥(𝑡))=𝑉₁(𝑥(𝑡))+𝑉₂(𝑥(𝑡))+𝑉₃(𝑥(𝑡))+𝑉₄(𝑥(𝑡))+𝑉₅(𝑥(𝑡)), (13) where 𝑉₁(𝑥(𝑡))=𝑒^2𝑘𝑡𝑥^𝑇(𝑡)(𝑡),𝑉𝑃𝑥₂(𝑥(𝑡))=2𝑒^{𝑛2𝑘𝑡}_𝑖=1𝜆_𝑖^𝑥𝑖₀^(𝑡)𝑔_𝑖(𝑠)𝑉𝑑𝑠,₃(𝑥(𝑡))=^𝑡_{(𝑡)𝑡−𝜏}𝑒^2𝑘𝑠𝑥^𝑇(𝑠)𝑅₁𝑥(𝑠)+𝑑𝑠_{𝑡(𝑡)𝑡−𝜏}𝑒^2𝑘𝑠𝑔^𝑇(𝑠)𝑅₂𝑔(𝑠)+𝑑𝑠_{𝑡𝑡−𝜏}𝑒^2𝑘𝑠𝑥^𝑇(𝑠)𝑅₃𝑥(𝑠)𝑉𝑑𝑠,₄(𝑥(𝑡))=^𝐾_𝑖=1^𝑡_{𝑡−𝜌𝑖𝜏(𝑡)}𝑒^2𝑘𝑠𝑥^𝑇(𝑠)𝑄_𝑖𝑥(𝑠)𝑉𝑑𝑠,₅(𝑥(𝑡))=𝜏⁰_−𝜏^𝑡_𝑡+𝜃𝑒^2𝑘𝑠𝑥̇^𝑇(𝑠)𝑍𝑥̇(𝑠)𝑑𝑠, (14) where 𝑃=𝑃^𝑇>0, Λ=diag(𝜆_𝑖)>0, 𝑅₁=𝑅^𝑇₁>0, 𝑅₂=𝑅^𝑇₂>0, 𝑅₃=𝑅^𝑇₃>0, 𝑄_𝑖=𝑄^𝑇_𝑖>0, 𝑍=𝑍^𝑇>0, and 𝑖=0,1,2,…,𝐾.
Let the parameters 𝜌₀=0,  𝜌_𝐾+1=1; calculating the time derivatives 𝑉_𝑖(𝑖=1,2,3,4,5) along the trajectories of system (5) yields 𝑉̇₁(𝑥(𝑡))=2𝑘𝑒^2𝑘𝑡𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+2𝑒^2𝑘𝑡𝑥^𝑇(𝑡)𝑃𝑥̇(𝑡),𝑉̇₂(𝑥(𝑡))=4𝑘𝑒^{𝑛2𝑘𝑡}_𝑖=1𝜆_𝑖^𝑥𝑖₀^(𝑡)𝑔_𝑖(𝑠)𝑑𝑠+2𝑒^2𝑘𝑡𝑔^𝑇(𝑥(𝑡Λ𝑥̇(𝑡),𝑉̇))₃(𝑥(𝑡))=𝑒^2𝑘𝑡𝑥^𝑇(𝑡)𝑅₁𝑥(𝑡)−𝑒^{−2𝑘𝜏(𝑡)}𝑥^𝑇(𝑡−𝜏(𝑡))𝑅₁𝑥((𝑡×(𝜏̇(𝑡+𝑡−𝜏))1−))𝑔^𝑇(𝑥(𝑡))𝑅₂𝑔(𝑥(𝑡))−𝑒^{−2𝑘𝜏(𝑡)}𝑔^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑅₂𝑔(𝑥((𝑡×(𝜏̇(𝑡𝑡−𝜏)))1−))+𝑥^𝑇(𝑡)𝑅₃𝑥(𝑡)−𝑒^−2𝑘𝜏𝑔^𝑇(𝑥(𝑡−𝜏))𝑅₃𝑔(𝑥(,𝑉̇𝑡−𝜏))₄(𝑥(𝑡))=^𝐾_𝑖=1𝑒^2𝑘𝑡𝑥^𝑇(𝑡)𝑄_𝑖𝑥(𝑡)−𝑒^{−2𝑘𝜌𝑖𝜏(𝑡)}𝑥^𝑇𝑡−𝜌_𝑖𝜏(𝑡)×𝑄_𝑖𝑥𝑡−𝜌_𝑖𝜏(𝑡)1−𝜌_𝑖𝜏̇(𝑡),𝑉̇₅(𝑥(𝑡))=𝜏²𝑒^2𝑘𝑡𝑥̇^𝑇(𝑡)𝑍𝑥̇(𝑡)−𝜏^𝑡_𝑡−𝜏𝑒^2𝑘𝑠𝑥̇^𝑇(𝑠)𝑍𝑥̇(𝑠)𝑑𝑠. (15) It is clear that the following inequality is true: 𝑥̇^𝑇(𝑡)𝑍𝑥̇(𝑡)≤−𝐷−𝐴𝐿₀𝑥(𝑡)+𝐴𝑔(𝑥(𝑡))+𝐵𝑔(𝑥(𝑡−𝜏(𝑡)))+𝐵𝑥(𝑡−𝜏(𝑡))^𝑇×𝑍−𝐷−𝐴𝐿₀𝑥(𝑡)+𝐴𝑔(𝑥(𝑡))+𝐵𝑔(𝑥(𝑡−𝜏(𝑡)))+𝐵𝑥(𝑡−𝜏(𝑡)), (16) According to (4), for some diagonal matrices 𝑋₁>0,  𝑋₂>0,  𝑌₁>0,  𝑌₂>0, we have 𝑔^𝑇(𝑥(𝑡))𝑋₁𝑔(𝑥(𝑡))≤𝑥^𝑇(𝑡)𝐿𝑋₁𝐿𝑥(𝑡),𝑔^𝑇(𝑥(𝑡))𝑋₂𝑔(𝑥(𝑡))≤𝑥^𝑇(𝑡)𝐿𝑋₁𝑔(𝑥(𝑡)),𝑔^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑌₁𝑔(𝑥(𝑡−𝜏(𝑡)))≤𝑥^𝑇(𝑡−𝜏(𝑡))𝐿𝑌₁𝐿𝑥(𝑡−𝜏(𝑡)),𝑔^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑌₂𝑔(𝑥(𝑡−𝜏(𝑡)))≤𝑥^𝑇(𝑡−𝜏(𝑡))𝐿𝑌₂𝑔(𝑥(𝑡−𝜏(𝑡))). (17) Using Lemma 2, we have −^{𝑡−𝜌𝑖𝜏(𝑡)}_{𝑡−𝜌𝑖+1𝜏(𝑡)}𝑥̇^𝑇(𝑠)𝑍𝑥̇(𝑠)𝑑𝑠≤𝜁^𝑇_𝑖(𝑡)𝑀^𝑇_2𝑖+1+𝑀_2𝑖+1−𝑀^𝑇_2𝑖+1+𝑀_2𝑖+2∗−𝑀^𝑇_2𝑖+2−𝑀_2𝑖+2𝜁_𝑖(𝑡)+𝜌_𝑖+1−𝜌_𝑖𝜏(𝑡)𝜁^𝑇_𝑖(𝑡)𝐻^𝑇_𝑖𝑍⁻¹𝐻_𝑖𝜁_𝑖(𝑡)−^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝑥̇^𝑇(𝑠)𝑍𝑥̇(𝑠)𝑑𝑠≤𝜁^𝑇_𝐾+1(𝑡)𝑀^𝑇_2𝐾+3+𝑀_2𝐾+3−𝑀^𝑇_2𝐾+3+𝑀_2𝐾+4∗−𝑀^𝑇_2𝐾+4−𝑀_2𝐾+4𝜁_𝐾+1(𝑡), (18) where 𝐻_𝑖=𝑀_2𝑖+1𝑀_2𝑖+2∈𝑅^𝑛×2𝑛, 𝜁_𝑖(𝑡)=[𝑥^𝑇(𝑡−𝜌_𝑖𝜏(𝑡))𝑥^𝑇(𝑡−𝜌_𝑖+1𝜏(𝑡))]^𝑇, 𝜁_𝐾+1(𝑡)=[𝑥^𝑇(𝑡−𝜏(𝑡))𝑥^𝑇(𝑡−𝜏)]^𝑇,𝑖=0,1,…,𝐾.
From (15)–(17), and using (18), we finally get 𝑉̇(𝑥(𝑡))≤𝑒^2𝑘𝑡Ω, (19) where Ω=𝜁^𝑇(𝑡)Σ𝜁(𝑡)+Ω₀+Ω₁+⋯+Ω_𝐾, Σ is defined in (11). Ω_𝑖=𝜌_𝑖+1−𝜌_𝑖𝜏²𝑒^−2𝑘𝜏𝜁^𝑇_𝑖(𝑡)𝐻^𝑇_𝑖𝑍⁻¹𝐻_𝑖𝜁_𝑖(𝑡),𝜁𝑖=0,1,…,𝐾,_𝑗(𝑡)=𝑥^𝑇𝑡−𝜌_𝑗𝜏(𝑡)𝑥^𝑇𝑡−𝜌_𝑗+1𝜏(𝑡)^𝑇,𝜁(𝑡)=𝑥𝑗=0,1,…,𝐾+1._𝑇(𝑡),𝑥^𝑇𝑡−𝜌₁𝜏(𝑡),…,𝑥^𝑇𝑡−𝜌_𝐾𝜏(𝑡,𝑥_𝑇)(𝑡−𝜏(𝑡)),𝑥^𝑇(𝑡−𝜏),𝑔^𝑇(𝑥(𝑡)),𝑔^𝑇(𝑥(𝑡−𝜏(𝑡)))^𝑇. (20) Obviously, if Ω<0, it implies 𝑉̇(𝑥(𝑡))<0 for any 𝜁(𝑡)≠0.
And 𝑉(𝑥(0))=𝑥^𝑇(0)𝑃𝑥(0)+2^𝑛_𝑖=1𝜆_𝑖^𝑥𝑖₀⁽⁰⁾𝑔_𝑖(𝑠)+𝑑𝑠_0(0)−𝜏𝑒^2𝑘𝑠𝑥^𝑇(𝑠)𝑅₁𝑥(𝑠)+𝑑𝑠_0(0)−𝜏𝑒^2𝑘𝑠𝑔^𝑇(𝑥(𝑠))𝑅₂𝑔(𝑥(𝑠+))𝑑𝑠_0−𝜏𝑒^2𝑘𝑠𝑥^𝑇(𝑠)𝑅₃𝑥(𝑠)+𝑑𝑠_{𝐾𝑖=10−𝜌𝑖𝜏(0)}𝑒^2𝑘𝑠𝑥^𝑇(𝑠)𝑄_𝑖𝑥(𝑠)𝑑𝑠+𝜏⁰_−𝜏⁰_𝜃𝑒^2𝑘𝑠𝑥̇^𝑇(𝑠)𝑍𝑥̇(𝑠)𝑑𝑠. (21) On the other hand, the following inequality holds: 𝑥̇^𝑇(𝑠)𝑥̇(𝑠)≤−𝐷−𝐴𝐿₀𝑥(𝑠)+𝐴𝑔(𝑥(𝑠))+𝐵𝑔(𝑥(𝑡−𝜏(𝑠)))+𝐵𝑥(𝑡−𝜏(𝑠))^𝑇×−𝐷−𝐴𝐿₀𝑥(𝑠)+𝐴𝑔(𝑥(𝑠))+𝐵𝑔(𝑥(𝑡−𝜏(𝑠)))+𝐵𝑥(𝑡−𝜏(𝑠)). (22) Combining Lemma 3, we have 𝑥̇^𝑇(𝑠)𝑥̇(𝑠)≤4𝜆_𝑀𝐷−𝐴𝐿₀^𝑇𝐷−𝐴𝐿₀+𝜆_𝑀𝐿²𝜆_𝑀𝐴^𝑇𝐴+𝜆_𝑀𝐿²𝜆_𝑀𝐵^𝑇𝐵+𝜆_𝑀𝐿₀𝐵^𝑇𝐵𝐿₀𝜙. (23) Thus, 𝑉(𝑥(0))=𝜆_𝑀(𝑃)𝜙²+𝜆_𝑀𝐿²+𝜆_𝑀Λ^𝑇Λ𝜙²+𝜆_𝑀𝑅₁𝜏𝜙²+𝜆_𝑀𝑅₂𝜆_𝑀𝐿²𝜏𝜙²+𝜆_𝑀𝑅₃𝜏𝜙²+^𝐾_𝑖=1𝜆_𝑀𝑄_𝑖𝜌_𝑖𝜏𝜙²+2𝜏³𝜆_𝑀(𝑍)𝜆_𝑀𝐷−𝐴𝐿₀^𝑇𝐷−𝐴𝐿₀+𝜆_𝑀𝐿²𝜆_𝑀𝐴^𝑇𝐴+𝜆_𝑀𝐿²𝜆_𝑀𝐵^𝑇𝐵+𝜆_𝑀𝐿₀𝐵^𝑇𝐵𝐿₀𝜙, (24) where Δ=𝜆_𝑀(𝑃)+𝜆_𝑀𝐿²+𝜆_𝑀Λ^𝑇Λ+𝜆_𝑀𝑅₁𝜏+𝜆_𝑀𝑅₂𝜆_𝑀𝐿²𝜏+𝜆_𝑀𝑅₃𝜏+^𝐾_𝑖=1𝜆_𝑀𝑄_𝑖𝜌_𝑖𝜏𝜙²+2𝜏³𝜆_𝑀(𝑍)𝜆_𝑀𝐷−𝐴𝐿₀^𝑇𝐷−𝐴𝐿₀+𝜆_𝑀𝐿²𝜆_𝑀𝐴^𝑇𝐴+𝜆_𝑀𝐿²𝜆_𝑀𝐵^𝑇𝐵+𝜆_𝑀𝐿₀𝐵^𝑇𝐵𝐿₀. (25) On the other hand, we have 𝑉(𝑥(𝑡))≥𝑒^2𝑘𝑡𝜆_𝑚(𝑃)𝜙. (26) Therefore, ‖𝑥(𝑡)‖≤𝑒^−𝑘𝑡Δ𝜆_𝑚(𝑃)𝜙. (27) Thus, according to Definition 1, we can conclude that the equilibrium point 𝑥^∗ of system (5) is globally exponentially stable. This completes the proof.

Remark 5. Differential from the results in [22, 23, 29], we divide the time delay interval [0,𝜏(𝑡)] into 𝐾+1 dynamical subintervals, and a novel Lyapunov-Krasovskii functional is introduced. This brings more degrees of freedom to ensure the global exponential stability. Therefore, Theorem 4 is less conservative than some previous results.

Remark 6. In Theorem 4, by setting 𝑅₁=𝑅₂=𝑄_𝑖=0(𝑖=1,2,…,𝐾), similar to the proof of Theorem 4, we can derive a criterion to guarantee the global exponential stability of RNNs with time-varying delay when 𝜏̇(𝑡) is unknown or 𝜏(𝑡) is not differentiable.

4. Illustrating Example

In this section, an illustrating example is given to verify the effectiveness and advantage of the criteria proposed in this paper.

Example 7. Consider system (5) with the following parameters: ,,,.𝐷=2003.5𝐴=−10.50.5−1𝐵=−0.50.50.50.5𝐿=1000 (28)

At first, we suppose that the time delay interval [0,𝜏(𝑡)] is divided into 2 (𝐾=1) subintervals, and 𝜌=0.1. While the upper bound 𝜏=1, the exponential convergence rates for various 𝜇 obtained from Theorem 4 and those in [22, 23, 29] are listed in Table 1. In addition, while the exponential convergence rate of 𝑘=0.8, the upper bounds of 𝜏 for various 𝜇 from Theorem 4 and those in [22, 23, 29] are listed in Table 2.

Table 1 

Allowable exponential convergence rate 𝑘 for variable 𝜇 and 𝜏=1.

Table 2 

Allowable upper bound of 𝜏 for variable 𝜇 and 𝑘=0.8.

Thus, from Tables 1 and 2, we can say that, the result in this paper is much effective and less conservative than those in [22, 23, 29]. Figure 1 shows the state response of Example 7 with constant delay 𝜏=1, when the initial value is [0.8,−0.6]^𝑇.

Figure 1 

State response curves with 𝜏=1 and 𝜇=0 for Example 7.

5. Conclusion

In this paper, we consider the global exponential stability for RNNs with time-varying delay. By dividing the time delay interval [0,𝜏(𝑡)] into 𝐾+1 dynamical subintervals, a novel Lyapunov-Krasovskii functional is introduced. A less conservative LMI-based delay-dependent stability criterion is derived based on Lyapunov stability theory. Furthermore, an illustrating example is given to show the effectiveness of the proposed result.

Acknowledgments

This work is supported by Guangxi Science Foundation Grant (0832067), the Foundation Grant of Guangxi Key Laboratory of Automobile Components and Vehicle Technology (13-A-03-01), and the Opening Project of Guangxi Key Laboratory of Automobile Components and Vehicle Technology (2012KFZD03).