Stochastic Stability of Neural Networks with Both Markovian Jump Parameters and Continuously Distributed Delays

Abstract

The problem of stochastic stability is investigated for a class of neural networks with both Markovian jump parameters and continuously distributed delays. The jumping parameters are modeled as a continuous-time, finite-state Markov chain. By constructing appropriate Lyapunov-Krasovskii functionals, some novel stability conditions are obtained in terms of linear matrix inequalities (LMIs). The proposed LMI-based criteria are computationally efficient as they can be easily checked by using recently developed algorithms in solving LMIs. A numerical example is provided to show the effectiveness of the theoretical results and demonstrate the LMI criteria existed in the earlier literature fail. The results obtained in this paper improve and generalize those given in the previous literature.

1. Introduction

In recent years, neural networks (especially recurrent neural networks, Hopfield neural networks, and cellular neural networks) have been successfully applied in many areas such as signal processing, image processing, pattern recognition, fault diagnosis, associative memory, and combinatorial optimization; see, for example, [1–5]. One of the best important works in these applications is to study the stability of the equilibrium point of neural networks. A major purpose that is concerned with is to find stability conditions (i.e., the conditions for the stability of the equilibrium point of neural networks). To do this, existensive literature has been presented; see, for example, [6–22] and references therein. It should be noted that the methods in the literature have seldom considered the case that the systems have Markovian jump parameters due to the difficulty of mathematics. However, neural networks in real life often have a phenomenon of information latching. It is recognized that a way for dealing with this information latching problem is to extract finite state representations (also called modes or clusters). In fact, such a neural network with information latching may have finite modes, and the modes may switch (or jump) from one to another at different times, and the switching (or jumping) between two arbitrarily different modes can be governed by a Markov chain. Hence, the neural networks with Markovian jump parameters are of great significance in modeling a class of neural networks with finite modes.

On the other hand, the time delay is frequently a major source of instability and poor performance in neural networks (e.g., see [6, 23, 24]), and so the stability analysis for neural networks with time delays is an important research topic. The existing works on neural networks with time delays can be classified into three categories: constant delays, time-varying delays, and distributed delays. It is noticed that most works in the literature have focused on the former two simple cases: constant delays or time-varying delays (e.g., see [6, 8–10, 12–16, 19–22]). However, as pointed out in [18], neural networks usually have a spatial nature due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths, and so it is desired to model them by introducing continuously distributed delays on a certain duration of time such that the distant past has less influence than the recent behavior of the state. But discussions about the neural networks with continuously distributed delays are only a few researchers [18, 25]. Therefore, there is enough room to develop novel stability conditions for improvement.

Motivated by the above discussion, the objective of this paper is to study the stability for a class of neural networks with both Markovian jump parameters and continuously distributed delays. Moreover, to make the model more general and practical, the factor of noise disturbance is considered in this paper since noise disturbance is also a major source leading to instability [7]. To the best of the authors' knowledge, up to now, the stability analysis problem for a class of stochastic neural networks with both Markovian jump parameters and continuously distributed delays is still an open problem that has not been properly studied. Therefore, this paper is the first attempt to introduce and investigate the problem of stochastic stability for a class of neural networks with both Markovian jump parameters and continuously distributed delays. By utilizing the Lyapunov stability theory and linear matrix inequality (LMI) technique, some novel delay-dependent conditions are obtained to guarantee the stochastically asymptotic stability of the equilibrium point. The proposed LMI-based criteria are computationally efficient as they can be easily checked by using recently developed standard algorithms such as interior point methods [24] in solving LMIs. Finally, a numerical example is provided to illustrate the effectiveness of the theoretical results and demonstrate the LMI criteria existed in the earlier literature fail. The results obtained in this paper improve and generalize those given in the previous literature.

The remainder of this paper is organized as follows. In Section 2, the model of a class of stochastic neural networks with both Markovian jump parameters and continuously distributed delays is introduced, and some assumptions needed in this paper are presented. By means of Lyapunov-Krasovskii functional approach, our main results are established in Section 3. In Section 4, a numerical example is given to show the effectiveness of the obtained results. Finally, in Section 5, the paper is concluded with some general conclusions.

Notation. Throughout this paper, the following notations will be used. ℝ^𝑛 and ℝ^𝑛×𝑛 denote the 𝑛-dimensional Euclidean space and the set of all 𝑛×𝑛 real matrices, respectively. The superscript “𝑇" denotes the transpose of a matrix or vector. Trace(⋅) denotes the trace of the corresponding matrix, and 𝐼 denotes the identity matrix with compatible dimensions. For square matrices 𝑀₁ and 𝑀₂, the notation 𝑀₁>(≥,<,≤)𝑀₂ denotes that 𝑀₁−𝑀₂ is positive-definite (positive-semidefinite, negative, negative-semidefinite) matrix. Let 𝑤(𝑡)=(𝑤₁,…,𝑤_𝑛)^𝑇 be an 𝑛-dimensional Brownian motion defined on a complete probability space (Ω,ℱ,𝑃) with a natural filtration {ℱ_𝑡}_𝑡≥0. Also, let 𝜏>0 and 𝐶([−𝜏,0];ℝ^𝑛) denote the family of continuous function 𝜙 from [−𝜏,0] to ℝ^𝑛 with the uniform norm ‖𝜙‖=sup_{−𝜏≤𝜃≤0}|𝜙(𝜃)|. Denote by 𝐿²_ℱ𝑡([−𝜏,0];ℝ^𝑛) the family of all ℱ_𝑡 measurable, 𝐶([−𝜏,0];ℝ^𝑛)-valued stochastic variables 𝜉={𝜉(𝜃)∶−𝜏≤𝜃≤0} such that ∫⁰_−𝜏𝐸|𝜉(𝑠)|²𝑑𝑠<∞, where 𝐸[⋅] stands for the correspondent expectation operator with respect to the given probability measure 𝑃.

2. Model Description and Problem Formulation

Let {𝑟(𝑡),𝑡≥0} be a right-continuous Markov chain on a complete probability space (Ω,ℱ,𝑃) taking values in a finite state space 𝑆={1,2,…,𝑁} with generator 𝑄=(𝑞_𝑖𝑗)_𝑁×𝑁 given by 𝑞𝑃{𝑟(𝑡+Δ𝑡)=𝑗∣𝑟(𝑡)=𝑖}=^𝑖𝑗Δ𝑡+𝑜(Δ𝑡),if𝑖≠𝑗,1+𝑞_𝑖𝑖Δ𝑡+𝑜(Δ𝑡),if𝑖=𝑗, (2.1) where Δ𝑡>0 and lim_Δ𝑡→0(𝑜(Δ𝑡)/Δ𝑡)=0. Here, 𝑞_𝑖𝑗≥0 is the transition rate from 𝑖 to 𝑗 if 𝑖≠𝑗 while 𝑞_𝑖𝑖=−∑_𝑗≠𝑖𝑞_𝑖𝑗.

In this paper we consider a class of neural networks with both Markovian jump parameters and continuously distributed delays, which is described by the following integro-differential equation: 𝑥̇(𝑡)=−𝐷(𝑟(𝑡))𝑥(𝑡)+𝐴(𝑟(𝑡))𝑓(𝑥(𝑡))+𝐵(𝑟(𝑡))𝑔(𝑥(𝑡−𝜏))+𝐶(𝑟(𝑡))^𝑡_−∞𝑅(𝑡−𝑠)ℎ(𝑥(𝑠))𝑑𝑠+𝑉, (2.2) where 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)]^𝑇 is the state vector associated with the 𝑛 neurons, and the diagonal matrix 𝐷(𝑟(𝑡))=diag(𝑑₁(𝑟(𝑡)),𝑑₂(𝑟(𝑡)),…,𝑑_𝑛(𝑟(𝑡))) has positive entries 𝑑_𝑖(𝑟(𝑡))>0(𝑖=1,2,…,𝑛). The matrices 𝐴(𝑟(𝑡))=(𝑎_𝑖𝑗(𝑟(𝑡)))_𝑛×𝑛, 𝐵(𝑟(𝑡))=(𝑏_𝑖𝑗(𝑟(𝑡)))_𝑛×𝑛, and 𝐶(𝑟(𝑡))=(𝑐_𝑖𝑗(𝑟(𝑡)))_𝑛×𝑛 are, respectively, the connection weight matrix, the discretely delayed connection weight matrix, and the distributively delayed connection weight matrix. 𝑓(𝑥(𝑡))=[𝑓₁(𝑥₁(𝑡)),𝑓₂(𝑥₂(𝑡)),…,𝑓_𝑛(𝑥_𝑛(𝑡))]^𝑇, 𝑔(𝑥(𝑡))=[𝑔₁(𝑥₁(𝑡)),𝑔₂(𝑥₂(𝑡)),…,𝑔_𝑛(𝑥_𝑛(𝑡))]^𝑇, and ℎ(𝑥(𝑡))=[ℎ₁(𝑥₁(𝑡)),ℎ₂(𝑥₂(𝑡)),…,ℎ_𝑛(𝑥_𝑛(𝑡))]^𝑇 denote the neuron activation functions, and 𝑉=[𝑉₁,𝑉₂,…,𝑉_𝑛]^𝑇 denotes a constant external input vector. The constant 𝜏>0 denotes the time delay, and 𝑅=[𝑅₁,𝑅₂,…,𝑅_𝑛]^𝑇 denotes the delay kernel vector, where 𝑅_𝑖 is a real value nonnegative continuous function defined on [0,∞) and such that ∫^∞₀𝑅_𝑖(𝑠)𝑑𝑠=1 for 𝑖=1,2,…,𝑛.

In this paper we will investigate a more general model in which the environmental noise is considered on system (2.2), and so this model can be written as the following integrodifferential equation: 𝑑𝑥(𝑡)=[−𝐷(𝑟(𝑡))𝑥(𝑡)+𝐴(𝑟(𝑡))𝑓(𝑥(𝑡))+𝐵(𝑟(𝑡))𝑔(𝑥(𝑡−𝜏))+𝐶(𝑟(𝑡))^𝑡_−∞𝑅(𝑡−𝑠)ℎ(𝑥(𝑠))𝑑𝑠+𝑉]𝑑𝑡+𝜎(𝑥(𝑡),𝑥(𝑡−𝜏),𝑡,𝑟(𝑡))𝑑𝑤(𝑡), (2.3) where 𝜎∶ℝ^𝑛×ℝ^𝑛×ℝ⁺×𝑆→ℝ^𝑛×𝑛 is the noise perturbation.

Throughout this paper, the following conditions are supposed to hold.

Assumption 2.1. There exist six diagonal matrices 𝑈⁻=diag(𝑢⁻₁,𝑢⁻₂,…,𝑢⁻_𝑛), 𝑈⁺=diag(𝑢⁺₁,𝑢⁺₂,…,𝑢⁺_𝑛), 𝑉⁻=diag(𝑣⁻₁,𝑣⁻₂,…,𝑣⁻_𝑛), 𝑉⁺=diag(𝑣⁺₁,𝑣⁺₂,…,𝑣⁺_𝑛), 𝑊⁻=diag(𝑤⁻₁,𝑤⁻₂,…,𝑤⁻_𝑛), and 𝑊⁺=diag(𝑤⁺₁,𝑤⁺₂,…,𝑤⁺_𝑛) satisfying 𝑢⁻_𝑖≤𝑓_𝑖(𝛼)−𝑓_𝑖(𝛽)𝛼−𝛽≤𝑢⁺_𝑖,𝑣⁻_𝑖≤𝑔_𝑖(𝛼)−𝑔_𝑖(𝛽)𝛼−𝛽≤𝑣⁺_𝑖,𝑤⁻_𝑖≤ℎ_𝑖(𝛼)−ℎ_𝑖(𝛽)𝛼−𝛽≤𝑤⁺_𝑖 (2.4) for all 𝛼,𝛽∈ℝ, 𝛼≠𝛽, 𝑖=1,2,…,𝑛.

Assumption 2.2. There exist two positive definite matrices Σ_1𝑖 and Σ_2𝑖 such that trace[𝜎^𝑇(𝑥,𝑦,𝑡,𝑟(𝑡))𝜎(𝑥,𝑦,𝑡,𝑟(𝑡))]≤𝑥^𝑇Σ_1𝑖𝑥+𝑦^𝑇Σ_2𝑖𝑦 (2.5) for all 𝑥,𝑦∈ℝ^𝑛 and 𝑟(𝑡)=𝑖, 𝑖∈𝑆.

Assumption 2.3. 𝜎(0,0,𝑡,𝑟(𝑡))≡0.

Under Assumptions 2.1 and 2.2, it is well known (see, e.g., Mao [16]) that for any initial data 𝑥(𝜃)=𝜉(𝜃) on −𝜏≤𝜃≤0 in 𝐿²_ℱ𝑡([−𝜏,0];ℝ^𝑛), (2.3) has a unique equilibrium point. Now, let 𝑥^∗=(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛) be the unique equilibrium point of (2.3), and set 𝑦(𝑡)=𝑥(𝑡)−𝑥^∗. Then we can rewrite system (2.3) as 𝑑𝑦(𝑡)=[−𝐷(𝑟(𝑡))𝑦(𝑡)+𝐴(𝑟(𝑡))𝑓^′(𝑦(𝑡))+𝐵(𝑟(𝑡))𝑔^′(𝑦(𝑡−𝜏))+𝐶(𝑟(𝑡))^𝑡_−∞𝑅(𝑡−𝑠)ℎ^′(𝑦(𝑠))𝑑𝑠]𝑑𝑡+𝜎(𝑦(𝑡),𝑦(𝑡−𝜏),𝑡,𝑟(𝑡))𝑑𝑤(𝑡), (2.6) where 𝑓^′(𝑦)=[𝑓^′₁(𝑦₁),𝑓^′₂(𝑦₂),…,𝑓^′_𝑛(𝑦_𝑛)]^𝑇, 𝑔^′(𝑦)=[𝑔^′₁(𝑦₁),𝑔^′₂(𝑦₂),…,𝑔^′_𝑛(𝑦_𝑛)]^𝑇, ℎ^′(𝑦)=[ℎ^′₁(𝑦₁),ℎ^′₂(𝑦₂),…,ℎ^′_𝑛(𝑦_𝑛)]^𝑇, and 𝑓^′_𝑖(𝑦_𝑖)=𝑓_𝑖(𝑦_𝑖+𝑥^∗_𝑖)−𝑓_𝑖(𝑥^∗_𝑖), 𝑔^′_𝑖(𝑦_𝑖)=𝑔_𝑖(𝑦_𝑖+𝑥^∗_𝑖)−𝑔_𝑖(𝑥^∗_𝑖), ℎ^′_𝑖(𝑦_𝑖)=ℎ_𝑖(𝑦_𝑖+𝑥^∗_𝑖)−ℎ_𝑖(𝑥^∗_𝑖)(𝑖=1,2,…,𝑛).

Noting the facts that 𝑓(0)=𝑔(0)=ℎ(0)=0 and 𝜎(0,0,𝑡,𝑟(𝑡))=0, the trivial solution of system (2.6) exists. Hence, to prove the stability of 𝑥^∗ of (2.3), it is sufficient to prove the stability of the trivial solution of system (2.6). On the other hand, by Assumption 2.1 we have 𝑢⁻_𝑖≤𝑓^′_𝑖(𝛼)−𝑓^′_𝑖(𝛽)𝛼−𝛽≤𝑢⁺_𝑖𝑣_−𝑖,≤𝑔^′_𝑖(𝛼)−𝑔^′_𝑖(𝛽)𝛼−𝛽≤𝑣⁺_𝑖𝑤_−𝑖,≤ℎ^′_𝑖(𝛼)−ℎ^′_𝑖(𝛽)𝛼−𝛽≤𝑤⁺_𝑖 (2.7)(2.8)(2.9) for all 𝛼,𝛽∈ℝ, 𝛼≠𝛽, 𝑖=1,2,…,𝑛.

Let 𝑦(𝑡;𝜉) denote the state trajectory from the initial data 𝑦(𝜃)=𝜉(𝜃) on −𝜏≤𝜃≤0 in 𝐿²_ℱ𝑡([−𝜏,0];ℝ^𝑛). Clearly, system (2.6) admits a trivial solution 𝑦(𝑡;0)≡0 corresponding to the initial data 𝜉=0. For simplicity, we write 𝑦(𝑡;𝜉)=𝑦(𝑡). Let 𝐶²₁(ℝ⁺×ℝ^𝑛×𝑆;ℝ⁺) denote the family of all nonnegative functions 𝑉(𝑡,𝑦,𝑖) on ℝ⁺×ℝ^𝑛×𝑆 which are continuously twice differentiable in 𝑦 and differentiable in 𝑡. If 𝑉∈𝐶²₁(ℝ⁺×ℝ^𝑛×𝑆;ℝ^𝑛), then along the trajectory of system (2.6) we define an operator ℒ𝑉 from ℝ⁺×ℝ^𝑛×𝑆 to ℝ by ℒ𝑉(𝑡,𝑦(𝑡),𝑖)=𝑉_𝑡(𝑡,𝑦(𝑡),𝑖)+𝑉_𝑦(𝑡,𝑦(𝑡),𝑖)[−𝐷(𝑟(𝑡))𝑦(𝑡)+𝐴(𝑟(𝑡))𝑓^′(𝑦(𝑡))+𝐵(𝑟(𝑡))𝑔^′(𝑦(𝑡−𝜏))+𝐶(𝑟(𝑡))^𝑡_−∞𝑅(𝑡−𝑠)ℎ^′+1(𝑦(𝑠))𝑑𝑠]2trace[𝜎^𝑇(𝑦(𝑡),𝑦(𝑡−𝜏),𝑡,𝑟(𝑡))𝑉_𝑦𝑦+(𝑡,𝑦(𝑡),𝑖)𝜎(𝑦(𝑡),𝑦(𝑡−𝜏),𝑡,𝑟(𝑡))]_𝑁𝑗=1𝑞_𝑖𝑗𝑉(𝑡,𝑦(𝑡),𝑗), (2.10) where 𝑉_𝑡(𝑡,𝑦(𝑡),𝑖)=𝜕𝑉(𝑡,𝑦(𝑡),𝑖)𝜕𝑡,𝑉_𝑦(𝑡,𝑦(𝑡),𝑖)=(𝜕𝑉(𝑡,𝑦(𝑡),𝑖)𝜕𝑦₁,…,𝜕𝑉(𝑡,𝑦(𝑡),𝑖)𝜕𝑦_𝑛𝑉),_𝑦𝑦𝜕(𝑡,𝑦(𝑡),𝑖)=(²𝑉(𝑡,𝑦(𝑡),𝑖)𝜕𝑦_𝑖𝜕𝑦_𝑗)_𝑛×𝑛. (2.11) Now we give the definition of stochastic asymptotically stability for system (2.6).

Definition 2.4. The equilibrium point of (2.6) (or (2.3) equivalently) is said to be stochastic asymptotically stable in the mean square if, for every 𝜉∈𝐿²_ℱ0([−𝜏,0];ℝ^𝑛), the following equality holds: lim_𝑡→∞𝐸|𝑦(𝑡;𝜉)|²=0. (2.12)

In the sequel, for simplicity, when 𝑟(𝑡)=𝑖, the matrices 𝐶(𝑟(𝑡)), 𝐴(𝑟(𝑡)), and 𝐵(𝑟(𝑡)) will be written as 𝐶_𝑖, 𝐴_𝑖 and 𝐵_𝑖, respectively.

3. Main Results and Proofs

In this section, the stochastic asymptotically stability in the mean square of the equilibrium point for system (2.6) is investigated under Assumptions 2.1–2.3.

Theorem 3.1. Under Assumptions 2.1–2.3, the equilibrium point of (2.6) (or (2.3) equivalently) is stochastic asymptotically stable, if there exist positive scalars 𝜆_𝑖, positive definite matrices 𝐺, 𝐻, 𝑀, 𝑃_𝑖, 𝐾_𝑖, 𝐿_𝑖(𝑖∈𝑆), and four positive diagonal matrices 𝑄₁, 𝑄₂, 𝑄₃, and 𝐹=diag(𝑓₁,𝑓₂,…,𝑓_𝑛) such that the following LMIs hold: Γ₁₁0𝑃_𝑖𝐴_𝑖000𝑃_𝑖𝐵_𝑖𝑃_𝑖𝐶_𝑖⋆Γ₂₂000000⋆⋆Γ₃₃00000⋆⋆⋆Γ₄₄0000⋆⋆⋆⋆−𝑄₃+𝐹000⋆⋆⋆⋆⋆−𝐾_𝑖00⋆⋆⋆⋆⋆⋆−𝐿_𝑖0𝑃⋆⋆⋆⋆⋆⋆⋆−𝐹<0,_𝑖≤𝜆_𝑖𝐼,𝐻≥∑^𝑁_𝑗=1𝑞_𝑖𝑗𝐾_𝑗,𝑀≥∑^𝑁_𝑗=1𝑞_𝑖𝑗𝐿_𝑗, (3.1)(3.2)(3.3)(3.4) where the symbol “⋆" denotes the symmetric term of the matrix: Γ₁₁=−2𝑃_𝑖𝐷_𝑖+𝐺+𝜆_𝑖Σ_1𝑖+𝑈𝑄₁𝑈+𝑉𝑄₂𝑉+𝑊𝑄₃𝑊+^𝑁_𝑗=1𝑞_𝑖𝑗𝑃_𝑗,Γ₂₂=−𝐺+𝜆_𝑖Σ_2𝑖,Γ₃₃=−𝑄₁+𝐾_𝑖+𝜏𝐻,Γ₄₄=−𝑄₂+𝐿_𝑖+𝜏𝑀,𝑈=diag(𝑢₁,𝑢₂,…,𝑢_𝑛),𝑉=diag(𝑣₁,𝑣₂,…,𝑣_𝑛),𝑊=diag(𝑤₁,𝑤₂,…,𝑤_𝑛𝑢),_𝑖=max{|𝑢⁻_𝑖|,|𝑢⁺_𝑖|},𝑣_𝑖=max{|𝑣⁻_𝑖|,|𝑣⁺_𝑖|},𝑤_𝑖=max{|𝑤⁻_𝑖|,|𝑤⁺_𝑖|}(𝑖=1,2,…,𝑛). (3.5)

Proof. Fixing 𝜉∈𝐿²_ℱ𝑡([−𝜏,0];ℝ^𝑛) arbitrarily and writing 𝑦(𝑡;𝜉)=𝑦(𝑡), consider the following Lyapulov-Krasovskii functional: 𝑉(𝑡,𝑦(𝑡),𝑖)=⁷_𝑘=1𝑉_𝑘(𝑡,𝑦(𝑡),𝑖), (3.6) where 𝑉₁(𝑡,𝑦(𝑡),𝑖)=𝑦^𝑇(𝑡)𝑃_𝑖𝑉𝑦(𝑡),₂(𝑡,𝑦(𝑡),𝑖)=^𝑡_𝑡−𝜏𝑓^′𝑇(𝑦(𝑠))𝐾_𝑖𝑓^′𝑉(𝑦(𝑠))𝑑𝑠,₃(𝑡,𝑦(𝑡),𝑖)=⁰_−𝜏𝑑𝜃^𝑡_𝑡+𝜃𝑓^′𝑇(𝑦(𝑠))𝐻𝑓^′𝑉(𝑦(𝑠))𝑑𝑠,₄(𝑡,𝑦(𝑡),𝑖)=^𝑛_𝑗=1𝑓_𝑗^∞₀𝑅_𝑗(𝜃)^𝑡_𝑡−𝜃ℎ_𝑗^′2(𝑦_𝑗𝑉(𝑠))𝑑𝑠𝑑𝜃,₅(𝑡,𝑦(𝑡),𝑖)=^𝑡_𝑡−𝜏𝑦^𝑇𝑉(𝑠)𝐺𝑦(𝑠)𝑑𝑠,₆(𝑡,𝑦(𝑡),𝑖)=^𝑡_𝑡−𝜏𝑔^′𝑇(𝑦(𝑠))𝐿_𝑖𝑔^′𝑉(𝑦(𝑠))𝑑𝑠,₇(𝑡,𝑦(𝑡),𝑖)=⁰_−𝜏𝑑𝜃^𝑡_𝑡+𝜃𝑔^′𝑇(𝑦(𝑠))𝑀𝑔^′(𝑦(𝑠))𝑑𝑠. (3.7) For simplicity, denote 𝜎(𝑦(𝑡),𝑦(𝑡−𝜏),𝑡,𝑟(𝑡)) by 𝜎(𝑡). Then it follows from (2.10) and (2.6) that ℒ𝑉₁(𝑡,𝑦(𝑡),𝑖)=2𝑦^𝑇(𝑡)𝑃_𝑖[−𝐷_𝑖𝑦(𝑡)+𝐴_𝑖𝑓^′(𝑦(𝑡))+𝐵_𝑖𝑔^′(𝑦(𝑡−𝜏))+𝐶_𝑖^𝑡_−∞𝑅(𝑡−𝑠)ℎ^′(𝑦(𝑠))𝑑𝑠]+^𝑁_𝑗=1𝑞_𝑖𝑗𝑦^𝑇(𝑡)𝑃_𝑗𝑦(𝑡)+trace[𝜎^𝑇(𝑡)𝑃_𝑖𝜎(𝑡)]=𝑦^𝑇(𝑡)(−2𝑃_𝑖𝐷_𝑖)𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑃_𝑖𝐴_𝑖𝑓^′(𝑦(𝑡))+2𝑦^𝑇(𝑡)𝑃_𝑖𝐵_𝑖𝑔^′(𝑦(𝑡−𝜏))+2𝑦^𝑇(𝑡)𝑃_𝑖𝐶_𝑖^𝑡_−∞𝑅(𝑡−𝑠)ℎ^′(𝑦(𝑠))𝑑𝑠+𝑦^𝑇(𝑡)^𝑁_𝑗=1𝑞_𝑖𝑗𝑃_𝑗𝑦(𝑡)+trace[𝜎^𝑇(𝑡)𝑃_𝑖𝜎(𝑡)]. (3.8) On the other hand, by Assumption 2.2 and condition (3.2) we obtain trace[𝜎^𝑇(𝑡)𝑃_𝑖𝜎(𝑡)]≤𝜆_𝑖trace[𝜎^𝑇(𝑡)𝜎(𝑡)]≤𝜆_𝑖𝑦^𝑇(𝑡)Σ_1𝑖𝑦(𝑡)+𝜆_𝑖𝑦^𝑇(𝑡−𝜏)Σ_2𝑖𝑦(𝑡−𝜏), (3.9) which together with (3.8) gives ℒ𝑉₁(𝑡,𝑦(𝑡),𝑖)≤𝑦^𝑇(𝑡)(−2𝑃_𝑖𝐷_𝑖)𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑃_𝑖𝐴_𝑖𝑓^′(𝑦(𝑡))+2𝑦^𝑇(𝑡)𝑃_𝑖𝐵_𝑖𝑔^′(𝑦(𝑡−𝜏))+2𝑦^𝑇(𝑡)𝑃_𝑖𝐶_𝑖^𝑡_−∞𝑅(𝑡−𝑠)ℎ^′(𝑦(𝑠))𝑑𝑠+𝑦^𝑇(𝑡)^𝑁_𝑗=1𝑞_𝑖𝑗𝑃_𝑗𝑦(𝑡)+𝜆_𝑖𝑦^𝑇(𝑡)Σ_1𝑖𝑦(𝑡)+𝜆_𝑖𝑦^𝑇(𝑡−𝜏)Σ_2𝑖𝑦(𝑡−𝜏). (3.10) Also, from direct computations, it follows that ℒ𝑉₂(𝑡,𝑦(𝑡),𝑖)=𝑓^′𝑇(𝑦(𝑡))𝐾_𝑖𝑓^′(𝑦(𝑡))−𝑓^′𝑇(𝑦(𝑡−𝜏))𝐾_𝑖𝑓^′+(𝑦(𝑡−𝜏))_{𝑡𝑡−𝜏}𝑓^′𝑇(𝑦(𝑠))(^𝑁_𝑗=1𝑞_𝑖𝑗𝐾_𝑗)𝑓^′(𝑦(𝑠))𝑑𝑠,ℒ𝑉₃(𝑡,𝑦(𝑡),𝑖)=𝜏𝑓^′𝑇(𝑦(𝑡))𝐻𝑓^′(𝑦(𝑡))−^𝑡_𝑡−𝜏𝑓^′𝑇(𝑦(𝑠))𝐻𝑓^′(𝑦(𝑠))𝑑𝑠,ℒ𝑉₄(𝑡,𝑦(𝑡),𝑖)=^𝑛_𝑗=1𝑓_𝑗^∞₀𝑅_𝑗(𝜃)ℎ_𝑗^′2(𝑦_𝑗(𝑡))𝑑𝜃−^𝑛_𝑗=1𝑓_𝑗^∞₀𝑅_𝑗(𝜃)ℎ_𝑗^′2(𝑦_𝑗(𝑡−𝜃))𝑑𝜃=ℎ^′𝑇(𝑦(𝑡))𝐹ℎ^′(𝑦(𝑡))−^𝑛_𝑗=1𝑓_𝑗^∞₀𝑅_𝑗(𝜃)𝑑𝜃^∞₀𝑅_𝑗(𝜃)ℎ_𝑗^′2(𝑦_𝑗(𝑡−𝜃))𝑑𝜃≤ℎ^′𝑇(𝑦(𝑡))𝐹ℎ^′(𝑦(𝑡))−^𝑛_𝑗=1𝑓_𝑗[^∞₀𝑅_𝑗(𝜃)ℎ^′_𝑗(𝑦_𝑗(𝑡−𝜃))𝑑𝜃]²=ℎ^′𝑇(𝑦(𝑡))𝐹ℎ^′(𝑦(𝑡))−(^𝑡_−∞𝑅(𝑡−𝑠)ℎ^′(𝑦(𝑠))𝑑𝑠)^𝑇𝐹^𝑡_−∞𝑅(𝑡−𝑠)ℎ^′(𝑦(𝑠))𝑑𝑠,ℒ𝑉₅(𝑡,𝑦(𝑡),𝑖)=𝑦^𝑇(𝑡)𝐺𝑦(𝑡)−𝑦^𝑇(𝑡−𝜏)𝐺𝑦(𝑡−𝜏),ℒ𝑉₆(𝑡,𝑦(𝑡),𝑖)=𝑔^′𝑇(𝑦(𝑡))𝐿_𝑖𝑔^′(𝑦(𝑡))−𝑔^′𝑇(𝑦(𝑡−𝜏))𝐿_𝑖𝑔^′+(𝑦(𝑡−𝜏))_{𝑡𝑡−𝜏}𝑔^′𝑇(𝑦(𝑠))(^𝑁_𝑗=1𝑞_𝑖𝑗𝐿_𝑗)𝑔^′(𝑦(𝑠))𝑑𝑠,ℒ𝑉₇(𝑡,𝑦(𝑡),𝑖)=𝜏𝑔^′𝑇(𝑦(𝑡))𝑀𝑔^′(𝑦(𝑡))−^𝑡_𝑡−𝜏𝑔^′𝑇(𝑦(𝑠))𝑀𝑔^′(𝑦(𝑠))𝑑𝑠. (3.11) It should be mentioned that the calculation of ℒ𝑉₄(𝑡,𝑦(𝑡),𝑖) has applied the following inequality: [^∞₀𝑙₁(𝜃)𝑙₂(𝜃)𝑑𝜃]²≤^∞₀𝑙²₁(𝜃)𝑑𝜃^∞₀𝑙²₂(𝜃)𝑑𝜃 (3.12) with 𝑙₁(𝜃)∶=[𝑅(𝜃)]^1/2 and 𝑙₂(𝜃)∶=[𝑅(𝜃)]^1/2ℎ^′(𝑦(𝑡−𝜃)).
Furthermore, it follows from the conditions (3.3) and (3.4) that ^𝑡_𝑡−𝜏𝑓^′𝑇(𝑦(𝑠))(^𝑁_𝑗=1𝑞_𝑖𝑗𝐾_𝑗)𝑓^′(𝑦(𝑠))𝑑𝑠−^𝑡_𝑡−𝜏𝑓^′𝑇(𝑦(𝑠))𝐻𝑓^′(𝑦(𝑠))𝑑𝑠≤0,_{𝑡𝑡−𝜏}𝑔^′𝑇(𝑦(𝑠))(^𝑁_𝑗=1𝑞_𝑖𝑗𝐿_𝑗)𝑔^′(𝑦(𝑠))𝑑𝑠−^𝑡_𝑡−𝜏𝑔^′𝑇(𝑦(𝑠))𝑀𝑔^′(𝑦(𝑠))𝑑𝑠≤0. (3.13)
On the other hand, by Assumption 2.1 we have 𝑓^′𝑇(𝑦(𝑡))𝑄₁𝑓^′(𝑦(𝑡))≤𝑦^𝑇(𝑡)𝑈𝑄₁𝑔𝑈𝑦(𝑡),_′𝑇(𝑦(𝑡))𝑄₂𝑔^′(𝑦(𝑡))≤𝑦^𝑇(𝑡)𝑉𝑄₂ℎ𝑉𝑦(𝑡),_′𝑇(𝑦(𝑡))𝑄₃ℎ^′(𝑦(𝑡))≤𝑦^𝑇(𝑡)𝑊𝑄₃𝑊𝑦(𝑡). (3.14)
Hence, by (3.8)–(3.14), we get ℒ𝑉(𝑡,𝑦(𝑡),𝑖)≤𝜁^𝑇(𝑡)Π_𝑖𝜁(𝑡), (3.15) where 𝜁^𝑇(𝑡)=[𝑦^𝑇(𝑡)𝑦^𝑇(𝑡−𝜏)𝑓^′𝑇(𝑦(𝑡))𝑔^′𝑇(𝑦(𝑡))ℎ^′𝑇𝑓(𝑦(𝑡))_′𝑇(𝑦(𝑡−𝜏))𝑔^′𝑇(𝑦(𝑡−𝜏))(^𝑡_−∞𝑅(𝑡−𝑠)ℎ^′(𝑦(𝑠))𝑑𝑠)^𝑇Π],_𝑖=Γ₁₁0𝑃_𝑖𝐴_𝑖000𝑃_𝑖𝐵_𝑖𝑃_𝑖𝐶_𝑖⋆Γ₂₂000000⋆⋆Γ₃₃00000⋆⋆⋆Γ₄₄0000⋆⋆⋆⋆−𝑄₃+𝐹000⋆⋆⋆⋆⋆−𝐾_𝑖00⋆⋆⋆⋆⋆⋆−𝐿_𝑖0,Γ⋆⋆⋆⋆⋆⋆⋆−𝐹₁₁=−2𝑃_𝑖𝐷_𝑖+𝐺+𝜆_𝑖Σ_1𝑖+𝑈𝑄₁𝑈+𝑉𝑄₂𝑉+𝑊𝑄₃𝑊+^𝑁_𝑗=1𝑞_𝑖𝑗𝑃_𝑗,Γ₂₂=−𝐺+𝜆_𝑖Σ_2𝑖,Γ₃₃=−𝑄₁+𝐾_𝑖+𝜏𝐻,Γ₄₄=−𝑄₂+𝐿_𝑖+𝜏𝑀. (3.17) By condition (3.1), there must exist a scalar 𝛽_𝑖>0(𝑖∈𝑆) such that Π_𝑖+𝛽_𝑖𝐼<0. Setting 𝛽=min_𝑖∈𝑆𝛽_𝑖, it is clear that 𝛽>0. Taking the mathematical expectation on both sides of (3.15), we obtain 𝐄ℒ𝑉(𝑡,𝑦(𝑡),𝑖)≤𝐄𝜁^𝑇(𝑡)Π_𝑖𝜁(𝑡)≤−𝛽_𝑖𝐄|𝑦(𝑡;𝜉)|²≤−𝛽𝐄|𝑦(𝑡;𝜉)|². (3.18) Applying the Dynkin formula and from (3.18), it follows that 𝐄𝑉(𝑡,𝑥(𝑡),𝑖)−𝐄𝑉(0,𝑥(0),𝑟(0))=^𝑡₀𝐄ℒ𝑉(𝑠,𝑥(𝑠),𝑟(𝑠))𝑑𝑠<−𝛽^𝑡₀𝐄|𝑥(𝑠)|²𝑑𝑠, (3.19) and so ^𝑡₀𝐄|𝑥(𝑠)|²1𝑑𝑠<−𝛽1𝐄𝑉(𝑡,𝑥(𝑡),𝑖)+𝛽1𝐄𝑉(0,𝑥(0),𝑟(0))<𝛽𝐄𝑉(0,𝑥(0),𝑟(0)), (3.20) which implies that the equilibrium point of (2.3) (or (2.2) equivalently) is stochastically asymptotic stability in the mean square. This completes the proof

Remark 3.2. Theorem 3.1 provides a sufficient condition for the generalized neural network (2.3) to ascertain the stochastic asymptotically stability in the mean square of the equilibrium point. The condition is easy to be verified and can be applied in practice as it can be checked by using recently developed algorithms in solving LMIs.

Remark 3.3. The generalized neural network (2.3) is quite general since it considers the effects of many factors including noise perturbations, Markovian jump parameters, and continuously distributed delays. Furthermore, the constants 𝑢⁻_𝑖, 𝑣⁻_𝑖, 𝑤⁻_𝑖, 𝑢⁺_𝑖, 𝑣⁺_𝑖, 𝑤⁺_𝑖 in Assumption 2.1 are allowed to be positive, negative, or zero. To the best of our knowledge, the generalized neural network (2.3) has never been considered in the previous literature. Hence, the LMI criteria existed in all the previous literature fail in our results.

Remark 3.4. If we take 𝐴_𝑖=0(𝑖∈𝑆), then system (2.3) can be written as 𝑑𝑥(𝑡)=[−𝐷(𝑟(𝑡))𝑥(𝑡)+𝐵(𝑟(𝑡))𝑔(𝑥(𝑡−𝜏))+𝐶(𝑟(𝑡))^𝑡_−∞𝑅(𝑡−𝑠)ℎ(𝑥(𝑠))𝑑𝑠+𝑉]𝑑𝑡+𝜎(𝑥(𝑡),𝑥(𝑡−𝜏),𝑡,𝑟(𝑡))𝑑𝑤(𝑡). (3.21) If we take 𝐵_𝑖=0(𝑖∈𝑆), then system (2.3) can be written as 𝑑𝑥(𝑡)=[−𝐷(𝑟(𝑡))𝑥(𝑡)+𝐴(𝑟(𝑡))𝑓(𝑥(𝑡))+𝐶(𝑟(𝑡))^𝑡_−∞𝑅(𝑡−𝑠)ℎ(𝑥(𝑠))𝑑𝑠+𝑉]𝑑𝑡+𝜎(𝑥(𝑡),𝑥(𝑡−𝜏),𝑡,𝑟(𝑡))𝑑𝑤(𝑡). (3.22) If we do not consider noise perturbations, then system (2.3) can be written as 𝑑𝑥(𝑡)=[−𝐷(𝑟(𝑡))𝑥(𝑡)+𝐴(𝑟(𝑡))𝑓(𝑥(𝑡))+𝐵(𝑟(𝑡))𝑔(𝑥(𝑡−𝜏))+𝐶(𝑟(𝑡))^𝑡_−∞𝑅(𝑡−𝑠)ℎ(𝑥(𝑠))𝑑𝑠+𝑉]𝑑𝑡. (3.23) To the best of our knowledge, even systems (3.21)–(3.23) still have not been investigated in the previous literature.

Remark 3.5. We now illustrate that the neural network (2.3) generalizes some neural networks considered in the earlier literature. For example, if we take 𝑅_𝑖(𝑠)=0𝑠≥𝜏^′,1𝑠<𝜏^′,𝜏^′>0,𝑖=1,2,…,𝑛, (3.24) then system (2.3) can be written as 𝑑𝑥(𝑡)=[−𝐷(𝑟(𝑡))𝑥(𝑡)+𝐴(𝑟(𝑡))𝑓(𝑥(𝑡))+𝐵(𝑟(𝑡))𝑔(𝑥(𝑡−𝜏))+𝐶(𝑟(𝑡))^𝑡_{𝑡−𝜏′}ℎ(𝑥(𝑠))𝑑𝑠+𝑉]𝑑𝑡+𝜎(𝑥(𝑡),𝑥(𝑡−𝜏),𝑡,𝑟(𝑡))𝑑𝑤(𝑡). (3.25) System (3.25) was discussed by Liu et al. [26] and Wang et al. [27], although the delays are time-varyings in [27]. Nevertheless, we point out that system (3.25) can be generalized to the neural networks with time-varying delays without any difficulty. If we take 𝐶_𝑖=0(𝑖∈𝑆) and do not consider noise perturbations, then system (2.3) can be written as 𝑑𝑥(𝑡)=[−𝐷(𝑟(𝑡))𝑥(𝑡)+𝐴(𝑟(𝑡))𝑓(𝑥(𝑡))+𝐵(𝑟(𝑡))𝑔(𝑥(𝑡−𝜏))+𝑉]𝑑𝑡. (3.26) The stability analysis for system (3.26) was investigated by Wang et al. [20]. In [15], Lou and Cui also consider system (3.26) with 𝐴_𝑖=0(𝑖∈𝑆).

The next five corollaries follow directly from Theorem 3.1, and so we omit their proofs.

Corollary 3.6. Under Assumptions 2.1–2.3, the equilibrium point of (3.21) is stochastic asymptotically stable, if there exist positive scalars 𝜆_𝑖, positive definite matrices 𝐺, 𝑀, 𝑃_𝑖, 𝐿_𝑖(𝑖∈𝑆), and three positive diagonal matrices 𝑄₂, 𝑄₃, and 𝐹=diag(𝑓₁,𝑓₂,…,𝑓_𝑛) such that the following LMIs hold: Γ₁₁000𝑃_𝑖𝐵_𝑖𝑃_𝑖𝐶_𝑖⋆Γ₂₂0000⋆⋆Γ₄₄000⋆⋆⋆−𝑄₃+𝐹00⋆⋆⋆⋆−𝐿_𝑖0𝑃⋆⋆⋆⋆⋆−𝐹<0,_𝑖≤𝜆_𝑖𝐼,𝑀≥^𝑁_𝑗=1𝑞_𝑖𝑗𝐿_𝑗, (3.27) where the symbol “⋆" denotes the symmetric term of the matrix: Γ₁₁=−2𝑃_𝑖𝐷_𝑖+𝐺+𝜆_𝑖Σ_1𝑖+𝑉𝑄₂𝑉+𝑊𝑄₃𝑊+^𝑁_𝑗=1𝑞_𝑖𝑗𝑃_𝑗,Γ₂₂=−𝐺+𝜆_𝑖Σ_2𝑖,Γ₄₄=−𝑄₂+𝐿_𝑖+𝜏𝑀,𝑉=diag(𝑣₁,𝑣₂,…,𝑣_𝑛),𝑊=diag(𝑤₁,𝑤₂,…,𝑤_𝑛𝑣),_𝑖=max{|𝑣⁻_𝑖|,|𝑣⁺_𝑖|},𝑤_𝑖=max{|𝑤⁻_𝑖|,|𝑤⁺_𝑖|}(𝑖=1,2,…,𝑛). (3.28)

Corollary 3.7. Under Assumptions 2.1–2.3, the equilibrium point of (3.22) is stochastic asymptotically stable, if there exist positive scalars 𝜆_𝑖, positive definite matrices 𝐺, 𝐻, 𝑃_𝑖, 𝐾_𝑖(𝑖∈𝑆), and three positive diagonal matrices 𝑄₁, 𝑄₃, and 𝐹=diag(𝑓₁,𝑓₂,…,𝑓_𝑛) such that the following LMIs hold: Γ₁₁0𝑃_𝑖𝐴_𝑖00𝑃_𝑖𝐶_𝑖⋆Γ₂₂0000⋆⋆Γ₃₃000⋆⋆⋆−𝑄₃+𝐹00⋆⋆⋆⋆−𝐾_𝑖0𝑃⋆⋆⋆⋆⋆−𝐹<0,_𝑖≤𝜆_𝑖𝐼,𝐻≥^𝑁_𝑗=1𝑞_𝑖𝑗𝐾_𝑗, (3.29) where the symbol “⋆" denotes the symmetric term of the matrix: Γ₁₁=−2𝑃_𝑖𝐷_𝑖+𝐺+𝜆_𝑖Σ_1𝑖+𝑈𝑄₁𝑈+𝑊𝑄₃𝑊+^𝑁_𝑗=1𝑞_𝑖𝑗𝑃_𝑗,Γ₂₂=−𝐺+𝜆_𝑖Σ_2𝑖,Γ₃₃=−𝑄₁+𝐾_𝑖+𝜏𝐻,𝑈=diag(𝑢₁,𝑢₂,…,𝑢_𝑛),𝑊=diag(𝑤₁,𝑤₂,…,𝑤_𝑛𝑢),_𝑖=max{|𝑢⁻_𝑖|,|𝑢⁺_𝑖|},𝑤_𝑖=max{|𝑤⁻_𝑖|,|𝑤⁺_𝑖|}(𝑖=1,2,…,𝑛). (3.30)

Corollary 3.8. Under Assumptions 2.1–2.3, the equilibrium point of (3.23) is stochastic asymptotically stable, if there exist positive scalars 𝜆_𝑖, positive definite matrices 𝐺, 𝐻, 𝑀, 𝑃_𝑖, 𝐾_𝑖, 𝐿_𝑖(𝑖∈𝑆), and three positive diagonal matrices 𝑄₁, 𝑄₂, and 𝐹=diag(𝑓₁,𝑓₂,…,𝑓_𝑛) such that the following LMIs hold: Γ₁₁0𝑃_𝑖𝐴_𝑖000𝑃_𝑖𝐵_𝑖𝑃_𝑖𝐶_𝑖⋆−𝐺000000⋆⋆Γ₃₃00000⋆⋆⋆Γ₄₄0000⋆⋆⋆⋆−𝑄₃+𝐹000⋆⋆⋆⋆⋆−𝐾_𝑖00⋆⋆⋆⋆⋆⋆−𝐿_𝑖0⋆⋆⋆⋆⋆⋆⋆−𝐹<0,𝐻≥^𝑁_𝑗=1𝑞_𝑖𝑗𝐾_𝑗,𝑀≥^𝑁_𝑗=1𝑞_𝑖𝑗𝐿_𝑗, (3.31) where the symbol “⋆" denotes the symmetric term of the matrix: Γ₁₁=−2𝑃_𝑖𝐷_𝑖+𝐺+𝑈𝑄₁𝑈+𝑉𝑄₂𝑉+𝑊𝑄₃𝑊+^𝑁_𝑗=1𝑞_𝑖𝑗𝑃_𝑗,Γ₃₃=−𝑄₁+𝐾_𝑖+𝜏𝐻,Γ₄₄=−𝑄₂+𝐿_𝑖+𝜏𝑀,𝑈=diag(𝑢₁,𝑢₂,…,𝑢_𝑛),𝑉=diag(𝑣₁,𝑣₂,…,𝑣_𝑛),𝑊=diag(𝑤₁,𝑤₂,…,𝑤_𝑛𝑢),_𝑖=max{|𝑢⁻_𝑖|,|𝑢⁺_𝑖|},𝑣_𝑖=max{|𝑣⁻_𝑖|,|𝑣⁺_𝑖|},𝑤_𝑖=max{|𝑤⁻_𝑖|,|𝑤⁺_𝑖|}(𝑖=1,2,…,𝑛). (3.32)

Corollary 3.9. Under Assumptions 2.1–2.3, the equilibrium point of (3.25) is stochastic asymptotically stable, if there exist positive scalars 𝜆_𝑖, positive definite matrices 𝐺, 𝐻, 𝑀, 𝑃_𝑖, 𝐾_𝑖, 𝐿_𝑖(𝑖∈𝑆), and four positive diagonal matrices 𝑄₁, 𝑄₂, 𝑄₃, and 𝐹=diag(𝑓₁,𝑓₂,…,𝑓_𝑛) such that the following LMIs hold: Γ₁₁0𝑃_𝑖𝐴_𝑖000𝑃_𝑖𝐵_𝑖𝑃_𝑖𝐶_𝑖⋆Γ₂₂000000⋆⋆Γ₃₃00000⋆⋆⋆Γ₄₄0000⋆⋆⋆⋆−𝑄₃+𝐹000⋆⋆⋆⋆⋆−𝐾_𝑖00⋆⋆⋆⋆⋆⋆−𝐿_𝑖0𝑃⋆⋆⋆⋆⋆⋆⋆−𝐹<0,_𝑖≤𝜆_𝑖𝐼,𝐻≥^𝑁_𝑗=1𝑞_𝑖𝑗𝐾_𝑗,𝑀≥^𝑁_𝑗=1𝑞_𝑖𝑗𝐿_𝑗, (3.33) where the symbol “⋆" denotes the symmetric term of the matrix: Γ₁₁=−2𝑃_𝑖𝐷_𝑖+𝐺+𝜆_𝑖Σ_1𝑖+𝑈𝑄₁𝑈+𝑉𝑄₂𝑉+𝑊𝑄₃𝑊+^𝑁_𝑗=1𝑞_𝑖𝑗𝑃_𝑗,Γ₂₂=−𝐺+𝜆_𝑖Σ_2𝑖,Γ₃₃=−𝑄₁+𝐾_𝑖+𝜏𝐻,Γ₄₄=−𝑄₂+𝐿_𝑖+𝜏𝑀,𝑈=diag(𝑢₁,𝑢₂,…,𝑢_𝑛),𝑉=diag(𝑣₁,𝑣₂,…,𝑣_𝑛),𝑊=diag(𝑤₁,𝑤₂,…,𝑤_𝑛𝑢),_𝑖=max{|𝑢⁻_𝑖|,|𝑢⁺_𝑖|},𝑣_𝑖=max{|𝑣⁻_𝑖|,|𝑣⁺_𝑖|},𝑤_𝑖=max{|𝑤⁻_𝑖|,|𝑤⁺_𝑖|}(𝑖=1,2,…,𝑛). (3.34)

Corollary 3.10. Under Assumption 2.1, the equilibrium point of (3.26) is stochastic asymptotically stable, if there exist positive scalars 𝜆_𝑖, positive definite matrices 𝐺, 𝐻, 𝑀, 𝑃_𝑖, 𝐾_𝑖, 𝐿_𝑖(𝑖∈𝑆), and three positive diagonal matrices 𝑄₁, 𝑄₂, and 𝐹=diag(𝑓₁,𝑓₂,…,𝑓_𝑛) such that the following LMIs hold: Γ₁₁0𝑃_𝑖𝐴_𝑖00𝑃_𝑖𝐵_𝑖⋆−𝐺0000⋆⋆Γ₃₃000⋆⋆⋆Γ₄₄00⋆⋆⋆⋆−𝐾_𝑖0⋆⋆⋆⋆⋆−𝐿_𝑖<0,𝐻≥^𝑁_𝑗=1𝑞_𝑖𝑗𝐾_𝑗,𝑀≥^𝑁_𝑗=1𝑞_𝑖𝑗𝐿_𝑗, (3.35) where the symbol “⋆" denotes the symmetric term of the matrix: Γ₁₁=−2𝑃_𝑖𝐷_𝑖+𝐺+𝑈𝑄₁𝑈+𝑉𝑄₂𝑉+^𝑁_𝑗=1𝑞_𝑖𝑗𝑃_𝑗,Γ₃₃=−𝑄₁+𝐾_𝑖+𝜏𝐻,Γ₄₄=−𝑄₂+𝐿_𝑖+𝜏𝑀,𝑈=diag(𝑢₁,𝑢₂,…,𝑢_𝑛),𝑉=diag(𝑣₁,𝑣₂,…,𝑣_𝑛𝑢),_𝑖=max{|𝑢⁻_𝑖|,|𝑢⁺_𝑖|},𝑣_𝑖=max{|𝑣⁻_𝑖|,|𝑣⁺_𝑖|}(𝑖=1,2,…,𝑛). (3.36)

Remark 3.11. As discussed in Remark 3.4, Corollaries 3.6–3.8 are “new" since they have never been considered in the previous literature. Corollary 3.9 was discussed by Liu et al. [26] and Wang et al. [27], although the delays are time-varyings in [27]. Nevertheless, we point out that our results can be generalized to the neural networks with time-varying delays without any difficulty. Corollary 3.10 has been discussed by Wang et al. [20] and Lou and Cui [15], but our conditions are weaker than those in [15, 20], for the constants 𝑢⁻_𝑖, 𝑣⁻_𝑖, 𝑢⁺_𝑖, 𝑣⁺_𝑖 in Corollary 3.10 are allowed to be positive, negative, or zero.

4. Illustrative Example

In this section, a numerical example is given to illustrate the effectiveness of the obtained results.

Example 4.1. Consider a two-dimensional stochastic neural network with both Markov jump parameters and continuously distributed delays: 𝑑𝑥(𝑡)=[−𝐷(𝑟(𝑡))𝑥(𝑡)+𝐴(𝑟(𝑡))𝑓(𝑥(𝑡))+𝐵(𝑟(𝑡))𝑔(𝑥(𝑡−𝜏))+𝐶(𝑟(𝑡))^𝑡_−∞𝑒^{−(𝑡−𝑠)}ℎ(𝑥(𝑠))𝑑𝑠+𝑉]𝑑𝑡+𝜎(𝑥(𝑡),𝑥(𝑡−𝜏),𝑡,𝑟(𝑡))𝑑𝑤(𝑡), (4.1) where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡))^𝑇, 𝑉=(0,0)^𝑇, 𝑤(𝑡) is a two dimensional Brownian motion, and 𝑟(𝑡) is a right-continuous Markov chain taking values in 𝑆={1,2} with generator 𝑄=−885−5. (4.2) Let 𝑓_𝑖(𝑥_𝑖)=0.01tanℎ𝑥_𝑖=0.01(𝑒^𝑥𝑖−𝑒^−𝑥𝑖)(𝑒^𝑥𝑖+𝑒^−𝑥𝑖),𝑔_𝑖(𝑥_𝑖)=ℎ_𝑖(𝑥_𝑖)=0.005(|𝑥_𝑖+1|−|𝑥_𝑖−1|)(𝑖=1,2), (4.3) then system (4.1) satisfies Assumption 2.1 with 𝑈⁻=𝑉⁻=𝑊⁻=−0.01𝐼,𝑈⁺=𝑉⁺=𝑊⁺=0.01𝐼. Take 𝜎(𝑥(𝑡),𝑥(𝑡−𝜏),𝑡,𝑖)=0.2𝑥_𝑖(𝑡)0.4𝑥_𝑖(𝑡−𝜏)0.4𝑥_𝑖(𝑡)0.2𝑥_𝑖(𝑡−𝜏)(𝑖=1,2), (4.4) then system (4.1) satisfies Assumptions 2.2 and 2.3 with Σ_1𝑖=Σ_2𝑖=0.2𝐼(𝑖=1,2).

Other parameters of the network (4.1) are given as follows: 𝐷₁=diag(0.3,0.1),𝐴₁=0.2−0.20.40.1,𝐵₁=0.1−0.20.30.2,𝐶₁=,𝐷0.10.2−0.30.3₂=diag(0.4,0.3),𝐴₂=0.30.2−0.10.2,𝐵₂=0.20.1−0.20.1,𝐶₂=.0.20.2−0.30.1 (4.5) Here we let 𝜏=2. By using the Matlab LMI toolbox, we can obtain the following feasible solution for the LMIs (3.1)–(3.4): ,𝑄𝐺=18.711.411.414.68,𝐻=39.69−6.32−6.3244.13𝑀=29.85−1.07−1.0733.32,𝐸=diag(55.09,55.09),₁=224.6500224.65,𝑄₂=,𝑄163.8400163.84₃=𝑃72.690072.69,𝐹=diag(31.88,31.88),₁=57.771.911.9139.25,𝑃₂=56.591.661.6637.41,𝐾₁=46.70−4.19−4.1949.63,𝜆₁𝐾=83.25,₂=46.40−4.17−4.1749.33,𝐿₁=64.953.063.0655.17,𝐿₂=64.412.992.9954.64,𝜆₂=79.06. (4.6) Therefore, it follows from Theorem 3.1 that the network (4.1) is stochastic asymptotically stable.

By using the Euler-Maruyama numerical scheme, simulation results are as follows: 𝑇=50 and step size 𝛿𝑡=0.02. Figure 1 is the state response of model 1 (i.e., the network (4.1) when 𝑟(𝑡)=1) with the initial condition [0.5,−0.7]^𝑇, for −2≤𝑡≤0, and Figure 2 is the state response of model 2 (i.e., the network (4.1) when 𝑟(𝑡)=2) with the initial condition [−0.6,0.4]^𝑇, for −2≤𝑡≤0.

Figure 1 

The state response of the model 1 in Example 4.1.

Figure 2 

The state response of the model 2 in Example 4.1.

Remark 4.2. As discussed in Remarks 3.2–3.11, the LMI criteria existed in all the previous literature (e.g., Liu et al. [26], Wang et al. [20, 27], Lou and Cui [15], etc.) fail in Example 4.1 since many factors including noise perturbations, Markovian jump parameters, and continuously distributed delays are considered in Example 4.1.

5. Concluding Remarks

In this paper we have investigated the stochastic stability analysis problem for a class of neural networks with both Markovian jump parameters and continuously distributed delays. It is worth mentioning that our obtained stability condition is delay-dependent, which is less conservative than delay-independent criteria when the delay is small. Furthermore, the obtained stability criteria in this paper are expressed in terms of LMIs, which can be solved easily by recently developed algorithms. A numerical example is given to show the less conservatism and effectiveness of our results. The results obtained in this paper improve and generalize those given in the previous literature. On the other hand, it should be noted that the explicit rate of convergence for the considered system is not given in this paper since it is difficult to deal with continuously distributed delays. Therefore, investigating the explicit rate of convergence for the considered system remains an open issue. Finally, we point out that it is possible to generalize our results to a class of neural networks with uncertainties. Research on this topic is in progress.

Acknowledgments

The authors would like to thank the editor and five anonymous referees for their helpful comments and valuable suggestions regarding this paper. This work was jointly supported by the National Natural Science Foundation of China (10801056, 60874088), the Natural Science Foundation of Guangdong Province (06300957), K. C. Wong Magna Fund in Ningbo University, and the Specialized Research Fund for the Doctoral Program of Higher Education (20070286003).