Delay-Dependent Stability Analysis for Recurrent Neural Networks with Time-Varying Delays

Abstract

This paper concerns the problem of delay-dependent stability criteria for recurrent neural networks with time varying delays. By taking more information of states and activation functions as augmented vectors, a new class of the Lyapunov functional is proposed. Then, some less conservative stability criteria are obtained in terms of linear matrix inequalities (LMIs). Finally, two numerical examples are given to illustrate the effectiveness of the proposed method.

1. Introduction

In the past few decades, the stability analysis for recurrent neural networks has been extensively investigated because of their successful applications in various scientific fields, such as pattern recognition, image processing, associative memories, and fixed-point computations. 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, much more attention has been paid to recurrent delayed neural networks. Many interesting stability conditions, including delay-independent results [1, 2] and delay-dependent results [3–41], have been obtained for neural networks with time delays. Generally speaking, the delay-dependent stability criteria are less conservative than delay-independent ones when the size of time delay is small. For the delay-dependent case, some criteria have been derived by using Lyapunov-Krasovskii functional (LKF). It is well known that the construction of an appropriate LKF is crucial for obtaining less conservative stability conditions. Thus, some new methods have been developed for reducing conservatism, such as free-weighting matrix method [4–8], augmented LKF [9], discretized LKF [10], delay-partitioning method [12–18], and delay-slope-dependent method [19]. Some less conservative stability criteria were proposed in [6] by considering some useful terms which have been usually neglected in the previous literature and using the free-weighting matrices method. Recently, a novel method was proposed for Hopfield neural networks in [12], which divides the constant time delay interval [0,ℎ] into subintervals with the same size. This method utilizes more information about the delay interval [0,ℎ] to reduce the conservativeness. Very recently, by proposing the idea of dividing the delay interval with the weighted parameters, the weighting-delay-based stability criteria for neural networks with time-varying delay were investigated in [15]. The delay-partitioning method proved to be less conservative than most of the previous results, and the conservatism can be notably reduced by thinning the delay partitioning. However, the above methods suffer two common shortcomings. First, many matrix variables are introduced in the obtained results, which brings a large computational burden. Second, the information of neuron activation function is not adequately considered, which may lead to much conservatism.

In this paper, the problem of delay-dependent stability analysis for neural networks with time-varying delays is investigated. Different from the previous methods of [4–8, 12–18], no delay-partitioning methods or free-weighting matrix methods are utilized. Instead, by taking more information of states and activation functions as augmented vectors, an augmented Lyapunov-Krasovskii functional is proposed. Then, inspired by the results of [19, 42], a less conservative condition is derived to guarantee the asymptotical stablity of the considered systems. Finally, two numerical examples are given to indicate significant improvements over some existing results.

2. Problem Formulation

Consider the following neural networks with time-varying delay: 𝑥̇(𝑡)=−𝐶𝑥(𝑡)+𝐴𝑔(𝑥(𝑡))+𝐵𝑔(𝑥(𝑡−𝜏(𝑡)))+𝜇, (2.1) where 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)]^𝑇∈ℛ^𝑛 is the neuron state vector, 𝑔(𝑥(⋅))=[𝑔₁(𝑥₁(⋅)),𝑔₂(𝑥₂(⋅)),…,𝑔_𝑛(𝑥_𝑛(⋅))]^𝑇∈ℛ^𝑛 denotes the neuron activation function, and 𝜇=(𝜇₁,𝜇₂,…,𝜇_𝑛)^𝑇∈ℛ^𝑛 is a constant input vector. 𝐴,𝐵∈ℛ^𝑛×𝑛 are the connection weight matrix and the delayed connection weight matrix, respectively. 𝐶=diag(𝐶₁,𝐶₂,…,𝐶_𝑛) with 𝐶_𝑖>0,𝑖=1,2,…,𝑛. 𝜏(𝑡) is a time-varying continuous function that satisfies 0⩽𝜏(𝑡)⩽ℎ, 𝜏̇(𝑡)⩽𝑢, where ℎ and 𝑢 are constants. In addition, it is assumed that each neuron activation function in (2.1), 𝑔_𝑖(⋅),𝑖=1,2,…,𝑛, is bounded and satisfies the following condition: 𝑘⁻_𝑖⩽𝑔_𝑖(𝑥)−𝑔_𝑖𝑦𝑥−𝑦⩽𝑘⁺_𝑖,∀𝑥,𝑦∈𝑅,𝑥≠𝑦,𝑖=1,2,…,𝑛, (2.2) where 𝑘⁻_𝑖,𝑘⁺_𝑖,𝑖=1,2,…,𝑛 are constants.

Assuming that 𝑥^∗=[𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛]^𝑇 is the equilibrium point of (2.1) whose uniqueness has been given in [29] and using the transformation 𝑧(⋅)=𝑥(⋅)−𝑥^∗, system (2.1) can be converted to the following system: 𝑧̇(𝑡)=−𝐶𝑧(𝑡)+𝐴𝑓(𝑧(𝑡))+𝐵𝑓(𝑧(𝑡−𝜏(𝑡))), (2.3) where 𝑧(𝑡)=[𝑧₁(𝑡),𝑧₂(𝑡),…,𝑧_𝑛(𝑡)]^𝑇, 𝑓(𝑧(⋅))=[𝑓₁(𝑧₁(⋅)),𝑓₂(𝑧₂(⋅)),…,𝑓_𝑛(𝑧_𝑛(⋅))]^𝑇 and 𝑓_𝑖(𝑧_𝑖(⋅))=𝑔_𝑖(𝑧_𝑖(⋅)+𝑥^∗_𝑖)−𝑔_𝑖(𝑥^∗_𝑖),𝑖=1,2,…,𝑛. According to the inequality (2.2), one can obtain that 𝑘⁻_𝑖⩽𝑓_𝑖𝑧_𝑖(𝑡)𝑧_𝑖(𝑡)⩽𝑘⁺_𝑖𝑓_𝑖(0)=0,𝑖=1,2,…,𝑛. (2.4) Thus, under this assumption, the following inequality holds for any diagonal matrix 𝑄₄>0, 𝑧^𝑇(𝑡)𝐾𝑄₄𝐾𝑧(𝑡)−𝑓^𝑇(𝑧(𝑡))𝑄₄𝑓(𝑧(𝑡))⩾0, (2.5) where 𝐾=diag(𝑘₁,𝑘₂,…,𝑘_𝑛), 𝑘_𝑖=max(|𝑘⁻_𝑖|,|𝑘⁺_𝑖|).

Lemma 2.1 (see[43]). For any constant matrix 𝑍∈ℛ^𝑛×𝑛,𝑍=𝑍^𝑇>0, scalars ℎ₂>ℎ₁>0, such that the following integrations are well defined, then −ℎ₂−ℎ₁^𝑡−ℎ1_𝑡−ℎ2𝑥^𝑇(𝑠)𝑍𝑥(𝑠)𝑑𝑠⩽−^𝑡−ℎ1_𝑡−ℎ2𝑥^𝑇(𝑠)𝑑𝑠𝑍^𝑡−ℎ1_𝑡−ℎ2𝑥(𝑠)𝑑𝑠. (2.6)

3. Main Results

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

Theorem 3.1. For given scalars ℎ⩾0,𝑢, diagonal matrices 𝐾₁=diag(𝑘⁻₁,𝑘⁻₂,…,𝑘⁻_𝑛), 𝐾₂=diag(𝑘⁺₁,𝑘⁺₂,…,𝑘⁺_𝑛), the system (2.3) is globally asymptotically stable if there exist symmetric positive matrices 𝑃=[^𝑃𝑖𝑗]_4×4,𝑄=[^𝑄𝑖𝑗]_2×2,𝑋=[^𝑋𝑖𝑗]_2×2,𝑌=[^𝑌𝑖𝑗]_2×2,𝑄_𝑖(𝑖=1,2,3), positive diagonal matrices 𝑇₁,𝑇₂,𝑇₃,𝑄₄,Δ=diag(𝛿₁,𝛿₂,…,𝛿_𝑛),Λ=diag(𝜆₁,𝜆₂,…,𝜆_𝑛), and any matrices 𝑆_𝑖(𝑖=1,2,…,5) with appropriate dimensions, such that the following LMIs hold: 𝑌₁₁𝑌₁₂𝑆₁𝑆₂∗𝑌₂₂𝑆₃𝑆₄∗∗𝑌₁₁𝑌₁₂∗∗∗𝑌₂₂𝑄>0,₁𝑆₅∗𝑄₁>0,E𝒜^𝑇𝑅∗−𝑅<0, (3.1)(3.2)(3.3) where E𝐸=¹¹E₁₂E₁₃E₁₄E₁₅−𝑃₁₃E₁₇E₁₈E₁₉E_1,10E_1,11∗E₂₂E₂₃0E₂₅000000∗∗E₃₃00E₃₆−𝑃₂₂−𝑃₂₂−𝑃₂₃−𝑃₂₃−𝑃₂₄∗∗∗E₄₄E₄₅0E₄₇E₄₈E₄₉E_4,10E_4,11∗∗∗∗E₅₅0𝐵^𝑇𝑃₁₂𝐵^𝑇𝑃₁₂𝐵^𝑇𝑃₁₃𝐵^𝑇𝑃₁₃𝐵^𝑇𝑃₁₄∗∗∗∗∗E₆₆−𝑃^𝑇₂₃−𝑃^𝑇₂₃−𝑃₃₃−𝑃₃₃−𝑃₃₄∗∗∗∗∗∗E₇₇E₇₈E_7,9E_7,10−𝑃₄₄∗∗∗∗∗∗∗E₈₈E₈₉E_8,10−𝑃₄₄∗∗∗∗∗∗∗∗−𝑌₂₂−𝑆₄0∗∗∗∗∗∗∗∗∗−𝑌₂₂0∗∗∗∗∗∗∗∗∗∗−𝑄₃,𝒜=−𝐶00𝐴𝐵000000,𝑅=ℎ²𝑄₁+ℎ⁴4𝑄₂,E₁₁=−𝑃₁₁𝐶−𝐶𝑃₁₁+𝑃₁₂+𝑃^𝑇₁₂+ℎ𝑃₁₄+𝑃^𝑇₁₄+𝑄₁₁+2𝐾₁Λ𝐶−2𝐾₂Δ𝐶+𝑋₁₁+ℎ²𝑌₁₁−𝑄₁−ℎ²𝑄₂+𝐾𝑄₄𝐾−2𝐾₂𝑇₁𝐾₁+ℎ⁴4𝑄₃,E₁₂=𝑄₁−𝑆₅,E₁₃=−𝑃₁₂+𝑆₅,E₁₄=𝑃₁₁𝐴+𝑃₁₃+𝑄₁₂−𝐶Λ−𝐾₁Λ𝐴+𝐾₂Δ𝐴+𝐶Δ+𝑋₁₂+ℎ²𝑌₁₂+𝑇₁𝐾₁+𝐾₂,E₁₅=𝑃₁₁𝐵−𝐾₁Λ𝐵+𝐾₂EΔ𝐵,₁₇=−𝐶𝑃₁₂+𝑃₂₂+ℎ𝑃^𝑇₂₄−𝑃₁₄+ℎ𝑄₂,E₁₈=−𝐶𝑃₁₂+𝑃₂₂+ℎ𝑃^𝑇₂₄−𝑃₁₄+ℎ𝑄₂,E₁₉=−𝐶𝑃₁₃+𝑃₂₃+ℎ𝑃^𝑇₃₄,E_1,10=−𝐶𝑃₁₃+𝑃₂₃+ℎ𝑃^𝑇₃₄,E_1,11=−𝐶𝑃₁₄+𝑃₂₄+ℎ𝑃₄₄,E₂₂=−(1−𝑢)𝑋₁₁−2𝑄₁+𝑆₅+𝑆^𝑇₅−(1−𝑢)𝐾𝑄₄𝐾−2𝐾₂𝑇₂𝐾₁,E₂₃=𝑄₁−𝑆₅,E₂₅=−(1−𝑢)𝑋₁₂+𝑇₂𝐾₁+𝐾₂,E₃₃=−𝑄₁₁−𝑄₁−2𝐾₂𝑇₃𝐾₁,E₃₆=−𝑄₁₂+𝑇₃𝐾₁+𝐾₂,E₄₄=𝑄₂₂+Λ𝐴+𝐴^𝑇Λ−Δ𝐴−𝐴^𝑇Δ+𝑋₂₂+ℎ²𝑌₂₂−𝑄₄−2𝑇₁,E₄₅E=Λ𝐵−Δ𝐵,₄₇=𝐴^𝑇𝑃₁₂+𝑃^𝑇₂₃,E₄₈=𝐴^𝑇𝑃₁₂+𝑃^𝑇₂₃,E₄₉=𝐴^𝑇𝑃₁₃+𝑃₃₃,E_4,10=𝐴^𝑇𝑃₁₃+𝑃₃₃,E_4,11=𝐴^𝑇𝑃₁₄+𝑃₃₄,E₅₅=−(1−𝑢)𝑋₂₂+(1−𝑢)𝑄₄−2𝑇₂,E₆₆=−𝑄₂₂−2𝑇₃,E₇₇=−𝑃₂₄−𝑃^𝑇₂₄−𝑌₁₁−𝑄₂,E₇₈=−𝑃₂₄−𝑃^𝑇₂₄−𝑆₁−𝑄₂,E₇₉=−𝑃^𝑇₃₄−𝑌₁₂,E_7,10=−𝑃^𝑇₃₄−𝑆₂,E₈₈=−𝑃₂₄−𝑃^𝑇₂₄−𝑌₁₁−𝑄₂,E₈₉=−𝑃^𝑇₃₄−𝑆^𝑇₃,E_8,10=−𝑃^𝑇₃₄−𝑌₁₂. (3.4)

Proof. Construct a new Lyapunov functional as follow: 𝑉𝑧_𝑡=⁸_𝑖=1𝑉_𝑖𝑧_𝑡, (3.5) where 𝑉₁𝑧_𝑡=𝑧(𝑡)∫^𝑡_𝑡−ℎ𝑧(𝑠)∫𝑑𝑠_{𝑡𝑡−ℎ}𝑓(𝑧(𝑠∫))𝑑𝑠_0−ℎ∫^𝑡_𝑡+𝜃𝑧(𝑠)𝑑𝑠𝑑𝜃^𝑇𝑃₁₁𝑃₁₂𝑃₁₃𝑃₁₄∗𝑃₂₂𝑃₂₃𝑃₂₄∗∗𝑃₃₃𝑃₃₄∗∗∗𝑃₄₄𝑧(𝑡)∫^𝑡_𝑡−ℎ𝑧(𝑠)∫𝑑𝑠_{𝑡𝑡−ℎ}𝑓(𝑧(𝑠∫))𝑑𝑠_0−ℎ∫^𝑡_𝑡+𝜃𝑧(𝑠),𝑉𝑑𝑠𝑑𝜃₂𝑧_𝑡=^𝑡_𝑡−ℎ𝑧(𝑠)𝑓(𝑧(𝑠))^𝑇𝑄₁₁𝑄₁₂∗𝑄₂₂𝑧(𝑠)𝑓(𝑧(𝑠𝑉))𝑑𝑠,₃𝑧_𝑡=^𝑡_{(𝑡)𝑡−𝜏}𝑧(𝑠)𝑓(𝑧(𝑠))^𝑇𝑋₁₁𝑋₁₂∗𝑋₂₂𝑧(𝑠)𝑓(𝑧(𝑠𝑉))𝑑𝑠,₄𝑧_𝑡=2^𝑛_𝑖=1^𝑧𝑖₀^(𝑡)𝜆_𝑖𝑓_𝑖(𝑠)−𝑘⁻_𝑖𝑠𝑑𝑠+^𝑧𝑖₀^(𝑡)𝛿_𝑖𝑘⁺_𝑖𝑠−𝑓_𝑖(𝑠),𝑉𝑑𝑠₅𝑧_𝑡=ℎ⁰_−ℎ^𝑡_𝑡+𝜃𝑧(𝑠)𝑓(𝑧(𝑠))^𝑇𝑌₁₁𝑌₁₂∗𝑌₂₂𝑧(𝑠)𝑓(𝑧(𝑠𝑉))𝑑𝑠𝑑𝜃,₆𝑧_𝑡=ℎ⁰_−ℎ^𝑡_𝑡+𝜃𝑧̇^𝑇(𝑠)𝑄₁𝑧̇(𝑠)𝑉𝑑𝑠𝑑𝜃,₇𝑧_𝑡=ℎ²2⁰_−ℎ⁰_𝜃^𝑡_𝑡+𝜆𝑧̇^𝑇(𝑠)𝑄₂𝑧̇(𝑠)𝑉𝑑𝑠𝑑𝜆𝑑𝜃,₈𝑧_𝑡=ℎ²2⁰_−ℎ⁰_𝜃^𝑡_𝑡+𝜆𝑧^𝑇(𝑠)𝑄₃𝑧(𝑠)𝑉𝑑𝑠𝑑𝜆𝑑𝜃,₉𝑧_𝑡=^𝑡_{(𝑡)𝑡−𝜏}𝑧^𝑇(𝑠)𝐾𝑄₄𝐾𝑧(𝑠)−𝑓^𝑇(𝑧(𝑠))𝑄₄𝑓(𝑧(𝑠))𝑑𝑠. (3.6)

Remark 3.2. Since the terms 2∑^𝑛_𝑖=1𝛿_𝑖∫^𝑧𝑖₀^(𝑡)(𝑘⁺_𝑖𝑠−𝑓_𝑖(𝑠))𝑑𝑠 in 𝑉₄(𝑧_𝑡) and ∫^𝑡_{𝑡−𝜏(𝑡)}[𝑧^𝑇(𝑠)𝐾𝑄₄𝐾𝑧(𝑠)−𝑓^𝑇(𝑧(𝑠))𝑄₄𝑓(𝑧(𝑠))]𝑑𝑠 are 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 (2.4) is given by: 𝑉̇𝑧_𝑡=⁸_𝑖=1𝑉̇_𝑖𝑧_𝑡, (3.7) where 𝑉̇₁𝑧_𝑡𝑧(𝑡)∫=2^{𝑡𝑡−ℎ}𝑧(𝑠)∫𝑑𝑠_{𝑡𝑡−ℎ}𝑓(𝑧(𝑠∫))𝑑𝑠_0−ℎ∫^𝑡_𝑡+𝜃𝑧(𝑠)𝑑𝑠𝑑𝜃^𝑇𝑃₁₁𝑃₁₂𝑃₁₃𝑃₁₄∗𝑃₂₂𝑃₂₃𝑃₂₄∗∗𝑃₃₃𝑃₃₄∗∗∗𝑃₄₄𝑧̇(𝑡)𝑧(𝑡)()𝑓(𝑧(𝑡(𝑧((𝑡)−𝑧𝑡−ℎ))−𝑓𝑡−ℎ))ℎ𝑧−∫^𝑡_𝑡−ℎ𝑧(𝑠),𝑉̇𝑑𝑠₂𝑧_𝑡=𝑧(𝑡)𝑓(𝑧(𝑡))^𝑇𝑄₁₁𝑄₁₂∗𝑄₂₂𝑧(𝑡)𝑓(𝑧(𝑡−𝑧()))𝑓(𝑧(𝑡−ℎ𝑡−ℎ))^𝑇𝑄₁₁𝑄₁₂∗𝑄₂₂𝑧()𝑓(𝑧(,𝑉̇𝑡−ℎ𝑡−ℎ))₃𝑧_𝑡⩽𝑧(𝑡)𝑓(𝑧(𝑡))^𝑇𝑋₁₁𝑋₁₂∗𝑋₂₂𝑧(𝑡)𝑓(𝑧(𝑡−()𝑧((𝑡𝑓(𝑧((𝑡))1−𝑢𝑡−𝜏))𝑡−𝜏)))^𝑇𝑋₁₁𝑋₁₂∗𝑋₂₂𝑧((𝑡𝑓(𝑧((𝑡,𝑉̇𝑡−𝜏))𝑡−𝜏)))₄𝑧_𝑡=2𝑓(𝑧(𝑡))−𝐾₁𝑧(𝑡)^𝑇Λ𝑧̇(𝑡)+2𝐾₂𝑧(𝑡)−𝑓(𝑧(𝑡))^𝑇Δ𝑧̇(𝑡),=2𝑓(𝑧(𝑡))−𝐾₁𝑧(𝑡)^𝑇Λ(𝑡)(𝑧(𝑡(𝑧((𝑡𝐾−𝐶𝑧+𝐴𝑓))+𝐵𝑓𝑡−𝜏)))+2₂𝑧(𝑡)−𝑓(𝑧(𝑡))^𝑇Δ−𝐶𝑧(𝑡)+𝐴𝑓(𝑧(𝑡))+𝐵𝑓(𝑧(𝑡−𝜏(𝑡))). (3.8)(3.9)(3.10)(3.11) By the use of Lemma 2.1 and Theorem 3.1 in [42], one can obtain 𝑉̇₅𝑧_𝑡=ℎ²𝑧(𝑡)𝑓(𝑧(𝑡))^𝑇𝑌₁₁𝑌₁₂∗𝑌₂₂𝑧(𝑡)𝑓(𝑧(𝑡))−ℎ^𝑡_𝑡−ℎ𝑧(𝑠)𝑓(𝑧(𝑠))^𝑇𝑌₁₁𝑌₁₂∗𝑌₂₂𝑧(𝑠)𝑓(𝑧(𝑠))𝑑𝑠=ℎ²𝑧(𝑡)𝑓(𝑧(𝑡))^𝑇𝑌₁₁𝑌₁₂∗𝑌₂₂𝑧(𝑡)𝑓(𝑧(𝑡))−ℎ^𝑡_{𝑡−𝜏(𝑡)}𝑧(𝑠)𝑓(𝑧(𝑠))^𝑇𝑌₁₁𝑌₁₂∗𝑌₂₂𝑧(𝑠)𝑓(𝑧(𝑠))𝑑𝑠−ℎ^{𝑡−𝜏(𝑡)}_𝑡−ℎ𝑧(𝑠)𝑓(𝑧(𝑠))^𝑇𝑌₁₁𝑌₁₂∗𝑌₂₂𝑧(𝑠)𝑓(𝑧(𝑠))𝑑𝑠⩽ℎ²𝑧(𝑡)𝑓(𝑧(𝑡))^𝑇𝑌₁₁𝑌₁₂∗𝑌₂₂𝑧(𝑡)𝑓(𝑧(𝑡−ℎ))𝜏(𝑡)^𝑡_{(𝑡)𝑡−𝜏}𝑧(𝑠)𝑓(𝑧(𝑠))^𝑇𝑌×𝑑𝑠¹¹𝑌₁₂∗𝑌₂₂^{𝑡(𝑡)𝑡−𝜏}𝑧(𝑠)𝑓(𝑧(𝑠−ℎ))𝑑𝑠ℎ−𝜏(𝑡)^{𝑡−𝜏(𝑡)}_𝑡−ℎ𝑧(𝑠)𝑓(𝑧(𝑠))^𝑇𝑌𝑑𝑠¹¹𝑌₁₂∗𝑌₂₂^{𝑡−𝜏(𝑡)𝑡−ℎ}𝑧(𝑠)𝑓(𝑧(𝑠))𝑑𝑠⩽ℎ²𝑧(𝑡)𝑓(𝑧(𝑡))^𝑇𝑌₁₁𝑌₁₂∗𝑌₂₂𝑧(𝑡)𝑓(𝑧(𝑡−∫))^{𝑡(𝑡)𝑡−𝜏}𝑧(𝑠)∫𝑑𝑠_{𝑡(𝑡)𝑡−𝜏}𝑓(𝑧(𝑠∫))𝑑𝑠_{𝑡−𝜏(𝑡)𝑡−ℎ}𝑧(𝑠)∫𝑑𝑠_{𝑡−𝜏(𝑡)𝑡−ℎ}𝑓(𝑧(𝑠))𝑑𝑠^𝑇×𝑌₁₁𝑌₁₂𝑆₁𝑆₂∗𝑌₂₂𝑆₃𝑆₄∗∗𝑌₁₁𝑌₁₂∗∗∗𝑌₂₂∫^𝑡_{(𝑡)𝑡−𝜏}𝑧(𝑠)∫𝑑𝑠_{𝑡(𝑡)𝑡−𝜏}𝑓(𝑧(𝑠∫))𝑑𝑠_{𝑡−𝜏(𝑡)𝑡−ℎ}𝑧(𝑠)∫𝑑𝑠_{𝑡−𝜏(𝑡)𝑡−ℎ}𝑓(𝑧(𝑠,))𝑑𝑠 (3.12)(3.13) where ^{𝑌11𝑌12𝑆1𝑆2∗𝑌22𝑆3𝑆4}_{∗∗𝑌11𝑌12∗∗∗𝑌22}>0 should be satisfied. Similar to (3.12), one can obtain 𝑉̇₆𝑧_𝑡⩽ℎ²𝑧̇^𝑇(𝑡)𝑄₁𝑧̇(𝑡)−∫^𝑡_{(𝑡)𝑡−𝜏}𝑧̇(𝑠)∫𝑑𝑠_{𝑡−𝜏(𝑡)𝑡−ℎ}𝑧̇(𝑠)𝑑𝑠^𝑇𝑄₁𝑆₅∗𝑄₁∫^𝑡_{(𝑡)𝑡−𝜏}𝑧̇(𝑠)∫𝑑𝑠_{𝑡−𝜏(𝑡)𝑡−ℎ}𝑧̇(𝑠),𝑑𝑠 (3.14) where ^𝑄1𝑆5_∗𝑄1>0 should be satisfied.

Consider the following: 𝑉̇₇𝑧_𝑡=ℎ⁴4𝑧̇^𝑇(𝑡)𝑄₂𝑧̇(𝑡)−ℎ²2⁰_−ℎ^𝑡_𝑡+𝜃𝑧̇^𝑇(𝑠)𝑄₂𝑧̇(𝑠)⩽ℎ𝑑𝑠𝑑𝜃₄4𝑧̇^𝑇(𝑡)𝑄₂𝑧̇(𝑡)−⁰_−ℎ^𝑡_𝑡+𝜃𝑧̇(𝑠)𝑑𝑠𝑑𝜃^𝑇𝑄₂⁰_−ℎ^𝑡_𝑡+𝜃𝑧̇(𝑠),=ℎ𝑑𝑠𝑑𝜃₄4𝑧̇^𝑇(𝑡)𝑄₂𝑧̇(𝑡)−ℎ𝑧(𝑡)−^𝑡_𝑡−ℎ𝑧(𝑠)𝑑𝑠^𝑇𝑄₂ℎ𝑧(𝑡)−^𝑡_𝑡−ℎ𝑧(𝑠),𝑉̇𝑑𝑠₈𝑧_𝑡=ℎ⁴4𝑧^𝑇(𝑡)𝑄₃𝑧(𝑡)−ℎ²2⁰_−ℎ^𝑡_𝑡+𝜃𝑧^𝑇(𝑠)𝑄₃𝑧(𝑠)⩽ℎ𝑑𝑠𝑑𝜃₄4𝑧^𝑇(𝑡)𝑄₃𝑧(𝑡)−⁰_−ℎ^𝑡_𝑡+𝜃𝑧(𝑠)𝑑𝑠𝑑𝜃^𝑇𝑄₃⁰_−ℎ^𝑡_𝑡+𝜃𝑧(𝑠)𝑉̇𝑑𝑠𝑑𝜃₉𝑧_𝑡⩽𝑧^𝑇(𝑡)𝐾𝑄₄𝐾𝑧(𝑡)−𝑓^𝑇(𝑧(𝑡))𝑄₄𝑓(𝑧(𝑡))−(1−𝑢)𝑧^𝑇(𝑡−𝜏(𝑡))𝐾𝑄₄((𝑡+()𝑓𝐾𝑧𝑡−𝜏))1−𝑢^𝑇(𝑧(𝑡−𝜏(𝑡)))𝑄₄𝑓(𝑧(𝑡−𝜏(𝑡))). (3.15)(3.16)(3.17) Furthermore, there exist positive diagonal matrices 𝑇₁,𝑇₂,𝑇₃, such that the following inequalities hold based on (2.4): −2𝑓^𝑇(𝑧(𝑡))𝑇₁𝑓(𝑧(𝑡))+2𝑧^𝑇(𝑡)𝑇₁𝐾₁+𝐾₂𝑓(𝑧(𝑡))−2𝑧^𝑇(𝑡)𝐾₂𝑇₁𝐾₁𝑧(𝑡)⩾0,−2𝑓^𝑇(𝑧(𝑡−𝜏(𝑡)))𝑇₂𝑓(𝑧(𝑡−𝜏(𝑡)))+2𝑧^𝑇(𝑡−𝜏(𝑡))𝑇₂𝐾₁+𝐾₂𝑓(𝑧(𝑡−𝜏(𝑡)))−2𝑧^𝑇(𝑡−𝜏(𝑡))𝐾₂𝑇₂𝐾₁𝑧(𝑡−𝜏(𝑡))⩾0,−2𝑓^𝑇(𝑧(𝑡−ℎ))𝑇₃𝑓(𝑧(𝑡−ℎ))+2𝑧^𝑇(𝑡−ℎ)𝑇₃𝐾₁+𝐾₂𝑓(𝑧(𝑡−ℎ)),−2𝑧^𝑇(𝑡−ℎ)𝐾₂𝑇₃𝐾₁𝑧(𝑡−ℎ)⩾0. (3.18)(3.19)(3.20) From (3.8)–, one can obtain that 𝑉̇𝑧_𝑡⩽𝜁^𝑇(𝑡)𝐸+𝒜^𝑇𝑅𝒜𝜁(𝑡), (3.21) where 𝜁^𝑇(𝑡)=𝑧^𝑇(𝑡)𝑧^𝑇(𝑡−𝜏(𝑡))𝑧^𝑇(𝑡−ℎ)𝑓^𝑇(𝑧(𝑡))𝑓𝑧^𝑇(𝑡−𝜏(𝑡))𝑓𝑧^𝑇(𝑡−ℎ_{𝑡(𝑡)𝑡−𝜏})𝑧^𝑇(𝑠)𝑑𝑠^{𝑡−𝜏(𝑡)}_𝑡−ℎ𝑧^𝑇(𝑠)𝑑𝑠^𝑡_{𝑡−𝜏(𝑡)}𝑓^𝑇(𝑧(𝑠))d𝑠^{𝑡−𝜏(𝑡)}_𝑡−ℎ𝑓^𝑇(𝑧(𝑠))𝑑𝑠_{0−ℎ𝑡𝑡+𝜃}𝑧^𝑇(𝑠)𝑑𝑠𝑑𝜃. (3.22)

If 𝐸+𝒜^𝑇𝑅𝒜<0, then there exists a scalar 𝜀>0, such that 𝑉̇𝑧_𝑡⩽−𝜀𝜁^𝑇(𝑡)𝜁(𝑡)⩽−𝜀𝑧^𝑇(𝑡)𝑧(𝑡)<0,∀𝑧(𝑡)≠0. (3.23) Thus, according to [44], system (2.1) is globally asymptotically stable. By Schur complement, 𝐸+𝒜^𝑇𝑅𝒜<0 is equivalent to (3.3), this completes the proof.

Remark 3.3. By taking the states ∫^𝑡_{𝑡−𝜏(𝑡)}𝑓^𝑇(𝑧(𝑠))𝑑𝑠,∫^{𝑡−𝜏(𝑡)}_𝑡−ℎ𝑓^𝑇(𝑧(𝑠))𝑑𝑠, as augmented variables, the stability condition in Theorem 3.1 utilizes more information about 𝑓(𝑧(𝑡)) on state variables, which may lead to less conservative results.

Remark 3.4. Recently, the reciprocally convex optimization technique [42] is used to reduce the conservatism of stability criteria for systems with time-varying delays. Motivated by this work, the proposed method of [42] was utilized in (3.12) and (3.14), which have potential to yield less conservative conditions. However, an augmented vector with ∫^𝑡_{𝑡−𝜏(𝑡)}𝑧^𝑇(𝑠)𝑑𝑠,∫^{𝑡−𝜏(𝑡)}_𝑡−ℎ𝑧^𝑇(𝑠)𝑑𝑠,∫^𝑡_{𝑡−𝜏(𝑡)}𝑓^𝑇(𝑧(𝑠))𝑑𝑠,∫^{𝑡−𝜏(𝑡)}_𝑡−ℎ𝑓^𝑇(𝑧(𝑠))𝑑𝑠,∫⁰_−ℎ∫^𝑡_𝑡+𝜃𝑧^𝑇(𝑠)𝑑𝑠𝑑𝜃 was used, which is different from the method of [42].
In many cases, 𝑢 is unknown. Considering this situation, a rate-independent corollary for the delay 𝜏(𝑡) satisfying 0≤𝜏(𝑡)≤ℎ is derived by setting 𝑋=0,𝑄₄=0 in the proof of Theorem 3.1.

Corollary 3.5. For given scalar ℎ⩾0, diagonal matrices 𝐾₁=diag(𝑘⁻₁,𝑘⁻₂,…,𝑘⁻_𝑛), 𝐾₂=diag(𝑘⁺₁,𝑘⁺₂,…,𝑘⁺_𝑛), the system (2.3) is globally asymptotically stable if there exist symmetric positive matrices 𝑃=[^𝑃𝑖𝑗]_4×4,𝑄=[^𝑄𝑖𝑗]_2×2,𝑌=[^𝑌𝑖𝑗]_2×2,𝑄_𝑖(𝑖=1,2,3), positive diagonal matrices 𝑇₁,𝑇₂,𝑇₃,Δ=diag(𝛿₁,𝛿₂,…,𝛿_𝑛),Λ=diag(𝜆₁,𝜆₂,…,𝜆_𝑛), and any matrices 𝑆_𝑖(𝑖=1,2,…,5) with appropriate dimensions, such that (3.1), (3.2) and the following LMI hold: E𝒜^𝑇𝑅∗−𝑅<0, (3.24) where E=E₁₁E₁₂E₁₃E₁₄E₁₅−𝑃₁₃E₁₇E₁₈E₁₉E_1,10E_1,11∗E₂₂E₂₃0E₂₅000000∗∗E₃₃00E₃₆−𝑃₂₂−𝑃₂₂−𝑃₂₃−𝑃₂₃−𝑃₂₄∗∗∗E₄₄E₄₅0E₄₇E₄₈E₄₉E_4,10E_4,11∗∗∗∗−2𝑇₂0𝐵^𝑇𝑃₁₂𝐵^𝑇𝑃₁₂𝐵^𝑇𝑃₁₃𝐵^𝑇𝑃₁₃𝐵^𝑇𝑃₁₄∗∗∗∗∗E₆₆−𝑃^𝑇₂₃−𝑃^𝑇₂₃−𝑃₃₃−𝑃₃₃−𝑃₃₄∗∗∗∗∗∗E₇₇E₇₈E_7,9E_7,10−𝑃₄₄∗∗∗∗∗∗∗E₈₈E₈₉E_8,10−𝑃₄₄∗∗∗∗∗∗∗∗−𝑌₂₂−𝑆₄0∗∗∗∗∗∗∗∗∗−𝑌₂₂0∗∗∗∗∗∗∗∗∗∗−𝑄₃,E₁₁=−𝑃₁₁𝐶−𝐶𝑃₁₁+𝑃₁₂+𝑃^𝑇₁₂+ℎ𝑃₁₄+𝑃^𝑇₁₄+𝑄₁₁+2𝐾₁Λ𝐶−2𝐾₂Δ𝐶+ℎ²𝑌₁₁−𝑄₁−ℎ²𝑄₂+𝐾𝑄₄𝐾−2𝐾₂𝑇₁𝐾₁+ℎ⁴4𝑄₃,E₁₄=𝑃₁₁𝐴+𝑃₁₃+𝑄₁₂−𝐶Λ−𝐾₁Λ𝐴+𝐾₂Δ𝐴+𝐶Δ+ℎ²𝑌₁₂+𝑇₁𝐾₁+𝐾₂,E₂₂=−2𝑄₁+𝑆₅+𝑆^𝑇₅−2𝐾₂𝑇₂𝐾₁,E₂₅=𝑇₂𝐾₁+𝐾₂,E₄₄=𝑄₂₂+Λ𝐴+𝐴^𝑇Λ−Δ𝐴−𝐴^𝑇Δ+ℎ²𝑌₂₂−𝑄₄−2𝑇₁. (3.25) The other E_𝑖𝑗 is defined in Theorem 3.1.

4. Numerical Examples

In this section, two numerical examples are given to demonstrate the effectiveness of the proposed method.

Example 4.1. Consider the system (2.3) with the following parameters: 𝐶=,1.276900000.623100000.923000000.4480,𝐴=−0.03730.4852−0.33510.2336−1.60330.5988−0.32241.23520.3394−0.0860−0.3824−0.5785−0.13110.3253−0.9534−0.5015𝐵=0.8674−1.2405−0.53250.02200.0474−0.91640.03600.98161.84952.6117−0.37880.8428−2.04130.51791.1734−0.2775,𝐾₁{},𝐾=diag0,0,0,0₂{},𝑓=diag0.1137,0.1279,0.7994,0.2368₁(𝑠)=0.05685(|𝑠+1|−|𝑠−1|),𝑓₂(𝑠)|−|,𝑓=0.06395(|𝑠+1𝑠−1|)₃(𝑠)|−|,𝑓=0.3997(|𝑠+1𝑠−1|)₄(𝑠)=0.1184(|𝑠+1|−|𝑠−1|). (4.1) The upper bounds of ℎ for different 𝑢 are derived by Theorem 3.1 in our paper and the results in [13–16] are listed in Table 1. According to Table 1, this example shows that the stability condition in this paper gives much less conservative results than those in the literature. For ℎ=2.7098, the global asymptotic stability with the initial state(−0.2,0.3,−0.4,0.2)^𝑇 is shown in Figure 1.

Table 1 

Allowable upper bound of ℎ for different 𝑢.

Figure 1 

The dynamical behavior of 𝑧(𝑡).

Example 4.2. Consider the system (2.3) with the following parameters: ()𝐾𝐶=diag2,2,𝐴=11−1−1,𝐵=0.88111₁,diag{0,0},𝐾₂=diag{0.4,0.8}. (4.2)(4.3)
Our purpose is to estimate the allowable upper bounds delay ℎ under different 𝑢 such that the system (2.3) is globally asymptotically stable. According to the Table 2, this example is given to indicate significant improvements over some existing results.

Table 2 

Allowable upper bound of ℎ for different 𝑢.

5. Conclusions

In this paper, a new Lyapunov functional was proposed to investigate the stability of neural networks with time-varying delays. Some improved generalized delay-dependent stability criteria have been established. The obtained criteria are less conservative because a convex optimization approach is considered. Finally, two numerical examples have shown that these new stability criteria are less conservative than some existing ones in the literature.

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 the Fundamental Research Funds for the Central Universities under Grant ZYGX2011YB029.