Global 𝜇-Stability Analysis for Impulsive Stochastic Neural Networks with Unbounded Mixed Delays

Abstract

We investigate the global 𝜇-stability in the mean square of impulsive stochastic neural networks with unbounded time-varying delays and continuous distributed delays. By choosing an appropriate Lyapunov-Krasovskii functional, a novel robust stability condition, in the form of linear matrix inequalities, is derived. These sufficient conditions can be tested by MATLAB LMI software packages. The results extend and improve the earlier publication. Two numerical examples are provided to illustrate the effectiveness of the obtained theoretical results.

1. Introduction

Since 1982 when American California Engineering institute physicist Hopfield proposed Hopfield neural networks models [1, 2], artificial neural networks theory and applied research have attracted more and more attentions. The application of artificial neural networks is very extensive. It can process a variety of information, such as artificial intelligence, secure communications, network optimization, military information, pattern recognition, and so forth. Information processing of neural networks greatly depends on the system's dynamic characteristics. The stability is one of very important dynamic characteristics of neural network.

External perturbations can affect the stability of a real system, so it is necessary to consider the stochastic effects for the stability property of neural networks. Recently, some results to guarantee the global asymptotic stability or exponential stability for stochastic neural networks are obtained, see [3–8]. For example, in [4], Blythe et al. investigated stochastic stability of neural networks with constant delay. In [6], Wang et al. analysed the stability in the mean square for stochastic Cohen-Grossberg neural networks with time-varying delays and continuous distributed delays.

As we known, artificial neural networks are often subject to impulsive perturbations which can affect systems’ stability, see [9–11]. Impulsive perturbations often make stable systems unstable. Therefore, we should consider systems’ stability in impulsive effects. Very recently, some results on stability of stochastic neural networks with impulses have been proposed and studied, see [12, 13].

The neural network may be disturbed by environmental noises, which cause the uncertainty of the connection weights of the neurons and affect the dynamical behaviors of system. Therefore, parameter uncertainties should be taken into account in system model. There are some global stability results of neural networks under the influence of parameter uncertainties [14–20]. In [14], the authors kept a watchful eye on robust exponential stability for uncertain stochastic neural networks with discrete interval and distributed time-varying delays by Lyapunov-Krasovski functional method and the random analysis theory.

The time delays happen frequently in the neurotransmission. Many researchers have a large amount interest in time delay neural networks. There are many sufficient conditions have been proposed to guarantee the stability of neural networks with various type of time delays [21–26]. For example, in [22], Lou and Cui investigated the global asymptotic stability of the Hopfield neural networks with bounded time delay by constructing suitable Lyapunov-krasovski functional and employing LMI method. In [24], Raja et al. further obtained some results of stochastic neural networks with mixed time delays by using Lyapunov stability theory and LMI technology. However, most of the results have only focused on bounded delays. As we know, in many engineering time delays depend on the histories heavily and may be unbounded [27, 28]. In this case, the conditions on delays of those existing results are too restrictive.

Recently, Chen and Wang [29], Liu and Chen [30] proposed a new concept of 𝜇-stability, which can be applied to neural networks with unbounded time-varying delay. Moreover, few results have been reported in the literature concerning the problem of 𝜇-stability for parameter impulsive stochastic neural networks with unbounded time-varying delays and continuously distributed delays, which inspire our interests.

This paper is concerned with the global stability analysis in the mean square for impulsive stochastic neural networks with unbounded time-varying delays and continuously distributed delays. By choosing an appropriate Lyapunov-Krasovskii functional, a novel robust stability condition, in the form of linear matrix inequalities, is derived. These sufficient conditions can be tested by Matlab LMI software packages. Our results extend and improve the some results [29, 30].

The paper is organized as following. In Section 2, the basic definitions and assumptions are given together with the statement of the problem. In Section 3, the sufficient conditions of 𝜇-stability in the mean square of impulsive stochastic neural networks with unbounded time-varying delays and continuously distributed delays are obtained. In Section 4, global robust 𝜇-stability criteria in the mean square are derived for uncertain neural networks model. Then, two numerical examples are provided to demonstrate the effectiveness of our results in Section 5. Finally, concluding remarks are given in Section 6.

Notations. Throughout this paper, 𝑅 and 𝑅^𝑛 denote, respectively, the set of real numbers and 𝑛-dimensional real spaces equipped with the Euclidean norm |⋅|; 𝑍₊ denotes the set of positive integers; 𝑅^𝑛×𝑚 denotes the set of all 𝑛×𝑚. Let 𝐴≥0 (𝐴≤0) denote that the matrix 𝐴 is a symmetric and positive semidefinite (negative semidefinite) matrix; 𝐴^𝑇 denotes the transpose of the matrix 𝐴; 𝜆_max(⋅) (𝜆_min) denotes the maximum eigenvalue (minimum eigenvalue) of matrix 𝐴. ℒ₂[0,∞) is the space of square integrable vector. Moreover, let (Ω,ℱ,(ℱ_𝑡)_𝑡≥0,𝒫) be a probability space with a filtration (ℱ_𝑡)_𝑡≥0 satisfying the usual conditions (It is right continuous and ℱ₀ contains all 𝒫-null sets). 𝐿²_ℱ0(−∞,0];𝑅^𝑛 denotes the family of all ℱ₀-measurable. 𝐸(⋅) stands for the mathematical expectation operator with respect to the given probability measure 𝒫.

2. Model Description and Some Assumptions

Consider the following neural network model: 𝑥̇(𝑡)=−𝐴₀𝑥(𝑡)+𝐴₁𝑓(𝑥(𝑡))+𝐴₂𝑓(𝑥(𝑡−𝜏(𝑡)))+𝐴₃^∞₀ℎ(𝑠)𝑓(𝑥(𝑡−𝑠))𝑑𝑠+𝐽,𝑡≠𝑡_𝑘𝑡,𝑡>0,Δ𝑥_𝑘=𝑥𝑡_𝑘−𝑥𝑡⁻_𝑘=𝐽_𝑘𝑥𝑡⁻_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍₊, (1) where 𝑥=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 is the neuron state vector of the neural network; 𝐴₀=diag(𝑎₁⁽⁰⁾,𝑎₂⁽⁰⁾,…,𝑎_𝑛⁽⁰⁾),𝑎_𝑖⁽⁰⁾>0,𝑖=1,2,…,𝑛, is decay rate; 𝐴₁=(𝑎⁽¹⁾_𝑖𝑗)_𝑛×𝑛, 𝐴₂=(𝑎⁽²⁾_𝑖𝑗)_𝑛×𝑛 and 𝐴₃=(𝑎⁽³⁾_𝑖𝑗)_𝑛×𝑛 are connection weight matrix; 𝑓(⋅)=(𝑓₁(⋅),𝑓₂(⋅),…,𝑓_𝑛(⋅))^𝑇 is the neuron activation function; 𝜏(𝑡) represents the transmission delay of neural networks; ℎ(⋅)=diag(ℎ₁(⋅),ℎ₂(⋅),…,ℎ_𝑛(⋅)) is the delay kernel function; 𝐽 is an input constant matrix, and 𝐽_𝑘 is the impulsive function.

Throughout the paper, we make the following assumptions.

Assumption 1. The neuron activation functions 𝑓_𝑗(⋅),𝑗=1,2,…,𝑛, are bounded, continuous differentiable, and satisfy 𝑓0≤^𝑗(𝑢)−𝑓_𝑗(𝑣)𝑢−𝑣≤𝑙_𝑗,𝑢≠𝑣,𝑢,𝑣∈𝑅, (2) where 𝑙_𝑗,𝑗=1,2,…,𝑛 are some positive constants.

Assumption 2. 𝜏(𝑡) is a nonnegative and continuous differentiable time-varying delay and satisfies 𝜏̇(𝑡)≤𝜌<1, 𝜌 is a positive constant.

Assumption 3. The delay kernels ℎ_𝑖,𝑖=1,2,…,𝑛, are some real nonnegative continuous functions defined in [0,∞) and satisfy ^∞₀ℎ_𝑖(𝑠)𝑑𝑠=1,𝑖=1,2,…,𝑛. (3)

Assumption 4. The impulse times 𝑡_𝑘 satisfy 0=𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯,lim_𝑡→∞𝑡_𝑘=+∞.

If the functions 𝑓_𝑗(⋅),𝑗=1,2,…,𝑛 satisfy Assumption 1, the system (1) exists an equilibrium point, see [14]. Assume that 𝑥^∗=(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)^𝑇 is an equilibrium point of system (1), and the impulsive function of system (1) characterized by 𝐽_𝑘(𝑥(𝑡⁻_𝑘))=−𝑀_𝑘(𝑥(𝑡⁻_𝑘)−𝑥^∗), where 𝐽_𝑘(𝑥(𝑡⁻_𝑘))=(𝐽_1𝑘(𝑥(𝑡⁻_𝑘)),𝐽_2𝑘(𝑥(𝑡⁻_𝑘)),…,𝐽_𝑛𝑘(𝑥(𝑡⁻_𝑘)))^𝑇, 𝑀_𝑘 is a real matrix.

Letting 𝑦_𝑖=𝑥_𝑖−𝑥^∗, one can switch the equilibrium point of system (1) to the origin and have 𝑦̇(𝑡)=−𝐴₀𝑦(𝑡)+𝐴₁𝑔𝑦(𝑡)+𝐴₂𝑔𝑦(𝑡−𝜏(𝑡))+𝐴₃^∞₀ℎ(𝑠)𝑔𝑦(𝑡−𝑠)𝑑𝑠,𝑡≠𝑡_𝑘𝑡,𝑡>0,Δ𝑦_𝑘=𝑦𝑡_𝑘−𝑦𝑡⁻_𝑘=−𝑀_𝑘𝑦𝑡⁻_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍₊, (4) where 𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))^𝑇, 𝑔(𝑦(𝑠))=(𝑔₁(𝑦₁(𝑠)),𝑔₂(𝑦₂(𝑠)),…,𝑔_𝑛(𝑦_𝑛(𝑠)))^𝑇, and 𝑔_𝑖(𝑦_𝑖(𝑠))=𝑓_𝑖(𝑥_𝑖(𝑠)+𝑥^∗_𝑖)−𝑓_𝑖(𝑥^∗_𝑖). By the definition of g_𝑖(⋅),𝑖=1,2,…,𝑛 and Assumption 1, 𝑔_𝑖 satisfies the sector condition 𝑔0≤^𝑖(𝑢)𝑢≤𝑙_𝑖,𝑢∈𝑅 (5) and 𝑔_𝑖(0)=0,𝑖=1,2,…,𝑛, where 𝑙_𝑗,𝑗=1,2,…,𝑛 are some positive constants.

In the practical application, a neural system is affected by external perturbations. It is necessary to consider the effect of randomness to neural networks. We obtain stochastic impulsive neural networks with delays: 𝑑𝑦(𝑡)=−𝐴₀𝑦(𝑡)+𝐴₁𝑔𝑦(𝑡)+𝐴₂𝑔𝑦(𝑡−𝜏(𝑡))+𝐴₃^∞₀ℎ(𝑠)𝑔𝑦()(𝑡)((𝑡(𝑡)𝑡−𝑠𝑑𝑠𝑑𝑡+𝜎𝑡,𝑦,𝑦𝑡−𝜏))𝑑𝜔,𝑡≠𝑡_𝑘𝑡,𝑡>0,Δ𝑦_𝑘=𝑦𝑡_𝑘−𝑦𝑡⁻_𝑘=−𝑀_𝑘𝑦𝑡⁻_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍₊, (6) where 𝜔(𝑡)=(𝜔₁(𝑡),𝜔₂(𝑡),…,𝜔_𝑛(𝑡))^𝑇 is 𝑛 dimensional Brownian motions defined on a complete probability space (Ω,ℱ,(ℱ_𝑡)_𝑡≥0,𝒫).

Assumption 5. Assume that 𝜎∶𝑅₊×𝑅^𝑛×𝑅^𝑛→𝑅^𝑛×𝑛 is local Lipschitz continuous and satisfies the linear growth condition, that is, 𝜎(𝑡,0,0)=0 for all 𝑡∈𝑅⁺. Moreover, there exist 𝑛×𝑛 dimension matrices Γ₀>0,Γ₁>0, such that trace𝜎𝑡,𝑦(𝑡),𝑦(𝑡−𝜏(𝑡))^𝑇𝜎𝑡,𝑦(𝑡),𝑦(𝑡−𝜏(𝑡))≤𝑦^𝑇(𝑡)Γ₀𝑦(𝑡)+𝑦^𝑇(𝑡−𝜏(𝑡))Γ₁𝑦(𝑡−𝜏(𝑡)). (7)

The initial conditions for system (6) are 𝑦(𝑠)=𝜑(𝑠),𝑠∈(−∞,0],𝜑∈𝐶²_ℱ0((−∞,0],𝑅^𝑛), where 𝐶²_ℱ0 denotes the family of all bounded ℱ₀-measurable, 𝐶((−∞,0],𝑅^𝑛)-valued variables, satisfying ‖𝜑‖=sup_𝑠≤0𝐸|𝜑(𝑠)|²<∞.

For completeness, we first give the following definition and lemmas.

Definition 1 (see [29]). Suppose that 𝜇(𝑡) is a nonnegative continuous function and satisfieslim_𝑡→∞𝜇(𝑡)=∞. If there exists a scalar 𝑀>0 such that 𝐸𝑦(𝑡≤𝑀)𝜇(𝑡),𝑡≥0, (8) then the system (6) is said to be globally stochastically 𝜇-stable in the mean square.

Obviously, the definition of global stochastic 𝜇-stability in the mean square includes the globally stochastically asymptotical stability and exponential stability in the mean square.

Lemma 2 (see [31]). Let 𝑊₁,𝑊₂,𝑄=𝑄^𝑇>0 be real matrices of appropriate dimensions, and 𝜀>0 be a scalar, then one has 𝑊^𝑇₁𝑊₂+𝑊^𝑇₂𝑊₁≤𝜀𝑊^𝑇₁Q𝑊₁+𝜀⁻¹𝑊^𝑇₂𝑄⁻¹𝑊₂. (9)

Lemma 3 (see [32]). Let 𝑈,𝑉,𝑊 and 𝐹(𝑡) be the real matrices of appropriate dimensions with 𝑊 satisfying 𝑊^𝑇=𝑊, then 𝑊+𝑈𝐹(𝑡)𝑉+𝑉^𝑇𝐹^𝑇(𝑡)𝑈<0,∀𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼, (10) if and only if there exists a scalar 𝜀>0, such that 𝑊+𝜀𝑈𝑈^𝑇+𝜀⁻¹𝑉^𝑇𝑉<0. (11)

Lemma 4 (see [32]). For a given matrix 𝑆𝑆=¹¹𝑆₁₂𝑆₂₁𝑆₂₂>0, (12) where 𝑆^𝑇₁₁=𝑆₁₁, 𝑆^𝑇₂₂=𝑆₂₂ is equivalent to any one of the following conditions:(1)𝑆₂₂>0,𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂>0; (2)S₁₁>0,𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂>0.

3. Stochastic Stability Analysis of Neural Networks

Let 𝐶^2,1(𝑅^𝑛×𝑅⁺→𝑅⁺) denote the family of all nonnegative functions 𝑉(𝑦,𝑡) on 𝑅^𝑛×𝑅⁺, which are continuous one differentiable in 𝑡 and twice differentiable in 𝑥. For every such 𝑉, we define an operator 𝐿𝑉 associated with system (6), 𝐿𝑉=𝑉_𝑡𝑦,𝑡+𝑉_𝑦𝑦,𝑡−𝐴₀𝑦(𝑡)+𝐴₁𝑔𝑦(𝑡)+𝐴₂𝑔𝑦(𝑡−𝜏(𝑡))+𝐴₃^∞₀ℎ(𝑠)𝑔𝑦()+1𝑡−𝑠𝑑𝑠2trace𝜎^𝑇𝑉_𝑦𝑦𝑦,𝑡𝜎, (13) where 𝑉_𝑡𝑦,𝑡=𝜕𝑉𝑦,𝑡,𝑉𝜕𝑡_𝑦𝑦,𝑡=𝜕𝑉(𝑦,𝑡)𝜕𝑦₁,…,𝜕𝑉(𝑦,𝑡)𝜕𝑦_𝑛^𝑇,𝑉_𝑦𝑦𝑦,𝑡=𝜕𝑉(𝑦,𝑡)𝜕𝑦_𝑖𝑦_{𝑗𝑛×𝑛},𝑗=1,2,…,𝑛. (14)

Theorem 5. Assume that Assumptions 1–5 hold. Then, the zero solution of system (6) is globally stochastically 𝜇-stable in the mean sense if there exist diagonal matrices 𝑃>0, 𝑄_𝑖>0,𝑖=1,2 and 𝑄₃=diag(𝑞₁⁽³⁾,…,𝑞_𝑛⁽³⁾), some constants 𝛼₁≥0, 𝛼₂>0, 𝛼₃>0, 𝛿_𝑗>0,𝑗=0,1,2, 𝑚_𝑖𝑘∈[0,2],𝑖=1,2,…,𝑛,𝑘∈𝑍₊, a nonnegative continuous differential function 𝜇(𝑡) defined on [0,∞), and a constant 𝑇>0 such that, for 𝑡≥𝑇, 𝜇̇(𝑡)𝜇(𝑡)≤𝛼₁,𝜇(𝑡−𝜏(𝑡))𝜇(𝑡)≥𝛼₂,∫^∞₀ℎ_𝑖(𝑠)𝜇(𝑡+𝑠)𝑑𝑠𝜇(𝑡)≤𝛼₃,𝑖=1,2,…,𝑛, (15) and the following LMI hold: Ξ^∗=Π₀𝛿₀𝑃𝐴₁𝛿₁1−𝜌𝑃𝐴₂𝛿₂𝑃𝐴₃∗−𝑄₁00∗∗−𝑄₂0∗∗∗−𝑄₃<0, (16) where Π₀=𝛼₁𝑃−𝑃𝐴₀−𝐴^𝑇₀𝑃+𝜆Γ₀+(1/(1−𝜌))𝛼₂⁻¹𝜆Γ₁+𝐿[𝛿₀⁻¹𝑄₁+𝛿₁⁻¹𝛼₂⁻¹𝑄₂+𝛿₂⁻¹𝛼₃𝑄₃]𝐿
𝐿=diag(𝑙₁,𝑙₂,…,𝑙_𝑛), 𝜆=𝜆_max(𝑃), and impulsive operator 𝑀_𝑘=diag(𝑚_1𝑘,…,𝑚_𝑛𝑘).

Proof. By Lemma 4, (16) is equal to the following inequality: 𝛼₁𝑃−𝑃𝐴₀−𝐴^𝑇₀𝑃+𝛿₀𝑃𝐴₁𝑄₁⁻¹𝐴^𝑇₁𝑃+𝛿₁11−𝜌𝑃𝐴₂𝑄₂⁻¹𝐴^𝑇₂𝑃+𝛿₂𝑃𝐴₃𝑄₃⁻¹𝐴^𝑇₃𝑃+𝜆Γ₀+1𝛼1−𝜌²⁻¹𝜆Γ₁+𝐿𝛿₀⁻¹𝑄₁+𝛿₁⁻¹𝛼₂⁻¹𝑄₂+𝛿₂⁻¹𝛼₃𝑄₃𝐿<0. (17) Based on the system (6), we construct the following Lyapunov-Krasovski functional: 𝑉𝑦(𝑡),𝑡=𝑉₁𝑦(𝑡),𝑡+𝑉₂𝑦(𝑡),𝑡+𝑉₃𝑦(𝑡),𝑡, (18) where 𝑉₁𝑦(𝑡),𝑡=𝜇(𝑡)𝑦^𝑇(𝑡)(𝑡),𝑉𝑃𝑦₂𝑦(𝑡)=1,𝑡𝛼1−𝜌²⁻¹𝜆^𝑡_{𝑡−𝜏(𝑡)}𝜇(𝑠)𝑦^𝑇(𝑠)Γ₁𝑦(𝑠)𝑑𝑠+𝛿₁⁻¹𝛼₂^−1𝑡_{(𝑡)𝑡−𝜏}𝜇(𝑠)𝑔^𝑇𝑦(𝑠)𝑄₂𝑔𝑦(𝑠)𝑉𝑑𝑠,₃𝑦(𝑡),𝑡=^𝑛_𝑖=1𝑞_𝑖⁽³⁾𝛿₂^−1∞₀ℎ_𝑖(𝜎)^𝑡_𝑡−𝜎𝜇(𝑠+𝜎)𝑔²_𝑖×𝑦_𝑖(𝑠)𝑑𝑠𝑑𝜎. (19) By Itô's formula, we can obtain the following stochastic differential: 𝑑𝑉𝑦(𝑡),𝑡=𝐿𝑉𝑦(𝑡),𝑡𝑑𝑡+𝑉_𝑦𝑦,𝑡𝜎𝑡,𝑦(𝑡),𝑦(𝑡−𝜏(𝑡))𝑑𝜔(𝑡), (20) where 𝐿𝑉𝑦(𝑡),𝑡=𝐿𝑉₁𝑦(𝑡),𝑡+𝐿𝑉₂𝑦(𝑡),𝑡+𝐿𝑉₃𝑦(𝑡),𝑡. (21) Along the trajectories of system (6), we have that 𝐿𝑉₁𝑦(𝑡),𝑡=𝜇̇𝑦^𝑇(𝑡)𝑃𝑦(𝑡)+2𝜇(𝑡)𝑦^𝑇(𝑡)𝑃×−𝐴₀𝑦(𝑡)+𝐴₁𝑔𝑦(𝑡)+𝐴₂𝑔𝑦(𝑡−𝜏(𝑡))+𝐴₃^∞₀ℎ(𝑠)𝑔𝑦()+1𝑡−𝑠𝑑𝑠2trace𝜎^𝑇𝑉_𝑦𝑦𝑦,𝑡𝜎. (22) By Lemma 2, we get that 2𝑦^𝑇(𝑡)𝑃𝐴₁𝑔𝑦(𝑡)≤𝛿₀𝑦^𝑇(𝑡)𝑃𝐴₁𝑄₁⁻¹𝐴^𝑇₁𝑃𝑦(𝑡)+𝛿₀⁻¹𝑔^𝑇𝑦(𝑡)𝑄₁𝑔𝑦(𝑡),2𝑦^𝑇(𝑡)𝑃𝐴₂𝑔𝑦(𝑡−𝜏(𝑡))≤𝛿₁1−𝜌⁻¹𝑦^𝑇(𝑡)𝑃𝐴₂𝑄₂⁻¹𝐴^𝑇₂𝑃𝑦(𝑡)+𝛿₁⁻¹1−𝜌𝑔^𝑇𝑦(𝑡−𝜏(𝑡))𝑄₂𝑔𝑦(𝑡−𝜏(𝑡)),2𝑦^𝑇(𝑡)𝑃𝐴₃^∞₀ℎ(𝑠)𝑔𝑦(𝑡−𝑠)𝑑𝑠≤𝛿₂𝑦^𝑇(𝑡)𝑃𝐴₃𝑄₃⁻¹𝐴^𝑇₃𝑃𝑦(𝑡)+𝛿₂^−1∞₀ℎ(𝑠)𝑔𝑦(𝑡−𝑠)𝑑𝑠𝑄₃×^∞₀ℎ(𝑠)𝑔𝑦(𝑡−𝑠)𝑑𝑠. (23) From Assumption 5, we obtain that 12trace𝜎^𝑇𝑉_𝑦𝑦𝜎(𝑡)𝑦,𝑡=𝜇trace𝜎^𝑇(𝑡)𝜆𝑃𝜎≤𝜇_max(𝑃)trace𝜎^𝑇𝜎≤𝜇(𝑡)𝜆𝑦^𝑇(𝑡)Γ₀𝑦(𝑡)+𝑦^𝑇(𝑡−𝜏(𝑡))Γ₁𝑦(𝑡−𝜏(𝑡)). (24) Therefore, 𝐿𝑉₁𝑦(𝑡),𝑡≤𝜇(𝑡)𝑦^𝑇(𝑡)𝛼₁𝑃−𝑃𝐴₀−𝐴^𝑇₀𝑃+𝛿₀𝑃𝐴₁𝑄₁⁻¹𝐴^𝑇₁𝑃+𝛿₁1−𝜌⁻¹𝑃𝐴₂𝑄₂⁻¹𝐴^𝑇₂𝑃+𝛿₂𝑃𝐴₃𝑄₃⁻¹𝐴^𝑇₃𝑃+𝜆Γ₀𝑦(𝑡)+𝜇(𝑡)𝛿₀⁻¹𝑔^𝑇𝑦(𝑡)𝑄₁𝑔𝑦(𝑡)+𝜇(𝑡)𝜆𝑦^𝑇(𝑡−𝜏(𝑡))Γ₁𝑦((𝑡(𝑡)𝛿𝑡−𝜏))+𝜇₁⁻¹1−𝜌𝑔^𝑇𝑦×(𝑡−𝜏(𝑡))𝑄₂𝑔𝑦((𝑡(𝑡)𝛿𝑡−𝜏))+𝜇₂^−1∞₀ℎ(𝑠)𝑔𝑦(𝑡−𝑠)𝑑𝑠𝑄₃×^∞₀ℎ(𝑠)𝑔𝑦(𝑡−𝑠)𝑑𝑠,𝐿𝑉₂𝑦(𝑡)≤1,𝑡𝛼1−𝜌²⁻¹𝜆𝜇(𝑡)𝑦^𝑇(𝑡)Γ₁𝑦(𝑡)−𝜇(𝑡−𝜏(𝑡))𝑦^𝑇(𝑡−𝜏(𝑡))×Γ₁𝑦^𝑇(𝑡−𝜏(𝑡))1−𝜏̇(𝑡)+𝛿₁⁻¹𝛼₂⁻¹𝜇(𝑡)𝑔^𝑇𝑦(𝑡)𝑄₂𝑔𝑦(𝑡)−𝜇(𝑡−𝜏(𝑡))𝑔^𝑇𝑦(𝑡−𝜏(𝑡))×𝑄₂𝑔𝑦((𝑡𝜏̇(𝑡≤1𝑡−𝜏))1−)𝛼1−𝜌²⁻¹𝜆𝜇(𝑡)𝑦^𝑇(𝑡)Γ₁𝑦(𝑡)−𝛼₂⁻¹𝜆𝜇(𝑡−𝜏(𝑡))𝑦^𝑇(𝑡−𝜏(𝑡))×Γ₁𝑦^𝑇(𝑡−𝜏(𝑡))+𝛿₁⁻¹𝛼₂⁻¹×𝜇(𝑡)𝑔^𝑇𝑦(𝑡)𝑄₂𝑔𝑦(𝑡)−𝜇(𝑡−𝜏(𝑡))𝑔^𝑇×𝑦(𝑡−𝜏(𝑡))𝑄₂𝑔×𝑦(𝑡−𝜏(𝑡))1−𝜌. (25) From (15), we have 𝐿𝑉₂𝑦(𝑡)≤1,𝑡𝛼1−𝜌²⁻¹𝜆𝜇(𝑡)𝑦^𝑇(𝑡)Γ₁𝑦(𝑡)−𝜆𝜇(𝑡)𝑦^𝑇(𝑡−𝜏(𝑡))Γ₁𝑦^𝑇(𝑡−𝜏(𝑡))+𝛿₁⁻¹𝛼₂⁻¹𝜇(𝑡)𝑔^𝑇𝑦(𝑡)𝑄₂𝑔𝑦(𝑡)−𝛿₁⁻¹𝛼₂⁻¹𝜇(𝑡−𝜏(𝑡))𝑔^𝑇𝑦(𝑡−𝜏(𝑡))×𝑄₂𝑔𝑦(𝑡−𝜏(𝑡))1−𝜌,𝐿𝑉₃𝑦(𝑡),𝑡=𝛿₂^𝑛−1_𝑖=1𝑞_𝑖⁽³⁾𝑔²_𝑖𝑦_𝑖(𝑡)^∞₀ℎ_𝑖(𝜎)𝜇()(𝑡)𝑡+𝜎𝑑𝜎−𝜇^𝑛_𝑖=1𝑞_𝑖⁽³⁾𝛿₂^−1∞₀ℎ_𝑖(𝜎)𝑔²_𝑖𝑦_𝑖(𝑡−𝜎)𝑑𝜎≤𝛿₂⁻¹𝜇(𝑡)∫^∞₀ℎ_𝑖(𝜎)𝜇(𝑡+𝜎)𝑑𝜎𝜇(𝑡)^𝑛_𝑖=1𝑞_𝑖⁽³⁾𝑔²_𝑖𝑦_𝑖(𝑡)−𝜇(𝑡)^𝑛_𝑖=1𝑞_𝑖⁽³⁾𝛿₂^−1∞₀ℎ_𝑖(𝜎)𝑔²_𝑖𝑦_𝑖(𝑡−𝜎)𝑑𝜎≤𝛿₂⁻¹𝜇(𝑡)𝛼₃^𝑛_𝑖=1𝑞_𝑖⁽³⁾𝑔²_𝑖𝑦_𝑖(𝑡)−𝜇(𝑡)^𝑛_𝑖=1𝑞_𝑖⁽³⁾𝛿₂^−1∞₀ℎ_𝑖(𝜎)𝑔²_𝑖𝑦_𝑖(𝑡−𝜎)𝑑𝜎. (26) We use Cauchy's inequality (∫𝑝(𝑠)𝑞(𝑠))²≤(∫𝑝²(𝑠)𝑑𝑠)(∫𝑞²(𝑠)𝑑𝑠) and get 𝜇(𝑡)^𝑛_𝑖=1𝑞_𝑖⁽³⁾𝛿₂^−1∞₀ℎ_𝑖(𝜎)𝑑𝜎^∞₀ℎ_𝑖(𝜎)𝑔²_𝑖𝑦_𝑖((𝑡)𝛿𝑡−𝜎)𝑑𝜎≥𝜇₂^−1∞₀ℎ(𝜎)𝑔𝑦(𝑡−𝜎)𝑑𝜎^𝑇×𝑄₃^∞₀ℎ(𝜎)𝑔𝑦(𝑡−𝜎)𝑑𝜎. (27) Thus, 𝐿𝑉₃𝑦(𝑡),𝑡≤𝛿₂⁻¹𝜇(𝑡)𝛼₃𝑔^𝑇𝑦(𝑡)𝑄₃𝑔𝑦(𝑡(𝑡)𝛿)−𝜇₂^−1∞₀ℎ(𝜎)𝑔𝑦(𝑡−𝜎)𝑑𝜎^𝑇,𝑄₃^∞₀ℎ(𝜎)𝑔𝑦(𝑡−𝜎)𝑑𝜎. (28) Substituting (25), (26), and (28) to (21), we get 𝑦(𝑡)(𝑡)𝑦𝐿𝑉,𝑡≤𝜇^𝑇(𝑡)𝛼₁𝑃−𝑃𝐴₀−𝐴^𝑇₀𝑃+𝛿₀𝑃𝐴₁𝑄₁⁻¹𝐴^𝑇₁𝑃+𝛿₁1−𝜌⁻¹𝑃𝐴₂𝑄₂⁻¹𝐴^𝑇₂𝑃+𝛿₂𝑃𝐴₃𝑄₃⁻¹𝐴^𝑇₃𝑃+𝜆Γ₀+1−𝜌⁻¹𝛼₂⁻¹𝜆Γ₁𝑦(𝑡)+𝜇(𝑡)𝑔^𝑇𝑦(𝑡×𝛿₀₋₁)𝑄₁+𝛿₁⁻¹𝛼₂⁻¹𝑄₂+𝛿₂⁻¹𝛼₃𝑄₃𝑔𝑦(𝑡(𝑡)𝑦)≤𝜇^𝑇(𝑡)𝛼₁𝑃−𝑃𝐴₀−𝐴^𝑇₀𝑃+𝛿₀𝑃𝐴₁𝑄₁⁻¹𝐴^𝑇₁𝑃+𝛿₁1−𝜌⁻¹𝑃𝐴₂𝑄₂⁻¹𝐴^𝑇₂𝑃+𝛿₂𝑃𝐴₃𝑄₃⁻¹𝐴^𝑇₃𝑃+𝜆Γ₀+1−𝜌⁻¹𝛼₂⁻¹𝜆Γ₁+𝐿𝛿₀⁻¹𝑄₁+𝛿₁⁻¹𝛼₂⁻¹𝑄₂+𝛿₂⁻¹𝛼₃𝑄₃𝐿𝑦(𝑡)=𝜇(𝑡)𝑦^𝑇(𝑡)Ξ𝑦(𝑡), (29) where Ξ=𝛼₁𝑃−𝑃𝐴₀−𝐴^𝑇₀𝑃+𝛿₀𝑃𝐴₁𝑄₁⁻¹𝐴^𝑇₁𝑃+𝛿₁1−𝜌⁻¹𝑃𝐴₂𝑄₂⁻¹𝐴^𝑇₂𝑃+𝛿₂𝑃𝐴₃𝑄₃⁻¹𝐴^𝑇₃𝑃+𝜆Γ₀+1−𝜌⁻¹𝛼₂⁻¹𝜆Γ₁+𝐿𝛿₀⁻¹𝑄₁+𝛿₁⁻¹𝛼₂⁻¹𝑄₂+𝛿₂⁻¹𝛼₃𝑄₃𝐿. (30) So, we have 𝑑𝑉𝑦(𝑡),𝑡≤𝜇(𝑡)𝑦^𝑇(𝑡)Ξ𝑦(𝑡)+𝑉_𝑦𝑦(𝑡),𝑡𝜎𝑡,𝑦(𝑡),𝑦(𝑡−𝜏(𝑡))𝑑𝜔(𝑡). (31) Taking the mathematical expectation, we get dEV𝑦(𝑡),𝑡(𝑡)𝑦𝑑𝑡≤𝜇^𝑇(𝑡)Ξ𝑦(𝑡). (32) By Ξ<0, we get dEV𝑦(𝑡),𝑡𝑡𝑑𝑡≤0,𝑡∈_𝑘−1,𝑡_𝑘∩[𝑇,+∞),𝑘∈𝑍₊. (33) In addition, 𝑉₁𝑡_𝑘=𝜇𝑡_𝑘𝑦^𝑇𝑡_𝑘𝑃𝑦𝑡_𝑘=𝜇𝑡⁻_𝑘𝑦𝑡⁻_𝑘−𝑀_𝑘𝑦𝑡⁻_𝑘^𝑇×𝑃𝑦^𝑇𝑡⁻_𝑘−𝑀_𝑘𝑦^𝑇𝑡⁻_𝑘=𝜇𝑡⁻_𝑘𝐼−𝑀_𝑘𝑦𝑡⁻_𝑘^𝑇𝑃𝐼−𝑀_𝑘𝑦𝑡⁻_𝑘=𝜇𝑡⁻_𝑘𝑦^𝑇𝑡⁻_𝑘𝐼−𝑀_𝑘^𝑇𝑃𝐼−𝑀_𝑘𝑦𝑡⁻_𝑘. (34) Noting 𝑀_𝑘=diag(𝑚_1𝑘,𝑚_2𝑘,…,𝑚_𝑛𝑘),𝑚_𝑖𝑘∈[0,2],𝑖=1,2,…,𝑛,𝑘∈𝑍₊, we have 𝑉₁𝑡_𝑘≤𝜇𝑡⁻_𝑘𝑦^𝑇𝑡⁻_𝑘𝑃𝑦𝑡⁻_𝑘=𝑉₁𝑡⁻_𝑘. (35) It is obvious that 𝑉₂(𝑡_𝑘)=𝑉₂(𝑡⁻_𝑘), 𝑉₃(𝑡_𝑘)=𝑉₃(𝑡⁻_𝑘), therefore 𝑉𝑡_𝑘≤𝑉𝑡⁻_𝑘,𝑘∈𝑍₊. (36) By (33) and (36), we know that 𝑉 is monotonically nonincreasing for 𝑡∈[𝑇,∞), which implies that 𝐸𝑉𝑦(𝑡),𝑡≤𝐸𝑉𝑦(𝑇),𝑇,𝑡≥𝑇. (37) From the definition of 𝑉, we can deduce that 𝜇(𝑡)𝜆_min(𝑃)𝐸𝑦(𝑡)²≤𝜇(𝑡)𝐸𝑦^𝑇(𝑡)(𝑡)𝑦(𝑡)𝑃𝐸𝑦≤𝐸𝑉,𝑡≤𝑀<∞,𝑡≥0, (38) where 𝑀=max_{0≤𝑠≤𝑇}𝐸𝑉(𝑦,𝑠). It implies that 𝐸𝑦(𝑡)²≤𝑀𝜇(𝑡)𝜆_min(𝑃),𝑡≥0. (39) This completes the proof of Theorem 5.

Remark 6. Theorem 5 provides a global stochastic 𝜇-stability in the mean sense for a impulsive stochastic differential system (6). It should be noted that our results depend on the upper bound of the derivative of time-varying delay, the influence of randomness, the delay kernels of continuous distribution, and have nothing to do with the range of time-varying delay. Therefore, our results can be applied to the impulsive stochastic neural networks with unbounded time-varying delay and continuous distributed delay.

Remark 7. In [29, 30], different approaches were used to study 𝜇-stability of neural networks with unbounded time-varying. However, the impulsive effect and unbounded continuous distributed delay are not considered in their models. It is known that neural networks with unbounded continuous distributed delays in impulsive effect are of great importance in many practically problems. Hence, our results in this paper are more general than those reported in [29, 30],

If the diffusion coefficient 𝜎(𝑡,𝑦(𝑡),𝑦(𝑡−𝜏(𝑡)))=0, the model (6) becomes the model (4), then the following result can be obtained.

Corollary 8. Assume that Assumptions 1–4 hold. Then, the zero solution of system (4) is globally 𝜇-stable if there exist diagonal matrices 𝑃>0, 𝑄_𝑖>0,𝑖=1,2 and 𝑄₃=diag(𝑞₁⁽³⁾,…,𝑞_𝑛⁽³⁾), some constants 𝛼₁≥0, 𝛼₂>0, 𝛼₃>0, 𝛿_𝑗>0,𝑗=0,1,2, 𝑚_𝑖𝑘∈[0,2],𝑖=1,2,…,𝑛,𝑘∈𝑍₊, a nonnegative continuous differential function 𝜇(𝑡) defined on [0,∞), and a constant 𝑇>0 such that, for 𝑡≥𝑇, 𝜇̇(𝑡)𝜇(𝑡)≤𝛼₁,𝜇(𝑡−𝜏(𝑡))𝜇(𝑡)≥𝛼₂,∫^∞₀ℎ_𝑖(𝑠)𝜇(𝑡+𝑠)𝑑𝑠𝜇(𝑡)≤𝛼₃,𝑖=1,2,…,𝑛, (40) and the following LMI hold: Ξ^∗=Π₀𝛿₀𝑃𝐴₁𝛿₁1−𝜌𝑃𝐴₂𝛿₂𝑃𝐴₃∗−𝑄₁00∗∗−𝑄₂0∗∗∗−𝑄₃<0, (41) where Π₀=𝛼₁𝑃−𝑃𝐴₀−𝐴^𝑇₀𝑃+𝐿[𝛿₀⁻¹𝑄₁+𝛿₁⁻¹𝛼₂⁻¹𝑄₂+𝛿₂⁻¹𝛼₃𝑄₃]𝐿, 𝐿=diag(𝑙₁,𝑙₂,…,𝑙_𝑛), 𝜆=𝜆_max(𝑃), and impulsive operator 𝑀_𝑘=diag(𝑚_1𝑘,…,𝑚_𝑛𝑘).

If we take 𝑃=𝑄₁=𝑄₂=𝑄₃=𝐼 in Theorem 5, then we obtain the following testable condition.

Corollary 9. Assume that Assumptions 1–5 hold. Then, the zero solution of system (6) is globally stochastically 𝜇-stable in the mean square if there exist some constants 𝛼₁≥0,𝛼₂>0,𝛼₃>0,𝛿_𝑗>0,𝑗=0,1,2,𝑚_𝑖𝑘∈[0,2],𝑖=1,2,…,𝑛,𝑘∈𝑍₊, a nonnegative continuous differential function 𝜇(𝑡) defined on [0,∞), and a constant 𝑇>0 such that, for 𝑡≥𝑇, 𝜇̇(𝑡)𝜇(𝑡)≤𝛼₁,𝜇(𝑡−𝜏(𝑡))𝜇(𝑡)≥𝛼₂,∫^∞₀ℎ_𝑖(𝑠)𝜇(𝑡+𝑠)𝑑𝑠𝜇(𝑡)≤𝛼₃,𝑖=1,2,…,𝑛, (42) and the following LMI hold: Ξ^∗=Π₀𝛿₀𝐴₁𝛿₁𝐴1−𝜌²𝛿₂𝐴₃∗−𝐼00∗∗−𝐼0∗∗∗−𝐼<0, (43) where Π₀=𝛼₁−𝐴₀−𝐴^𝑇₀+𝜆Γ₀+(1/(1−𝜌))𝛼₂⁻¹𝜆Γ₁+𝐿[𝛿₀⁻¹+𝛿₁⁻¹𝛼₂⁻¹+𝛿₂⁻¹𝛼₃]𝐿, 𝐿=diag(𝑙₁,𝑙₂,…,𝑙_𝑛), 𝜆=𝜆_max(𝑃), and impulsive operator 𝑀_𝑘=diag(𝑚_1𝑘,…,𝑚_𝑛𝑘).

4. Robust Stochastic Stability Analysis of Neural Networks

Consider the following uncertain stochastic neural networks: 𝑦̇(𝑡)=−𝐴₀(𝑡)𝑦(𝑡)+𝐴₁(𝑡)𝑔𝑦(𝑡)+𝐴₂(𝑡)𝑔𝑦(𝑡−𝜏(𝑡))+𝐴₃(𝑡)^∞₀ℎ(𝑠)𝑔𝑦()(𝑡)((𝑡(𝑡)𝑡−𝑠𝑑𝑠+𝜎𝑡,𝑦,𝑦𝑡−𝜏))𝑑𝜔,𝑡≠𝑡_𝑘𝑡,𝑡>0,Δ𝑦_𝑘=𝑦𝑡_𝑘−𝑦𝑡⁻_𝑘=−𝑀_𝑘𝑦𝑡⁻_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍₊, (44) where some parameters and variables are introduced in Section 2, the uncertainties are described as follows: 𝐴_𝑖(𝑡)=𝐴_𝑖+Δ𝐴_𝑖(𝑡),𝑖=1,2,3, (45) where Δ𝐴_𝑖(𝑡),𝑖=1,2,3 are perturbed matrices satisfying Δ𝐴₀(𝑡),Δ𝐴₁(𝑡),Δ𝐴₂(𝑡),Δ𝐴₃(𝑡(𝑡𝑇)=𝐻𝐹)₀,𝑇₁,𝑇₂,𝑇₃, (46) where 𝐻,𝑇_𝑖,𝑖=1,2,3 are some appropriate dimensions constant matrices, 𝐹(𝑡) is an unknown real time-varying function with appropriate dimensions and satisfies 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼, (47) where 𝐼 is appropriate dimensions unit matrix.

Theorem 10. Assume that Assumptions 1–5 hold. Then, the zero solution of system (44) is globally robustly stochastically 𝜇-stable in the mean square if there exist diagonal matrices 𝑃>0,𝑄_𝑖>0, 𝑖=1,2 and 𝑄₃=diag(𝑞₁⁽³⁾,…,𝑞_𝑛⁽³⁾), some constants 𝛼₁≥0, 𝛼₂>0, 𝛼₃>0, 𝛿_𝑗>0, 𝑗=0,1,2,𝜀_𝑗>0, 𝑗=0,1,2,3,𝑚_𝑖𝑘∈[0,2], 𝑖=1,2,…,𝑛, 𝑘∈𝑍₊, a nonnegative continuous differential function 𝜇(𝑡) defined on [0,∞), and a constant 𝑇>0 such that, for 𝑡≥𝑇, 𝜇̇(𝑡)𝜇(𝑡)≤𝛼₁,𝜇(𝑡−𝜏(𝑡))𝜇(𝑡)≥𝛼₂,∫^∞₀ℎ_𝑖(𝑠)𝜇(𝑡+𝑠)𝑑𝑠𝜇(𝑡)≤𝛼₃,𝑖=1,2,…𝑛, (48) and the following LMI hold: Ξ^∗∗=Π₁𝛿₀𝑃𝐴₁𝛿₁1−𝜌𝑃𝐴₂𝛿₂𝑃𝐴₃∗𝑅₁00∗∗𝑅₂0∗∗∗𝑅₃<0, (49) where Π₁=𝛼₁𝑃−𝑃𝐴₀−𝐴^𝑇₀𝑃+𝜆Γ₀+(1/(1−𝜌))𝛼₂⁻¹𝜆Γ₁+𝐿[𝛿₀⁻¹𝑄₁+𝛿₁⁻¹𝛼₂⁻¹𝑄₂+𝛿₂⁻¹𝛼₃𝑄₃]𝐿+𝜀₀⁻¹𝑃𝐻𝐻^𝑇𝑃+𝜀₀𝑇^𝑇₀𝑇₀+𝜀₁𝛿₀𝑇^𝑇₁𝑇₁+𝜀₂(𝛿₁/(1−𝜌))𝑇^𝑇₂𝑇₂+𝜀₃𝛿₂𝑇^𝑇₃𝑇₃, 𝐿=diag(𝑙₁,𝑙₂,…,𝑙_𝑛), 𝜆=𝜆_max(𝑃), 𝑅₁=−𝑄₁+𝜀₁⁻¹𝑃𝐻𝐻^𝑇𝑃, 𝑅₂=−𝑄₂+𝜀₂⁻¹𝑃𝐻𝐻^𝑇𝑃, 𝑅₃=−𝑄₃+𝜀₃⁻¹𝑃𝐻𝐻^𝑇𝑃, and impulsive operator 𝑀_𝑘=diag(𝑚_1𝑘,…,𝑚_𝑛𝑘).

Proof. Let the Lyapunov-Krasovskii functional as defined in Theorem 5, then by Itô's formula, we haveΞ^∗∗=Ξ^∗+−𝑃Δ𝐴₀(𝑡)−Δ𝐴^𝑇₀(𝑡)𝑃𝛿₀𝑃Δ𝐴₁(𝑡)𝛿₁1−𝜌𝑃Δ𝐴₂(𝑡)𝛿₂𝑃Δ𝐴₃(𝑡)∗000∗∗00∗∗∗0=Ξ^∗+−𝑃𝐻𝐹(𝑡)𝑇₀−𝑇^𝑇₀𝐹^𝑇(𝑡)𝐻^𝑇𝑃𝛿₀𝑃𝐻𝐹(𝑡)𝑇₁𝛿₁(𝑡)𝑇1−𝜌𝑃𝐻𝐹₂𝛿₂𝑃𝐻𝐹(𝑡)𝑇₃∗000∗∗00∗∗∗0=Ξ^∗+−𝑇^𝑇₀000𝐹^𝑇(𝑡)𝐻^𝑇𝑃000+𝐻^𝑇𝑃000^𝑇𝐹(𝑡)−𝑇^𝑇₀000^𝑇+𝛿₀𝑇^𝑇₁000𝐹^𝑇(𝑡)0𝐻^𝑇𝑃00+0𝐻^𝑇𝑃00^𝑇𝐹(𝑡)𝛿₀𝑇^𝑇₁000^𝑇+𝛿₁𝑇1−𝜌^𝑇2000𝐹^𝑇(𝑡)00𝐻^𝑇𝑃0+00𝐻^𝑇𝑃0^𝑇𝐹(𝑡)𝛿₁𝑇1−𝜌^𝑇2000^𝑇+𝛿₂𝑇^𝑇₃000𝐹^𝑇(𝑡)000𝐻^𝑇𝑃+000𝐻^𝑇𝑃^𝑇𝐹(𝑡)𝛿₂𝑇^𝑇₃000^𝑇. (50)
By Lemma 3, we get Ξ^∗∗≤Ξ^∗+𝜀₀⁻¹000𝐻𝑃𝐻_𝑇𝑃000+𝜀₀−𝑇^𝑇₀000−𝑇₀000+𝜀₁⁻¹000𝑃𝐻0𝐻^𝑇𝑃00+𝜀₁𝛿₀𝑇^𝑇₁000𝛿₀𝑇₁000+𝜀₂⁻¹000𝑃𝐻00𝐻^𝑇𝑃0+𝜀₂𝛿₁𝑇1−𝜌^𝑇2000𝛿₁𝑇1−𝜌²000+𝜀₃⁻¹000𝑃𝐻000𝐻^𝑇𝑃+𝜀₃𝛿₂𝑇^𝑇₃000𝛿₂𝑇₃000=Ξ^∗+Π₂000∗𝜀₁⁻¹𝑃𝐻𝐻^𝑇𝑃00∗∗𝜀₂⁻¹𝑃𝐻𝐻^𝑇𝑃0∗∗∗𝜀₃⁻¹𝑃𝐻𝐻^𝑇𝑃, (51)whereΠ₂=𝜀₀⁻¹𝑃𝐻𝐻^𝑇𝑃+𝜀₀𝑇^𝑇₀𝑇₀+𝜀₁𝛿₀𝑇^𝑇₁𝑇₁+𝜀₂(𝛿₁/(1−𝜌))𝑇^𝑇₂𝑇₂+𝜀₃𝛿₂𝑇^𝑇₃𝑇₃.
This completes the proof of Theorem 10.

Remark 11. The neural network can be disturbed by environmental noises, which cause system parameters' uncertainty. It can lead to system instability. As far as we know, there is no literature that is published on robust 𝜇-stability on uncertain neural networks with unbounded continuous distributed delays in impulsive perturbations.

There exist only parameter uncertainties and no stochastic perturbations. So the uncertain neural networks can be described as 𝑦̇(𝑡)=−𝐴₀(𝑡)𝑦(𝑡)+𝐴₁(𝑡)𝑔𝑦(𝑡)+𝐴₂(𝑡)𝑔𝑦(𝑡−𝜏(𝑡))+𝐴₃(𝑡)^∞₀ℎ(𝑠)𝑔𝑦(𝑡−𝑠)𝑑𝑠,𝑡≠𝑡_𝑘𝑡,𝑡>0,Δ𝑦_𝑘=𝑦𝑡_𝑘−𝑦𝑡⁻_𝑘=−𝑀_𝑘𝑦𝑡⁻_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍₊. (52)

Corollary 12. Assume that Assumptions 1–5 hold. Then, the zero solution of system (52) is globally robustly 𝜇-stable if there exist diagonal matrices 𝑃>0, 𝑄_𝑖>0, 𝑖=1,2 and 𝑄₃=diag(𝑞₁⁽³⁾,…,𝑞_𝑛⁽³⁾), some constants 𝛼₁≥0, 𝛼₂>0, 𝛼₃>0, 𝛿_𝑗>0, 𝑗=0,1,2, 𝜀_𝑗>0,𝑗=0,1,2,3, 𝑚_𝑖𝑘∈[0,2],𝑖=1,2,…,𝑛,𝑘∈𝑍₊, a nonnegative continuous differential function 𝜇(𝑡) defined on [0,∞], and a constant 𝑇>0 such that, for 𝑡≥𝑇, 𝜇̇(𝑡)𝜇(𝑡)≤𝛼₁,𝜇(𝑡−𝜏(𝑡))𝜇(𝑡)≥𝛼₂,∫^∞₀ℎ_𝑖(𝑠)𝜇(𝑡+𝑠)𝑑𝑠𝜇(𝑡)≤𝛼₃,𝑖=1,2,…𝑛, (53) and the following LMI hold: Ξ^∗∗=Π₁𝛿₀𝑃𝐴₁𝛿₁1−𝜌𝑃𝐴₂𝛿₂𝑃𝐴₃∗𝑅₁00∗∗𝑅₂0∗∗∗𝑅₃<0, (54) where Π₁=𝛼₁𝑃−𝑃𝐴₀−𝐴^𝑇₀𝑃+𝐿[𝛿₀⁻¹𝑄₁+𝛿₁⁻¹𝛼₂⁻¹𝑄₂+𝛿₂⁻¹𝛼₃𝑄₃]𝐿+𝜀₀⁻¹𝑃𝐻𝐻^𝑇𝑃+𝜀₀𝑇^𝑇₀𝑇₀+𝜀₁𝛿₀𝑇^𝑇₁𝑇₁+𝜀₂(𝛿₁/(1−𝜌))𝑇^𝑇₂𝑇₂+𝜀₃𝛿₂𝑇^𝑇₃𝑇₃, 𝐿=diag(𝑙₁,𝑙₂,…,𝑙_𝑛), 𝜆=𝜆_max(𝑃), 𝑅₁=−𝑄₁+𝜀₁⁻¹𝑃𝐻𝐻^𝑇𝑃, 𝑅₂=−𝑄₂+𝜀₂⁻¹𝑃𝐻𝐻^𝑇𝑃, 𝑅₃=−𝑄₃+𝜀₃⁻¹𝑃𝐻𝐻^𝑇𝑃, and impulsive operator 𝑀_𝑘=diag(𝑚_1𝑘,…,𝑚_𝑛𝑘).

5. Illustrative Examples

In this section, we will give two examples to show the validity of the results obtained.

Example 13. Consider the stochastic impulsive neural network with two neurons: 𝑑𝑥(𝑡)=−𝐴₀𝑥(𝑡)+𝐴₁𝑔(𝑥(𝑡))+𝐴₂𝑔(𝑥(𝑡−𝜏(𝑡)))+𝐴₃^∞₀ℎ(𝑠)𝑔(𝑥(((𝑡)((𝑡(𝑡)𝑡−𝑠))𝑑𝑠𝑑𝑡+𝜎𝑡,𝑥,𝑥𝑡−𝜏)))𝑑𝜔,𝑡≠𝑡_𝑘𝑡,𝑡>0,Δ𝑥_𝑘=𝑥𝑡_𝑘−𝑥𝑡⁻_𝑘=−𝑀_𝑘𝑥𝑡⁻_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍₊, (55) where the activation function is described by 𝑔_𝑖(𝑠)=tanh(𝑠), 𝑖=1,2,…,𝑛, 𝜏(𝑡)=0.4𝑡, ℎ_𝑖(𝑠)=𝑒^−𝑠. It is obvious that (0,0)^𝑇 is an equilibrium point of system (55). Let 𝜇(𝑡)=𝑡 and choose 𝛼₁=0.1, 𝛼₂=0.6, 𝛼₃=1.2, 𝑀_𝑘=diag(0.5,0.5), and the parameter matrices are, respectively, given by 𝐴₀=2.8003.2,𝐴₁=,𝐴0.2−0.10.10.1₂=0.060.07−0.070.03,𝐴₃=,Γ0.02−0.030.010.01₀=Γ₁=.1.1001.1 (56)

In this case, we get 𝐿=diag(1,1),𝜌=0.4. Let 𝑄₁=𝑄₂=𝑄₃=𝐼, 𝛿₀=𝛿₁=𝛿₂=1, and 𝑃=diag(10,9). We can get 𝜆_max(Ξ)=−0.9590<0 via MATLAB LMI toolbox. By Theorem 5, the equilibrium point of model (55) with unbounded time-varying delay and continuously distributed delay is globally stochastically 𝜇-stable in the mean square. The numerical simulation is shown in Figure 1.

(a)

(b)

(c)

(d)

Figure 1 

(a) Time-series of the 𝐸(𝑥1)2 of model (55) without impulsive effects for 𝑡∈[0,5]. (b) Time-series of the 𝐸(𝑥2)2 of model (55) without impulsive effects for 𝑡∈[0,5]. (c) Time-series of the 𝐸(𝑥1)2 of model (55) with impulsive effects for 𝑡∈[0,5]. (d) Time-series of the 𝐸(𝑥2)2 of model (55) with impulsive effects for 𝑡∈[0,5].

Example 14. Consider the uncertain stochastic impulsive neural network with two neurons: 𝑑𝑥(𝑡)=−𝐴₀(𝑡)𝑥(𝑡)+𝐴₁𝑔(𝑥(𝑡))+𝐴₂(𝑡)𝑔(𝑥(𝑡−𝜏(𝑡)))+𝐴₃(𝑡)^∞₀ℎ(𝑠)𝑔(𝑥(((𝑡)((𝑡(𝑡)𝑡−𝑠))𝑑𝑠𝑑𝑡+𝜎𝑡,𝑥,𝑥𝑡−𝜏)))𝑑𝜔,𝑡≠𝑡_𝑘𝑡,𝑡>0,Δ𝑥_𝑘=𝑥𝑡_𝑘−𝑥𝑡⁻_𝑘=−𝑀_𝑘𝑥𝑡⁻_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑍₊, (57) where the activation function is described by 𝑔_𝑖(𝑠)=tanh(𝑠),𝑖=1,2,…,𝑛, 𝜏(𝑡)=0.5𝑡, ℎ_𝑖(𝑠)=𝑒^−𝑠. It is obvious that (0,0)^𝑇 is an equilibrium point of system (57). Let 𝜇(𝑡)=𝑡 and choose 𝛼₁=0.1, 𝛼₂=0.5, 𝛼₃=1.2, 𝑀_𝑘=diag(1.5,1.5), and the parameter matrices are, respectively, given by 𝐴₀=9.80011.2,𝐴₁=,𝐴0.2−0.20.40.3₂=0.60.7−0.10.3,𝐴₃=,Γ0.12000.21₀=Γ₁=.0.5000.5 (58) In this case, we get 𝐿=diag(1,1),𝜌=0.5. Let 𝐻=𝑇₀=𝑇₁=𝑇₂=𝑇₃=0.2𝐼, 𝛿₀=𝛿₁=𝛿₂=𝜀₀=𝜀₁=𝜀₂=𝜀₃=1, we can get the following feasible solution via MATLAB LMI toolbox: 𝑃=0.0863000.0863,𝑄₁=,𝑄0.1416000.1416₂=0.0527000.0527,𝑄₃=,0.1461000.1461 (59) and obtain 𝜆_max(Ξ^∗∗)=−1.0865<0. By Theorem 10, the equilibrium point of model (24) is globally robustly stochastically 𝜇-stable in the mean square.

6. Concluding Remarks

This paper firstly investigated the global stochastic 𝜇-stability in the mean square of a class of impulsive stochastic neural network with unbounded mixed delays. We obtained some sufficient conditions by using Lyapunov-Krasovskii functional method, the random analysis theory, and the linear matrix inequality (LMI) technique. Secondly, we researched global robust 𝜇-stability in the mean square for a class of uncertain impulsive neural network with unbounded mixed delays. Our results improve and generalize some earlier works reported in the literature. Finally, two examples and their numerical simulations are given to illustrate the effectiveness of obtained results.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61174217), the Natural Science Foundation of Shandong Province of China (ZR2010AL016 and ZR2011AL007), and the Doctoral Foundation of University of Jinan (XBS1244).