Stability Analysis for Uncertain Neural Networks of Neutral Type with Time-Varying Delay in the Leakage Term and Distributed Delay

Abstract

The stability problem is investigated for a class of uncertain networks of neutral type with leakage, time-varying discrete, and distributed delays. Both the parameter uncertainty and the generalized activation functions are considered in this paper. New stability results are achieved by constructing an appropriate Lyapunov-Krasovskii functional and employing the free weighting matrices and the linear matrix inequality (LMI) method. Some numerical examples are given to show the effectiveness and less conservatism of the proposed results.

1. Introduction

Neural networks are a complex large-scale power system and have very rich dynamic property. In the past years, neural networks have some applications in areas of associative memory [1], pattern recognition [2], and optimization problems [3–5]. Recently, the stability problem for neural networks has been widely investigated and some results have been reported in [6–14].

Time delays are often encountered in various engineering, biological, and economical systems [15–23]. Due to the finite speed of information processing, the existence of time delays frequently causes oscillation, divergence, or instability in neural networks. Recently, the stability of neural networks with time delays has drawn considerable attention, and many results on stability of neural networks with time delays have been reported in the literature [24–30]. In practice, the time-varying delay often belongs to a given interval, and the lower bound of the delays may not be zero. Recently, some papers investigated the stability conditions for neural networks with interval time-varying delay in the literature, for example [31]. Furthermore, due to the presence of parallel pathways of different axonal sizes and lengths, there may exist a spatial extent in neural networks, which may cause distributed time delays [32–35]. Recently, the stability problem of neural networks with neutral-type was studied in [36, 37].

More recently, more and more attention has been paid to time delay in the leakage (or “forgetting”) term [38, 39], since there exist some theoretical and technical difficulties when handling the leakage delay [40]. It has been pointed out that the leakage delay has a great impact on the dynamics of neural networks [41, 42]. The authors in [43] showed that time delay in the stabilizing negative feedback term has a tendency to destabilize a system. More recently, the authors in [44] focused on recurrent neural networks with time delay in the leakage term and showed that all the results mentioned about the existence and uniqueness of the equilibrium point are independent of time delays and initial conditions. Then, it can be seen that time delays in leakage terms do not affect the existence and uniqueness of the equilibrium point. Recently, some results about stability analysis for neural networks with leakage delay have been reported in [38, 39, 41]. To mention a few, the work in [14] investigated the problems of delay-dependent stability analysis and strict (𝑄,𝑆,𝑅)-𝛼-dissipativity analysis for cellular neural networks with distributed delay. However, it should be mentioned that there are few results about stability analysis for uncertain neural networks of neutral type with time-varying delay in the leakage term and distributed delay, which motivates this study.

In this paper, the stability problem is investigated for uncertain neural networks of neutral type with time-varying delay in the leakage term and time-varying distributed delay. Firstly, by constructing a new type of Lyapunov functional and developing some novel techniques to handle the delays considered in this paper, some novel robust stability criteria are proposed. Secondly, the proposed conditions can be expressed in terms of linear matrix inequalities (LMIs), which can be easily solved via standard software. Finally, some numerical examples are given to demonstrate the effectiveness and less conservatism of the proposed results.

Notation. Throughout this paper, the notations are standard. ℝ^𝑛 and ℝ^𝑛×𝑚 denote the 𝑛-dimensional Euclidean space and the set of all 𝑛×𝑚 real matrices, respectively. In this paper, the superscript 𝑇 denotes matrix transposition. For real symmetric matrices 𝑋 and 𝑌, the natation 𝑋≥𝑌 (𝑋>𝑌 resp.) means that the 𝑋−𝑌 is positive-semidefinite (positive-definite resp.). The notation diag{} stands for a block-diagonal matrix. 𝐼 is the identity matrix with appropriate dimensions. Matrices if not explicitly stated, where the symbol “∗” stands for the symmetric term in a matrix, are assumed to have compatible dimensions.

2. Problem Formulation

Consider the following uncertain neural network of neutral type with time delay in the leakage term and distributed delay: 𝑥̇(𝑡)=−𝑊₁+Δ𝑊₁(𝑡)𝑥(𝑡−𝛿)+𝑊₂+Δ𝑊₂(𝑡)𝑓(𝑥(𝑡+𝑊))₃+Δ𝑊₃(𝑡)𝑓((𝑡+𝑊𝑡−𝜏))₄+Δ𝑊₄(𝑡)^𝑡_{(𝑡)𝑡−𝑟}𝑓(𝑥(𝑠+𝑊))𝑑𝑠₅+Δ𝑊₅(𝑡)𝑥̇(𝑡−ℎ(𝑡)), (1) where 𝑥(𝑡)=𝑥₁(𝑡)𝑥₂(𝑡)⋯𝑥_𝑛(𝑡)^𝑇∈ℝ^𝑛 is the network state vector at time 𝑡; 𝑓(𝑡)∶=𝑓(𝑥(𝑡))=𝑓₁(𝑥₁(𝑡))𝑓₂(𝑥₂(𝑡))⋯𝑓_𝑛(𝑥_𝑛(𝑡))^𝑇∈ℝ^𝑛 denotes the activation function at time 𝑡; 𝑊₁ is a positive diagonal matrix; 𝑊₁=(𝑊_1𝑖𝑗)_𝑛×𝑛, 𝑊₂=(𝑊_2𝑖𝑗)_𝑛×𝑛, 𝑊₃=(𝑊_3𝑖𝑗)_𝑛×𝑛, 𝑊₄=(𝑊_4𝑖𝑗)_𝑛×𝑛, and 𝑊₅=(𝑊_5𝑖𝑗)_𝑛×𝑛 are known constant matrices; Δ𝑊₁(𝑡), Δ𝑊₂(𝑡), Δ𝑊₃(𝑡), Δ𝑊₄(𝑡), and Δ𝑊₅(𝑡), are unknown matrices; 𝛿 is leakage delay, 𝜏(𝑡) is time-varying discrete delay; ℎ(𝑡) is neutral delay; and 𝑟(𝑡) is time-varying distributed delay. They satisfy the following conditions: 0≤𝛿,0<𝜏₁≤𝜏(𝑡)≤𝜏₂,𝜏̇(𝑡)≤𝜏_𝑑,0<ℎ(𝑡)≤ℎ,ℎ̇(𝑡)≤ℎ_𝑑<1,0<𝑟(𝑡)≤𝑟, (2) where 𝛿, 𝜏₁, 𝜏₂, 𝜏_𝑑, ℎ, and ℎ_𝑑 are constants. The time-varying parameter uncertainties Δ𝑊₁(𝑡), Δ𝑊₂(𝑡), Δ𝑊₃(𝑡), Δ𝑊₄(𝑡), and Δ𝑊₅(𝑡) are assumed to be of the form Δ𝑊₁(𝑡)Δ𝑊₂(𝑡)Δ𝑊₃(𝑡)Δ𝑊₄(𝑡)Δ𝑊₅(𝑡)=𝐻𝐺(𝑡)𝐸₁𝐸₂𝐸₃𝐸₄𝐸₅, (3) where 𝐻, 𝐸₁, 𝐸₂, 𝐸₃, 𝐸₄, 𝐸₅ are known constant matrices and 𝐺(𝑡) is an unknown time-varying matrix satisfying 𝐺(𝑡)^𝑇𝐺(𝑡)≤𝐼. (4) Throughout this paper, we make the following assumption:

(𝐻1) for any 𝑖=1,2,…,𝑛, 𝑓_𝑖(0)=0, there exist constants 𝐹⁻_𝑖, 𝐹⁺_𝑖, for all 𝛼₁, 𝛼₂∈𝑅^𝑛 with 𝛼₁≠𝛼₂ such that 𝐹⁻_𝑖≤𝑓_𝑖𝛼₁−𝑓_𝑖𝛼₂𝛼₁−𝛼₂≤𝐹⁺_𝑖. (5)

Lemma 1 (see [45]). For any positive symmetric constant matrices 𝑀∈ℝ^𝑛×𝑛, scalar 𝜏>0, vector function 𝜔(𝑠)∈ℝ^𝑛 such that the integrations concerned are well defined; then −^𝑡_𝑡−𝜏𝜔^𝑇(𝑠)(𝑠)1𝑀𝜔𝑑𝑠≤−𝜏^𝑡_𝑡−𝜏𝜔(𝑠)𝑑𝑠^𝑇𝑀^𝑡_𝑡−𝜏𝜔(𝑠)𝑑𝑠. (6)

Lemma 2 ([46] Schur complement). Given constant matrices Ω₁, Ω₂, and Ω₃ with appropriate dimensions, where Ω^𝑇₁=Ω₁ and Ω^𝑇₂=Ω₂>0, then Ω₁+Ω^𝑇₃Ω₂⁻¹Ω₃<0 if and only if Ω₁Ω^𝑇₃∗−Ω₂<0𝑜𝑟−Ω₂Ω₃∗Ω₁<0. (7)
In order to present novel stability criteria for neural networks (1), the following notations are defined: 𝐹₁=diag𝐹⁻₁,𝐹⁻₂,…,𝐹⁻_𝑛,𝐹₂=diag𝐹⁺₁,𝐹⁺₂,…,𝐹⁺_𝑛,𝐹₃=diag𝐹⁻₁𝐹⁺₁,𝐹⁻₂𝐹⁺₂,…,𝐹⁻_𝑛𝐹⁺_𝑛,𝐹₄𝐹=diag⁻¹+𝐹⁺₁2,𝐹⁻₂+𝐹⁺₂2𝐹,…,^−𝑛+𝐹⁺_𝑛2. (8)

3. Main Results

In this section, by considering the delay in the leakage term and distributed delay and using some new techniques, novel stability criteria will be proposed for uncertain neural network of neutral-type with time-varying delay in (1). Firstly, considering neutral-type delay, the delay in the leakage term, and distributed delay, we have the following theorem.

Theorem 3. For given scalars 0≤𝛿, 0<𝜏₁<𝜏₂, 0<ℎ, 0<𝑟, ℎ_𝑑<1, and 𝜏_𝑑, neural network (1) under Assumption (𝐻1) is robustly asymptotically stable, if there exist matrices 𝑃_𝑖>0(𝑖=1,2,…,5), 𝑄=[𝑄_𝑖𝑗]_2×2>0, 𝐾=[𝐾_𝑖𝑗]_2×2>0, 𝑍_𝑖>0(𝑖=1,2,…,7), 𝐷=diag{𝑑₁,𝑑₂,…,𝑑_𝑛}>0, 𝐿=diag{𝑙₁,𝑙₂,…,𝑙_𝑛}>0, positive diagonal matrices 𝑅 and 𝑆, and appropriately dimensioned matrices 𝑁_𝑘, 𝑀_𝑘(𝑘=1,2,…,14), such that the following LMIs hold: 𝑍₃𝑇∗𝑍₃>0,Ξ+Π+Π^𝑇√ℎ𝑁𝑀𝐻𝜀𝐸^𝑇∗−𝑍₇00∗∗−𝜀𝐼0∗∗∗−𝜀𝐼<0, (9)(10) whereΘ₁₂=0𝑃₁−𝐹^𝑇₁𝐷^𝑇+𝐹^𝑇₂𝐿^𝑇0𝐹₄𝑅0𝑊₁𝑃₁𝑊₁00000𝐹₄𝑆00000000000000000000000000000000000−𝑊₁𝑃₁𝑊₁0Θ,Ξ=¹¹Θ₁₂∗Θ₂₂,Θ₂₂=−𝑍₅000000∗Ψ₉₉0𝐷−𝐿0−𝑃₁𝑊₁0∗∗−1−ℎ_𝑑𝑍₆0000∗∗∗𝑃₅+𝑟²𝑍₄−𝑅000∗∗∗∗−1−𝜏_𝑑𝑃₅−𝑆00∗∗∗∗∗−𝑃₃0∗∗∗∗∗∗−𝑍₄,Θ₁₁=Ψ₁₁0𝑍₂0𝑍₁+𝑄₁₂𝐾₁₂𝑃₁𝑊₁∗Ψ₂₂𝑍₃−𝑇^𝑇𝑍₃−𝑇000∗∗𝑃₄−𝑍₂−𝑍₃−𝑄₂₂𝑇−𝑄^𝑇₁₂00∗∗∗−𝐾₂₂−𝑍₃0−𝐾^𝑇₁₂0∗∗∗∗−𝑍₁+𝑄₂₂−𝑄₁₁00∗∗∗∗∗𝐾₂₂−𝐾₁₁0∗∗∗∗∗∗−𝑃₂,Ψ₁₁=−𝑃₁𝑊₁−𝑊^𝑇₁𝑃^𝑇₁+𝑃₂+𝛿²𝑃₃+𝑄₁₁+𝐾₁₁−𝑍₁−𝑍₂−𝐹₃𝑅+𝑍₅,Ψ₂₂=−1−𝜏_𝑑𝑃₄−𝑍₃−𝑍^𝑇₃−𝐹₃𝑆+𝑇+𝑇^𝑇,Ψ₉₉=𝜏²₁4𝑍₁+𝜏²₁𝑍₂+𝜏²₁₂𝑍₃+𝑍₆+ℎ𝑍₇,Π=[𝑁,0,0,0,0,0,0,−𝑁,0,0,0,0,0,0]+𝑀0,0,0,0,0,0,−𝑊₁,0,−𝐼,𝑊₅,𝑊_2,𝑊₃,0,𝑊₄,𝑁=𝑁^𝑇₁,𝑁^𝑇₂,𝑁^𝑇₃,𝑁^𝑇₄,𝑁^𝑇₅,𝑁^𝑇₆,𝑁^𝑇₇,𝑁^𝑇₈,𝑁^𝑇₉,𝑁^𝑇₁₀,𝑁^𝑇₁₁,𝑁^𝑇₁₂,𝑁^𝑇₁₃,𝑁^𝑇₁₄^𝑇,𝑀=𝑀^𝑇₁,𝑀^𝑇₂,𝑀^𝑇₃,𝑀^𝑇₄,𝑀^𝑇₅,𝑀^𝑇₆,𝑀^𝑇₇,𝑀^𝑇₈,𝑀^𝑇₉,𝑀^𝑇₁₀,𝑀^𝑇₁₁,𝑀^𝑇₁₂,𝑀^𝑇₁₃,𝑀^𝑇₁₄^𝑇,𝐸=0,0,0,0,0,0,−𝐸₁,0,0,𝐸₅,𝐸₂,𝐸₃,0,𝐸₄. (11)

Proof. By using Newton-Leibniz formulation and considering neural network (1), the following equalities hold for appropriately dimensioned matrices 𝑁 and 𝑀: 2𝜂^𝑇(𝑡)𝑁𝑥(𝑡)−𝑥(𝑡−ℎ(𝑡))−^𝑡_{(𝑡)𝑡−ℎ}𝑥̇(𝑡)𝑑𝑠=0,2𝜂^𝑇(𝑡)𝑀−𝑊₁+Δ𝑊₁(𝑡)𝑥(𝑡−𝛿)+𝑊₂+Δ𝑊₂(𝑡)×𝑓(𝑥(𝑡))+𝑊₃+Δ𝑊₃(𝑡)𝑓((𝑡+𝑊𝑡−𝜏))₄+Δ𝑊₄(𝑡)^𝑡_{(𝑡)𝑡−𝑟}𝑓(𝑥(𝑠+𝑊))𝑑𝑠₅+Δ𝑊₅(𝑡)𝑥̇(𝑡−ℎ(𝑡))−𝑥̇(𝑡)=0, (12) with 𝜂^𝑇(𝑡)=𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−𝜏(𝑡)),𝑥^𝑇𝑡−𝜏₁,𝑥^𝑇𝑡−𝜏₂,𝑥^𝑇𝜏𝑡−¹2𝑥^𝑇𝜏𝑡−²2,𝑥^𝑇(𝑡−𝛿),𝑥^𝑇((𝑡,𝑥̇𝑡−ℎ))_𝑇(𝑡),𝑥̇^𝑇(𝑡−ℎ(𝑡)),𝑓^𝑇(𝑥(𝑡)),𝑓^𝑇(𝑥(𝑡−𝜏(𝑡)))_{𝑡𝑡−𝛿},𝑥(𝑠)𝑑𝑠^𝑇,^𝑡_{(𝑡)𝑡−𝑟}𝑓(𝑥(𝑠))𝑑𝑠^𝑇𝑇. (13) For positive diagonal matrices 𝑅 and 𝑆, based on Assumption (𝐻1), it can be seen that the following inequalities hold: 𝑥(𝑡)𝑓(𝑥(𝑡))^𝑇𝐹₃𝑅−𝐹₄𝑅−𝐹₄𝑥(𝑡)𝑓(𝑥(𝑡𝑥((𝑡𝑓(𝑥((𝑡𝑅𝑅))≤0,𝑡−𝜏))𝑡−𝜏)))^𝑇𝐹₃𝑆−𝐹₄𝑆−𝐹₄𝑥((𝑡𝑆𝑆𝑓(𝑥((𝑡𝑡−𝜏))𝑡−𝜏)))≤0. (14)
Now, choosing the following Lyapunov-Krasovskii functional: 𝑉(𝑡)=⁷_𝑖=1𝑉_𝑖(𝑡), (15) where 𝑉₁(𝑡)=𝑥(𝑡)−𝑊₁^𝑡_𝑡−𝛿𝑥(𝑠)𝑑𝑠^𝑇𝑃₁𝑥(𝑡)−𝑊₁^𝑡_𝑡−𝛿𝑥(𝑠),𝑉𝑑𝑠₂(𝑡)=2^𝑛_𝑖=1𝑑_𝑖^{𝑥𝑖(𝑡)}₀𝑓_𝑖(𝑠)−𝐹⁻_𝑖𝑠𝑑𝑠+2^𝑛_𝑖=1𝑙_𝑖^{𝑥𝑖(𝑡)}₀𝐹⁺_𝑖𝑠−𝑓_𝑖(𝑠)𝑉𝑑𝑠,₃(𝑡)=^𝑡_𝑡−𝛿𝑥^𝑇(𝑠)𝑃₂𝑥(𝑠)𝑑𝑠+𝛿⁰_−𝛿^𝑡_𝑡+𝜃𝑥^𝑇(𝑠)𝑃₃𝑥(𝑠)𝑉𝑑𝑠𝑑𝜃,₄(𝑡)=^𝑡_{𝜏𝑡−1/2}𝜔^𝑇₁(𝑠)𝑄𝜔₁(𝑠)𝑑𝑠+^𝑡_{𝜏𝑡−2/2}𝜔^𝑇₂(𝑠)𝐾𝜔₂(𝑠)+𝑑𝑠_{𝑡−𝜏1(𝑡)𝑡−𝜏}𝑥^𝑇(𝑠)𝑃₄𝑥(𝑠)+𝑑𝑠_{𝑡(𝑡)𝑡−𝜏}𝑓^𝑇(𝑥(𝑠))𝑃₅𝑓(𝑥(𝑠𝑉))𝑑𝑠,₅(𝑡)=⁰_−𝜏1/2^𝑡_𝑡+𝜃𝜏₁2𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)+𝑑𝑠𝑑𝜃_{0−𝜏1𝑡𝑡+𝜃}𝜏₁𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)+𝑑𝑠𝑑𝜃_{−𝜏1−𝜏2𝑡𝑡+𝜃}𝜏₁₂𝑥̇^𝑇(𝑠)𝑍₃𝑥̇(𝑠)𝑉𝑑𝑠𝑑𝜃,₆(𝑡)=𝑟⁰_−𝑟^𝑡_𝑡+𝜃𝑓^𝑇(𝑥(𝑠))𝑍₄𝑓(𝑥(𝑠𝑉))𝑑𝑠𝑑𝜃,₇(𝑡)=^𝑡_{(𝑡)𝑡−ℎ}𝑥^𝑇(𝑠)𝑍₅𝑥(𝑠)𝑑𝑠+^𝑡_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑍₆𝑥̇(𝑠)+𝑑𝑠_{0−ℎ𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)𝑍₇𝑥̇(𝑠)𝑑𝑠𝑑𝜃, (16) where 𝜏₁₂=𝜏₂−𝜏₁, and 𝜔_𝑖(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇𝜏𝑡−^𝑖2^𝑇,𝑖=1,2. (17) Then, the derivatives of 𝑉_𝑖(𝑡), (𝑖=1,…,7) with time 𝑡 can be obtained as 𝑉̇₁(𝑡)=2𝑥(𝑡)−𝑊₁^𝑡_𝑡−𝛿𝑥(𝑠)𝑑𝑠^𝑇×𝑃₁𝑥̇(𝑡)−𝑊₁𝑥(𝑡)+𝑊₁𝑥(),𝑉̇𝑡−𝛿₂(𝑡)=2𝑥̇^𝑇(𝑡)𝐷𝑓(𝑥(𝑡))−𝐹₁𝑥(𝑡)+2𝑥̇^𝑇(𝑡)𝐿𝐹₂𝑥(𝑡)(𝑥(𝑡,𝑉̇−𝑓))₃(𝑡)=𝑥^𝑇(𝑡)𝑃₂𝑥(𝑡)+𝑥^𝑇(𝑡−𝛿)𝑃₂𝑥(𝑡−𝛿)+𝛿²𝑥^𝑇(𝑡)𝑃₃𝑥(𝑠)−𝛿^𝑡_𝑡−𝛿𝑥^𝑇(𝑠)𝑃₃𝑥(𝑠)𝑉̇𝑑𝑠,₄(𝑡)=𝜔^𝑇₁(𝑡)𝑄𝜔₁(𝑡)−𝜔^𝑇₁𝜏𝑡−¹2𝑄𝜔₁𝜏𝑡−¹2+𝜔^𝑇₂(𝑡)𝐾𝜔₂(𝑡)−𝜔^𝑇₂𝜏𝑡−²2𝐾𝜔₂𝜏𝑡−²2+𝑥^𝑇𝑡−𝜏₁𝑃₄𝑥𝑡−𝜏₁−(1−𝜏̇(𝑡))𝑥^𝑇(𝑡−𝜏(𝑡))𝑃₄𝑥(𝑡−𝜏(𝑡))+𝑓^𝑇(𝑥(𝑡))𝑃₅𝑓(𝑥(𝑡−(𝜏̇(𝑡𝑓))1−))^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑃₅𝑓(𝑥((𝑡,𝑉̇𝑡−𝜏)))₅(𝑡)=𝜏²₁4𝑥̇^𝑇(𝑡)𝑍₁𝑥̇(𝑡)−^𝑡_{𝜏𝑡−1/2}𝜏₁2𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)𝑑𝑠+𝜏²₁𝑥̇^𝑇(𝑡)𝑍₂𝑥̇(𝑡)−^𝑡_𝑡−𝜏1𝜏₁𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠+𝜏²₁₂𝑥̇^𝑇(𝑡)𝑍₃𝑥̇(𝑡)−^𝑡−𝜏1_𝑡−𝜏2𝜏₁₂𝑥̇^𝑇(𝑠)𝑍₃𝑥̇(𝑠)𝑉̇𝑑𝑠,₆(𝑡)=𝑟²𝑓^𝑇(𝑥(𝑡))𝑍₄𝑓(𝑥(𝑡−))_{𝑡𝑡−𝑟}𝑟𝑓^𝑇(𝑥(𝑠))𝑍₄𝑓(𝑥(𝑠𝑉̇))𝑑𝑠,₇(𝑡)=𝑥^𝑇(𝑡)𝑍₅𝑥(𝑡)−𝑥^𝑇(𝑡−ℎ(𝑡))𝑍₅𝑥((𝑡+𝑥̇𝑡−ℎ))_𝑇(𝑡)𝑍₆𝑥̇(𝑡)−1−ℎ̇(𝑡)𝑥̇^𝑇(𝑡−ℎ(𝑡))×𝑍₆𝑥̇((𝑡𝑥̇𝑡−ℎ))+ℎ^𝑇(𝑡)𝑍₇𝑥̇(𝑡)−^𝑡_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑍₇𝑥̇(𝑠)𝑑𝑠. (18) By Lemma 1, one can have −^𝑡_{𝜏𝑡−1/2}𝜏₁2𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)𝑥(𝑡)𝜏𝑑𝑠≤−−𝑥𝑡−¹2^𝑇𝑍₁𝑥(𝑡)𝜏−𝑥𝑡−¹2,−^𝑡_𝑡−𝜏1𝜏₁𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑥(𝑡)𝑑𝑠≤−−𝑥𝑡−𝜏₁^𝑇𝑍₂𝑥(𝑡)−𝑥𝑡−𝜏₁,−^𝑡−𝜏2_𝑡−𝜏1𝜏₁₂𝑥̇^𝑇(𝑠)𝑍₃𝑥̇(𝑠)𝑑𝑠=−^𝑡−𝜏1_{(𝑡)𝑡−𝜏}𝜏₁₂𝑥̇^𝑇(𝑠)𝑍₃𝑥̇(𝑠)−𝑑𝑠_{𝑡−𝜏(𝑡)𝑡−𝜏2}𝜏₁₂𝑥̇^𝑇(𝑠)𝑍₃𝑥̇(𝑠)𝜏𝑑𝑠≤−¹²𝜏(𝑡)−𝜏₁𝜂^𝑇(𝑡)𝑒₃−𝑒₂𝑍₃𝑒₃−𝑒₂^𝑇𝜂(𝑡)−𝜏₁₂𝜏₂(𝑡)𝜂−𝜏^𝑇(𝑡)𝑒₂−𝑒₄𝑍₃𝑒₂−𝑒₄^𝑇𝜂(𝑡)1=−𝛼₁𝜂^𝑇(𝑡)𝑒₃−𝑒₂𝑍₃𝑒₃−𝑒₂^𝑇𝜂(𝑡)−1𝛼₂𝜂^𝑇(𝑡)𝑒₂−𝑒₄𝑍₃𝑒₂−𝑒₄^𝑇𝜂(𝑡)=−𝜂^𝑇(𝑡)𝑒₃−𝑒₂𝑍₃𝑒₃−𝑒₂^𝑇𝜂(𝑡)−𝜂^𝑇(𝑡)𝑒₂−𝑒₄𝑍₃𝑒₂−𝑒₄^𝑇𝜂(𝑡)−𝛼₂𝛼₁𝜂^𝑇(𝑡)𝑒₃−𝑒₂𝑍₃𝑒₃−𝑒₂^𝑇𝜂(𝑡)−𝛼₁𝛼₂𝜂^𝑇(𝑡)𝑒₂−𝑒₄𝑍₃𝑒₂−𝑒₄^𝑇𝜂(𝑡), (19)(20)(21) where 𝛼₁=(𝜏(𝑡)−𝜏₁)/𝜏₁₂ and 𝛼₂=(𝜏₂−𝜏(𝑡))/𝜏₁₂.
It can be seen from the condition (9) that −𝛼₂𝛼₁𝜂^𝑇(𝑡)𝑒₃−𝑒₂𝑍₃𝑒₃−𝑒₂^𝑇𝜂(𝑡)−𝛼₁𝛼₂𝜂^𝑇(𝑡)𝑒₂−𝑒₄𝑍₃𝑒₂−𝑒₄^𝑇𝜂(𝑡)+𝜂^𝑇(𝑡)𝑒₃−𝑒₂𝑇𝑒₂−𝑒₄^𝑇𝜂(𝑡)+𝜂^𝑇(𝑡)𝑒₂−𝑒₄𝑇^𝑇𝑒₃−𝑒₂^𝑇𝜂(𝑡)=−𝜂^𝑇(𝑡)𝛼₂𝛼₁𝑒^𝑇₃−𝑒^𝑇₂−𝛼₁𝛼₂𝑒^𝑇₂−𝑒^𝑇₄^𝑇𝑍₃𝑇𝑇^𝑇𝑍₃×𝛼₂𝛼₁𝑒^𝑇₃−𝑒^𝑇₂−𝛼₁𝛼₂𝑒^𝑇₂−𝑒^𝑇₄𝜂(𝑡)≤0, (22) which means −𝛼₂𝛼₁𝜂^𝑇(𝑡)𝑒₃−𝑒₂𝑍₃𝑒₃−𝑒₂^𝑇𝜂(𝑡)−𝛼₁𝛼₂𝜂^𝑇(𝑡)𝑒₂−𝑒₄𝑍₃𝑒₂−𝑒₄^𝑇𝜂(𝑡)≤−𝜂^𝑇(𝑡)𝑒₃−𝑒₂𝑇𝑒₂−𝑒₄^𝑇𝜂(𝑡)−𝜂^𝑇(𝑡)×𝑒₂−𝑒₄𝑇^𝑇𝑒₃−𝑒₂^𝑇𝜂(𝑡). (23) It follows from (21) and (23) that −^𝑡−𝜏2_𝑡−𝜏1𝜏₁₂𝑥̇^𝑇(𝑠)𝑍₃𝑥̇(𝑠)𝑑𝑠≤−𝜂^𝑇(𝑡)𝑒₂−𝑒₄𝑇^𝑇𝑒^𝑇₃−𝑒^𝑇₂𝜂(𝑡)−𝜂^𝑇(𝑡)𝑒₃−𝑒₂𝑇𝑒^𝑇₂−𝑒^𝑇₄𝜂(𝑡)−𝜂^𝑇(𝑡)𝑒₃−𝑒₂𝑍₃𝑒₃−𝑒₂^𝑇𝜂(𝑡)−𝜂^𝑇(𝑡)𝑒₂−𝑒₄𝑍₃𝑒₂−𝑒₄^𝑇𝜂(𝑡)=−𝜂^𝑇(𝑡)𝑒^𝑇₃−𝑒^𝑇₂𝑒^𝑇₂−𝑒^𝑇₄^𝑇𝑍₃𝑇𝑇^𝑇𝑍₃×𝑒^𝑇₃−𝑒^𝑇₂𝑒^𝑇₂−𝑒^𝑇₄𝜂(𝑡). (24) Therefore, it is straight forward to obtain that 𝑉̇(𝑡)≤2𝑥(𝑡)−𝑊₁^𝑡_𝑡−𝛿𝑥(𝑠)𝑑𝑠^𝑇×𝑃₁𝑥̇(𝑡)−𝑊₁𝑥(𝑡)+𝑊₁𝑥()𝑥̇𝑡−𝛿+2^𝑇(𝑡)𝐷𝑓(𝑥(𝑡))−𝐹₁𝑥(𝑡)+2𝑥̇^𝑇(𝑡)𝐿𝐹₂𝑥(𝑡)−𝑓(𝑥(𝑡))+𝑥^𝑇(𝑡)𝑃₂𝑥(𝑡)−𝑥^𝑇(𝑡−𝛿)𝑃₂𝑥(𝑡−𝛿)+𝛿²𝑥^𝑇(𝑡)𝑃₃𝑥(𝑠)𝑑𝑠−^𝑡_𝑡−𝛿𝑥(𝑠)𝑑𝑠^𝑇𝑃₃^𝑡_𝑡−𝛿𝑥(𝑠)𝑑𝑠+𝜔^𝑇₁(𝑡)𝑄𝜔₁(𝑡)−𝜔^𝑇₁𝜏𝑡−¹2𝑄𝜔₁𝜏𝑡−¹2+𝜔^𝑇₂(𝑡)𝐾𝜔₂(𝑡)−𝜔^𝑇₂𝜏𝑡−²2𝐾𝜔₂𝜏𝑡−²2+𝑥^𝑇𝑡−𝜏₁𝑃₄𝑥𝑡−𝜏₁−1−𝜏_𝑑𝑥^𝑇(𝑡−𝜏(𝑡))𝑃₄𝑥(𝑡−𝜏(𝑡))+𝑓^𝑇(𝑥(𝑡))𝑃₅𝑓(𝑥(𝑡−))1−𝜏_𝑑𝑓^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑃₅𝑓(𝑥((𝑡+𝜏𝑡−𝜏)))₂₁4𝑥̇^𝑇(𝑡)𝑍₁𝑥̇(𝑡)−𝑥(𝑡)𝜏−𝑥𝑡−¹2^𝑇𝑍₁𝑥(𝑡)𝜏−𝑥𝑡−¹2+𝜏²₁𝑥̇^𝑇(𝑡)𝑍₂𝑥̇(𝑡)−𝑥(𝑡)−𝑥𝑡−𝜏₁^𝑇𝑍₂𝑥(𝑡)−𝑥𝑡−𝜏₁+𝜏²₁₂𝑥̇^𝑇(𝑡)𝑍₃𝑥̇(𝑡)−𝜂^𝑇(𝑡)𝑒^𝑇₃−𝑒^𝑇₂𝑒^𝑇₂−𝑒^𝑇₄^𝑇𝑍₃𝑇𝑇^𝑇𝑍₃𝑒^𝑇₃−𝑒^𝑇₂𝑒^𝑇₂−𝑒^𝑇₄𝜂(𝑡)+𝑟²𝑓^𝑇(𝑥(𝑡))𝑍₄𝑓(𝑥(𝑡−))_{𝑡𝑡−𝑟}𝑟𝑓^𝑇(𝑥(𝑠))𝑍₄𝑓(𝑥(𝑠))𝑑𝑠+𝑥^𝑇(𝑡)𝑍₅𝑥(𝑡)−𝑥^𝑇(𝑡−ℎ(𝑡))𝑍₅𝑥((𝑡+𝑥̇𝑡−ℎ))_𝑇(𝑡)𝑍₆𝑥̇(𝑡)−1−ℎ_𝑑𝑥̇^𝑇(𝑡−ℎ(𝑡))𝑍₆𝑥̇((𝑡𝑥̇𝑡−ℎ))+ℎ^𝑇(𝑡)𝑍₇𝑥̇(𝑡)−^𝑡_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑍₇𝑥̇(𝑠)−𝑥(𝑡)𝑓(𝑥(𝑡𝑑𝑠))^𝑇𝐹₃𝑅−𝐹₄𝑅−𝐹₄𝑥(𝑡)𝑓(𝑥(𝑡−𝑥((𝑡𝑓(𝑥((𝑡𝑅𝑅))𝑡−𝜏))𝑡−𝜏)))^𝑇𝐹₃𝑆−𝐹₄𝑆−𝐹₄×𝑥((𝑡𝑓(𝑥((𝑡𝑆𝑆𝑡−𝜏))𝑡−𝜏)))+2𝜂^𝑇(𝑡)𝑀−𝑊₁+Δ𝑊₁(𝑡)𝑥()+𝑊𝑡−𝛿₂+Δ𝑊₂(𝑡)𝑓(𝑥(𝑡+𝑊))₃+Δ𝑊₃(𝑡)𝑓((𝑡+𝑊𝑡−𝜏))₄+Δ𝑊₄(𝑡)^𝑡_{(𝑡)𝑡−𝑟}𝑓(𝑥(𝑠+𝑊))𝑑𝑠₅+Δ𝑊₅(𝑡)𝑥̇(𝑡−ℎ(𝑡))−𝑥̇(𝑡)≤𝜂^𝑇(𝑡)Ξ+Π+Π^𝑇+ℎ𝑁𝑍₇⁻¹𝑁^𝑇𝜂(𝑡)−+ΔΞ_{𝑡(𝑡)𝑡−ℎ}𝑁^𝑇𝜂(𝑡)+𝑍₇𝑥̇(𝑠)^𝑇×𝑍₇⁻¹𝑁^𝑇𝜂(𝑡)+𝑍₇𝑥̇(𝑠)𝑑𝑠≤𝜂^𝑇(𝑡)Ξ+Π+Π^𝑇+ℎ𝑁𝑍₇⁻¹𝑁^𝑇+ΔΞ𝜂(𝑡)≤𝜂^𝑇(𝑡)Ξ+Π+Π^𝑇+ℎ𝑁𝑍₇⁻¹𝑁^𝑇+𝜀⁻¹𝑀𝐻𝐻^𝑇𝑀^𝑇+𝜀𝐸^𝑇𝐸𝜂(𝑡), (25) where ΔΞ=𝑀𝐻𝐹(𝑡)𝐸+(𝑀𝐻𝐹(𝑡)𝐸)^𝑇. By Schur complement, it can be seen from the condition (10) that 𝑉̇(𝑡)<0, which means that neural network (1) under Assumption (𝐻1) is robustly asymptotically stable. This completes this proof.

When neural network (1) without uncertainties, we have 𝑥̇(𝑡)=−𝑊₁𝑥(𝑡−𝛿)+𝑊₂𝑓(𝑥(𝑡))+𝑊₃𝑓(𝑡−𝜏(𝑡))+𝑊₄^𝑡_{(𝑡)𝑡−𝑟}𝑓(𝑥(𝑠))𝑑𝑠+𝑊₅𝑥̇(𝑡−ℎ(𝑡)). (26)

The stability condition can be easily obtained from Theorem 3 in the following corollary.

Corollary 4. Given scalars 0≤𝛿, 0≤𝜏₁<𝜏₂, 0<ℎ, 0<𝑟, ℎ_𝑑<1, and 𝜏_𝑑, neural network (26) is asymptotically stable, if there exist matrices 𝑃_𝑖>0(𝑖=1,2,…,5), 𝑄=[𝑄_𝑖𝑗]_2×2>0, 𝐾=[𝐾_𝑖𝑗]_2×2>0, 𝑍_𝑖>0(𝑖=1,2,…,7), 𝐷=diag{𝑑₁,𝑑₂,…,𝑑_𝑛}>0, 𝐿=diag{𝑙₁,𝑙₂,…,𝑙_𝑛}>0, positive diagonal matrices 𝑅 and 𝑆, and appropriately dimensioned matrices 𝑁_𝑘, 𝑀_𝑘(𝑘=1,2,…,14), such that the following LMIs hold: 𝑍₃𝑇𝑇^𝑇𝑍₃>0,Ξ+Π+Π^𝑇√ℎ𝑁∗−𝑍₇<0, (27) where Ξ, Π and 𝑁 have been defined in Theorem 3.

Remark 5. For neural network (1), if time delay in the leakage term is not considered in this paper, the following model can be obtained: 𝑥̇(𝑡)=−𝑊₁𝑥(𝑡)+𝑊₂𝑓(𝑥(𝑡))+𝑊₃𝑓(𝑡−𝜏(𝑡))+𝑊₄^𝑡_{(𝑡)𝑡−𝑟}𝑓(𝑥(𝑠))𝑑𝑠+𝑊₅𝑥̇(𝑡−ℎ(𝑡)). (28) In order to present stability criterion for neural network (28), we choose the following Lyapunov-Krasovskii functional: 𝑉(𝑡)=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+⁷_𝑖=2𝑉_𝑖(𝑡), (29) where 𝑉_𝑖 (𝑖=2,3,4,5,6,7) is defined in (15). From the proof of Theorem 3, the following stability condition for neural network (28) can be obtained.

Corollary 6. For given scalars 0≤𝜏₁<𝜏₂, 0<ℎ, 0<𝑟, ℎ_𝑑<1, and 𝜏_𝑑, neural network (1) under Assumption (𝐻1) is asymptotically stable, if there exist matrices 𝑃_𝑖>0(𝑖=1,4,…,5), 𝑄=[𝑄_𝑖𝑗]_2×2>0, 𝐾=[𝐾_𝑖𝑗]_2×2>0, 𝑍_𝑖>0(𝑖=1,2,…,7), 𝐷=diag{𝑑₁,𝑑₂,…,𝑑_𝑛}>0, 𝐿=diag{𝑙₁,𝑙₂,…,𝑙_𝑛}>0, positive diagonal matrices 𝑅 and 𝑆, and appropriately dimensioned matrices 𝑁_𝑘, 𝑀_𝑘(𝑘=1,2,…,12), such that the following LMIs hold: 𝑍₃𝑇𝑇^𝑇𝑍₃Π>0,Ξ+Π+^𝑇√ℎ𝑁∗−𝑍₇<0, (30) whereΘΞ=¹¹Θ₁₂∗Θ₂₂,Θ₁₂=0𝑃₁−𝐹^𝑇₁𝐷^𝑇+𝐹^𝑇₂𝐿^𝑇0𝐹₄𝑅000000𝐹₄,Θ𝑆0000000000000000000000000₁₁=Ψ₁₁0𝑍₂0𝑍₁+𝑄₁₂𝐾₁₂∗Ψ₂₂𝑍₃−𝑇^𝑇𝑍₃−𝑇00∗∗𝑃₄−𝑍₂−𝑍₃−𝑄₂₂𝑇−𝑄^𝑇₁₂0∗∗∗−𝐾₂₂−𝑍₃0−𝐾^𝑇₁₂∗∗∗∗−𝑍₁+𝑄₂₂−𝑄₁₁0∗∗∗∗∗𝐾₂₂−𝐾₁₁,Ψ₁₁=𝑄₁₁+𝐾₁₁−𝑍₁−𝑍₂−𝐹₃𝑅+𝑍₅,Ψ₂₂=−1−𝜏_𝑑𝑃₄−𝑍₃−𝑍^𝑇₃−𝐹₃𝑆+𝑇+𝑇^𝑇,Ψ₈₈=𝜏²₁4𝑍₁+𝜏²₁𝑍₂+𝜏²₁₂𝑍₃+𝑍₆+ℎ𝑍₇,Π=𝑁,0,0,0,0,0,−𝑁,0,0,0,0,0+𝑀−𝑊₁,0,0,0,0,0,0,−𝐼,𝑊₅,𝑊_2,𝑊₃,𝑊₄,Θ₂₂=−𝑍₅∗Ψ00000₈₈0𝐷−𝐿00∗∗−1−ℎ_𝑑𝑍₆000∗∗∗𝑃₅+𝑟²𝑍₄−𝑅00∗∗∗∗−1−𝜏_𝑑𝑃₅−𝑆0∗∗∗∗∗−𝑍₄,𝑁=𝑁^𝑇₁,𝑁^𝑇₂,𝑁^𝑇₃,𝑁^𝑇₄,𝑁^𝑇₅,𝑁^𝑇₆,𝑁^𝑇₇,𝑁^𝑇₈,𝑁^𝑇₉,𝑁^𝑇₁₀,𝑁^𝑇₁₁,𝑁^𝑇₁₂^𝑇,𝑀=𝑀^𝑇₁,𝑀^𝑇₂,𝑀^𝑇₃,𝑀^𝑇₄,𝑀^𝑇₅,𝑀^𝑇₆,𝑀^𝑇₇,𝑀^𝑇₈,𝑀^𝑇₉,𝑀^𝑇₁₀,𝑀^𝑇₁₁,𝑀^𝑇₁₂^𝑇. (31)

Remark 7. When neural network (1) without time-varying neutral delay ℎ(𝑡), the following model can be obtained: 𝑥̇(𝑡)=−𝑊₁𝑥(𝑡−𝛿)+𝑊₂𝑓(𝑥(𝑡))+𝑊₃𝑓(𝑡−𝜏(𝑡))+𝑊₄^𝑡_{(𝑡)𝑡−𝑟}𝑓(𝑥(𝑠))𝑑𝑠. (32)
In order to present stability criterion for neural network (32), we choose the following Lyapunov-Krasovskii functional: 𝑉(𝑡)=⁶_𝑖=1𝑉_𝑖(𝑡), (33) where 𝑉_𝑖 (𝑖=1,…,6) are defined in (15). Following the same line of Theorem 3, the following corollary can be presented.

Corollary 8. Given scalars 0≤𝛿, 0≤𝜏₁<𝜏₂, 0<𝑟, and 𝜏_𝑑, the neural network (32) under Assumptions (𝐻1)-(𝐻2) is asymptotically stable, if there exist matrices 𝑃_𝑖>0(𝑖=1,2,…,5), 𝑄=[𝑄_𝑖𝑗]_2×2>0, 𝐾=[𝐾_𝑖𝑗]_2×2>0, 𝑍_𝑖>0(𝑖=1,2,…,4), 𝐷=diag{𝑑₁,𝑑₂,…,𝑑_𝑛}>0, 𝐿=diag{𝑙₁,𝑙₂,…,𝑙_𝑛}>0, positive diagonal matrices 𝑅 and 𝑆, and appropriately dimensioned matrices 𝑀̌_𝑘(𝑘=1,2,…,12), such that the following LMIs hold: 𝑍₃𝑇𝑇^𝑇𝑍₃>0,Ξ̌+Π̌+Π̌^𝑇<0, (34) whereΨ̌₁₁=−𝑃₁𝑊₁−𝑊^𝑇₁𝑃^𝑇₁+𝑃₂+𝛿²𝑃₃+𝑄₁₁+𝐾₁₁−𝑍₁−𝑍₂−𝐹₃Ψ̌𝑅,₂₂=−1−𝜏_𝑑𝑃₄−𝑍₃−𝑍^𝑇₃−𝐹₃𝑆+𝑇+𝑇^𝑇,Ψ̌₈₈=𝜏²₁4𝑍₁+𝜏²₁𝑍₂+𝜏²₁₂𝑍₃,Π=𝑀0,0,0,0,0,0,−𝑊₁,−𝐼,𝑊_2,𝑊₃,0,𝑊₄,𝑀=𝑀^𝑇₁,𝑀^𝑇₂,𝑀^𝑇₃,𝑀^𝑇₄,𝑀^𝑇₅,𝑀^𝑇₆,𝑀^𝑇₇,𝑀^𝑇₈,𝑀^𝑇₉,𝑀^𝑇₁₀,𝑀^𝑇₁₁,𝑀^𝑇₁₂^𝑇,Θ̌₁₁=Ψ̌₁₁0𝑍₂0𝑍₁+𝑄₁₂𝐾₁₂∗Ψ̌₂₂𝑍₃−𝑇^𝑇𝑍₃−𝑇00∗∗𝑃₄−𝑍₂−𝑍₃−𝑄₂₂𝑇−𝑄^𝑇₁₂0∗∗∗−𝐾₂₂−𝑍₃0−𝐾^𝑇₁₂∗∗∗∗−𝑍₁+𝑄₂₂−𝑄₁₁0∗∗∗∗∗𝐾₂₂−𝐾₁₁,Θ̌₁₂=𝑃₁𝑊₁𝑃₁−𝐹^𝑇₁𝐷^𝑇+𝐹^𝑇₂𝐿^𝑇𝐹₄𝑅0𝑊₁𝑃₁𝑊₁0000𝐹₄,̌Θ̌𝑆00000000000000000000000000Ξ=¹¹Θ̌₁₂∗Θ̌₂₂,Θ̌₂₂=−𝑃₂000−𝑊₁𝑃₁𝑊₁0∗Ψ̌₈₈𝐷−𝐿0−𝑃₁𝑊₁0∗∗𝑃₅+𝑟²𝑍₄−𝑅000∗∗∗−1−𝜏_𝑑𝑃₅−𝑆00∗∗∗∗−𝑃₃0∗∗∗∗∗−𝑍₄. (35)
Consider the following neural network with time-varying delay: 𝑥̇(𝑡)=−𝑊₁𝑥(𝑡)+𝑊₂𝑓(𝑥(𝑡))+𝑊₃𝑓(𝑡−𝜏(𝑡)). (36)
By choosing the Lyapunov-Krasovskii functional 𝑉(𝑡)=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝑉₂(𝑡)+𝑉₄(𝑡)+𝑉₆(𝑡), (37) where 𝑉_𝑖 (𝑖=2,4,6) is defined in (15). The novel stability condition for neural network with time-varying delay (36) can be presented from Theorem 3 in the following corollary.

Corollary 9. For given scalars 0≤𝜏₁<𝜏₂, and 𝜏_𝑑, neural network (36) under Assumptions (𝐻1)-(𝐻2) is asymptotically stable, if there exist matrices 𝑃_𝑖>0(𝑖=1,4,5), 𝑄=[𝑄_𝑖𝑗]_2×2>0, 𝐾=[𝐾_𝑖𝑗]_2×2>0, 𝑍_𝑖>0(𝑖=1,2,3), 𝐷=diag{𝑑₁,𝑑₂,…,𝑑_𝑛}>0, 𝐿=diag{𝑙₁,𝑙₂,…,𝑙_𝑛}>0, positive diagonal matrices 𝑅 and 𝑆, and appropriately dimensioned matrices 𝑀_𝑘(𝑘=1,2,…,9), such that the following LMIs hold: 𝑍₃𝑇𝑇^𝑇𝑍₃>0,Ξ+Π+Π^𝑇<0, (38) whereΘ₁₁=Ψ₁₁0𝑍₂0𝑍₁+𝑄₁₂∗Ψ₂₂𝑍₃−𝑇^𝑇𝑍₃−𝑇0∗∗𝑃₄−𝑍₂−𝑍₃−𝑄₂₂𝑇−𝑄^𝑇₁₂∗∗∗−𝐾₂₂−𝑍₃0∗∗∗∗−𝑍₁+𝑄₂₂−𝑄₁₁,Θ₁₂=𝐾₁₂𝑃₁−𝐹^𝑇₁𝐷^𝑇+𝐹^𝑇₂𝐿^𝑇𝐹₄𝑅0000𝐹₄𝑆0000−𝐾^𝑇₁₂,0000000Ψ₇₇=𝜏²₁4𝑍₁+𝜏²₁𝑍₂+𝜏²₁₂𝑍₃,Θ₂₂=𝐾₂₂−𝐾₁₁∗000Ψ₇₇𝐷−𝐿0∗∗𝑃₅−𝑅0∗∗∗−1−𝜏_𝑑𝑃₅,−𝑆Ξ=Θ₁₁Θ₁₂∗Θ₂₂,Ψ₁₁=𝑄₁₁+𝐾₁₁−𝑍₁−𝑍₂−𝐹₃𝑅,Ψ₂₂=−1−𝜏_𝑑𝑃₄−𝑍₃−𝑍^𝑇₃−𝐹₃𝑆+𝑇+𝑇^𝑇,Π=𝑀−𝑊₁,0,0,0,0,0,−𝐼,𝑊_2,𝑊₃,𝑀=𝑀^𝑇₁,𝑀^𝑇₂,𝑀^𝑇₃,𝑀^𝑇₄,𝑀^𝑇₅,𝑀^𝑇₆,𝑀^𝑇₇,𝑀^𝑇₈,𝑀^𝑇₉^𝑇. (39)

4. Numerical Examples

In this section, two numerical examples are provided to show the effectiveness of the proposed results.

Example 1. Consider uncertain neural network of neutral type with leakage delay time-varying delay, and distributed delay as follows: 𝑥̇(𝑡)=−𝑊₁+Δ𝑊₁(𝑡)𝑥(𝑡−𝛿)+𝑊₂+Δ𝑊₂(𝑡)𝑓(𝑥(𝑡+𝑊))₃+Δ𝑊₃(𝑡)𝑓((𝑡+𝑊𝑡−𝜏))₄+Δ𝑊₄(𝑡)^𝑡_{(𝑡)𝑡−𝑟}𝑓(𝑥(𝑠+𝑊))𝑑𝑠₅+Δ𝑊₅(𝑡)𝑥̇(𝑡−ℎ(𝑡)), (40) where 𝑊₁{},𝑊=diag1.6305,1.9221,2.5973,1.3775₂=,𝑊−2.5573−1.38131.9574−1.1398−1.0226−0.88450.5045−0.21111.03781.55320.66451.1902−0.38960.7079−0.3398−2.2543₃=,𝑊0.2853−0.07930.46940.5354−0.59551.3352−0.90360.5529−0.1497−0.6065−0.1641−0.2037−0.4348−1.3474−0.8275−2.2543₄=,𝑊0.02650.11570.0578−0.09300.3186−0.1363−0.08590.07420.2037−0.2049−0.01120.1475−0.3161−0.2469−0.0736−2.2543₅=,𝐹−0.30540.36820.1761−0.0235−0.0546−0.2089−0.07540.26680.45630.00230.14400.6928−0.0115−0.23490.20040.1574₁{},𝐹=diag0,0,0,0₂=diag{1.0275,0.9960,0.3223,0.2113},𝐻=0.1×𝐼,𝐸₁=0.1×𝐼,𝐸₂𝐸=0.2×𝐼,₃=0.3×𝐼,𝐸₄=0.4×𝐼,𝐸₅=0.5×𝐼. (41)

By using Matlab LMI Toolbox, from Theorem 3, it can be found that the uncertain neural network (1) under Assumption (𝐻1) is robustly asymptotically stable for 0<𝛿=𝜏_𝑑=ℎ_𝑑<0.7. To calculate the maximum allowable 𝜏₂=ℎ=𝑟 for different 𝛿=𝜏_𝑑=ℎ_𝑑 in this paper, the numerical results in Table 1 illustrate the effectiveness of the proposed results.

Table 1 

Allowable upper bound 𝜏₂=ℎ=𝑟 for different 𝛿=𝜏_𝑑=ℎ_𝑑.

Example 2. Consider the neural network with time-varying delay as follows: 𝑥̇(𝑡)=−𝑊₁𝑥(𝑡)+𝑊₂𝑓(𝑥(𝑡))+𝑊₃𝑓((𝑡.𝑊𝑡−𝜏))₁=2002,𝑊₂=11−1−1,𝑊₃=,𝐹0.88111₂=0.4000.8,𝐹₁=diag{0,0}. (42)(43)
By using Matlab LMI Toolbox, it can be found from Corollary 9 that the maximum allowable 𝜏₂ can be obtained to guarantee the stability of neural network with time-varying delay in (42) for different 𝜏_𝑑. Compared with the previous results proposed in [28], it is clear that the new stability condition in Corollary 9 is less conservative than the one in [28] (see Table 2).

Table 2 

Allowable upper bound 𝜏₂ for different 𝜏_𝑑.

5. Conclusion

In this paper, the problem of stability analysis for neural networks of neutral type with time-varying delay in the leakage term and distributed delay has been studied. By constructing appropriate Lyapunov-Krasovskii functional and employing some advanced methods, some novel stability criteria have been proposed in terms of LMIs, which can be easily solved by standard software. Two examples have been given to illustrate the effectiveness and merit of the proposed results. It should be mentioned that the leakage delay handling method proposed in this paper can also be used to investigate the systems with the delay in the leakage term, for example, the fault-tolerant control systems [47, 48].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was partially supported by the National Natural Science Foundation of China (61304003, 11226138, and 61304002).