Improved Stability Analysis for Neural Networks with Time-Varying Delay

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 [1–3], and delay-dependent ones [4–37]. 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 [10–12] and time-varying delay [13–16], 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 0⩽𝜏(𝑡)⩽ℎ. When the delay interval [0,𝜏(𝑡)] is divided into some subintervals, the delay interval [0,ℎ] 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: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, 𝜏.(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:0⩽𝑔^𝑖(𝑥)−𝑔_𝑖(𝑦)𝑥−𝑦⩽𝑘^𝑖,∀𝑥,𝑦∈𝑅,𝑥≠𝑦,𝑖=1,2,…,𝑛,(2.2) where 𝑘_𝑖,𝑖=1,2,…,𝑛 are positive constants.

Assuming that 𝑥^∗=[𝑥^∗1,𝑥^∗2,…,𝑥^∗𝑛]^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: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 that0⩽𝑓^𝑖𝑧_𝑖(𝑡)𝑧_𝑖(𝑡)⩽𝑘^𝑖,𝑓_𝑖(0)=0,𝑖=1,2,…,𝑛.(2.4) Thus, under this assumption, the following inequality holds for any diagonal matrix 𝑄>0,𝑧^𝑇(𝑡)𝐾𝑄𝐾𝑧(𝑡)−𝑓^𝑇(𝑧(𝑡))𝑄𝑓(𝑧(𝑡))⩾0,(2.5) where 𝐾=diag(𝑘₁,𝑘₂,…,𝑘_𝑛).

Lemma 2.1 (see [38]). 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 new delay-dependent stability criterion is obtained.

Theorem 3.1. For given scalars 𝐾=diag(𝑘₁,𝑘₂,…,𝑘_𝑛), ℎ⩾0,𝑢,and0<𝛼<1, the system (2.3) is globally asymptotically stable if there exist symmetric positive matrices 𝑃=[^𝑃_𝑖𝑗]_3×3,𝑄_𝑖(𝑖=1,2,…,6), 𝑅_𝑖(𝑖=1,2,…,6), positive diagonal matrices 𝑇₁,𝑇₂,𝑄,Δ=diag(𝛿₁,𝛿₂,…,𝛿_𝑛),Λ=diag(𝜆₁,𝜆₂,…,𝜆_𝑛), and any matrices 𝑃₁,𝑃₂,𝑁_𝑖,𝑀_𝑖,𝐿_𝑖,𝐻_𝑖,𝑍_𝑖,𝑆_𝑖,𝑈_𝑖,𝑉_𝑖,𝑊_𝑖(𝑖=1,2,3) with appropriate dimensions, such that the following LMIs hold: 𝐸₁=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝐸𝛼ℎ𝑁𝛼²ℎ²2𝐻1−𝛼²ℎ²∗−𝛼ℎ𝑅₃2𝑀∗∗−𝛼₂00ℎ²2𝑅⁵∗∗∗−1−𝛼0₂ℎ²2𝑅⁶𝐸⎤⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.1)₂=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐸(1−𝛼)𝛼ℎ𝐿𝛼²ℎ𝑆𝛼²ℎ²2𝐻1−𝛼²ℎ²∗−(1−𝛼)𝛼ℎ𝑅₃2𝑀∗∗−𝛼₂000∗∗∗ℎ𝑅₃−𝛼₂00ℎ²2𝑅⁵∗∗∗∗−1−𝛼0₂ℎ²2𝑅⁶Φ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.2)₁=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣Φ(1−𝛼)𝛼ℎ𝑈(1−𝛼)ℎ𝑊𝛼²ℎ²2𝐻1−𝛼²ℎ²∗−(1−𝛼)𝛼ℎ𝑅₃2𝑀∗∗−(1−0𝛼)ℎ𝑅₄00∗∗∗−𝛼₂00ℎ²2𝑅⁵∗∗∗∗−1−𝛼0₂ℎ²2𝑅⁶Φ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.3)₂=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣Φ(1−𝛼)𝛼ℎ𝑉(1−𝛼)ℎ𝑍𝛼²ℎ²2𝐻1−𝛼²ℎ²∗−(1−𝛼)𝛼ℎ𝑅₃2𝑀∗∗−(1−0𝛼)ℎ𝑅₄00∗∗∗−𝛼₂00ℎ²2𝑅⁵∗∗∗∗−1−𝛼0₂ℎ²2𝑅⁶⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.4) where 𝐸=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐸¹¹𝐸₁₂𝐸₁₃𝐸₁₄0−𝑃₁₃𝐸₁₇𝑃₁𝐴+𝐾𝑇₁𝑃₁𝐵𝑃₂₂−𝐻₁𝑃₂₃−𝑀₁∗𝐸₂₂𝐸₂₃−𝑁₂00000−𝐻₂−𝑀₂∗∗𝐸₃₃−𝑁₃0000𝐾𝑇₂−𝐻₃−𝑀₃∗∗∗𝐸₄₄0𝑅⁴(1−𝛼)ℎ000−𝑃²²+𝑃^𝑇23−𝑃₂₃+𝑃₃₃∗∗∗∗∗∗∗∗−∗𝑄₅𝐸0₆₆00000000−𝑃^𝑇23−𝑃₃₃∗∗∗∗∗∗𝐸₇₇𝑃₂𝐴+Λ−Δ𝑃₂𝐵𝑃₁₂𝑃₁₃∗∗∗∗∗∗∗∗∗∗∗∗∗∗𝑄₆−2∗𝑇₁−𝑄𝐸0₉₉00∗∗∗∗∗∗∗∗∗−0𝑅₁0∗∗∗∗∗∗∗∗∗∗−0𝑅₂Φ=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝Φ₁₁⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,Φ₁₂Φ₁₃Φ₁₄𝑉₁+𝑅³𝛼²ℎ−𝑃¹³−𝑊₁Φ₁₇𝑃₁𝐴+𝐾𝑇₁𝑃₁𝐵𝑃₂₂−𝐻₁𝑃₂₃−𝑀₁∗Φ₂₂Φ₂₃−𝑈₂+𝑍₂𝑉₂−𝑊₂000−𝐻₂−𝑀₂∗∗Φ₃₃−𝑈₃+𝑍₃𝑉₃−𝑊₃00𝐾𝑇₂−𝐻₃−𝑀₃∗∗∗−𝑄₃00000−𝑃₂₂+𝑃^𝑇23−𝑃₂₃+𝑃₃₃∗∗∗∗−𝑄₅−𝑅³∗∗∗∗∗𝛼²ℎ−0𝑄₄00000000−𝑃^𝑇23−𝑃₃₃∗∗∗∗∗∗Φ₇₇𝑃₂𝐴+Λ−Δ𝑃₂𝐵𝑃₁₂𝑃₁₃∗∗∗∗∗∗∗∗∗∗∗∗∗∗𝑄₆−2∗𝑇₁−𝑄Φ0₉₉00∗∗∗∗∗∗∗∗∗−0𝑅₁0∗∗∗∗∗∗∗∗∗∗−0𝑅₂𝐸₁₁⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=𝑃₁₂+𝑃^𝑇12+𝑄₁+𝑄₂+𝑄₃+𝑄₄+𝑄₅+𝛼²ℎ²𝑅₁+(1−𝛼)²ℎ²𝑅₂+𝑆₁+𝑆^𝑇1−𝑃₁𝐶−𝐶𝑃^𝑇1𝐸+𝛼ℎ𝐻₁+𝐻^𝑇1+(1−𝛼)ℎ𝑀₁+𝑀^𝑇1+𝐾𝑄𝐾,₁₂𝐸₁₃=𝐿₁+𝑆^𝑇2−𝑆₁+𝛼ℎ𝐻^𝑇2+(1−𝛼)ℎ𝑀^𝑇2,𝐸₁₄=𝑁₁−𝐿₁+𝑆^𝑇3+𝛼ℎ𝐻^𝑇3+(1−𝛼)ℎ𝑀^𝑇3,𝐸₁₇=−𝑃₁₂+𝑃₁₃−𝑁₁,𝐸₂₂=𝑃₁₁+𝐾Δ−𝑃₁−𝐶𝑃^𝑇2,𝐸₂₃=−(1−𝛼𝑢)𝑄₁+𝐿₂+𝐿^𝑇2−𝑆₂−𝑆^𝑇2,𝐸₃₃=−𝐿₂+𝐿^𝑇3−𝑆^𝑇3+𝑁₂,𝐸=−(1−𝑢)𝑄₂+𝑁₃+𝑁^𝑇3−𝐿₃−𝐿^𝑇3−(1−𝑢)𝐾𝑄𝐾,₄₄=−𝑄₃−𝑅⁴𝐸(1−𝛼)ℎ₆₆,=−𝑄₄−𝑅⁴𝐸(1−𝛼)ℎ₇₇,=𝛼ℎ𝑅₃+(1−𝛼)ℎ𝑅₄+𝛼²ℎ²2𝑅⁵+1−𝛼²ℎ²2𝑅⁶𝐸₉₉−𝑃₂−𝑃^𝑇2,Φ=−(1−𝑢)𝑄₆−2𝑇₂+(1−𝑢)𝑄,₁₁=𝑃₁₂+𝑃^𝑇12+𝑄₁+𝑄₂+𝑄₃+𝑄₄+𝑄₅+𝛼²ℎ²𝑅₁+(1−𝛼)²ℎ²𝑅₂−𝑃₁𝐶−𝐶𝑃^𝑇1+𝛼ℎ𝐻₁+𝐻^𝑇1+(1−𝛼)ℎ𝑀₁+𝑀^𝑇1−𝑅³Φ𝛼²ℎ+𝐾𝑄𝐾,₁₂Φ₁₃=𝑈₁−𝑉₁+𝛼ℎ𝐻^𝑇2+(1−𝛼)ℎ𝑀^𝑇2,Φ₁₄=𝑊₁−𝑍₁+𝛼ℎ𝐻^𝑇3+(1−𝛼)ℎ𝑀^𝑇3,Φ₁₇=−𝑃₁₂+𝑃₁₃−𝑈₁+𝑍₁,Φ₂₂=𝑃₁₁+𝐾Δ−𝑃₁−𝐶𝑃^𝑇2,Φ₂₃=−(1−𝛼𝑢)𝑄₁+𝑈₂+𝑈^𝑇2−𝑉₂−𝑉^𝑇2,Φ₃₃=𝑈^𝑇3−𝑉^𝑇3+𝑊₂−𝑍₂,Φ=−(1−𝑢)𝑄₂+𝑊₃+𝑊^𝑇3−𝑍₃−𝑍^𝑇3−(1−𝑢)𝐾𝑄𝐾,₇₇=𝛼ℎ𝑅₃+(1−𝛼)ℎ𝑅₄+𝛼²ℎ²2𝑅⁵+1−𝛼²ℎ²2𝑅⁶Φ₉₉−𝑃₂−𝑃^𝑇2,𝑁==−𝑁(1_𝑇1−𝑢)𝑄₆−2𝑇₂+(1−𝑢)𝑄,𝐿=𝐿_𝑇1𝑁^𝑇2𝑁^𝑇300000000^𝑇,𝑆=𝑆_𝑇1𝐿^𝑇2𝐿^𝑇300000000^𝑇,𝐻=𝐻_𝑇1𝑆^𝑇2𝑆^𝑇300000000^𝑇,𝑀=𝑀_𝑇1𝐻^𝑇2𝐻^𝑇300000000^𝑇,𝑈=𝑈_𝑇1𝑀^𝑇2𝑀^𝑇300000000^𝑇,𝑉=𝑉_𝑇1𝑈^𝑇2𝑉^𝑇300000000^𝑇,𝑊=𝑊_𝑇1𝑉^𝑇2𝑉^𝑇300000000^𝑇,𝑍=𝑍_𝑇1𝑊^𝑇2𝑊^𝑇300000000^𝑇,𝑍^𝑇2𝑍^𝑇300000000^𝑇.(3.5)

Proof. Construct a new class of Lyapunov functional candidate as follows: 𝑉(𝑧(𝑡))=⁶_𝑖=1𝑉_𝑖(𝑧(𝑡)),(3.6) where 𝑉₁(𝑧(𝑡))=𝜉^𝑇(𝑡)𝑃𝜉(𝑡)+2^𝑛_𝑖=1𝜆_𝑖^{𝑧𝑖0(𝑡)}𝑓_𝑖(𝑠)𝑑𝑠+2^𝑛_𝑖=1𝛿_𝑖^{𝑧𝑖0(𝑡)}𝑘_𝑖𝑉𝑠−𝑓_𝑖(𝑠)𝑑𝑠,₂(𝑧(𝑡))=+^{𝑡𝑡−𝛼𝜏(𝑡)}𝑧^𝑇(𝑠)𝑄₁𝑧(𝑠)𝑑𝑠+^{𝑡𝑡−𝜏(𝑡)}𝑧^𝑇(𝑠)𝑄₂𝑧(𝑠)𝑑𝑠+^{𝑡𝑡−𝛼ℎ}𝑧^𝑇(𝑠)𝑄₃𝑧(𝑠)𝑑𝑠_{𝑡𝑡−ℎ}𝑉𝑧^𝑇(𝑠)𝑄₄𝑧(𝑠)𝑑𝑠+^{𝑡𝑡−𝛼}_2ℎ𝑧^𝑇(𝑠)𝑄₅𝑧(𝑠)𝑑𝑠+^{𝑡𝑡−𝜏(𝑡)}𝑓^𝑇(𝑧(𝑠))𝑄₆𝑓(𝑧(𝑠))𝑑𝑠,₃(𝑧(𝑡))=𝛼ℎ^0−𝛼ℎ^{𝑡𝑡+𝜃}𝑧^𝑇(𝑠)𝑅₁𝑧(𝑠)𝑑𝑠d𝜃+(1−𝛼)ℎ^{−𝛼ℎ−ℎ}^{𝑡𝑡+𝜃}𝑉𝑧^𝑇(𝑠)𝑅₂𝑧(𝑠)𝑑𝑠𝑑𝜃,₄(𝑧(𝑡))=^0−𝛼ℎ^{𝑡𝑡+𝜃}𝑧̇^𝑇(𝑠)𝑅₃𝑧̇(𝑠)𝑑𝑠𝑑𝜃+^{−𝛼ℎ−ℎ}^{𝑡𝑡+𝜃}𝑉𝑧̇^𝑇(𝑠)𝑅₄𝑧̇(𝑠)𝑑𝑠𝑑𝜃,₅(𝑧(𝑡))=^0−𝛼ℎ^0𝜃^{𝑡𝑡+𝜆}𝑧̇^𝑇(𝑠)𝑅₅𝑧̇(𝑠)𝑑𝑠𝑑𝜆𝑑𝜃+^{−𝛼ℎ−ℎ}^0𝜃^{𝑡𝑡+𝜆}𝑉𝑧̇^𝑇(𝑠)𝑅₆𝑧̇(𝑠)𝑑𝑠𝑑𝜆𝑑𝜃,₆(𝑧(𝑡))=^{𝑡𝑡−𝜏(𝑡)}𝑧^𝑇(𝑠)𝐾𝑄𝐾𝑧(𝑠)−𝑓^𝑇(𝑧(𝑠))𝑄𝑓(𝑧(𝑠))𝑑𝑠,(3.7) where 𝜉^𝑇(𝑡)=𝑧^𝑇(𝑡)^{𝑡𝑡−𝛼ℎ}𝑧^𝑇(𝑠)𝑑𝑠^{𝑡−𝛼ℎ𝑡−ℎ}𝑧^𝑇(𝑠)𝑑𝑠,0<𝛼<1.(3.8)

Remark 3.2. Since the term 2∑^𝑛𝑖=1𝛿_𝑖∫^{𝑧𝑖0(𝑡)}(𝑘_𝑖𝑠−𝑓_𝑖(𝑠))𝑑𝑠 in 𝑉₁(𝑧(𝑡)) and 𝑉₆(𝑧(𝑡))=∫^{𝑡𝑡−𝜏(𝑡)}[𝑧^𝑇(𝑠)𝐾𝑄𝐾𝑧(𝑠)−𝑓^𝑇(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 by𝑉̇(𝑧(𝑡))=⁶_𝑖=1𝑉_𝑖̇(𝑧(𝑡)),(3.9) where 𝑉𝑉₁̇̇(𝑧(𝑡))=2𝜉^𝑇(𝑡)𝑃⎡⎢⎢⎢⎢⎣𝑧(𝑧𝑡(−𝑡)𝛼−ℎ𝑧)𝑧̇(−(𝑡𝑡)𝑧−(𝑡𝛼−ℎ)ℎ)⎤⎥⎥⎥⎥⎦+2𝑓^𝑇(𝑧(𝑡))Λ𝑧̇(𝑡)+2(𝐾𝑧(𝑡))−𝑓(𝑧(𝑡))^𝑇Δ𝑧̇(𝑡),₂(𝑧(𝑡))=−𝑧^𝑇(1(𝑡−)𝛼𝑄𝑢₁)𝑧+_𝑇𝑄₂+𝑄₃+𝑄₄+𝑄₅𝑧(𝑡)+𝑓^𝑇(𝑧(𝑡))𝑄₆𝑓(𝑧(𝑡))−𝑧_𝑇(𝑡−𝛼𝜏(𝑡))𝑄₁𝑧(𝑡−𝛼𝜏(𝑡))−(1−𝑢)𝑧^𝑇(𝑡−𝜏(𝑡))𝑄₂𝑧(𝑡−𝜏(𝑡))−(1−(𝑡𝑢−)𝑓𝛼ℎ_𝑇)𝑄₃𝑧(𝑡−𝛼ℎ)−𝑧^𝑇(𝑡−ℎ)𝑄₄𝑧(𝑡−ℎ)−𝑧^𝑇𝑡−𝛼²ℎ𝑄₅𝑧𝑡−𝛼²ℎ(𝑧(𝑡−𝜏(𝑡)))𝑄₆𝑓(𝑧(𝑡−𝜏(𝑡))).(3.10) Using Lemma 2.1, one can obtain that 𝑉₃̇(𝑧(𝑡))=−𝛼²(ℎ1²−𝑧^𝑇𝛼()𝑡ℎ)𝑅₁𝑧_{𝑡−𝛼ℎ𝑡−ℎ}(𝑡)+(1−𝛼)²ℎ²𝑧^𝑇(𝑡)𝑅₂𝑧(𝑡)−𝛼ℎ^{𝑡𝑡−𝛼ℎ}𝑧^𝑇(𝑠)𝑅₁𝑧(𝑠)𝑑𝑠⩽𝛼₂𝑧^𝑇(𝑠)𝑅₂𝑧(𝑠)𝑑𝑠−ℎ²𝑧^𝑇(𝑡)𝑅₁𝑧(𝑡)+(1−𝛼)²ℎ²𝑧^𝑇(𝑡)𝑅₂𝑧(𝑡)_{𝑡𝑡−𝛼ℎ}𝑧(𝑠)𝑑𝑠^𝑇𝑅₁^{𝑡𝑡−𝛼ℎ}𝑧(𝑠)𝑑𝑠−^{𝑡−𝛼ℎ𝑡−ℎ}𝑧(𝑠)𝑑𝑠^𝑇𝑅₂^{𝑡−𝛼ℎ𝑡−ℎ}𝑉̇𝑧(𝑠)𝑑𝑠₄,𝑉̇(𝑧(𝑡))=𝑧̇^𝑇(𝑡)𝛼ℎ𝑅₃+(1−𝛼)ℎ𝑅₄𝑧̇(𝑡)−^{𝑡𝑡−𝛼ℎ}𝑧̇^𝑇(𝑠)𝑅₃𝑧̇(𝑠)𝑑𝑠−^{𝑡−𝛼ℎ𝑡−ℎ}𝑧̇^𝑇(𝑠)𝑅₄𝑧̇(𝑠)𝑑𝑠,₅(𝑧(𝑡))=𝑧̇^𝑇(𝑡)𝛼²ℎ²2𝑅⁵+1−𝛼²ℎ²2𝑅⁶𝑧̇(𝑡)−^0−𝛼ℎ^{𝑡𝑡+𝜃}−𝑧̇^𝑇(𝑠)𝑅₅𝑧̇(𝑠)𝑑𝑠𝑑𝜃_{−𝛼ℎ−ℎ}^{𝑡𝑡+𝜃}𝑉̇𝑧̇^𝑇(𝑠)𝑅₆𝑧̇(𝑠)𝑑𝑠𝑑𝜃,₆(𝑧(𝑡))⩽+𝑧^𝑇(1(𝑡−)𝐾𝑢𝑄)𝑓𝐾_𝑇𝑧(𝑡)−𝑓^𝑇(𝑧(𝑡))𝑄𝑓(𝑧(𝑡))−(1−𝑢)𝑧^𝑇(𝑡−𝜏(𝑡))𝐾𝑄𝐾𝑧(𝑡−𝜏(𝑡))(𝑧(𝑡−𝜏(𝑡)))𝑄𝑓(𝑧(𝑡−𝜏(𝑡))).(3.11)(1) For the case of 0⩽𝜏(𝑡)⩽𝛼ℎ, then it gets −^{𝑡𝑡−𝛼ℎ}𝑧̇^𝑇(𝑠)𝑅₃𝑧̇(𝑠)𝑑𝑠=−−^{𝑡−𝜏(𝑡)𝑡−𝛼ℎ}𝑧̇^𝑇(𝑠)𝑅₃𝑧̇(𝑠)𝑑𝑠−^{𝑡−𝛼𝜏(𝑡)𝑡−𝜏(𝑡)}𝑧̇^𝑇(𝑠)𝑅₃𝑧̇(𝑠)𝑑𝑠_{𝑡𝑡−𝛼𝜏(𝑡)}𝑧̇^𝑇(𝑠)𝑅₃𝑧̇(𝑠)𝑑𝑠.(3.12) Similar to [8], the following equalities hold: 22𝜁𝜁^𝑇_𝑇(𝑡)𝑁𝑧(𝑡−𝜏(𝑡))−𝑧(𝑡−𝛼ℎ)−^{𝑡−𝜏(𝑡)𝑡−𝛼ℎ}𝑧̇(𝑠)𝑑𝑠=0,(3.13)2𝜁_𝑇(𝑡)𝐿𝑧(𝑡−𝛼𝜏(𝑡))−𝑧(𝑡−𝜏(𝑡))−^{𝑡−𝛼𝜏(𝑡)𝑡−𝜏(𝑡)}𝑧̇(𝑠)𝑑𝑠=0,(3.14)2𝑧(𝑡)𝑆𝑧(𝑡)−𝑧(𝑡−𝛼𝜏(𝑡))−^{𝑡𝑡−𝛼𝜏(𝑡)}𝑧̇(𝑠)𝑑𝑠=0,(3.15)_𝑇2𝜁_𝑇(𝑡)𝑃₁+𝑧̇^𝑇(𝑡)𝑃₂[−𝑧̇(𝑡)−𝐶𝑧(𝑡)+𝐴𝑓(𝑧(𝑡))+𝐵𝑓(𝑧(𝑡−𝜏(𝑡)))]=0,(3.16)(𝑡)𝐻𝛼ℎ𝑧(𝑡)−^{𝑡𝑡−𝛼ℎ}𝑧(𝑠)𝑑𝑠−^0−𝛼ℎ^{𝑡𝑡+𝜃}2𝜁_𝑇𝑧̇(𝑠)𝑑𝑠𝑑𝜃=0,(3.17)(𝑡)𝑀(1−𝛼)ℎ𝑧(𝑡)−^{𝑡−𝛼ℎ𝑡−ℎ}𝑧(𝑠)𝑑𝑠−^{−𝛼ℎ−ℎ}^{𝑡𝑡+𝜃}𝜁𝑧̇(𝑠)𝑑𝑠𝑑𝜃=0,(3.18)_𝑇(𝑡)=𝑧𝑓^𝑇(𝑡)𝑧^𝑇(𝑡−𝛼𝜏(𝑡))𝑧^𝑇(𝑡−𝜏(𝑡))𝑧^𝑇(𝑡−𝛼ℎ)𝑧^𝑇𝑡−𝛼²ℎ𝑧^𝑇(𝑡−ℎ)𝑧̇^𝑇(𝑡)_𝑇(𝑧(𝑡))𝑓^𝑇(𝑧(𝑡−𝜏(𝑡)))^{𝑡𝑡−𝛼ℎ}𝑧^𝑇(𝑠)𝑑𝑠^{𝑡−𝛼ℎ𝑡−ℎ}𝑧^𝑇(𝑠)𝑑𝑠.(3.19) It is easy to see that −−22𝜁𝜁^𝑇_𝑇(𝑡)𝑁^{𝑡−𝜏(𝑡)𝑡−𝛼ℎ}𝑧̇(𝑠)𝑑𝑠⩽(𝛼ℎ−𝜏(𝑡))𝜁^𝑇(𝑡)𝑁𝑅³⁻¹𝑁^𝑇𝜁(𝑡)+^{𝑡−𝜏(𝑡)𝑡−𝛼ℎ}𝑧̇^𝑇(𝑠)𝑅₃𝑧̇(𝑠)𝑑𝑠,(3.20)−2𝜁_𝑇(𝑡)𝐿^{𝑡−𝛼𝜏(𝑡)𝑡−𝜏(𝑡)}𝑧̇(𝑠)𝑑𝑠⩽(1−𝛼)𝜏(𝑡)𝜁^𝑇(𝑡)𝐿𝑅³⁻¹𝐿^𝑇𝜁(𝑡)+^{𝑡−𝛼𝜏(𝑡)𝑡−𝜏(𝑡)}𝑧̇^𝑇(𝑠)𝑅₃𝑧̇(𝑠)𝑑𝑠,(3.21)−2𝜁_𝑇𝑆^{𝑡𝑡−𝛼𝜏(𝑡)}𝑧̇(𝑠)𝑑𝑠⩽𝛼𝜏(𝑡)𝜁^𝑇(𝑡)𝑆𝑅³⁻¹𝑆^𝑇𝜁(𝑡)+^{𝑡𝑡−𝛼𝜏(𝑡)}𝑧̇^𝑇(𝑠)𝑅₃𝑧̇(𝑠)𝑑𝑠,(3.22)𝐻^0−𝛼ℎ^{𝑡𝑡+𝜃}𝑧̇(𝑠)𝑑𝑠𝑑𝜃⩽𝛼²ℎ²2𝜁^𝑇(𝑡)𝐻𝑅⁵⁻¹𝐻^𝑇𝜁(𝑡)+^0−𝛼ℎ^{𝑡𝑡+𝜃}−2𝜁_𝑇𝑧̇^𝑇(𝑠)𝑅₅𝑧̇(𝑠)𝑑𝑠𝑑𝜃,(3.23)𝑀^{−𝛼ℎ−ℎ}^{𝑡𝑡+𝜃}𝑧̇(𝑠)𝑑𝑠𝑑𝜃⩽1−𝛼²ℎ²2𝜁^𝑇(𝑡)𝑀𝑅⁶⁻¹𝑀^𝑇𝜁(𝑡)+^{−𝛼ℎ−ℎ}^{𝑡𝑡+𝜃}𝑧̇^𝑇(𝑠)𝑅₆𝑧̇(𝑠)𝑑𝑠𝑑𝜃.(3.24) Using Lemma 2.1, it is easy to obtain that −^{𝑡−𝛼ℎ𝑡−ℎ}𝑧̇^𝑇(𝑠)𝑅₄𝑧̇(𝑠)𝑑𝑠⩽−[𝑧(𝑡−𝛼ℎ)−𝑧(𝑡−ℎ)]^𝑇𝑅₄(1−𝛼)ℎ[𝑧(𝑡−𝛼ℎ)−𝑧(𝑡−ℎ)].(3.25) Furthermore, there exist positive diagonal matrices 𝑇₁,𝑇₂, such that the following inequalities hold based on (2.4): −2𝑓_𝑇−2𝑓^𝑇(𝑧(𝑡))𝑇₁𝑓(𝑧(𝑡))+2𝑧^𝑇(𝑡)𝐾𝑇₁𝑓(𝑧(𝑡))⩾0,(3.26)(𝑧(𝑡−𝜏(𝑡)))𝑇₂𝑓(𝑧(𝑡−𝜏(𝑡)))+2𝑧^𝑇(𝑡−𝜏(𝑡))𝐾𝑇₂𝑓(𝑧(𝑡−𝜏(𝑡)))⩾0.(3.27) From (3.10)–(3.27), one can obtain that 𝑉̇(𝑧(𝑡))⩽𝜁^𝑇(𝑡)Σ₁𝜁(𝑡),(3.28) where Σ₁=𝐸+(𝛼ℎ−𝜏(𝑡))𝑁𝑅³⁻¹𝑁^𝑇+(1−𝛼)𝜏(𝑡)𝐿𝑅³⁻¹𝐿^𝑇+𝛼𝜏(𝑡)𝑆𝑅³⁻¹𝑆^𝑇+𝛼²ℎ²2𝐻𝑅⁵⁻¹𝐻^𝑇+1−𝛼²ℎ²2𝑀𝑅⁶⁻¹𝑀^𝑇.(3.29) Note that 0⩽𝜏(𝑡)⩽𝛼ℎ, (𝛼ℎ−𝜏(𝑡))𝑁𝑅³⁻¹𝑁^𝑇+(1−𝛼)𝜏(𝑡)𝐿𝑅³⁻¹𝐿^𝑇+𝛼𝜏(𝑡)𝑆𝑅³⁻¹𝑆^𝑇 can be seen as the convex combination of 𝑁𝑅³⁻¹𝑁^𝑇,𝐿𝑅³⁻¹𝐿^𝑇, and 𝑆𝑅³⁻¹𝑆^𝑇 on 𝜏(𝑡). Therefore, Σ₁<0 holds if and only if 𝐸+𝛼ℎ𝑁𝑅³⁻¹𝑁^𝑇+𝛼²ℎ²2𝐻𝑅⁵⁻¹𝐻^𝑇+1−𝛼²ℎ²2𝑀𝑅⁶⁻¹𝐸+(1−𝛼)𝛼ℎ𝐿𝑅₃₋₁𝑀^𝑇<0,(3.30)𝐿^𝑇+𝛼²ℎ𝑆𝑅³⁻¹𝑆^𝑇+𝛼²ℎ²2𝐻𝑅⁵⁻¹𝐻^𝑇+1−𝛼²ℎ²2𝑀𝑅⁶⁻¹𝑀^𝑇<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 𝛼ℎ⩽𝜏(𝑡)⩽ℎ, then it gets −^{𝑡𝑡−𝛼ℎ}𝑧̇^𝑇(𝑠)𝑅₃𝑧̇(𝑠)𝑑𝑠=−−^{𝑡−𝛼𝜏(𝑡)𝑡−𝛼ℎ}𝑧̇^𝑇(𝑠)𝑅₃𝑧̇(𝑠)𝑑𝑠−^{𝑡−𝛼2ℎ𝑡−𝛼𝜏(𝑡)}𝑧̇^𝑇(𝑠)𝑅₃𝑧̇(𝑠)𝑑𝑠_{𝑡𝑡−𝛼2ℎ}−𝑧̇^𝑇(𝑠)𝑅₃𝑧̇(𝑠)𝑑𝑠,_{𝑡−𝛼ℎ𝑡−ℎ}𝑧̇^𝑇(𝑠)𝑅₄𝑧̇(𝑠)𝑑𝑠=−^{𝑡−𝜏(𝑡)𝑡−ℎ}𝑧̇^𝑇(𝑠)𝑅₄𝑧̇(𝑠)𝑑𝑠−^{𝑡−𝛼ℎ𝑡−𝜏(𝑡)}𝑧̇^𝑇(𝑠)𝑅₄𝑧̇(𝑠)𝑑𝑠.(3.32) Similar to [8], the following equalities hold: 22𝜁𝜁^𝑇_𝑇(𝑡)𝑈𝑧(𝑡−𝛼𝜏(𝑡))−𝑧(𝑡−𝛼ℎ)−^{𝑡−𝛼𝜏(𝑡)𝑡−𝛼ℎ}𝑧̇(𝑠)𝑑𝑠=0,2𝜁(_𝑇𝑡)𝑉𝑧𝑡−𝛼²ℎ−𝑧(𝑡−𝛼𝜏(𝑡))−^{𝑡−𝛼2ℎ𝑡−𝛼𝜏(𝑡)}𝑧̇(𝑠)𝑑𝑠=0,2𝜁_𝑇(𝑡)𝑊𝑧(𝑡−𝜏(𝑡))−𝑧(𝑡−ℎ)−^{𝑡−𝜏(𝑡)𝑡−ℎ}𝑧̇(𝑠)𝑑𝑠=0,(𝑡)𝑍𝑧(𝑡−𝛼ℎ)−𝑧(𝑡−𝜏(𝑡))−^{𝑡−𝛼ℎ𝑡−𝜏(𝑡)}𝑧̇(𝑠)𝑑𝑠=0.(3.33) It is easy to obtain that −−22𝜁𝜁^𝑇_𝑇(𝑡)𝑈^{𝑡−𝛼𝜏(𝑡)𝑡−𝛼ℎ}𝑧̇(𝑠)𝑑𝑠⩽𝛼(ℎ−𝜏(𝑡))𝜁^𝑇(𝑡)𝑈𝑅³⁻¹𝑈^𝑇𝜁(𝑡)+^{𝑡−𝛼𝜏(𝑡)𝑡−𝛼ℎ}𝑧̇^𝑇(𝑠)𝑅₃𝑧̇(𝑠)𝑑𝑠,−2𝜁_𝑇(𝑡)𝑉^{𝑡−𝛼2ℎ𝑡−𝛼𝜏(𝑡)}𝑧̇(𝑠)𝑑𝑠⩽𝛼(𝜏(𝑡)−𝛼ℎ)𝜁^𝑇(𝑡)𝑉𝑅³⁻¹𝑉^𝑇𝜁(𝑡)+^{𝑡−𝛼2ℎ𝑡−𝛼𝜏(𝑡)}𝑧̇^𝑇(𝑠)𝑅₃𝑧̇(𝑠)𝑑𝑠,−2𝜁_𝑇(𝑡)𝑊^{𝑡−𝜏(𝑡)𝑡−ℎ}𝑧̇(𝑠)𝑑𝑠⩽(ℎ−𝜏(𝑡))𝜁^𝑇(𝑡)𝑊𝑅⁴⁻¹𝑊^𝑇𝜁(𝑡)+^{𝑡−𝜏(𝑡)𝑡−ℎ}𝑧̇^𝑇(𝑠)𝑅₄𝑧̇(𝑠)𝑑𝑠,(𝑡)𝑍^{𝑡−𝛼ℎ𝑡−𝜏(𝑡)}𝑧̇(𝑠)𝑑𝑠⩽(𝜏(𝑡)−𝛼ℎ)𝜁^𝑇(𝑡)𝑍𝑅⁴⁻¹𝑍^𝑇𝜁(𝑡)+^{𝑡−𝛼ℎ𝑡−𝜏(𝑡)}𝑧̇^𝑇(𝑠)𝑅₄𝑧̇(𝑠)𝑑𝑠.(3.34) Using Lemma 2.1, one can obtain that −^{𝑡𝑡−𝛼}_2ℎ𝑧̇^𝑇(𝑠)𝑅₄𝑧̇(𝑠)𝑑𝑠⩽−1𝛼²ℎ𝑧(𝑡)−𝑧𝑡−𝛼²ℎ^𝑇𝑅₃𝑧(𝑡)−𝑧𝑡−𝛼²ℎ.(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 𝑉̇(𝑧(𝑡))⩽𝜁^𝑇(𝑡)Σ₂𝜁(𝑡),(3.36) where Σ₂=Φ+𝛼(ℎ−𝜏(𝑡))𝑈𝑅³⁻¹𝑈^𝑇+𝛼(𝜏(𝑡)−𝛼ℎ)𝑉𝑅³⁻¹𝑉^𝑇+(ℎ−𝜏(𝑡))𝑊𝑅⁴⁻¹𝑊^𝑇+(𝜏(𝑡)−𝛼ℎ)𝑍𝑅⁴⁻¹𝑍^𝑇+𝛼²ℎ²2𝐻𝑅⁵⁻¹𝐻^𝑇+1−𝛼²ℎ²2𝑀𝑅⁶⁻¹𝑀^𝑇.(3.37) Note that 𝛼ℎ⩽𝜏(𝑡)⩽ℎ, 𝛼(ℎ−𝜏(𝑡))𝑈𝑅³⁻¹𝑈^𝑇+𝛼(𝜏(𝑡)−𝛼ℎ)𝑉𝑅³⁻¹𝑉^𝑇+(ℎ−𝜏(𝑡))𝑊𝑅⁴⁻¹𝑊^𝑇+(𝜏(𝑡)−𝛼ℎ)𝑍𝑅⁴⁻¹𝑍^𝑇 can be seen as the convex combination of 𝑈𝑅³⁻¹𝑈^𝑇,𝑉𝑅³⁻¹𝑉^𝑇,𝑊𝑅⁴⁻¹𝑊^𝑇, and 𝑍𝑅⁴⁻¹𝑍^𝑇 on 𝜏(𝑡). Therefore, Σ₂<0 holds if and only if Φ+𝛼(1−𝛼)ℎ𝑈𝑅³⁻¹𝑈^𝑇+(1−𝛼)ℎ𝑊𝑅⁴⁻¹𝑊^𝑇+𝛼²ℎ²2𝐻𝑅⁵⁻¹𝐻^𝑇+1−𝛼²ℎ²2𝑀𝑅⁶⁻¹Φ+𝛼(1−𝛼)ℎ𝑉𝑅₃₋₁𝑀^𝑇<0,(3.38)𝑉^𝑇+(1−𝛼)ℎ𝑍𝑅⁴⁻¹𝑍^𝑇+𝛼²ℎ²2𝐻𝑅⁵⁻¹𝐻^𝑇+1−𝛼²ℎ²2𝑀𝑅⁶⁻¹𝑀^𝑇<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 0⩽𝜏(𝑡)⩽ℎ.

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,ℎ]. Unlike [10, 25], the new Lyapunov functional in our paper which not only divides the delay interval [0,ℎ] into two ones [0,𝛼ℎ] and [𝛼ℎ,ℎ] but also divides the delay interval [0,ℎ] into three ones [0,𝛼𝜏(𝑡)], [𝛼𝜏(𝑡),𝜏(𝑡)], and [𝜏(𝑡),ℎ] 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(𝑡−𝛼ℎ),𝑧(𝑡−𝜏(𝑡)),𝑧(𝑡−𝛼²ℎ),𝑧(𝑡−ℎ), and 𝑧(𝑡−𝛼𝜏(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 𝑉₄̇(𝑧(𝑡)), different free-weighting matrices are introduced in two different intervals 0⩽𝜏(𝑡)⩽𝛼ℎ and 𝛼ℎ⩽𝜏(𝑡)⩽ℎ. These methods mentioned above may lead to obtain an improved feasible region for delay-dependent stability criteria.

Remark 3.5. In (3.28), 𝐸+(𝛼ℎ−𝜏(𝑡))𝑁𝑅³⁻¹𝑁^𝑇+(1−𝛼)𝜏(𝑡)𝐿𝑅³⁻¹𝐿^𝑇+𝛼𝜏(𝑡)𝑆𝑅³⁻¹𝑆^𝑇+(𝛼²ℎ²/2)𝐻𝑅⁵⁻¹𝐻^𝑇+((1−𝛼²)ℎ²/2)𝑀𝑅⁶⁻¹𝑀^𝑇 is not simply guaranteed by 𝐸+𝛼ℎ𝑁𝑅³⁻¹𝑁^𝑇+(1−𝛼)𝛼ℎ𝐿𝑅³⁻¹𝐿^𝑇+𝛼²ℎ𝑆𝑅³⁻¹𝑆^𝑇+(𝛼²ℎ²/2)𝐻𝑅⁵⁻¹𝐻^𝑇+((1−𝛼²)ℎ²/2)𝑀𝑅⁶⁻¹𝑀^𝑇 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 0≤𝜏(𝑡)≤ℎ 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,…,𝑘_𝑛), ℎ⩾0, and 0<𝛼<1, the system (2.3) is globally asymptotically stable if there exist symmetric positive matrices 𝑃=[^𝑃_𝑖𝑗]_3×3,𝑄₃,𝑄₄,𝑄₅, 𝑅_𝑖(𝑖=1,2,…,6), positive diagonal matrices 𝑇₁,𝑇₂,Λ=diag(𝜆₁,𝜆₂,…,𝜆_𝑛),Δ=diag(𝛿₁,𝛿₂,…,𝛿_𝑛), and any matrices  𝑃₁,𝑃₂,𝑁_𝑖,𝑀_𝑖,𝐿_𝑖,𝐻_𝑖,𝑍_𝑖,𝑆_𝑖,𝑈_𝑖,𝑉_𝑖,𝑊_𝑖(𝑖=1,2,3) with appropriate dimensions, such that the following LMIs hold: 𝐸₁=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝐸𝛼ℎ𝑁𝛼²ℎ²2𝐻1−𝛼²ℎ²∗−𝛼ℎ𝑅₃2𝑀∗∗−𝛼₂00ℎ²2𝑅⁵∗∗∗−1−𝛼0₂ℎ²2𝑅⁶⎤⎥⎥⎥⎥⎥⎥⎥⎦<0,𝐸₂=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐸(1−𝛼)𝛼ℎ𝐿𝛼²ℎ𝑆𝛼²ℎ²2𝐻1−𝛼²ℎ²∗−(1−𝛼)𝛼ℎ𝑅₃2𝑀∗∗−𝛼₂000∗∗∗ℎ𝑅₃−𝛼₂00ℎ²2𝑅⁵∗∗∗∗−1−𝛼0₂ℎ²2𝑅⁶⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,Φ₁=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣Φ(1−𝛼)𝛼ℎ𝑈(1−𝛼)ℎ𝑊𝛼²ℎ²2𝐻1−𝛼²ℎ²∗−(1−𝛼)𝛼ℎ𝑅₃2𝑀∗∗−(1−0𝛼)ℎ𝑅₄00∗∗∗−𝛼₂00ℎ²2𝑅⁵∗∗∗∗−1−𝛼0₂ℎ²2𝑅⁶⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,Φ₂=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣Φ(1−𝛼)𝛼ℎ𝑉(1−𝛼)ℎ𝑍𝛼²ℎ²2𝐻1−𝛼²ℎ²∗−(1−𝛼)𝛼ℎ𝑅₃2𝑀∗∗−(1−0𝛼)ℎ𝑅₄00∗∗∗−𝛼₂00ℎ²2𝑅⁵∗∗∗∗−1−𝛼0₂ℎ²2𝑅⁶⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.40) where 𝐸=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐸₁₁𝐸₁₂𝐸₁₃𝐸₁₄0−𝑃₁₃𝐸₁₇𝑃₁𝐴+𝐾𝑇₁𝑃₁𝐵𝑃₂₂−𝐻₁𝑃₂₃−𝑀₁∗𝐸₂₂𝐸₂₃−𝑁₂00000−𝐻₂−𝑀₂∗∗𝐸₃₃−𝑁₃0000𝐾𝑇₂−𝐻₃−𝑀₃∗∗∗𝐸₄₄0𝑅⁴(1−𝛼)ℎ000−𝑃²²+𝑃^𝑇23−𝑃₂₃+𝑃₃₃∗∗∗∗∗∗∗∗−∗𝑄₅𝐸0₆₆00000000−𝑃^𝑇23−𝑃₃₃∗∗∗∗∗∗𝐸₇₇𝑃₂𝐴+Λ−Δ𝑃₂𝐵𝑃₁₂𝑃₁₃∗∗∗∗∗∗∗∗∗∗∗∗∗∗−2∗𝑇₁−20𝑇₂00∗∗∗∗∗∗∗∗∗−0𝑅₁0∗∗∗∗∗∗∗∗∗∗−0𝑅₂⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,Φ=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝Φ₁₁Φ₁₂Φ₁₃Φ₁₄𝑉₁+𝑅³𝛼²ℎ−𝑃¹³−𝑊₁Φ₁₇𝑃₁𝐴+𝐾𝑇₁𝑃₁𝐵𝑃₂₂−𝐻₁𝑃₂₃−𝑀₁∗Φ₂₂Φ₂₃−𝑈₂+𝑍₂𝑉₂−𝑊₂000−𝐻₂−𝑀₂∗∗Φ₃₃−𝑈₃+𝑍₃𝑉₃−𝑊₃00𝐾𝑇₂−𝐻₃−𝑀₃∗∗∗−𝑄₃00000−𝑃₂₂+𝑃^𝑇23−𝑃₂₃+𝑃₃₃∗∗∗∗−𝑄₅−𝑅³∗∗∗∗∗𝛼²ℎ−0𝑄₄00000000−𝑃^𝑇23−𝑃₃₃∗∗∗∗∗∗Φ₇₇𝑃₂𝐴+Λ−Δ𝑃₂𝐵𝑃₁₂𝑃₁₃∗∗∗∗∗∗∗∗∗∗∗∗∗∗−2∗𝑇₁−20𝑇₂00∗∗∗∗∗∗∗∗∗−0𝑅₁0∗∗∗∗∗∗∗∗∗∗−0𝑅₂⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠𝐸₁₁=𝑃₁₂+𝑃^𝑇12+𝑄₃+𝑄₄+𝑄₅+𝛼²ℎ²𝑅₁+(1−𝛼)²ℎ²𝑅₂+𝑆₁+𝑆^𝑇1−𝑃₁𝐶−𝐶𝑃^𝑇1+𝛼ℎ𝐻₁+𝐻^𝑇1+(1−𝛼)ℎ𝑀₁+𝑀^𝑇1,𝐸₂₂=𝐿₂+𝐿^𝑇2−𝑆₂−𝑆^𝑇2,𝐸₃₃=𝑁₃+𝑁^𝑇3−𝐿₃−𝐿^𝑇3,Φ₁₁=𝑃₁₂+𝑃^𝑇12+𝑄₃+𝑄₄+𝑄₅+𝛼²ℎ²𝑅₁+(1−𝛼)²ℎ²𝑅₂−𝑃₁𝐶−𝐶𝑃^𝑇1+𝛼ℎ𝐻₁+𝐻^𝑇1+(1−𝛼)ℎ𝑀₁+𝑀^𝑇1−𝑅³𝛼²ℎ,Φ₂₂=𝑈₂+𝑈^𝑇2−𝑉₂−𝑉^𝑇2,Φ₃₃=𝑊₃+𝑊^𝑇3−𝑍₃−𝑍^𝑇3.(3.41) The other 𝐸_𝑖𝑗,Φ_𝑖𝑗 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),𝜎𝐴=⎡⎢⎢⎣−11−11⎤⎥⎥⎦,𝐵=⎡⎢⎢⎣0.18811⎤⎥⎥⎦,𝜎₁=0.4,₂=0.8,𝛾₁=0,𝛾₂=0.(4.1) 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 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 ℎ=2.9144 and obtain 𝑃𝑃₁₁=⎡⎢⎢⎣−00.0.0000052−00.0.0001082⎤⎥⎥⎦,𝑃₁₂=1.0𝑒⁻⁰⁰³×⎡⎢⎢⎣−00.1.0636633−00.0.1098415₁₃⎤⎥⎥⎦,𝑃=1.0𝑒⁻⁰⁰⁶×⎡⎢⎢⎣−00.3.1909032−00.0.2245928⎤⎥⎥⎦,𝑃₂₂=1.0𝑒⁻⁰⁰⁴×⎡⎢⎢⎣−00.1.2617582−00.5.201457₂₃2⎤⎥⎥⎦,𝑄=1.0𝑒⁻⁰⁰⁶×⎡⎢⎢⎣00..00242618−−00..03863868⎤⎥⎥⎦,𝑃₃₃=1.0𝑒⁻⁰⁰⁶×⎡⎢⎢⎣−00.2.2611446−00.5.241148₁6⎤⎥⎥⎦,𝑄=1.0𝑒⁻⁰⁰⁶×⎡⎢⎢⎣−00.2.2346584−00.4.2040584⎤⎥⎥⎦,𝑄₂=1.0𝑒⁻⁰⁰⁵×⎡⎢⎢⎣−00.0.0107847−00.1.090786₃7⎤⎥⎥⎦,=𝑄1.0𝑒⁻⁰⁰³×⎡⎢⎢⎣00..0052296600..02259461⎤⎥⎥⎦,𝑄₄=1.0𝑒⁻⁰⁰⁴×⎡⎢⎢⎣00..9591711000..54161309₅⎤⎥⎥⎦,𝑅=1.0𝑒⁻⁰⁰⁵×⎡⎢⎢⎣−00.0.0253621−00.1.0558641⎤⎥⎥⎦,𝑄₆=⎡⎢⎢⎣00..0000231500..00003557₁⎤⎥⎥⎦,𝑅=1.0𝑒⁻⁰⁰⁴×⎡⎢⎢⎣−00.4.6502230−00.8.6706200⎤⎥⎥⎦,𝑅₂=1.0𝑒⁻⁰⁰⁶×⎡⎢⎢⎣−00.1.2014599−00.5.221959₃9⎤⎥⎥⎦,𝑅=1.0𝑒⁻⁰⁰⁴×⎡⎢⎢⎣00..7396155000..34615100⎤⎥⎥⎦,𝑅₄=1.0𝑒⁻⁰⁰⁴×⎡⎢⎢⎣00..9570079100..54027112₅⎤⎥⎥⎦,=1.0𝑒Λ=⁻⁰⁰³×⎡⎢⎢⎣0⎡⎢⎢⎣.0−0000.01.01088073.2000−1040.3⎤⎥⎥⎦.17,87762⎤⎥⎥⎦Δ,=1𝑅.0₆𝑒=1₋₀₀₃.0𝑒⁻⁰⁰⁶×⎡⎢⎢⎣−00.0.1629686−00.2.1921606⎤⎥⎥⎦,𝑇×⎡⎢⎢⎣0.106860.00027₁⎤⎥⎥⎦,=⎡⎢⎢⎣0.000370.00039⎤⎥⎥⎦,𝑇₂=⎡⎢⎢⎣0.000090.00010⎤⎥⎥⎦,𝑄=⎡⎢⎢⎣0.000000.00038⎤⎥⎥⎦.(4.2) Therefore, it follows from Theorem 3.1 that the system (2.3) with given parameters is globally asymptotically stable.

Table 1 

Allowable upper bound of ℎ for different 𝑢 (Example 4.1).

Example 4.2. Consider the stability of system (2.3) with time-varying delay and 𝐶=diag(1.5,𝜎0.7),𝐴=⎡⎢⎢⎣00..0059083700..02405745⎤⎥⎥⎦,𝐵=⎡⎢⎢⎣00..2033881800..95302602₁⎤⎥⎥⎦,=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 ℎ=4.7444 and obtain𝑃𝑃₁₁=⎡⎢⎢⎣−10.0.7250268−10.2.7357288⎤⎥⎥⎦,𝑃₁₂=1.0𝑒⁻⁰⁰³×⎡⎢⎢⎣−−00..00064552−00.0.0411377₁₃⎤⎥⎥⎦,𝑃=1.0𝑒⁻⁰⁰³×⎡⎢⎢⎣−−00..21486379−00.6.3490241⎤⎥⎥⎦,𝑃₂₂=⎡⎢⎢⎣−00.0.0007230−00.0.000222₂₃0⎤⎥⎥⎦,𝑄=1.0𝑒⁻⁰⁰⁴×⎡⎢⎢⎣−−00..0401806800..41601865⎤⎥⎥⎦,𝑃₃₃=1.0𝑒⁻⁰⁰³×⎡⎢⎢⎣−00.1.1920942−00.1.102599₁2⎤⎥⎥⎦,𝑄=⎡⎢⎢⎣−00.0.0001132−00.0.0001122⎤⎥⎥⎦,𝑄₂=⎡⎢⎢⎣00..0025568600..01526668⎤⎥⎥⎦,𝑄₃=⎡⎢⎢⎣00..0060553100..00005510₄⎤⎥⎥⎦,=⎡⎢⎢⎣−0𝑅0.0.0131040−00.0.0833090⎤⎥⎥⎦,𝑄₅=⎡⎢⎢⎣−00.0.0001049−00.0.0000069⎤⎥⎥⎦,𝑄₆=⎡⎢⎢⎣00..0033400000..00330100₁⎤⎥⎥⎦,=𝑅⎡⎢⎢⎣−00.0.0009606−00.0.0006626⎤⎥⎥⎦,𝑅₂=1.0𝑒⁻⁰⁰³×⎡⎢⎢⎣−00.1.1934631−00.1.103366₃1⎤⎥⎥⎦,𝑅=⎡⎢⎢⎣−00.2.3488456−00.7.3384486⎤⎥⎥⎦,𝑅₄=⎡⎢⎢⎣−00.0.2810927−00.6.211292₅7⎤⎥⎥⎦,=Λ⎡⎢⎢⎣=−00.⎡⎢⎢⎣1.005.8502033118−000.0..007038042941⎤⎥⎥⎦⎤⎥⎥⎦,,Δ𝑅₆==11.0.0𝑒𝑒⁻⁰⁰³₋₀₀₃×⎡⎢⎢⎣−00.7.4695016−00.5.4892056⎤⎥⎥⎦,𝑇×⎡⎢⎢⎣0.007970.30593₁⎤⎥⎥⎦,=⎡⎢⎢⎣0.001360.10968⎤⎥⎥⎦,𝑇₂=⎡⎢⎢⎣0.300560.30423⎤⎥⎥⎦,𝑄=⎡⎢⎢⎣0.405670.00141⎤⎥⎥⎦.(4.4) Therefore, it follows from Theorem 3.1 that the system (2.3) with given parameters is globally asymptotically stable.

Table 2 

Allowable upper bound of ℎ 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.