Exponential Stability Results of Discrete-Time Stochastic Neural Networks with Time-Varying Delays

Abstract

An innovative stability analysis approach for a class of discrete-time stochastic neural networks (DSNNs) with time-varying delays is developed. By constructing a novel piecewise Lyapunov-Krasovskii functional candidate, a new sum inequality is presented to deal with sum items without ignoring any useful items, the model transformation is no longer needed, and the free weighting matrices are added to reduce the conservatism in the derivation of our results, so the improvement of computational efficiency can be expected. Numerical examples and simulations are also given to show the effectiveness and less conservatism of the proposed criteria.

1. Introduction

In the past decades, neural networks (NNs) have attracted considerable attention due to their potential applications in associative memory, pattern recognition, optimization and signal processing, and so forth [1–3]. It is well known to us that stability is one of the preconditions in the design of neural networks. For example, if a neural network is employed to solve some optimization problems, it is highly desirable for the NNs to have a unique globally stable equilibrium. Therefore, stability analysis of NNs is a very important issue and has been studied extensively [4–11]. It is worth noting that most of the NNs have been analyzed by using a continuous-time model. However, when it comes to the implementation of continuous-time networks for the sake of computer-based simulation, experimentation or computation, it is necessary to discretize the continuous-time networks to formulate a discrete-time system. Under mild or no restriction on the discretization of step size, the dynamic characteristics of the continuous-time counterpart can be inherited by the discrete-time analogue to a certain extent, and the discrete-time model also remains similar to some other properties of the continuous-time system.

On the other hand, as a result of the finite switching speed of amplifiers and the inherent communication time of neurons, time delays are frequently encountered in neural networks in electronic implementations. Time delays can change the dynamic behaviors of neural networks evidently, which is very often the sources of instability, oscillation, and poor performance. Therefore, stability analysis of neural networks with time delays has been studied extensively during the past years; see [12–14] and the references therein. In practice, when modeling real neural systems, stochastic disturbances are probably part of the main sources leading to unwilling behaviors of neural networks. It has been proved that certain stochastic inputs could make the neural network unstable. Therefore, it is necessary to take into account both time delay and stochastic external flucation when modeling neural networks.

Recently, stochastic discrete-time system has been studied extensively; [15] presented a necessary and sufficient condition for the existence of 𝐻₂/𝐻∞ control to transform 𝐻₂/𝐻∞ controller design into solving coupled matrix-valued equation for discrete-time system with state and disturbance-dependent noise. In [16], the robust filtering analysis and synthesis of nonlinear stochastic systems with state and exogenous disturbance-dependent noise are presented. Instead of solving the Hamilton-Jacobi inequalities, a more convenient algorithm for practical applications is given by solving several linear matrix inequalities, and a few examples have shown the effectiveness of the proposed methods. In [17], based on a nonlinear stochastic bounded real lemma and an exponential estimate formula, an exponential mean square 𝐻∞ filtering design of nonlinear stochastic time-delay systems is presented via solving a Hamilton-Jacobi inequality.

But as pointed out in some previous articles, the discretization cannot preserve the dynamics of continuous-time system even for a small sampling period. Therefore, the study on the dynamics of discrete-time neural networks is crucially needed; see [18–22] and the references therein.

Based on the discussions previously mentioned, the problem of stability analysis for discrete stochastic neural networks (DSNNs) with time-varying delays has been investigated recently. In [23], the problem of exponentially stability analysis for uncertain discrete-time stochastic neural networks with time-varying delays is investigated by utilizing Lyapunov-Krasovskii method of converting the addressed stability analysis problem into a convex optimization problem. In [24], combining with the free weighting matrix method and a new Lyapunov-Krasovskii functional candidate, a delay-dependent stability condition has been obtained, which proves to be less conservative than [23]. In [25], a new Lyapunov-Krasovskii functional candidate and the delay partition idea are used to solve the problem of asymptotic stability analysis in the mean square for a class of DSNNs with time-varying delay; the stability analysis problem is converted into a feasibility problem of LMIs, and a numerical example is provided to show the usefulness of the proposed condition.

In [26], the global exponential stability problem for a class of discrete-time uncertain stochastic neural networks with time-varying delays is studied, and an improved result is obtained.

In [27], the midpoint of the time delay variation interval is introduced, and the variation interval is divided into two subintervals; a new Lyapunov-Krasovskii functional candidate is constructed, and the variation in the two subintervals are checked by LMIs; some novel delay-dependent stability criteria for the addressed neural networks are derived, and less conservation is obtained.

In [28], a new Lyapunov functional candidate with the idea of delay partitioning is introduced; the effects of both variation range and distribution probability of the time delay are taken into account at the same time; the time-varying delay is characterized by introducing a Bernoulli stochastic variable; the distribution probability of time delay is translated into parameter matrices of the transferred DSNNs model, so the conservation has been reduced further. However, one of the main issues in the stability criteria is how to reduce the possible conservatism induced by introduction of the Lyapunov-Krasovskii functional candidate when dealing with time delay, which leaves much room for further research by using the latest analysis techniques.

In this paper, we develop an innovative stability analysis approach for a class of discrete-time stochastic neural networks with time-varying delays. By constructing a novel piecewise Lyapunov-Krasovskii functional, a new sum inequality is presented to deal with sum items without ignoring any useful items, and the model transformation is no longer needed in the derivation of our results. All results are expressed in the form of LMIs, whose feasibility can be easily checked by using the numerically efficient Matlab LMI toolbox, and no tuning of parameters is required, so the improvement of computational efficiency can be expected. Numerical examples are also given to show the effectiveness and less conservatism of the proposed criteria.

Notation. Throughout this paper, if not explicit, matrices are assumed to have compatible dimensions. The notation 𝑀>(≥,<,≤)0 means that the symmetric matrix 𝑀 is positive definite (positive semidefinite, negative, negative semidefinite). The superscript 𝑇 stands for the transpose of a matrix; the shorthand diag{⋯} denotes the block diagonal matrix; ‖⋅‖ represents the Euclidean norm for vector or the spectral norm of matrices; and 𝜆_𝑀(𝐴),𝜆_𝑚(𝐴) denote the maximal and minimal eigenvalue of matrix 𝐴, respectively. 𝐼 refers to an identity matrix of appropriate dimensions, 𝔼{⋅} stands for the mathematical expectation, and ∗ means the symmetric terms. Sometimes, the arguments of a function will be omitted in the analysis when no confusion can arise.

2. System Description

Consider the following 𝑛-neuron discrete stochastic neural networks with mixed time-varying delays:𝑥()(𝑘)(𝑥(𝑘(𝑥((𝑘((𝑘)((𝑘𝜔(𝑘),𝑘+1=𝐴𝑥+𝐵𝑓))+𝑊𝑔𝑘−𝜏)))+𝜎𝑘,𝑥,𝑥𝑘−𝜏))) (1) where 𝑥(𝑘)=[𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘)]^𝑇∈ℝ^𝑛 is the neuron state vector; 𝑓(𝑥(𝑘))=[𝑓₁(𝑥₁(𝑘)),𝑓₂(𝑥₂(𝑘)),…,𝑓_𝑛(𝑥_𝑛(𝑘))]^𝑇∈ℝ^𝑛 and 𝑔(𝑥(𝑘))=[𝑔₁(𝑥₁(𝑘)),𝑔₂(𝑥₂(𝑘)),…,𝑔_𝑛(𝑥_𝑛(𝑘))]^𝑇∈ℝ^𝑛 denotes the neuron activation function. 𝐴=diag{𝑎₁,…,𝑎_𝑛} with |𝑎_𝑖|<1(𝑖=1,2,…,𝑛). 𝐵,𝑊 are the connection weight matrix and the delayed connection weight matrix, respectively, 𝜏(𝑘) represents the transmission time-varying delay, 𝜎∶ℤ×ℝ^𝑛×ℝ^𝑛→ℝ^𝑛 is a continuous function, and 𝜔(𝑘) is a Brown Motion defined on the complete probability space (Ω,ℱ,{𝐹_𝑡}_𝑡) with 𝔼{𝜔(𝑘)}=0, 𝔼{𝜔²(𝑘)}=1,𝔼{𝑤(𝑖)𝑤(𝑗)}=0(𝑖≠𝑗).

For further discussion, we introduce the following assumptions and lemmas.

Assumption 1. There exist two positive constants 𝛽₁ and 𝛽₁ such that 𝜎(𝑘,𝑥(𝑘),𝑥(𝑘−𝜏(𝑘)))^𝑇𝜎(𝑘,𝑥(𝑘),𝑥(𝑘−𝜏(𝑘)))≤𝛽₁𝑥^𝑇(𝑘)𝑥(𝑘)+𝛽₂𝑥^𝑇(𝑘−𝜏(𝑘))𝑥(𝑘−𝜏(𝑘)). (2)

Assumption 2. For 𝑖∈1,2,…,𝑛, the neuron activation functions in the DSNNS in (1) satisfy 𝑙⁻_𝑖≤𝑓_𝑖(𝑥)−𝑓_𝑖𝑦𝑥−𝑦≤𝑙⁺_𝑖∀𝑥,𝑦∈ℝ,𝑥≠𝑦,𝑓_𝑖(0)𝜌=0,_−𝑖≤𝑔_𝑖(𝑥)−𝑔_𝑖𝑦𝑥−𝑦≤𝜌⁺_𝑖∀𝑥,𝑦∈ℝ,𝑥≠𝑦,𝑔_𝑖(0)=0, (3) where 𝑙⁻_𝑖,𝑙⁺_𝑖,𝜌⁺_𝑖,𝜌⁻_𝑖 are some constants.

Remark 3. Assumption 2 previously mentioned on the activation function was widely used in many papers; see [16–21, 23–26], for example.

Lemma 4 (see [27]). For any symmetric constant matrix 𝑄∈ℝ^𝑛×𝑛, 𝑄>0, integers 𝜏_𝑚<𝜏_𝑀, and vector valued function 𝑦(𝑘)=𝑥(𝑘+1)−𝑥(𝑘), one has −𝜏_𝑀−𝜏_𝑚^{𝑘−𝜏𝑚−1}_{𝑖=𝑘−𝜏𝑀}𝑦^𝑇(𝑖)(𝑖)≤𝑥𝑄𝑦𝑘−𝜏_𝑚𝑥𝑘−𝜏_𝑀^𝑇−𝑄𝑄𝑥𝑄−𝑄𝑘−𝜏_𝑚𝑥𝑘−𝜏_𝑀. (4)

Remark 5. Lemma 4 is called discrete Jensen inequality, which is a very important tool for us to get the main results in this paper. The Lemma has been used in some literatures such as [28, 29].

Lemma 6 (see [30]). For any constant matrix 𝑅_𝑖∈ℝ^𝑛×𝑛,𝑅_𝑖=𝑅^𝑇_𝑖≥0, a scalar 𝜆, a positive integer time-varying 𝜏(𝑘)∈[𝜏₁,𝜏₃], and vector function 𝜂∶[−𝜏₃,−𝜏₁]→ℝ^𝑛, such that the following sum is well defined, the following inequalities hold: Ω=−𝜏₃−𝜏₁^{𝑘−𝜏1−1}_{𝜃=𝑘−𝜏3}𝜂(𝜃)𝑅_𝑖𝜂(𝜃)≤𝜉^𝑇₁(𝑘)1+𝜋₁ℛ_𝑖𝜉₁(𝑘)+𝜉^𝑇₂(𝑘)1+𝜋₂ℛ_𝑖𝜉₂(𝑘). (5) Furthermore, if 𝜆+𝜉^𝑇₁(𝑘)3ℛ_𝑖𝜉₁(𝑘)+𝜉^𝑇₂(𝑘)ℛ_𝑖𝜉₂(𝑘)≤0,𝜆+𝜉^𝑇₁(𝑘)ℛ_𝑖𝜉₁(𝑘)+𝜉^𝑇₂(𝑘)3ℛ_𝑖𝜉₂(𝑘)≤0, (6) then 𝜆+𝜉^𝑇₁(𝑘)1+𝜋₁ℛ_𝑖𝜉₁(𝑘)+𝜉^𝑇₂(𝑘)1+𝜋₂ℛ_𝑖𝜉₂(𝑘)≤0, (7) where 𝜋₁=−1,𝜋₂=0,𝑤ℎ𝑒𝑛𝜏(𝑘)=𝜏₁,𝜋₁=𝜏₃−𝜏(𝑘)𝜏(𝑘)−𝜏₁,𝜋₂=1𝜋₁,𝑤ℎ𝑒𝑛𝜏₁<𝜏(𝑘)<𝜏₃,𝜋₁=0,𝜋₂=−1,𝑤ℎ𝑒𝑛𝜏(𝑘)=𝜏₃ (8) and 𝜉₁(𝑘)=[𝑥^𝑇(𝑘−𝜏₁)𝑥^𝑇(𝑘−𝜏(𝑘))]^𝑇, 𝜉₂(𝑘)=[𝑥^𝑇(𝑘−𝜏(𝑘))𝑥^𝑇(𝑘−𝜏₃)]^𝑇,??𝜂(𝑘)=𝑥(𝑘+1)−𝑥(𝑘), ℛ_𝑖=−𝑅_𝑖𝑅_𝑖𝑅_𝑖−𝑅_𝑖. (9)

Remark 7. Lemma 6 is given and proved in [29], which is an effective method to reduce the conservation when studying the time delay stability problem for discrete system; see the literature mentioned previously [30]. Our main study is based on Lemma 6. It is worth mentioning that if 𝜉^𝑇₁(𝑘)𝜋₁ℛ_𝑖𝜉₁(𝑘) and 𝜉^𝑇₂(𝑘)𝜋₂ℛ_𝑖𝜉₂(𝑘) in the proof of (6) are directly ignored, then the following inequality can be derived: Ω≤𝜉^𝑇₁(𝑘)ℛ_𝑖𝜉₁(𝑘)+𝜉^𝑇₂(𝑘)ℛ_𝑖𝜉₂(𝑘). (10)
Compared with the literature [12–18], none of useful items is ignored in (5); what is more, (5) provides a tighter bound to deal with sum terms than those based on (10). However, since (6) is related to time-varying delay items 𝜋₁ and 𝜋₂, it cannot be directly solved based on MATLAB LMI toolbox. To tackle this problem, (6) is used to transfer the time-varying matrix inequality to a set of solvable LMIs. That is, the additional information 𝜉^𝑇₁(𝑘)𝜋₁ℛ_𝑖𝜉₁(𝑘) and 𝜉^𝑇₂(𝑘)𝜋₂ℛ_𝑖𝜉₂(𝑘) in (10) is effectively expressed by (7), and then the less conservative results can be expected.

3. Main Results

For convenience of presentation, we use the following notations: Σ₁=diag𝑙⁻₁𝑙⁺₁,𝑙⁻₂𝑙⁺₂,…,𝑙⁻_𝑛𝑙⁺_𝑛,Σ₂𝑙=diag⁻¹+𝑙⁺₁2,𝑙⁻₂+𝑙⁺₂2𝑙,…,^−𝑛+𝑙⁺_𝑛2,Σ₃=diag𝜚⁻₁𝜚⁺₁,𝜚⁻₂𝜚⁺₂,…,𝜚⁻_𝑛𝜚⁺_𝑛,Σ₄𝜚=diag⁻¹+𝜚⁺₁2,𝜚⁻₂+𝜚⁺₂2𝜚,…,^−𝑛+𝜚⁺_𝑛2,𝜉(𝑘)=𝑥^𝑇(𝑘),𝑥^𝑇𝑘−𝜏_𝑚,𝑥^𝑇𝑘−𝜏₀,𝑥^𝑇((𝑘,𝑥𝑘−𝜏))_𝑇𝑘−𝜏_𝑀,𝑦^𝑇(𝑘),𝑓^𝑇(𝑥(𝑘)),𝑔^𝑇(𝑘−𝜏(𝑘))^𝑇. (11)

In this section, a new delay-dependent stability criteria is proposed for system (1) with time-varying delay satisfying (3); the sufficient conditions of stability are given as follows by Theorem 8.

Theorem 8. For given 𝜏_𝑚,𝜏₀ and 𝜏_𝑀, diagonal matrices Σ₁,Σ₂, Σ₃ and Σ₄, the system (1) is said to be exponentially stable, if there exist scalar 𝜆^∗, diagonal matrices Λ₁,Λ₂ and symmetric matrices 𝑃>0,𝑄_𝑖>0,𝑅_𝑖>0(𝑖=1,2,3,4) such that the following LMIs (12), (13) and (14) hold for 𝛼=0,1: 𝑃<𝜆^∗Ψ𝐼,₁₁₁(𝛼)Ψ<0,₂₁₁(𝛼)<0, (12)(13)(14) where Ψ¹₁₁(𝛼)=2(1−𝛼)Θ₁+2𝛼Θ₂+Θ₁+Θ₂+Θ₃,Ψ²₁₁(𝛼)=2(1−𝛼)Θ₁+2𝛼Θ₂+Θ₁+Θ₂+Θ₃,Θ₁=000000000−𝑅₂00000000000000000−𝑅₂00000𝑅₂,Θ000000000000000000000000000000₂=000000000000000000−𝑅₂0000000−𝑅₂−𝑅₂,000000000000000000000000000000000000Θ₁=000000000000000000−𝑅₃0000000𝑅₃−𝑅₃,000000000000000000000000000000000000Θ₂=000000000000000000000000000−𝑅₃0000000𝑅₃−𝑅₃,Θ000000000000000000000000000₃=Θ₁₁Θ₁₂Θ₁₃Θ₁₄Θ₁₅Θ₁₆Θ₁₇Θ₁₈∗Θ₂₂000Θ₂₆0Θ₂₈∗∗Θ₃₃Θ₃₄Θ₃₅Θ₃₆0Θ₃₈∗∗∗Θ₄₄0Θ₄₆Θ₄₇Θ₄₈∗∗∗∗Θ₅₅Θ₅₆0Θ₅₈∗∗∗∗∗Θ₆₆Θ₆₇Θ₆₈∗∗∗∗∗∗Θ₇₇Θ₇₈∗∗∗∗∗∗∗Θ₈₈,Θ₁₁=𝜏_𝑀−𝜏_𝑚+1𝑅₄−𝑅₁+⁴_𝑖=1𝑄_𝑖−1𝜏_𝑚𝑅₁−Λ₁Σ₁+𝜆^∗𝛽₁𝐼+𝑈₁(𝐼−𝐴)+(𝐼−𝐴)^𝑇𝑈^𝑇₁,Θ₁₂=𝑅₁+𝑈₂(𝐼−𝐴),Θ₁₃=𝑈₃(),Θ𝐼−𝐴₁₄=𝑈₄(𝐼−𝐴),Θ₁₅=𝑈₅(),Θ𝐼−𝐴₁₆=𝑈₆(𝐼−𝐴)+𝑈₁,Θ₁₇=𝑈₈(𝐼−𝐴)−𝑈₁𝐵+Λ₁Σ₂,Θ₁₈=𝑈₈(𝐼−𝐴)−𝑈₁𝑊,Θ₂₂=−𝑄₁−𝜏_𝑚𝑅₁,Θ₂₆=𝑈₂,Θ₂₇=−𝑈₂𝐵,Θ₂₈=−𝑈₂Θ𝑊,₃₃=𝑄₂−1𝜏_𝑀−𝜏₀𝑅₃,Θ₃₄=𝑅₂,Θ₃₅=1𝜏_𝑀−𝜏₀𝑅₃,Θ₃₆=𝑈₃,Θ₃₇=−𝑈₃𝐵,Θ₃₈=−𝑈₃Θ𝑊,₄₄=−𝑅₄−𝑄₃−Λ₂Σ₁+𝜆^∗𝛽₂Θ𝐼,₄₆=𝑈₄,Θ₄₇=−𝑈₄𝐵,Θ₄₈=−𝑈₄𝑊+Λ₂Σ₂,Θ₅₅=−𝑄₄−1𝜏_𝑀−𝜏₀𝑅₃,Θ₅₆=𝑈₅,Θ₅₇=−𝑈₅𝐵,Θ₅₈=−𝑈₅Θ𝑊,₆₆=𝑈₆+𝑈^𝑇₆1+𝑃+𝜏_𝑚𝑅₁+1𝜏₀−𝜏_𝑚𝑅₂+1𝜏_𝑀−𝜏₀𝑅₃,Θ₆₇=−𝑈₆𝐵,Θ₆₈=−𝑈₆Θ𝑊,₇₇=−𝑉₁−𝑈₇𝐵−𝑈^𝑇₇𝐵^𝑇,Θ₇₈=−𝑈₇𝑊−𝐵^𝑇𝑈^𝑇₈,Θ₈₈=−𝑉₂−𝑈₈𝑊−𝑈^𝑇₈𝑊^𝑇,Θ₃=Θ₁₁Θ₁₂Θ₁₃Θ₁₄Θ₁₅Θ₁₆Θ₁₇Θ₁₈∗Θ₂₂Θ₂₃0000Θ₂₈∗∗Θ₃₃Θ₃₄Θ₃₅Θ₃₆0Θ₃₈∗∗∗Θ₄₄0Θ₄₆Θ₄₇Θ₄₈∗∗∗∗Θ₅₅Θ₅₆0Θ₅₈∗∗∗∗∗Θ₆₆Θ₆₇Θ₆₈∗∗∗∗∗∗Θ₇₇Θ₇₈∗∗∗∗∗∗∗Θ₈₈, (15) where Θ₂₂=−𝑄₁−𝜏_𝑚𝑅₁−1𝜏₀−𝜏_𝑚𝑅₂,Θ₂₃1=−𝜏₀−𝜏_𝑚𝑅₂,Θ₃₃=𝑄₂−1𝜏₀−𝜏_𝑚𝑅₂,Θ₃₅=0,Θ₅₅=−𝑄₄. (16) The other terms in Θ₃ have the same expression as that in Θ₃.

Proof. Define 𝑦(𝑘)=𝑥(𝑘+1)−𝑥(𝑘),𝜂(𝑘)=[𝑥^𝑇(𝑘)𝑦^𝑇(𝑘)]^𝑇.
Construct the following Lyapunov-Krasovskii functional candidates: 𝑉(𝑘)=⁴_𝑖=1𝑉_𝑖(𝑘), (17) where 𝑉₁(𝑘)=𝑥^𝑇(𝑘)(𝑘),𝑉𝑃𝑥₂(𝑘)=^𝑘−1_{(𝑘)𝑖=𝑘−𝜏}𝑥^𝑇(𝑘)𝑅₄𝑥(𝑘)+^{𝑘−𝜏𝑚}_{𝑗=𝑘+1−𝜏𝑀}^𝑘−1_𝑗=𝑖𝑥^𝑇𝑗𝑅₄𝑥𝑗,𝑉₃(𝑘)=^𝑘−1_{𝑖=𝑘−𝜏𝑚}𝑥^𝑇(𝑖)𝑄₁𝑥(𝑖)+^𝑘−1_{𝑖=𝑘−𝜏0}𝑥^𝑇(𝑖)𝑄₂𝑥(𝑖)+^𝑘−1_{(𝑘)𝑖=𝑘−𝜏}𝑥^𝑇(𝑖)𝑄₃𝑥(𝑖)+^𝑘−1_{𝑖=𝑘−𝜏𝑀}𝑥^𝑇(𝑖)𝑄₄𝑥(𝑖),𝑉₄(𝑘)=⁻¹_{𝑖=−𝜏𝑚}^𝑘−1_{𝜃=𝑘+𝑖}𝑦^𝑇(𝜃)𝑅₁𝑦(𝜃)+^{−𝜏𝑚−1}_{𝑖=−𝜏0}^𝑘−1_{𝜃=𝑘+𝑖}𝑦^𝑇(𝜃)𝑅₂𝑦(𝜃)+^−𝜏0−1_{𝑖=−𝜏𝑀}^𝑘−1_{𝜃=𝑘+𝑖}𝑦^𝑇(𝜃)𝑅₃𝑦(𝜃), (18) where if (𝜏_𝑀−𝜏_𝑚)/2 is an integer, then 𝜏₀=(𝜏_𝑀+𝜏_𝑚)/2, else 𝜏₀=(𝜏_𝑀+𝜏_𝑚+1)/2.
By calculating the difference of 𝑉(𝑘) along the solution of the system (1), and taking the mathematical expectation, we have 𝔼{Δ𝑉(𝑘)}=𝔼⁴_𝑖=1Δ𝑉_𝑖(𝑘), (19) where 𝔼Δ𝑉₁(𝑘)=𝔼𝑥^𝑇(𝑘+1)𝑃𝑥^𝑇(𝑘+1)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)=𝔼2𝑥^𝑇(𝑘)𝑃𝑦(𝑘)+𝜎^𝑇((𝑘)((𝑘((𝑘)((𝑘𝑘,𝑥,𝑥𝑘−𝜏)))×𝑃𝜎𝑘,𝑥,𝑥𝑘−𝜏)))+𝑦^𝑇(𝑘)𝑃𝑦(𝑘). (20) From Assumption 2, we can obtain that 𝜎(𝑘,𝑥(𝑘),𝑥(𝑘−𝜏(𝑘)))^𝑇𝜎(𝑘,𝑥(𝑘),𝑥(𝑘−𝜏(𝑘)))≤𝜆^∗𝛽₁𝑥^𝑇(𝑘)𝑥(𝑘)+𝛽₂𝑥^𝑇(𝑘−𝜏(𝑘))𝑥(𝑘−𝜏(𝑘)). (21) So we can get that 𝔼Δ𝑉₁(𝑘)≤𝔼2𝑥^𝑇(𝑘)𝑃𝑦(𝑘)+𝑦^𝑇(𝑘)𝑃𝑦(𝑘)+𝜆^∗𝛽₁𝑥^𝑇(𝑘)𝑥(𝑘)+𝛽₂𝑥^𝑇((𝑘𝑥((𝑘,𝔼𝑘−𝜏))𝑘−𝜏))Δ𝑉₂(𝑘)=𝔼𝑥^𝑇(𝑘)𝑅₄𝑥(𝑘)−𝑥^𝑇(𝑘−𝜏(𝑘))×𝑅₄𝑥(𝑘−𝜏(𝑘))+^{𝑘−𝜏𝑚}_{𝑗=𝑘+1−𝜏𝑘+1}𝑥^𝑇(𝑘)𝑅₄𝑥(𝑘)+^𝑘−1_{𝑗=𝑘+1−𝜏𝑚}𝑥^𝑇(𝑘)(𝑘)−𝑅𝑥_{𝑘−1𝑗=𝑘+1−𝜏𝑘}𝑥^𝑇(𝑘)𝑅₄𝑥(𝑘)+^{𝑘+1−𝜏𝑚}_{𝑗=𝑘+2−𝜏𝑀}^𝑘_𝑗=𝑖𝑥^𝑇𝑗𝑅₄𝑥𝑗−^{𝑘−𝜏𝑚}_{𝑗=𝑘+1−𝜏𝑀}^𝑘−1_𝑗=𝑖𝑥^𝑇𝑗𝑅₄𝑥𝑗≤𝔼𝑥^𝑇(𝑘)𝑅₄𝑥(𝑘)−𝑥^𝑇(𝑘−𝜏(𝑘))𝑅₄𝑥((𝑘+𝑘−𝜏))_{𝑘−𝜏𝑚j=𝑘+1−𝜏𝑀}𝑥^𝑇(𝑘)𝑅₄𝑥(𝑘)+^{𝑘−𝜏𝑚}_{𝑗=𝑘+1−𝜏𝑀}𝑥^𝑇(𝑘)𝑅₄𝑥(𝑘)−𝑥^𝑇𝑗𝑅₄𝑥𝑗=𝔼𝜏_𝑀−𝜏_𝑚+1𝑥^𝑇(𝑘)𝑅₄𝑥(𝑘)−𝑥^𝑇(𝑘−𝜏(𝑘))𝑅₄𝑥((𝑘,𝔼𝑘−𝜏))Δ𝑉₃(𝑘)=𝔼𝑥^𝑇(𝑘)⁴_𝑖=1𝑄_𝑖𝑥(𝑘)−𝑥^𝑇𝑘−𝜏_𝑚𝑄₁𝑥𝑘−𝜏_𝑚−𝑥^𝑇𝑘−𝜏₀𝑄₂𝑥𝑘−𝜏₀−𝑥^𝑇(𝑘−𝜏(𝑘))𝑄₃𝑥(𝑘−𝜏(𝑘))−𝑥^𝑇𝑘−𝜏_𝑀𝑄₄𝑥𝑘−𝜏_𝑀,𝔼Δ𝑉₄(𝑘)=𝔼𝜏_𝑚𝑦^𝑇(𝑘)𝑅₁𝑦(𝑘)−^𝑘−1_{𝑗=𝑘−𝜏𝑚}𝑦^𝑇(𝑘)𝑅₁𝑦(𝑘)+𝜏₀−𝜏_𝑚𝑦^𝑇(𝑘)𝑅₂𝑦(𝑘)−^{𝑘−𝜏𝑚−1}_{𝑗=𝑘−𝜏0}𝑦^𝑇(𝑘)𝑅₂𝑦(𝑘)+𝜏_𝑀−𝜏₀𝑦^𝑇(𝑘)𝑅₃𝑦(𝑘)−^{𝑘−𝜏0−1}_{𝑗=𝑘−𝜏𝑀}𝑦^𝑇(𝑘)𝑅₃𝑦(𝑘). (22)(23)(24)(25)
Now, we are in the position to prove that 𝔼{Δ𝑉(𝑘)}<0 holds for both 𝜏_𝑀≥𝜏_𝑘≥𝜏₀ and 𝜏₀≥𝜏_𝑘≥𝜏_𝑚.
Case I (when 𝜏₀≥𝜏_𝑘≥𝜏_𝑚). Using Lemmas 4 and Lemma 6 to deal with the second sum items in the right side of (25), we have −^𝑘−1_{𝑗=𝑘−𝜏𝑚}𝑦^𝑇(𝑘)𝑅₁𝑦(𝑘)≤1𝜏_𝑚𝑥(𝑘)𝑥𝑘−𝜏_𝑚^𝑇−𝑅₁𝑅₁𝑅₁−𝑅₁𝑥(𝑘)𝑥𝑘−𝜏_𝑚−^{𝑘−𝜏0−1}_{𝑗=𝑘−𝜏𝑀}𝑦^𝑇(𝑘)𝑅₃𝑦(𝑘)≤1𝜏_𝑀−𝜏₀𝑥𝑘−𝜏₀𝑥𝑘−𝜏_𝑀^𝑇−𝑅₃𝑅₃𝑅₃−𝑅₃𝑥𝑘−𝜏₀𝑥𝑘−𝜏_𝑀−^{𝑘−𝜏𝑚−1}_{𝑗=𝑘−𝜏0}𝑦^𝑇(𝑘)𝑅₂𝑦(𝑘)≤1𝜏₀−𝜏_𝑚𝜉^𝑇₁(𝑘)1+𝜋₁ℛ₂𝜉₁(𝑘)+𝜉^𝑇₃(𝑘)1+𝜋₂ℛ₂𝜉₃(𝑘), (26)(27)(28) where 𝜉₁(𝑘)=𝑥^𝑇𝑘−𝜏_𝑚𝑥^𝑇(𝑘−𝜏(𝑘))^𝑇,𝜉₃(𝑘)=𝑥^𝑇𝑘−𝜏₀𝑥^𝑇(𝑘−𝜏(𝑘))^𝑇. (29)
At the same time, for any matrices 𝑈 of appropriate dimensions, we have 𝔼𝜉^𝑇(𝑘)𝑈+𝑈^𝑇)𝑥(𝑘)(𝑥(𝑘(𝑥((𝑘((𝑘)((𝑘𝜔(𝑘)(𝑘(𝐴−𝐼+𝐵𝑓))+𝑊𝑔𝑘−𝜏)))+𝜎𝑘,𝑥,𝑥𝑘−𝜏)))−𝑦)=0, (30) where 𝑈=𝑈^𝑇₁,𝑈^𝑇₂,𝑈^𝑇₃,𝑈^𝑇₄,𝑈^𝑇₅,𝑈^𝑇₆,𝑈^𝑇₇,𝑈^𝑇₈^𝑇. (31)
In addition, it can be deduced from Assumption 2 that there exist two positive diagonal matrices Λ_𝑖=diag{𝜀_𝑖,1,𝜀_𝑖,2,…,𝜀_𝑖,𝑛},(𝑖=1,2) such that 0≤−^𝑛_𝑖=1𝜀_1,𝑖𝑓_𝑖𝑥_𝑖(𝑘)−𝑙⁻_𝑖𝑥_𝑖(𝑘)𝑓_𝑖𝑥_𝑖(𝑘)−𝑙⁺_𝑖𝑥_𝑖(𝑘−_𝑛𝑖=1)𝜀_2,𝑖𝑔_𝑖𝑥_𝑖(𝑘−𝜏(𝑘))−𝑙⁻_𝑖𝑥_𝑖((𝑘×𝑔𝑘−𝜏))_𝑖𝑥_𝑖(𝑘−𝜏(𝑘))−𝑙⁺_𝑖𝑥_𝑖(𝑘−𝜏(𝑘))=−𝑥^𝑇(𝑘)Λ₁Σ₁𝑥(𝑘)+2𝑓^𝑇(𝑥(𝑘))Λ₁Σ₂𝑥(𝑘)−𝑓^𝑇(𝑥(𝑘))Λ₁𝑓(𝑥(𝑘))−𝑥^𝑇(𝑘−𝜏(𝑘))Λ₂Σ₃𝑥(𝑘−𝜏(𝑘))+2𝑔^𝑇(𝑥(𝑘−𝜏(𝑘)))×Λ₂Σ₄𝑥(𝑘−𝜏(𝑘))−𝑔^𝑇(𝑥(𝑘−𝜏(𝑘)))Λ₂𝑔(𝑥(𝑘−𝜏(𝑘))). (32)
By substituting (22)–(25) into (19), adding (30) and (32) into the right side of (19), and using (26) and (27), we can get that 𝔼{Δ𝑉(𝑘)}≤𝔼𝜆₁+1𝜏₀−𝜏_𝑚𝜉^𝑇₁(𝑘)1+𝜋₁ℛ₂𝜉₁(𝑘)+𝜉^𝑇₃(𝑘)1+𝜋₂ℛ₂𝜉₃(𝑘), (33) where 𝜆₁=𝜉^𝑇(𝑘)Θ₃𝜉(𝑘),??𝜋₁=(𝜏₀−𝜏(𝑘))/(𝜏(𝑘)−𝜏_𝑚) and 𝜋₁=1/𝜋₂, 𝜉(𝑘)=𝑥^𝑇(𝑘)𝑥^𝑇𝑘−𝜏_𝑚𝑥^𝑇𝑘−𝜏₀𝑥^𝑇(𝑘−𝜏(𝑘))𝑥^𝑇𝑘−𝜏_𝑀𝑓^𝑇(𝑥(𝑘))𝑓^𝑇(𝑥(𝑘−𝜏(𝑘)))^𝑇,𝜉(𝑘)=𝑥^𝑇(𝑘)𝑥^𝑇𝑘−𝜏_𝑚𝑥^𝑇𝑘−𝜏₀𝑥^𝑇(𝑘−𝜏(𝑘))𝑥^𝑇𝑘−𝜏_𝑀𝑦^𝑇(𝑥(𝑘))𝑓^𝑇(𝑥(𝑘))𝑓^𝑇(𝑥(𝑘−𝜏(𝑘)))^𝑇. (34)
So (12) and (13) imply that 𝜆₁+𝜉^𝑇₁(𝑘)3ℛ₂𝜉₁(𝑘)+𝜉^𝑇₂(𝑘)ℛ₂𝜉₂(𝑘)𝜆≤0,₁+𝜉^𝑇₁(𝑘)ℛ₂𝜉₁(𝑘)+𝜉^𝑇₂(𝑘)3ℛ₂𝜉₂(𝑘)≤0. (35) So from Lemma 6, (35) guarantees that 𝔼{Δ𝑉(𝑘)}≤0, so there exists a positive scalar 𝑐₁ such that 𝔼{Δ𝑉(𝑘)}≤𝑐₁𝔼‖𝑥(𝑘)‖². (36)
Case II (when 𝜏_𝑀≥𝜏_𝑘≥𝜏₀). Keeping (26) and by utilizing Lemmas 4 and 6 to deal with the accumulative items of (27) and (28), we can get −^𝑘−1_{𝑗=𝑘−𝜏𝑚}𝑦^𝑇(𝑘)𝑅₁𝑦(𝑘)≤1𝜏_𝑚𝑥(𝑘)𝑥𝑘−𝜏_𝑚^𝑇−𝑅₁𝑅₁𝑅₁−𝑅₁𝑥(𝑘)𝑥𝑘−𝜏_𝑚−^{𝑘−𝜏0−1}_{𝑗=𝑘−𝜏𝑀}𝑦^𝑇(𝑘)𝑅₃𝑦(𝑘)≤1𝜏_𝑀−𝜏₀𝜉^𝑇₂(𝑘)1+𝜋₁ℛ₃𝜉₂(𝑘)+𝜉^𝑇₃(𝑘)1+𝜋₂ℛ₃𝜉₃(𝑘),−^{𝑘−𝜏𝑚−1}_{𝑗=𝑘−𝜏0}𝑦^𝑇(𝑘)𝑅₂𝑦(𝑘)≤1𝜏₀−𝜏_𝑚𝑥𝑘−𝜏_𝑚𝑥𝑘−𝜏₀^𝑇−𝑅₂𝑅₂𝑅₂−𝑅₂𝑥𝑘−𝜏_𝑚𝑥𝑘−𝜏₀, (37) where 𝜉₂(𝑘)=𝑥^𝑇𝑘−𝜏₀𝑥^𝑇(𝑘−𝜏(𝑘))^𝑇,𝜉₃(𝑘)=𝑥^𝑇(𝑘−𝜏(𝑘))𝑥^𝑇𝑘−𝜏_𝑀^𝑇. (38)
So the whole difference of Lyapunov-Krasovskii functional candidate is given as follows: 𝔼{Δ𝑉(𝑘)}≤𝜆₂+𝜉^𝑇₂(𝑘)1+𝜋₁ℛ₃𝜉₂(𝑘)+𝜉^𝑇₃(𝑘)1+𝜋₂ℛ₃𝜉₃(𝑘), (39) where 𝜆₂=𝜉^𝑇(𝑘)[Θ₃𝜉(𝑘), 𝜋₁=(𝜏_𝑀−𝜏(𝑘))/(𝜏(𝑘)−𝜏₀) and??𝜋₂=1/𝜋₁. For 𝜏(𝑘)≠𝜏₀ and 𝜏(𝑘)≠𝜏_𝑀, we can also get that there exists a positive scalar 𝑐₂ such that 𝔼{Δ𝑉(𝑘)}≤𝑐₂𝔼‖𝑥(𝑘)‖². (40) Combining Cases I and II, we can conclude that (12), (13) and (14) guarantee that 𝔼{Δ𝑉(𝑘)}<−min𝑐₁,𝑐₂𝔼‖𝑥(𝑘)‖². (41) Defining a new function 𝕍(𝑘,𝑥(𝑘))=𝜇^𝑘𝑉(𝑘,𝑥(𝑘)) and then using the similar analysis method of Theorem 1 in [18], we can easily get that the system (1) is globally exponentially stable in the mean square sense. This completes the proof.

Remark 9. It can be seen from the proof previously mentioned that no model transformation has been employed to deal with the sum terms, and none of the useful items are ignored in the proof.
If we neglect the effect of the stochastic term 𝜔(𝑘) in (1), then 𝛽₁=𝛽₂=0 and (1) will reduce to 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵𝑓(𝑥(𝑘))+𝑊𝑓(𝑥(𝑘−𝜏(𝑘))). (42) For system (42), we can obtain the following corollary based on Theorem 8.

Corollary 10. For given 𝜏_𝑚, 𝜏₀, and 𝜏_𝑀 and diagonal matrices Σ₁,Σ₂, Σ₃, and Σ₄, the system (42) is said to be exponentially stable, if there exist diagonal matrices Λ₁,Λ₂, symmetric positive definite matrices 𝑃>0,𝑄_𝑖>0 and ?𝑅_𝑖>0(𝑖=1,2,3) such that (12) and the following LMIs hold for 𝛼=0,1: Θ₄=Θ₁₁Θ₁₂Θ₁₃Θ₁₄Θ₁₅Θ₁₆Θ₁₇Θ₁₈∗Θ₂₂000Θ₂₆0Θ₂₈∗∗Θ₃₃Θ₃₄Θ₃₅Θ₃₆0Θ₃₈∗∗∗Θ₄₄0Θ₄₆Θ₄₇Θ₄₈∗∗∗∗Θ₅₅Θ₅₆0Θ₅₈∗∗∗∗∗Θ₆₆Θ₆₇Θ₆₈∗∗∗∗∗∗Θ₇₇Θ₇₈∗∗∗∗∗∗∗Θ₈₈,Θ₄=Θ₁₁Θ₁₂Θ₁₃Θ₁₄Θ₁₅Θ₁₆Θ₁₇Θ₁₈∗Θ₂₂Θ₂₃00Θ₂₆0Θ₂₈∗∗Θ₃₃Θ₃₄Θ₃₅Θ₃₆0Θ₃₈∗∗∗Θ₄₄0Θ₄₆Θ₄₇Θ₄₈∗∗∗∗Θ₅₅Θ₅₆0Θ₅₈∗∗∗∗∗Θ₆₆Θ₆₇Θ₆₈∗∗∗∗∗∗Θ₇₇Θ₇₈∗∗∗∗∗∗∗Θ₈₈,Θ₁₁=𝜏_𝑀−𝜏_𝑚+1𝑅₄−𝑅₁+⁴_𝑖=1𝑄_𝑖−1𝜏_𝑚𝑅₁−Λ₁Σ₁+𝑈₁(𝐼−𝐴)+(𝐼−𝐴)^𝑇𝑈^𝑇₁,Θ₄₄=−𝑅₄−𝑄₃−Λ₂Σ₁. (43)

Remark 11. The system (42) has been studied by many researchers, many stability criteria have been proposed, and many improved analysis results have been obtained; see [20–22].

4. Numerical Example

In this section, three examples are given to demonstrate the benefits of the proposed method.

Example 1. Consider the following stochastic discrete neural networks [25, 26]: 𝑥()(𝑘)(𝑥(𝑘(𝑥((𝑘(𝑥(𝑘)((𝑘𝜔(𝑘)𝑘+1=𝐴𝑥+𝐵𝑓))+𝑊𝑔𝑘−𝜏)))+𝜎,𝑥𝑘−𝜏))) (44) with the following parameters: ,,𝛽𝐴=−0.100−0.2,𝐵=−0.10.1−0.10.5𝑊=0.050.10.50.5₁=0.2,𝛽₂𝑓=0.2,₁(𝑠)=sin(0.2𝑠)−0.6cos(𝑠),𝑓₂(𝑠)(),𝑔=tanh−0.4𝑠₁(𝑠)=tanh(0.83𝑠)+0.6cos(𝑠),𝑔₂(𝑠)=tanh(0.2𝑠). (45)
So it can be verified that Σ₁=diag{−0.64,0},Σ₂{},Σ=diag0,−0.2₃=diag{−0.6,0},Σ₄=diag{0.2,0.1}. (46)
For 𝜏_𝑚=2, by using Matlab LMI toolbox, the maximum allowable value is 𝜏_𝑀=11. Setting 𝜏_𝑚=2 and 𝜏_𝑀=5 in Theorem 8, we can solve a set of feasible solutions for the LMIs (12), (13) and (14) which are listed as follows: 𝑃=2.8554−0.1283−0.12831.3300,𝑅₁=,𝑅0.00440.00210.00210.0022₂=0.04210.00370.00370.0128,𝑅₃=,𝑄0.04890.00240.00240.0136₁=0.1793−0.0356−0.03560.1487,𝑄₂=,𝑄0.42790.00510.00510.0779₃=0.3817−0.0218−0.02180.1063,Λ₁=,Λ0.0651000.0904₂=0.0803000.3411,𝜆^∗=4.7129. (47)
At the same time, we set 𝜏_𝑀=5; the state curve can be obtained as Figure 1 by using Matlab simulation software. From Figure 1, we can see that the system (1) is asymptotically stable.
In our study, our purpose is to compare the maximum upper bound 𝜏_𝑀 for different 𝜏_𝑚. Now assume the different lower bound 𝜏_𝑚; by solving the LMIs (12), (13) and (14), we can get the maximum delays of upper bound, which are listed in Table 1. From Table 1, we can see that Theorem 8 is less conservative than the criterion proposed in [25, 26].

Table 1 

Allowable upper bound of 𝜏_𝑀 for various 𝜏_𝑚.

Figure 1 

Trajectories of 𝑥(𝑘) of system in Example 1.

Remark 12. According to Theorem 8, we can obtain that system (1) with the pervious parameters is mean square exponentially stable, and meantime it is very easy to check that our results have improved the conclusions in [26] by the Matlab LMI Toolbox. If we negelect the uncertainty effect and take 𝜌₁=0.2, 𝜌₂=0.2 then apply the criteria in [26], the value of 𝜏_𝑀 for mean square exponentially stable of system (1) is 55, respectively. While by using Theorem 8 in this paper, taking 𝜏_𝑀 62 into LMIs (12), (13) and (14), we find that LMIs (12), (13) and (14) are feasible, which shows that our result is less conservative than [26] under the identical conditions.

Example 2. Consider the neural networks (1) with the following parameters [23]: ,.𝐴=0.40000.50000.4,𝐵=0.3−0.10.20−0.30.2−0.1−0.1−0.2𝑊=0.20.10.1−0.20.30.10.3−0.30.1 (48)
Take the activation function as follows: 𝑓₁(𝑠)=tanh(0.6𝑠)−0.2sin(𝑠),𝑓₂(𝑠)(),𝑓=tanh−0.4𝑠₃(𝑠)(),𝑔=tanh−0.2𝑠₁(𝑠)()(𝑠),𝑔=tanh−0.4𝑠+0.2sin₂(𝑠)=tanh(0.2𝑠),𝑔₃(𝑠)=tanh(0.4𝑠). (49) From the previous parameters, it can be verified that Σ₁=−0.1600000000,Σ₂=,Σ−0.30000.20000.1₃=−0.1200000000,Σ₄=.0.2000−0.1000−0.2 (50)
For the Corollary 1 in [23], using the previous parameters, we find that it is unsolvable, but it is solvable in our criteria in Theorem 8. By virtue of the Matlab Toolbox, we can obtain the feasible solutions as follows: ,Λ𝑃=41.9042−0.0335−0.2435−0.033543.37110.3170−0.24350.317043.0218₁=,Λ5.765700011.348800011.7506₂=,𝑅19.868800025.456000022.0029₁=,𝑅2.25280.0223−0.02310.02231.81450.0270−0.02310.02701.8773₂=,𝑅0.50950.0818−0.05190.081825.4560−0.0313−0.0519−0.03130.6545₃=,𝜆0.54940.0367−0.02310.03670.5644−0.0065−0.0231−0.00650.4522_∗=46.4939. (51)

Example 3. Consider the neural networks (1) with the following parameters [23, 27]: 𝐴=0.40000.50000.4,𝐵=0.3−0.10.20−0.30.2−0.1−0.1−0.2,𝑊=0.20.10.1−0.20.30.10.3−0.30.1. (52)(53)
Take the activation function as follows: 𝑓₁(𝑠)=tanh(0.2𝑠),𝑓₂(𝑠)(),𝑓=tanh−0.4𝑠₃(𝑠)=tanh(−0.2𝑠),𝑔₁(𝑠)(),𝑔=tanh−0.12𝑠₂(𝑠)=tanh(0.2𝑠),𝑔₃(𝑠)=tanh(0.4𝑠). (54)
From the previous parameters, it can be verified that Σ₁=000000000,Σ₂=,Σ0.2000−0.40000.2₃=000000000,Σ₄=.−0.120000.20000.4 (55)
For this example, these conditions in [23] cannot be satisfied. For 𝜏_𝑚=2, it has been verified that the maximum allowable time delay for system (42) is 𝜏_𝑀=18 in [27]. By letting 𝜏_𝑚=2 and 𝜏_𝑀=26 in Example 3, we find that systems (42) are feasible, which implies that the exponential stability result proposed in Corollary 10 in this paper provides less conservatism than in [23, 27].

5. Conclusion

An effective sum inequality has been introduced to derive the delay-dependent stability criteria for a class of discrete stochastic neural networks system with an interval time-varying delay. By choosing piecewise Lyapunov-Krasovskii functional candidate and employing the proposed sum inequalities, significant performance improvement has been achieved with noticeably reducing the number of LMIs scalar decision variables. All results are given by the form of LMIs. Numerical examples show that the achieved results are less conservative than some existing literatures.