Exponential Stability for Discrete-Time Stochastic BAM Neural Networks with Discrete and Distributed Delays

Abstract

This paper deals with the stability analysis problem for a class of discrete-time stochastic BAM neural networks with discrete and distributed time-varying delays. By constructing a suitable Lyapunov-Krasovskii functional and employing M-matrix theory, we find some sufficient conditions ensuring the global exponential stability of the equilibrium point for stochastic BAM neural networks with time-varying delays. The conditions obtained here are expressed in terms of LMIs whose feasibility can be easily checked by MATLAB LMI Control toolbox. A numerical example is presented to show the effectiveness of the derived LMI-based stability conditions.

1. Introduction

Recently, the study of Bidirectional associative memory neural networks has attracted the attention of many researchers due to its applications in many fields such as pattern recognition, automatic control, associative memory, signal processing, and optimization; see, for example, [1–9]. The (BAM) neural networks model, proposed by Kosko [10, 11], is a two layer nonlinear feedback network model and it was described that the neurons in one layer are fully interconnected to the neurons in the other layer, while there are no interconnections among neurons in the same layer.

Furthermore, due to the finite switching speed of neuron amplifiers and the finite speed of signal propagation time delays are unavoidable in the implementation of neural networks [12–14]. According to the way it occurs, time delay can be classified as two types: discrete and distributed delays. Discrete time-delay is relatively easier to be identified in practice and hence the stability analysis for BAM with discrete delays has been an attractive subject of research in the past few years; see [15, 16]. On the other hand, due to the presence of an amount of parallel pathways of a variety of axon sizes and lengths, a neural network usually has a spatial nature. Therefore, it is necessary to introduce continuously distributed delays over a certain duration of time; see [17, 18].

Moreover, in implementations of neural networks, stochastic disturbances are inevitable owing to thermal noise in electronic devices. Practically, the stochastic phenomenon usually appears in the electrical circuit design of neural networks. The stochastic effects can have the ability to destabilize a neural system. Therefore, it is significant and of importance to consider stochastic effects to the stability property of the neural networks with delays. It is noted that most of the BAM neural networks have been assumed to act in a continuous-time manner. However, when it comes to the implementation of discrete-time BAM networks, there are only few works appeared in the literature; see [6, 19–24] and the references cited therein. Therefore, there is a crucial need to study the dynamics of discrete-time BAM neural networks and it becomes more significant from practical point of view. In [19], Gao and Cui discussed the global robust exponential stability of discrete-time interval BAM neural networks with time-varying delays, and in [24], the authors investigated the global exponential stability for discrete-time BAM neural network with time variable delay. In the above said references the stability problem for BAM neural networks is considered only with discrete delay, and distributed delay has not been taken into account and remains challenging. So, our main aim in this work is to make the first attempt to shorten such a gap.

Motivated by the above points, in this paper, we will study the exponential stability problem for a new class of discrete-time stochastic BAM neural networks with both discrete and distributed delays. The existence of the equilibrium point is proved under mild conditions on the activation functions. By constructing an appropriate Lyapunov-Krasovskii functional, a linear matrix inequality (LMI) approach is developed to establish sufficient conditions for the discrete-time BAM neural networks to be globally exponentially stable in the mean square. Here, we note that the LMIs can be easily solved by using Matlab LMI toolbox, and no tuning of parameters is involved. Finally, a numerical example is presented to show the usefulness of the derived LMI-based stability conditions.

Notations. Throughout this paper, ℝ^𝑛 and ℝ^𝑛×𝑚 denote, respectively, the 𝑛-dimensional Euclidean space and the set of all 𝑛×𝑚 real matrices. 𝐼 denotes the identity matrix with appropriate dimensions and diag(⋅) denotes the diagonal matrix. For real symmetric matrices 𝑋 and 𝑌, the notation 𝑋≥𝑌 (resp., 𝑋>𝑌) means that the matrix 𝑋−𝑌 is positive semidefinite (resp., positive definite). ℕ={1,2,…,𝑛} and ‖⋅‖ stands for the Euclidean norm in ℝ^𝑛. 𝜆_max(𝑋) (resp., 𝜆_min(𝑋)) stands for the maximum (resp., minimum) eigenvalue of the matrix 𝑋. The symbol ∗ within a matrix represents the symmetric term of the matrix.

2. Problem Description and Preliminaries

Consider the following discrete-time stochastic BAM neural networks with both discrete and distributed delays of the following form:𝑥_𝑖(𝑘+1)=𝑎_𝑖𝑥_𝑖(𝑘)+^𝑛_𝑗=1𝑐_𝑗𝑖𝑓_𝑗𝑦_𝑗(𝑘)+^𝑛_𝑗=1𝑤_𝑗𝑖𝑔_𝑗𝑦_𝑗𝑘−𝜏_𝑗𝑖(𝑘)+^𝑛_𝑗=1𝑚_{𝑗𝑖+∞ℳ}₌₁𝜇_ℳℎ_𝑗𝑦_𝑗(𝑘−ℳ)+𝐼_𝑖+𝛿_𝑗𝑖𝑦𝑥_𝑖(𝑘),𝑦_𝑗𝑘−𝜏_𝑗𝑖(𝑘),𝑘𝑤₁(𝑘),𝑖∈𝑁,_𝑗(𝑘+1)=𝑏_𝑗𝑦_𝑗(𝑘)+^𝑛_𝑖=1𝑑_𝑖𝑗𝑓_𝑖𝑥_𝑖(𝑘)+^𝑛_𝑖=1𝑣_𝑖𝑗𝑔̂_𝑖𝑥_𝑖𝑘−𝜎_𝑖𝑗(𝑘)+^𝑛_𝑖=1𝑛_{𝑖𝑗+∞𝒩}₌₁𝜌_𝒩ℎ_𝑖𝑥_𝑖(𝑘−𝒩)+𝐽_𝑗+𝜒_𝑖𝑗𝑦_𝑗(𝑘),𝑥_𝑖𝑘−𝜎_𝑖𝑗(𝑘),𝑘𝑤₂(𝑘),𝑗∈𝑁,(2.1) or, in an equivalent form,𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐶𝑓(𝑦(𝑘))+𝑊𝑔(𝑦(𝑘−𝜏(𝑘)))+𝑀^+∞_ℳ₌₁+𝛿(𝑥(𝑘),𝑦(𝑘−𝜏(𝑘)),𝑘)𝑤₁𝜇_ℳℎ(𝑦(𝑘−ℳ))+𝐼𝑦(𝑘+1)=𝐵𝑦(𝑘)+𝐷𝑓(𝑥(𝑘))+𝑉𝑔̂(𝑥((𝑘𝑘)−,𝜎(𝑘)))+𝑁_+∞𝒩₌₁+𝜒(𝑦(𝑘),𝑥(𝑘−𝜎(𝑘)),𝑘)𝑤₂𝜌_𝒩ℎ(𝑥(𝑘−𝒩))+𝐽(𝑘),(2.2) for 𝑘=1,2,…, where 𝑥(𝑘) and 𝑦(𝑘) are the neural state vector; 𝐴=diag{𝑎₁,𝑎₂,…,𝑎_𝑛} and 𝐵=diag{𝑏₁,𝑏₂,…,𝑏_𝑛} are the state feedback coefficient matrices; 𝐶=[𝑐_𝑖𝑗]_𝑛×𝑛,𝐷=[𝑑_𝑖𝑗]_𝑛×𝑛,𝑊=[𝑤_𝑖𝑗]_𝑛×𝑛,𝑉=[𝑣_𝑖𝑗]_𝑛×𝑛,𝑀=[𝑚_𝑖𝑗]_𝑛×𝑛, and 𝑁=[𝑛_𝑖𝑗]_𝑛×𝑛 are, respectively, the connection weight matrices, the discretely delayed connection weight matrices, and distributed delayed connection weight matrices; 𝜏(𝑘) and 𝜎(𝑘) denote the discrete time-varying delays satisfying𝜏_𝑚≤𝜏(𝑘)≤𝜏_𝑀,𝜎_𝑚≤𝜎(𝑘)≤𝜎_𝑀,(2.3) where 𝜏_𝑚,𝜏_𝑀,𝜎_𝑚, and 𝜎_𝑀 are known positive integer; 𝑀,𝑁 denotes the distributed time-varying delays.Then 𝑓𝑓((𝑥𝑦((𝑘𝑘))))==𝑓𝑓₁₁𝑦₁(𝑘),𝑓₂𝑦₂(𝑘),…,𝑓_𝑛𝑦_𝑛(𝑘)^𝑇,𝑔(𝑦(𝑘))=𝑔₁𝑥₁(𝑘),𝑓₂𝑥₂(𝑘),…,𝑓_𝑛𝑥_𝑛(𝑘)^𝑇,𝑔̂(𝑥(𝑘))=𝑔̂₁𝑦₁(𝑘),𝑔₂𝑦₂(𝑘),…,𝑔_𝑛𝑦_𝑛(𝑘)^𝑇,ℎ(𝑦(𝑘))=ℎ₁𝑥₁(𝑘),𝑔̂₂𝑥₂(𝑘),…,𝑔̂_𝑛𝑥_𝑛(𝑘)^𝑇,ℎ(𝑥(𝑘))=ℎ₁𝑦₁(𝑘),ℎ₂𝑦₂(𝑘),…,ℎ_𝑛𝑦_𝑛(𝑘)^𝑇,𝑥₁(𝑘),ℎ₂𝑥₂(𝑘),…,ℎ_𝑛𝑥_𝑛(𝑘)^𝑇,(2.4) denote the neuron activation functions. The constant vectors 𝐽=[𝐽₁,𝐽₂,…,𝐽_𝑛]^𝑇 and 𝐼=[𝐼₁,𝐼₂,…,𝐼_𝑛]^𝑇 are the external inputs from outside the system; 𝜇_ℳ,(ℳ=1,2,…) and 𝜌_𝒩,(𝒩=1,2,…) are scalar constants, where 𝑤₁(𝑘) and 𝑤₂(𝑘) are scalar Wiener process (Brownian motion) on the probability space (Ω,ℱ,𝔓) with𝔼𝔼𝑤𝑤₁(𝑘)=0,𝔼𝑤²¹(𝑘)=1,𝔼𝑤₁(𝑖)𝑤₁(𝑗)=0(𝑖≠𝑗),₂(𝑘)=0,𝔼𝑤²²(𝑘)=1,𝔼𝑤₂(𝑖)𝑤₂(𝑗)=0(𝑖≠𝑗),(2.5) with 𝔼(⋅) being the mathematical expectation operator; 𝛿∶ℝ^𝑛×ℝ^𝑛×ℝ→ℝ^𝑛 and 𝜒∶ℝ^𝑛×ℝ^𝑛×ℝ→ℝ^𝑛 are the nonlinear vector function representing the disturbance intensities.

In this paper, we make following assumptions for the neuron activation functions.

Assumption 1. For 𝑗,𝑖∈{1,2,…,𝑛}, the neuron activation functions 𝑓_𝑗(⋅), 𝑓_𝑖(⋅), 𝑔_𝑗(⋅), 𝑔̂_𝑖(⋅), ℎ_𝑗(⋅), and ℎ_𝑖(⋅) in (2.2) are continuous as well as bounded on ℝ.

Assumption 2. For 𝑗,𝑖∈{1,2,…,𝑛}, the neuron activation functions in (2.2) satisfy 𝑙^−𝑗≤𝑓^𝑗𝑠₁−𝑓_𝑗𝑠₂𝑠₁−𝑠₂≤𝑙^+𝑗,∀𝑠₁,𝑠₂∈ℝ,𝑢^−𝑖≤𝑓^𝑖𝑡₁−𝑓_𝑖𝑡₂𝑡₁−𝑡₂𝑚≤𝑢^+𝑖,∀𝑡₁,𝑡₂∈ℝ,_−𝑗≤𝑔^𝑗𝑠₁−𝑔_𝑗𝑠₂𝑠₁−𝑠₂≤𝑚^+𝑗,∀𝑠₁,𝑠₂∈ℝ,𝑣^−𝑖≤𝑔̂^𝑖𝑡₁−𝑔̂_𝑖𝑡₂𝑡₁−𝑡₂𝑛≤𝑣^+𝑖,∀𝑡₁,𝑡₂∈ℝ,_−𝑗≤ℎ^𝑗𝑠₁−ℎ_𝑗𝑠₂𝑠₁−𝑠₂≤𝑛^+𝑗,∀𝑠₁,𝑠₂∈ℝ,𝑤^−𝑖≤ℎ^𝑖𝑡₁−ℎ_𝑖𝑡₂𝑡₁−𝑡₂≤𝑤^+𝑖,∀𝑡₁,𝑡₂∈ℝ,(2.6) where 𝑙^−𝑗,𝑙^+𝑗,𝑚^−𝑗,𝑚^+𝑗,𝑛^−𝑗,𝑛^+𝑗,𝑢^−𝑖,𝑢^+𝑖,𝑣^−𝑖,𝑣^+𝑖,𝑤^−𝑖, and 𝑤^+𝑖 are some constants.

Remark 2.1. Assumption 2 was first introduced by Liu et al. [25]. The constants 𝑙^−𝑗,𝑙^+𝑗,𝑚^−𝑗,𝑚^+𝑗,𝑛^−𝑗,𝑛^+𝑗,𝑢^−𝑖,𝑢^+𝑖,𝑣^−𝑖,𝑣^+𝑖,𝑤^−𝑖, and 𝑤^+𝑖 in Assumption 2 are allowed to be positive, negative, or zero. So, the activation functions used in this paper may be nonmonotonic and more general than the usual sigmoid functions and Lipschitz functions. Such conditions are very rude in quantifying the lower and upper bounds of the activation functions; hence we use generalized activation functions, because it is very helpful for using LMI-based technique to reduce the possible conservatism.
In order to simplify our proof, we shift the equilibrium point 𝑥^∗=(𝑥^∗1,𝑥^∗2,…,𝑥^∗𝑛)^𝑇 and 𝑦^∗=(𝑦^∗1,𝑦^∗2,…,𝑦^∗𝑛)^𝑇 of system (2.2) to the origin. Let 𝑢(𝑘)=𝑥(𝑘)−𝑥^∗ and 𝑣(𝑘)=𝑦(𝑘)−𝑦^∗; then system (2.2) can be transformed to𝑢(𝑘+1)=𝐴𝑢(𝑘)+𝐶𝑓(𝑣(𝑘))+𝑊𝑔(𝑣(𝑘−𝜏(𝑘)))+𝑀^+∞_ℳ₌₁+𝛿(𝑢(𝑘),𝑣(𝑘−𝜏(𝑘)),𝑘)𝑤₁𝜇_ℳℎ(𝑣(𝑘−ℳ))𝑣(𝑘+1)=𝐵𝑣(𝑘)+𝐷𝑓(𝑢(𝑘))+𝑉𝑔̂(𝑢((𝑘𝑘)−,𝜎(𝑘)))+𝑁_+∞𝒩₌₁+𝜒(𝑣(𝑘),𝑢(𝑘−𝜎(𝑘)),𝑘)𝑤₂𝜌_𝒩ℎ(𝑢(𝑘−𝒩))(𝑘),(2.7) where𝑓(𝑣𝑢(𝑘(𝑘)))==𝑓𝑢₁₁(𝑘),𝑢₂(𝑘),…,𝑢_𝑛(𝑘)^𝑇,𝑣(𝑘)=𝑣₁(𝑘),𝑣₂(𝑘),…,𝑣_𝑛(𝑘)^𝑇,𝑔(𝑣(𝑘))=𝑔₁𝑣₁(𝑘),𝑓₂𝑣₂(𝑘),…,𝑓_𝑛𝑣_𝑛(𝑘)^𝑇=𝑓𝑦(𝑘)+𝑦^∗−𝑓𝑦^∗,ℎ(𝑣(𝑘))=ℎ₁𝑣₁(𝑘),𝑔₂𝑣₂(𝑘),…,𝑔_𝑛𝑣_𝑛(𝑘)^𝑇=𝑔𝑦(𝑘)+𝑦^∗−𝑔𝑦^∗,𝑓(𝑢(𝑘))=𝑓₁𝑣₁(𝑘),ℎ₂𝑣₂(𝑘),…,ℎ_𝑛𝑣_𝑛(𝑘)^𝑇=ℎ𝑦(𝑘)+𝑦^∗−ℎ𝑦^∗,𝑔̂(𝑢(𝑘))=𝑔̂₁𝑢₁(𝑘),𝑓₂𝑢₂(𝑘),…,𝑓_𝑛𝑢_𝑛(𝑘)^𝑇=𝑓𝑥(𝑘)+𝑥^∗−𝑓𝑥^∗,ℎ(𝑢(𝑘))=ℎ₁𝑢₁(𝑘),𝑔̂₂𝑢₂(𝑘),…,𝑔̂_𝑛𝑢_𝑛(𝑘)^𝑇=𝑔̂𝑥(𝑘)+𝑥^∗−𝑔̂𝑥^∗,𝑢₁(𝑘),ℎ₂𝑢₂(𝑘),…,ℎ_𝑛𝑢_𝑛(𝑘)^𝑇=ℎ𝑥(𝑘)+𝑥^∗−ℎ𝑥^∗.(2.8)

Assumption 3. Obviously, the activation functions 𝑓_𝑗,𝑓_𝑖,𝑔_𝑗,𝑔̂_𝑖,ℎ_𝑗,andℎ_𝑖(𝑖,𝑗∈ℕ) satisfy the following condition: l^−𝑗≤𝑓^𝑗(𝑠)𝑠≤𝑙^+𝑗,∀𝑠∈ℝ,𝑢^−𝑖≤𝑓^𝑖(𝑡)𝑡≤𝑢^+𝑖𝑚,∀𝑡∈ℝ,_−𝑗≤𝑔^𝑗(𝑠)𝑠≤𝑚^+𝑗,∀𝑠∈ℝ,𝑣^−𝑖≤𝑔̂^𝑖(𝑡)𝑡≤𝑣^+𝑖𝑛,∀𝑡∈ℝ,_−𝑗≤ℎ^𝑗(𝑠)𝑠≤𝑛^+𝑗,∀𝑠∈ℝ,𝑤^−𝑖≤ℎ^𝑖(𝑡)𝑡≤𝑤^+𝑖,∀𝑡∈ℝ.(2.9)

Assumption 4. The constants 𝜇_ℳ,𝜌_𝒩≥0 satisfy the following convergent conditions: ⁺^∞_ℳ=1𝜇_ℳ<+∞,^+∞_ℳ₌₁ℳ𝜇_ℳ<+∞,_+∞𝒩₌₁𝜌_𝒩<+∞,^+∞_𝒩₌₁𝒩𝜌_𝒩<+∞.(2.10)

Remark 2.2. Assumption 4 ensures that the terms 𝑀∑^+∞ℳ=1𝜇_ℳℎ(𝑣(𝑘−ℳ)) and 𝑁∑^+∞𝒩=1𝜌_𝒩ℎ(𝑢(𝑘−𝒩)) are convergent, which is significant for the subsequent analysis.

Assumption 5. There exist constant matrices 𝐺 and 𝐾 such that 𝜒𝛿^𝑇_𝑇(𝑥,𝑦,𝑘)𝛿(𝑥,𝑦,𝑘)≤||𝐺𝑥||²,∀𝑥,𝑦∈ℝ^𝑛,(𝑥,𝑦,𝑘)𝜒(𝑥,𝑦,𝑘)≤||𝐾𝑦||²,∀𝑥,𝑦∈ℝ^𝑛.(2.11)

The following definition and lemmas will be essential in employing the exponential stability conditions.

Definition 2.3. The delayed discrete-time stochastic BAM neural network (2.7) is said to be globally exponentially stable, if there exist two positive scalars 𝜈>0 and 0<𝒢<1 such that ‖𝑢(𝑘)‖+‖𝑣(𝑘)‖≤𝜈𝒢^𝑘sup_{−𝜎𝑀≤𝑠≤0}‖𝑢(𝑠)‖+sup_{−𝜏𝑀≤𝑠≤0}‖𝑣(𝑠)‖.(2.12)

Lemma 2.4. Let 𝑋 and 𝑌 be any n-dimensional real vectors and let 𝑃 be an 𝑛×𝑛 positive semidefinite matrix. Then, the following matrix inequality holds: 2𝑋^𝑇𝑃𝑌≤𝑋^𝑇𝑃𝑋+𝑌^𝑇𝑃𝑌.(2.13)

Lemma 2.5. Let 𝑀∈ℝ^𝑛×𝑛 be a positive semidefinite matrix, 𝑥_𝑖∈ℝ^𝑛, and 𝑎_𝑖≥0,(𝑖=1,2,…). If the series concerned are convergent, the following inequality holds: ^+∞_𝑖=1𝑎_𝑖𝑥_𝑖^𝑇𝑀^+∞_𝑖=1𝑎_𝑖𝑥_𝑖≤^+∞_𝑖=1𝑎_𝑖^+∞_𝑖=1𝑎_𝑖𝑥_𝑖𝑇𝑀𝑥_𝑖.(2.14)

In the rest of the paper, we will focus on the stability analysis of SBAMNN (2.7). By choosing an appropriate Lyapunov-Krasovskii functional, we aim to develop an LMI approach for deriving sufficient conditions under which the SBAMNN (2.7) is globally exponentially stable.

3. Main Results

Now, we are in a position to state our main results in the following theorem.

Theorem 3.1. Under Assumptions 1–5, the discrete-time stochastic BAM neural network (2.7) is globally exponentially stable in the mean square, if there exist constants 𝜆₀>0 and 𝜖₀>0 if there exist diagonal matrices Λ₁=diag{𝜆¹⁽¹⁾,𝜆²⁽¹⁾,…,𝜆^𝑛(1)}>0, Λ₂=diag{𝜆¹⁽²⁾,𝜆²⁽²⁾,…,𝜆^𝑛(2)}>0, Γ₁=diag{𝛾¹⁽¹⁾,𝛾²⁽¹⁾,…,𝛾^𝑛(1)}>0, Γ₂=diag{𝛾¹⁽²⁾,𝛾²⁽²⁾,…,𝛾^𝑛(2)}>0, Ω₁=diag{𝜔¹⁽¹⁾,𝜔²⁽¹⁾,…,𝜔^𝑛(1)}>0, and Ω₂=diag{𝜔¹⁽²⁾,𝜔²⁽²⁾,…,𝜔^𝑛(2)}>0 and positive definite matrices 𝑃₁>0,𝑃₂>0,𝑄₁>0,𝑄₂>0,𝑅₁>0,and𝑅₂>0, such that the following LMIs hold: Ξ𝑃₁<𝜆₀𝐼,₁=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣Π¹¹∗−0𝑄₂Λ₂𝑈₂Γ₂𝑉₂0Ω₂𝑊₂0∗∗Π0₃₃0000∗∗∗−0Γ₂000∗∗∗∗Π0₅₅00∗∗∗∗∗−0Ω₂0∗∗∗∗∗∗Π0₇₇𝑃⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,₂Ξ<𝜖₀𝐼,₂=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣Θ¹¹∗−0𝑄₁Λ₁𝐿₂Γ₁𝑀₂0Ω₁𝑁₂0∗∗Θ0₃₃0000∗∗∗−0Γ₁000∗∗∗∗Θ0₅₅00∗∗∗∗∗−0Ω₁0∗∗∗∗∗∗Θ0₇₇⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.1) where ΠΠ₁₁=𝐴^𝑇𝑃₁𝐴−2𝑃₁−Λ₂𝑈₁−Γ₂𝑉₁−Ω₂𝑊₁+1+𝜎_𝑀−𝜎_𝑚𝑄₂+𝜌𝑅₂+𝜆₀𝐺^𝑇𝐺,₃₃Π₇₇=𝐵𝐷𝑃₂𝐷^𝑇𝐵^𝑇−Λ₂+2𝑃₂,Π₅₅=𝐵𝑉𝑃₂𝑉^𝑇𝐵^𝑇+𝐷𝑉𝑃₂𝑉^𝑇𝐷^𝑇+𝑃₂,Θ₁₁=𝐵𝑁𝑃₂𝑁^𝑇𝐵^𝑇+𝐷𝑁𝑃₂𝑁^𝑇𝐷^𝑇+𝑉𝑁𝑃₂𝑁^𝑇𝑉^𝑇−𝜌⁻¹𝑅₂,Θ=𝐵^𝑇𝑃₂𝐵−2𝑃₂−Λ₁𝐿₁−Γ₁𝑀₁−Ω₁𝑁₁+1+𝜏_𝑀−𝜏_𝑚𝑄₁+𝜇𝑅₁+𝜖₀𝐾^𝑇𝐾,₃₃Θ₇₇=𝐴𝐶𝑃₁𝐶^𝑇𝐴^𝑇−Λ₁+2𝑃₁,Θ₅₅=𝐴𝑊𝑃₁𝑊^𝑇𝐴^𝑇+𝐶𝑊𝑃₁𝑊^𝑇𝐶^𝑇+𝑃₁,𝐿₁=𝐴𝑀𝑃₁𝑀^𝑇𝐴^𝑇+𝐶𝑀𝑃₁𝑀^𝑇𝐶^𝑇+𝑊𝑀𝑃₁𝑀^𝑇𝑊^𝑇−𝜇⁻¹𝑅₁,=diag𝑙⁺¹𝑙⁻¹,𝑙⁺²𝑙⁻²,…,𝑙^+𝑛𝑙^−𝑛,𝐿₂=diag𝑙⁺¹+𝑙⁻¹2,𝑙⁺²+𝑙⁻²2,…,𝑙^+𝑛+𝑙^−𝑛𝑀₁2,=diag𝑚⁺¹𝑚⁻¹,𝑚⁺²𝑚⁻²,…,𝑚^+𝑛𝑚^−𝑛,𝑀₂=diag𝑚⁺¹+𝑚⁻¹2,𝑚⁺²+𝑚⁻²2,…,𝑚^+𝑛+𝑚^−𝑛𝑁₁2,=diag𝑛⁺¹𝑛⁻¹,𝑛⁺²𝑛⁻²,…,𝑛^+𝑛𝑛^−𝑛,𝑁₂=diag𝑛⁺¹+𝑛⁻¹2,𝑛⁺²+𝑛⁻²2,…,𝑛^+𝑛+𝑛^−𝑛𝑈₁2,=diag𝑢⁺¹𝑢⁻¹,u⁺²𝑢⁻²,…,𝑢^+𝑛𝑢^−𝑛,𝑈₂=diag𝑢⁺¹+𝑢⁻¹2,𝑢⁺²+𝑢⁻²2,…,𝑢^+𝑛+𝑢^−𝑛𝑉₁2,=diag𝑣⁺¹𝑣⁻¹,𝑣⁺²𝑣⁻²,…,𝑣^+𝑛𝑣^−𝑛,𝑉₂=diag𝑣⁺¹+𝑣⁻¹2,𝑣⁺²+𝑣⁻²2,…,𝑣^+𝑛+𝑣^−𝑛𝑊₁2,=diag𝑤⁺¹𝑤⁻¹,𝑤⁺²𝑤⁻²,…,𝑤^+𝑛𝑤^−𝑛,𝑊₂=diag𝑤⁺¹+𝑤⁻¹2,𝑤⁺²+w⁻²2,…,𝑤^+𝑛+𝑤^−𝑛2,𝜇=^+∞_𝑘=1𝜇_𝑘,𝜌=^+∞_𝑘=1𝜌_𝑘.(3.2)

Proof. Let us choose the Lyapunov-Krasovskii functional as 𝑉𝑉₁(𝑘)=𝑢^𝑇(𝑘)𝑃₁𝑢(𝑘)+𝑣^𝑇(𝑘)𝑃₂𝑣(𝑘),₂(𝑘)=^𝑘−1_𝑖=𝑘_{−𝜏(𝑘)}𝑣^𝑇(𝑖)𝑄₁𝑣(𝑖)+^𝑘−1_𝑖=𝑘_{−𝜎(𝑘)}𝑉𝑢^𝑇(𝑖)𝑄₂𝑢(𝑖),₃(𝑘)=^{𝑘−𝜏𝑚}_{𝑗=𝑘−𝜏𝑀+1𝑘−1𝑖=𝑗}𝑣^𝑇(𝑖)𝑄₁𝑣(𝑖)+^{𝑘−𝜎𝑚}_{𝑗=𝑘−𝜎𝑀+1𝑘−1𝑖=𝑗}𝑉𝑢^𝑇(𝑖)𝑄₂𝑢(𝑖),₄(𝑘)=^+∞_𝑖=1𝜇_{𝑖𝑘−1𝑗}_=𝑘−𝑖𝑣^𝑇(𝑖)𝑅₁𝑣(𝑖)+^+∞_𝑖=1𝜌_{𝑖𝑘−1𝑗}_=𝑘−𝑖𝑢^𝑇(𝑖)𝑅₂𝑢(𝑖).(3.3)
In order to analyze the global exponential stability of the stochastic BAM neural network, we calculate differences Δ𝑉(𝑘) of the Lyapunov function 𝑉(𝑘), along with the trajectories of the BAM neural network (2.7); then we haveΔ𝑉(𝑘)=Δ𝑉₁(𝑘)+Δ𝑉₂(𝑘)+Δ𝑉₃(𝑘)+Δ𝑉₄(𝑘),(3.4) where Δ𝑉₁(𝑘)==𝑢^𝑇𝐴(𝑢𝑘(𝑘+)1+)𝑃𝐶₁𝑓𝑢((𝑣𝑘(+𝑘)1))+−𝑊𝑢^𝑇𝑔((𝑘𝑣)(𝑃𝑘₁−𝑢(𝜏𝑘()𝑘+)))𝑣+^𝑇(𝑀𝑘+_+∞1)𝑃₂𝑣(𝑘+1)−𝑣^𝑇(𝑘)𝑃₂𝑣(𝑘)_ℳ₌₁𝜇_ℳℎ(𝑣(𝑘−ℳ))^𝑇×𝑃₁𝐴𝑢(𝑘)+𝐶𝑓(𝑣(𝑘))+𝑊𝑔(𝑣(𝑘−𝜏(𝑘)))+𝑀^+∞_ℳ=1−𝑢_𝑇𝜇_ℳℎ(𝑣(𝑘−ℳ))+𝐵(𝑣𝑘()𝑘𝑃)₁+𝑢(𝐷𝑘𝑓)(𝑢(𝑘))+𝑉𝑔̂(𝑢(𝑘−𝜎(𝑘)))+𝑁_+∞𝒩₌₁𝜌_𝒩ℎ(𝑢(𝑘−𝒩))^𝑇×𝑃₂𝐵𝑣(𝑘)+𝐷𝑓(𝑢(𝑘))+𝑉𝑔̂(𝑢(𝑘−𝜎(𝑘)))+𝑁^+∞_𝒩=1−𝑣_𝑇𝜌_𝒩ℎ(𝑢(𝑘−𝒩))≤𝑢_𝑇(𝑘)𝑃₂𝑣(𝑘)+𝛿^𝑇(𝑢,𝑣,𝑘)𝑃₁𝛿(𝑢,𝑣,𝑘)+𝜒^𝑇(𝑢,𝑣,𝑘)𝑃₂𝜒(𝑢,𝑣,𝑘)+2(𝑢𝑘_𝑇)𝐴𝑃₁𝐴−𝑃₁𝑢(𝑘)+2𝑢^𝑇(𝑘)𝐴𝑃₁𝐶𝑓(𝑣(𝑘))+2𝑢^𝑇(𝑘)𝐴𝑃₁𝑊𝑔(𝑣(𝑘−𝜏(𝑘)))(𝑘)𝐴𝑃₁𝑀^+∞_ℳ₌₁×𝐶𝑃₁𝜇_ℳℎ(𝑣(𝑘−ℳ))+𝑓^𝑇(𝑣(𝑘))𝐶𝑃₁𝐶𝑓(𝑣(𝑘))+2𝑓^𝑇(𝑣(𝑘))𝑊𝑔(𝑣(𝑘−𝜏(𝑘)))+2𝑓^𝑇(𝑣(𝑘))𝐶𝑃₁𝑀^+∞_ℳ₌₁+𝑔_𝑇𝜇_ℳℎ(𝑣(𝑘−ℳ))×(𝑣_+∞(𝑘−𝜏(𝑘)))𝑊𝑃₁𝑊𝑔(𝑣(𝑘−𝜏(𝑘)))+2𝑔^𝑇(𝑣(𝑘−𝜏(𝑘)))𝑊𝑃₁𝑀_ℳ₌₁𝜇_ℳℎ(𝑣(𝑘−ℳ))+^+∞_ℳ₌₁×_+∞𝜇_ℳℎ(𝑣(𝑘−ℳ))^𝑇𝑀𝑃₁𝑀_ℳ₌₁+2𝑣_𝑇𝜇_ℳℎ(𝑣(𝑘−ℳ))+𝑣^𝑇(𝑘)𝐵𝑃₂𝐵−𝑃₂𝑣(𝑘)+2𝑣^𝑇(𝑘)𝐵𝑃₂𝐷𝑓(𝑢(𝑘))(𝑘)𝐵𝑃₂𝑉𝑓(𝑢(𝑘−𝜎(𝑘)))+2𝑣^𝑇(𝑘)𝐵𝑃₂𝑁^+∞_𝒩₌₁+𝑓𝜌_𝒩ℎ(𝑢(𝑘−𝒩))_𝑇×(_+∞𝑢(𝑘))𝐷𝑃₂𝐷𝑓(𝑢(𝑘))+2𝑓^𝑇(𝑢(𝑘))𝐷𝑃₂𝑉𝑔̂(𝑢(𝑘−𝜎(𝑘)))+2𝑓^𝑇(𝑢(𝑘))𝐷𝑃₂𝑁_𝒩₌₁+2𝑔̂_𝑇𝜌_𝒩ℎ(𝑢(𝑘−𝒩))+𝑔̂^𝑇(𝑢(𝑘−𝜎(𝑘)))𝑉𝑃₂𝑉𝑔̂(𝑢(𝑘−𝜎(𝑘)))(𝑢(𝑘−𝜎(𝑘)))𝑉𝑃₂𝑁^+∞_𝒩₌₁𝜌_𝒩ℎ(𝑢(𝑘−𝒩))+^+∞_𝒩₌₁×𝑃₂𝜌_𝒩ℎ(𝑢(𝑘−𝒩))𝑁𝑁^+∞_𝒩₌₁𝜌_𝒩ℎ(𝑢(𝑘−𝒩))+𝑢^𝑇(𝑘)𝜆₀𝐺^𝑇𝐺𝑢(𝑘)+𝑣^𝑇(𝑘)𝜖₀𝐾^𝑇𝐾𝑣(𝑘).(3.5) By using Lemma 2.4, we have 2𝑢^𝑇≤(𝑘𝑢_𝑇)𝐴𝑃₁𝐶𝑓(𝑣(𝑘))2𝑢_𝑇(𝑘)𝑃₁𝑢(𝑘)+𝑓^𝑇(𝑣(𝑘))𝐴𝐶𝑃₁𝐶^𝑇𝐴^𝑇𝑓(𝑣(𝑘)),≤(𝑘𝑢_𝑇)𝐴𝑃₁𝑊𝑔(𝑣(𝑘−𝜏(𝑘)))2𝑢_𝑇(𝑘)𝑃₁𝑢(𝑘)+𝑔^𝑇(𝑣(𝑘−𝜏(𝑘)))𝐴𝑊𝑃₁𝑊^𝑇𝐴^𝑇𝑔(𝑣(𝑘−𝜏(𝑘))),(𝑘)𝐴𝑃₁𝑀^+∞_ℳ₌₁+𝜇_ℳℎ(𝑣(𝑘−ℳ))≤𝑢^𝑇(𝑘)𝑃₁𝑢(𝑘)_+∞ℳ∑₌₁𝜇_ℳℎ(𝑣(𝑘−ℳ))^𝑇𝐴𝑀𝑃₁𝑀^𝑇𝐴^𝑇^+∞2𝑓_𝑇∑_ℳ=1𝜇_ℳℎ(𝑣(𝑘−ℳ)),+(𝑣𝑔(𝑘_𝑇))𝐶𝑃₁𝑊𝑔(𝑣(𝑘−𝜏(𝑘)))≤𝑓^𝑇(𝑣(𝑘))𝑃₁𝑓(𝑣(𝑘))2𝑓_𝑇(𝑣(𝑘−𝜏(𝑘)))𝐶𝑊𝑃₁𝑊^𝑇𝐶^𝑇𝑔(𝑣(𝑘−𝜏(𝑘))),(𝑣(𝑘))𝐶𝑃₁𝑀^+∞_ℳ₌₁≤𝑓_𝑇𝜇_ℳℎ(𝑣(𝑘−ℳ))(𝑣(𝑘))𝑃₁𝑓(𝑣(𝑘))+^+∞_ℳ₌₁𝜇_ℳℎ(𝑣(𝑘−ℳ))^𝑇×𝐶𝑀𝑃₁𝑀^𝑇𝐶^𝑇^+∞_ℳ=12𝑔_𝑇𝜇_ℳℎ(𝑣(𝑘−ℳ)),(𝑣(𝑘−𝜏(𝑘)))𝑊𝑃₁𝑀^+∞_ℳ₌₁≤𝑔_𝑇𝜇_ℳℎ(𝑣(𝑘−ℳ))(𝑣(𝑘−𝜏(𝑘)))𝑃₁𝑔(𝑣(𝑘−𝜏(𝑘)))+^+∞_ℳ₌₁𝜇_ℳℎ(𝑣(𝑘−ℳ))^𝑇×𝑊𝑀𝑃₁𝑀^𝑇𝑊^𝑇^+∞_ℳ=12𝑣_𝑇𝜇_ℳℎ(𝑣(𝑘−ℳ)),2𝑣_𝑇(𝑘)𝐵𝑃₂𝐷𝑓(𝑢(𝑘))≤𝑣^𝑇(𝑘)𝑃₂𝑣(𝑘)+𝑓^𝑇(𝑢(𝑘))𝐵𝐷𝑃₂𝐷^𝑇𝐵^𝑇𝑓(𝑢(𝑘)),≤(𝑣𝑘_𝑇)𝐵𝑃₂𝑉𝑔̂(𝑢(𝑘−𝜎(𝑘)))2𝑣_𝑇(𝑘)𝑃₂𝑣(𝑘)+𝑔̂^𝑇(𝑢(𝑘−𝜎(𝑘)))𝐵𝑉𝑃₂𝑉^𝑇𝐵^𝑇𝑔̂(𝑢(𝑘−𝜎(𝑘))),(𝑘)𝐵𝑃₂𝑁^+∞_𝒩₌₁≤𝑣_𝑇𝜌_𝒩ℎ(𝑢(𝑘−𝒩))(𝑘)𝑃₂𝑣(𝑘)+^+∞_𝒩₌₁𝜌_𝒩ℎ(𝑢(𝑘−𝒩))^𝑇𝐵𝑁𝑃₂𝑁^𝑇𝐵^𝑇^+∞_𝒩=12𝑓𝜌_𝒩ℎ(𝑢(𝑘−𝒩))_𝑇,+(𝑢𝑔(𝑘̂_𝑇))𝐷𝑃₂𝑉𝑔̂(𝑢(𝑘−𝜎(𝑘)))≤𝑓^𝑇(𝑢(𝑘))𝑃₂𝑓(𝑢(𝑘))2𝑓(𝑢(𝑘−𝜎(𝑘)))𝐷𝑉𝑃₂𝑉^𝑇𝐷^𝑇𝑔̂(𝑢(𝑘−𝜎(𝑘))),_𝑇(𝑢(𝑘))𝐷𝑃₂𝑁^+∞_𝒩₌₁𝜌_𝒩ℎ(𝑢(𝑘−𝒩))≤𝑓^𝑇(𝑢(𝑘))𝑃₂𝑓(𝑣(𝑘))+^+∞_𝒩₌₁𝜌_𝒩ℎ(𝑢(𝑘−𝒩))^𝑇×𝐷𝑁𝑃₂𝑁^𝑇𝐷^𝑇^+∞_𝒩=12𝑔̂_𝑇𝜌_𝒩ℎ(𝑢(𝑘−𝒩)),(𝑢(𝑘−𝜎(𝑘)))𝑉𝑃₂𝑁^+∞_𝒩₌₁≤𝑔̂_𝑇𝜌_𝒩ℎ(𝑢(𝑘−𝒩))+(𝑢(𝑘−𝜎(𝑘)))𝑃₂𝑔̂(𝑢(𝑘−𝜎(𝑘)))_+∞𝒩₌₁𝜌_𝒩ℎ(𝑢(𝑘−𝒩))^𝑇𝑉𝑁𝑃₂𝑁^𝑇𝑉^𝑇^+∞_𝒩=1Δ𝑉₂𝜌_𝒩ℎ(𝑢(𝑘−𝒩)),=(𝑘)=𝑉₂(𝑘+1)−𝑉₂(𝑘)_{𝑘𝑖=𝑘+1}_{−𝜏(𝑘+1)}𝑣^𝑇(𝑖)𝑄₁𝑣(𝑖)−^𝑘−1_𝑖=𝑘_{−𝜏(𝑘)}+𝑣^𝑇(𝑖)𝑄₁𝑣(𝑖)_{𝑘𝑖=𝑘+1}_{−𝜎(𝑘+1)}𝑢^𝑇(𝑖)𝑄₂𝑢(𝑖)−^𝑘−1_𝑖=𝑘_{−𝜎(𝑘)}=𝑣_𝑇𝑢^𝑇(𝑖)𝑄₂𝑢(𝑖)(𝑘)𝑄₁𝑣(𝑘)−𝑣^𝑇(𝑘−𝜏(𝑘))𝑄₁𝑣(𝑘−𝜏(𝑘))+^𝑘−1_𝑖=𝑘+1_{−𝜏(𝑘+1)}−𝑣^𝑇(𝑖)𝑄₁𝑣(𝑖)_{𝑘−1𝑖=𝑘−}_𝜏(𝑘)+1+𝑣^𝑇(𝑖)𝑄₁𝑣(𝑖)+𝑢^𝑇(𝑘)𝑄₂𝑢(𝑘)−𝑢^𝑇(𝑘−𝜎(𝑘))𝑄₂𝑢(𝑘−𝜎(𝑘))_{𝑘−1𝑖=𝑘+1}_{−𝜎(𝑘+1)}𝑢^𝑇(𝑖)𝑄₂𝑢(𝑖)−^𝑘−1_𝑖=𝑘−_𝜎(𝑘)+1=𝑣_𝑇𝑢^𝑇(𝑖)𝑄₂𝑢(𝑖)(𝑘)𝑄₁𝑣(𝑘)−𝑣^𝑇(𝑘−𝜏(𝑘))𝑄₁𝑣(𝑘−𝜏(𝑘))+^{𝑘−𝜏𝑚}_{𝑖=𝑘−𝜏}_(𝑘+1)+1+𝑣^𝑇(𝑖)𝑄₁𝑣(𝑖)_{𝑘−1𝑖=𝑘}_{−𝜏𝑚+1}𝑣^𝑇(𝑖)𝑄₁𝑣(𝑖)−^𝑘−1_𝑖=𝑘−_𝜏(𝑘)+1+𝑢_𝑇𝑣^𝑇(𝑖)𝑄₁𝑣(𝑖)(𝑘)𝑄₂𝑢(𝑘)−𝑢^𝑇(𝑘−𝜎(𝑘))𝑄₂𝑢(𝑘−𝜎(𝑘))+^{𝑘−𝜎𝑚}_{𝑖=𝑘−𝜎}_(𝑘+1)+1+𝑢^𝑇(𝑖)𝑄₂𝑢(𝑖)_{𝑘−1𝑖=𝑘}_{−𝜎𝑚+1}𝑢^𝑇(𝑖)𝑄₂𝑢(𝑖)−^𝑘−1_𝑖=𝑘−_𝜎(𝑘)+1=𝑣_𝑇𝑢^𝑇(𝑖)𝑄₂𝑢(𝑖)(𝑘)𝑄₁𝑣(𝑘)−𝑣^𝑇(𝑘−𝜏(𝑘))𝑄₁𝑣(𝑘−𝜏(𝑘))+^{𝑘−𝜏𝑚}_𝑖=𝑘_{−𝜏𝑀+1}+𝑢_𝑇𝑣^𝑇(𝑖)𝑄₁𝑣(𝑖)(𝑘)𝑄₂𝑢(𝑘)−𝑢^𝑇(𝑘−𝜎(𝑘))𝑄₂𝑢(𝑘−𝜎(𝑘))+^{𝑘−𝜎𝑚}_𝑖=𝑘_{−𝜎𝑀+1}𝑢^𝑇(𝑖)𝑄₂𝑢(𝑖),(3.6)Δ𝑉₃(𝑘)==𝑉₃(𝑘+1)−𝑉₃(𝑘)_{𝑘−𝜏𝑚+1𝑗=𝑘}_{−𝜏𝑀𝑘+2}_𝑖=𝑗𝑣^𝑇(𝑖)𝑄₁𝑣(𝑖)−^{𝑘−𝜏𝑚}_𝑗=𝑘_{−𝜏𝑀𝑘+1}_𝑖=𝑗+𝑣^𝑇(𝑖)𝑄₁𝑣(𝑖)_{𝑘−𝜎𝑚+1𝑗=𝑘}_{−𝜎𝑀𝑘+2}_𝑖=𝑗𝑢^𝑇(𝑖)𝑄₂𝑢(𝑖)−^{𝑘−𝜎𝑚}_𝑗=𝑘_{−𝜎𝑀𝑘+1}_𝑖=𝑗=𝑢^𝑇(𝑖)𝑄₂𝑢(𝑖)_{𝑘−𝜏𝑚𝑗=𝑘}_{−𝜏𝑀𝑘+1}_𝑖=𝑗𝑣^𝑇(𝑖)𝑄₁𝑣(𝑖)−^{𝑘−𝜏𝑚}_{𝑗=𝑘−𝜏𝑀+1𝑘−1𝑖=𝑗}+𝑣^𝑇(𝑖)𝑄₁𝑣(𝑖)_{𝑘−𝜎𝑚𝑗=𝑘}_{−𝜎𝑀𝑘+1}_𝑖=𝑗𝑢^𝑇(𝑖)𝑄₂𝑢(𝑖)−^{𝑘−𝜎𝑚}_{𝑗=𝑘−𝜎𝑀+1𝑘−1𝑖=𝑗}=𝑢^𝑇(𝑖)𝑄₂𝑢(𝑖)_{𝑘−𝜏𝑚𝑗=𝑘}_{−𝜏𝑀+1}𝑣^𝑇(𝑘)𝑄₁𝑣(𝑘)−𝑣^𝑇(𝑗)𝑄₁𝑣(𝑗)+^{𝑘−𝜎𝑚}_𝑗=𝑘_{−𝜎𝑀+1}𝑢^𝑇≤𝜏(𝑘)𝑄₂𝑢(𝑘)−𝑢^𝑇(𝑗)𝑄₂𝑢(𝑗)_𝑀−𝜏_𝑚𝑣^𝑇(𝑘)𝑄₁𝑣(𝑘)−^{𝑘−𝜏𝑚}_𝑗=𝑘_{−𝜏𝑀+1}−𝑣^𝑇(𝑖)𝑄₁𝑣(𝑖)+𝜎_𝑀−𝜎_𝑚𝑢^𝑇(𝑘)𝑄₂𝑢(𝑘)_{𝑘−𝜎𝑚𝑗=𝑘}_{−𝜎𝑀+1}𝑢^𝑇(𝑖)𝑄₂𝑢(𝑖),(3.7)Δ𝑉₄(𝑘)==𝑉₄(𝑘+1)−𝑉₄(𝑘)_+∞_𝑖=1𝜇_{𝑖𝑘𝑗=}_{𝑘−𝑖+1}𝑣^𝑇(𝑖)𝑅₁𝑣(𝑖)−^+∞_𝑖=1𝜇_{𝑖𝑘−1𝑗}_=𝑘−i+𝑣^𝑇(𝑖)𝑅₁𝑣(𝑖)_+∞_𝑖=1𝜌_{𝑖𝑘𝑗=}_{𝑘−𝑖+1}𝑢^𝑇(𝑖)𝑅₂𝑢(𝑖)−^+∞_𝑖=1𝜌_{𝑖𝑘−1𝑗}_=𝑘−𝑖=𝑢^𝑇(𝑖)𝑅₂𝑢(𝑖)_+∞_𝑖=1𝜇_𝑖𝑣^𝑇(𝑘)𝑅₁𝑣(𝑘)−𝑣^𝑇(𝑘−𝑖)𝑅₁𝑣(𝑘−𝑖)+^+∞_𝑖=1≤𝜌_𝑖𝑢^𝑇(𝑘)𝑅₂𝑢(𝑘)−𝑢^𝑇(𝑘−𝑖)𝑅₂𝑢(𝑘−𝑖)𝜇𝑣^𝑇(𝑘)𝑅₁𝑣(𝑘)−1𝜇^+∞_ℳ=1𝜇_ℳℎ(𝑣(𝑘−ℳ))^𝑇𝑅₁^+∞_ℳ=1+𝜇_ℳℎ(𝑣(𝑘−ℳ))𝜌𝑢^𝑇(𝑘)𝑅₂𝑢(𝑘)−1𝜌^+∞_𝒩=1𝜌_𝒩ℎ(𝑢(𝑘−𝒩))^𝑇𝑅₂^+∞_𝒩=1𝜌_𝒩ℎ(𝑢(𝑘−𝒩)).(3.8) It is clear from (2.9) that 𝑔𝑓^𝑗𝑣_𝑗(𝑘)−𝑙^+𝑗𝑣_𝑗(𝑘)𝑓_𝑗𝑣_𝑗(𝑘)−𝑙^−𝑗𝑣_𝑗(𝑘)≤0,(3.9)_𝑗ℎ𝑣_𝑗(𝑘)−𝑚^+𝑗𝑣_𝑗(𝑘)𝑔_𝑗𝑣_𝑗(𝑘)−𝑚^−𝑗𝑣_𝑗(𝑘)≤0,(3.10)_𝑗𝑓𝑣_𝑗(𝑘)−𝑛^+𝑗𝑣_𝑗(𝑘)ℎ_𝑗𝑣_𝑗(𝑘)−𝑛^−𝑗𝑣_𝑗(𝑘)≤0,(3.11)_𝑖𝑔̂_𝑖𝑢_𝑖(𝑘)−𝑢^+𝑖𝑢_𝑖(𝑘)𝑓_𝑖𝑢_𝑖(𝑘)−𝑢^−𝑖𝑢_𝑖(𝑘)≤0,(3.12)ℎ𝑢_𝑖(𝑘)−𝑣^+𝑖𝑢_𝑖(𝑘)𝑔̂_𝑖𝑢_𝑖(𝑘)−𝑣^−𝑖𝑢_𝑖(𝑘)≤0,(3.13)_𝑖𝑢_𝑖(𝑘)−𝑤^+𝑖𝑢_𝑖(𝑘)ℎ_𝑖𝑢_𝑖(𝑘)−𝑤^−𝑖𝑢_𝑖(𝑘)≤0,(3.14) which is equivalent to ⎡⎢⎢⎣𝑓𝑣(𝑣(𝑘(𝑘)))⎤⎥⎥⎦^𝑇⎡⎢⎢⎢⎣𝑙_+𝑗𝑙^−𝑗𝑒_𝑗𝑒^𝑇𝑗−𝑙^+𝑗+𝑙^−𝑗2𝑒^𝑗𝑒^𝑇𝑗−𝑙^+𝑗+𝑙^−𝑗2𝑒^𝑗𝑒^𝑇𝑗𝑒_𝑗𝑒^𝑇𝑗⎤⎥⎥⎥⎦⎡⎢⎢⎣𝑓𝑣(𝑣(𝑘(𝑘)))⎤⎥⎥⎦≤0,𝑘=1,2,…,𝑛,(3.15) where 𝑒_𝑘 denotes the unit column vector having “1” element on its 𝑘th row and zeros elsewhere.
Consequently,^𝑛_𝑗=1𝜆^𝑗(1)⎡⎢⎢⎣𝑓(𝑣𝑣((𝑘𝑘)))⎤⎥⎥⎦^𝑇⎡⎢⎢⎢⎣𝑙_+𝑗𝑙^−𝑗𝑒_𝑗𝑒^𝑇𝑗−𝑙^+𝑗+l^−𝑗2𝑒^𝑗𝑒^𝑇𝑗−𝑙^+𝑗+𝑙^−𝑗2𝑒^𝑗𝑒^𝑇𝑗𝑒_𝑗𝑒^𝑇𝑗⟹⎡⎢⎢⎣𝑓𝑣(𝑣(𝑘(𝑘)))⎤⎥⎥⎦_𝑇⎤⎥⎥⎥⎦⎡⎢⎢⎣𝑓𝑣(𝑣(𝑘(𝑘)))⎤⎥⎥⎦≤0,⎡⎢⎢⎣Λ₁𝐿₁−Λ₁𝐿₂−Λ₁𝐿₂Λ₁⎤⎥⎥⎦⎡⎢⎢⎣𝑓𝑣(𝑣(𝑘(𝑘)))⎤⎥⎥⎦≤0.(3.16) Similarly, from (3.10)–(3.14), we have ⎡⎢⎢⎣𝑔(𝑣𝑣(𝑘(𝑘)))⎤⎥⎥⎦^𝑇⎡⎢⎢⎣Γ₁𝑀₁−Γ₁𝑀₂−Γ₁𝑀₂Γ₁⎡⎢⎢⎣ℎ(𝑣𝑣(𝑘(𝑘)))⎤⎥⎥⎦_𝑇⎤⎥⎥⎦⎡⎢⎢⎣𝑔𝑣(𝑣(𝑘(𝑘)))⎤⎥⎥⎦≤0,(3.17)⎡⎢⎢⎣Ω₁𝑁₁−Ω₁𝑁₂−Ω₁𝑁₂Ω₁⎡⎢⎢⎣𝑓𝑢(𝑢(𝑘(𝑘)))⎤⎥⎥⎦_𝑇⎤⎥⎥⎦⎡⎢⎢⎣ℎ𝑣(𝑣(𝑘(𝑘)))⎤⎥⎥⎦≤0,(3.18)⎡⎢⎢⎣Λ₂𝑈₁−Λ₂𝑈₂−Λ₂𝑈₂Λ₂⎡⎢⎢⎣𝑔̂(𝑢𝑢(𝑘(𝑘)))⎤⎥⎥⎦_𝑇⎤⎥⎥⎦⎡⎢⎢⎣𝑓𝑢(𝑢(𝑘(𝑘)))⎤⎥⎥⎦≤0,(3.19)⎡⎢⎢⎣Γ₂𝑉₁−Γ₂𝑉₂−Γ₂𝑉₂Γ₂⎡⎢⎢⎣ℎ(𝑢𝑢(𝑘(𝑘)))⎤⎥⎥⎦_𝑇⎤⎥⎥⎦⎡⎢⎢⎣𝑔̂𝑢(𝑢(𝑘(𝑘)))⎤⎥⎥⎦≤0,(3.20)⎡⎢⎢⎣Ω₂𝑊₁−Ω₂𝑊₂−Ω₂𝑊₂Ω₂⎤⎥⎥⎦⎡⎢⎢⎣ℎ𝑢(𝑢(𝑘(𝑘)))⎤⎥⎥⎦≤0.(3.21) Then from (3.5)–(3.8) and (3.16)–(3.21), we obtain Δ𝑉(𝑘)≤+𝑢^𝑇𝑢_𝑇(𝑘)𝐴𝑃₁𝐴−2𝑃₁−Λ₂𝑈₁−Γ₂𝑉₁−Ω₂𝑊₁+1+𝜎_𝑀−𝜎_𝑚𝑄₂+𝜌𝑅₂+𝜆₀𝐺^𝑇𝐺𝑢(𝑘)×Ω₂(𝑘)Λ₂𝑈₂𝑓(𝑢(𝑘))+𝑢^𝑇(𝑘)Γ₂𝑉₂𝑔̂(𝑢(𝑘))+𝑢^𝑇(𝑘)×𝐵𝑊𝐷𝑃₂₂ℎ(𝑢(𝑘))−𝑢^𝑇(𝑘−𝜎(𝑘))𝑄₂𝑢(𝑘−𝜎(𝑘))+𝑓^𝑇(𝑢(𝑘))×𝐵𝑉𝑃₂𝐷^𝑇𝐵^𝑇−Λ₂+2𝑃₂𝑓(𝑢(𝑘))−𝑔̂^𝑇(𝑢(𝑘))Γ₂𝑔̂(𝑢(𝑘))+𝑔̂^𝑇(𝑢(𝑘−𝜎(𝑘)))+𝑉^𝑇𝐵^𝑇+𝐷𝑉𝑃₂𝑉^𝑇D^𝑇+𝑃₂𝑔̂(𝑢(𝑘−𝜎(𝑘)))−ℎ^𝑇(𝑢(𝑘))Ω₂ℎ(𝑢(𝑘))_+∞𝒩₌₁×_+∞𝜌_𝒩ℎ(𝑢(𝑘−𝒩))^𝑇𝐵𝑁𝑃₂𝑁^𝑇𝐵^𝑇+𝐷𝑁𝑃₂𝑁^𝑇𝐷^𝑇+𝑉𝑁𝑃₂𝑁^𝑇𝑉^𝑇−𝜌⁻¹𝑅₂_𝒩₌₁×𝐵𝑃₂𝜌_𝒩ℎ(𝑢(𝑘−𝒩))+𝑣^𝑇(𝑘)+𝑣_𝑇𝐵−2𝑃₂−Λ₁𝐿₁−Γ₁𝑀₁−Ω₁𝑁₁+1+𝜏_𝑀−𝜏_𝑚𝑄₁+𝜇𝑅₁+𝜖₀𝐾^𝑇𝐾−𝑣_𝑇(𝑘)Λ₁𝐿₂𝑓(𝑣(𝑘))+𝑣^𝑇(𝑘)Γ₁𝑀₂𝑔(𝑣(𝑘))+𝑣^𝑇(𝑘)Ω₁𝑁₂ℎ(𝑣(𝑘))−𝑔_𝑇(𝑘−𝜏(𝑘))𝑄₁𝑣(𝑘−𝜏(𝑘))+𝑓^𝑇(𝑣(𝑘))𝐴𝐶𝑃₁𝐶^𝑇𝐴^𝑇−Λ₁+2𝑃₁𝑓(𝑣(𝑘))×𝑔(𝑣(𝑣(𝑘(𝑘−))𝜏Γ(₁𝑘𝑔))()𝑣−(𝑘ℎ))_𝑇+𝑔^𝑇(𝑣(𝑘−𝜏(𝑘)))𝐴𝑊𝑃₁𝑊^𝑇𝐴^𝑇+𝐶𝑊𝑃₁𝑊^𝑇𝐶^𝑇+𝑃₁(𝑣(𝑘))Ω₁ℎ(𝑣(𝑘))+^+∞_ℳ₌₁𝜇_ℳℎ(𝑣(𝑘−ℳ))^𝑇××𝐴𝑀_+∞𝑃₁𝑀^𝑇𝐴^𝑇+𝐶𝑀𝑃₁𝑀^𝑇𝐶^𝑇+𝑊𝑀𝑃₁𝑀^𝑇𝑊^𝑇−𝜇⁻¹𝑅₁_ℳ₌₁𝜇_ℳℎ(𝑣(𝑘−ℳ))−⎡⎢⎢⎣𝑓𝑣(𝑣(𝑘(𝑘)))⎤⎥⎥⎦^𝑇⎡⎢⎢⎣Λ₁𝐿₁−Λ₁𝐿₂−Λ₁𝐿₂Λ₁−⎡⎢⎢⎣𝑔(𝑣𝑣(𝑘(𝑘)))⎤⎥⎥⎦_𝑇⎤⎥⎥⎦⎡⎢⎢⎣𝑓𝑣(𝑣(𝑘(𝑘)))⎤⎥⎥⎦⎡⎢⎢⎣Γ₁𝑀₁−Γ₁𝑀₂−Γ₁𝑀₂Γ₁−⎡⎢⎢⎣ℎ(𝑣𝑣(𝑘(𝑘)))⎤⎥⎥⎦_𝑇⎤⎥⎥⎦⎡⎢⎢⎣𝑔𝑣(𝑣(𝑘(𝑘)))⎤⎥⎥⎦⎡⎢⎢⎣Ω₁𝑁₁−Ω₁𝑁₂−Ω₁𝑁₂Ω₁−⎡⎢⎢⎣𝑓𝑢(𝑢(𝑘(𝑘)))⎤⎥⎥⎦_𝑇⎤⎥⎥⎦⎡⎢⎢⎣ℎ𝑣(𝑣(𝑘(𝑘)))⎤⎥⎥⎦⎡⎢⎢⎣Λ₂𝑈₁−Λ₂𝑈₂−Λ₂𝑈₂Λ₂−⎡⎢⎢⎣𝑔̂(𝑢𝑢(𝑘(𝑘)))⎤⎥⎥⎦_𝑇⎤⎥⎥⎦⎡⎢⎢⎣𝑓𝑢(𝑢(𝑘(𝑘)))⎤⎥⎥⎦⎡⎢⎢⎣Γ₂𝑉₁−Γ₂𝑉₂−Γ₂𝑉₂Γ₂−⎡⎢⎢⎣ℎ(𝑢𝑢(𝑘(𝑘)))⎤⎥⎥⎦_𝑇⎤⎥⎥⎦⎡⎢⎢⎣𝑔̂𝑢(𝑢(𝑘(𝑘)))⎤⎥⎥⎦⎡⎢⎢⎣Ω₂𝑊₁−Ω₂𝑊₂−Ω₂𝑊₂Ω₂=𝜉_𝑇⎤⎥⎥⎦⎡⎢⎢⎣ℎ𝑢(𝑢(𝑘(𝑘)))⎤⎥⎥⎦(𝑘)Ξ₁𝜉(𝑘)+𝜂^𝑇(𝑘)Ξ₂𝜂(𝑘),(3.22) where 𝜉[𝑣^𝑇_𝑇(𝑘)=[𝑢^𝑇(𝑘)𝑢^𝑇(𝑘−𝜎(𝑘))𝑓^𝑇(𝑢(𝑘))𝑔̂^𝑇(𝑢(𝑘))𝑔̂^𝑇(𝑢(𝑘−𝜎(𝑘)))ℎ^𝑇(𝑢(𝑘))(∑^+∞𝒩=1𝜌_𝒩ℎ(𝑢(𝑘−𝒩)))^𝑇]𝜂^𝑇(𝑘)=(𝑘)𝑣^𝑇(𝑘−𝜏(𝑘))𝑓^𝑇(𝑣(𝑘))𝑔^𝑇(𝑣(𝑘))𝑔^𝑇(𝑣(𝑘−𝜏(𝑘)))ℎ^𝑇(𝑣(𝑘))(∑^+∞ℳ=1𝜇_ℳℎ(𝑣(𝑘−ℳ)))^𝑇].
Therefore, if the LMIs (3.1) hold, it can be concluded that Δ𝑉(𝑘)≤0. It follows that 𝑉(𝑘)≤𝑉(0). By (3.22), the SBAMNN is globally asymptotically stable in the mean square.
Now, we are in a position to establish the exponential stability of the SBAMNN (2.7).
Then, there exists a scalar 𝛽>0 such thatΔ𝑉(𝑘)≤−𝛽‖𝑢(𝑘)‖²+‖𝑣(𝑘)‖².(3.23) From (3.3), it can be verified that 𝑉(𝑘)≤𝜆_max𝑃₁‖𝑢(𝑘)‖²+𝜆_max𝑃₂‖𝑣(𝑘)‖²+𝜆_max𝑄₁^𝑘−1_𝑖=_{𝑘−𝜏𝑀}‖𝑣(𝑖)‖²+𝜆_max𝑄₂^𝑘−1_𝑖=_{𝑘−𝜎𝑀}‖𝑢(𝑖)‖²+𝜏_𝑀−𝜏_𝑚𝜆_max𝑄₁^𝑘−1_𝑖=_{𝑘−𝜏𝑀}‖𝑣(𝑖)‖²+𝜎_𝑀−𝜎_𝑚𝜆_max𝑄₂^𝑘−1_𝑖=_{𝑘−𝜎𝑀}‖𝑣(𝑖)‖²=𝜆_max𝑃₁‖𝑢(𝑘)‖²+𝜆_max𝑃₂‖𝑣(𝑘)‖²+𝛽_{1𝑘−1𝑖=}_{𝑘−𝜎𝑀}‖𝑢(𝑖)‖²+𝛽_{2𝑘−1𝑖=}_{𝑘−𝜏𝑀}‖𝑣(𝑖)‖²,(3.24) where 𝛽₁=(1+𝜎_𝑀−𝜎_𝑚)𝜆_max(𝑄₁) and 𝛽₂=(1+𝜏_𝑀−𝜏_𝑚)𝜆_max(𝑄₂).
Choose a scalar 𝜃>1, satisfying−𝛽𝜃+(𝜃−1)𝜆_max𝑃₁+𝜆_max𝑃₂+(𝜃−1)𝛽₁𝜏_𝑀𝜃^𝜏𝑀+𝛽₂𝜎_𝑀𝜃^𝜎𝑀=0.(3.25) Then by (3.23) and (3.24), we have 𝜃^𝑘+1𝑉(𝑘+1)−𝜃^𝑘𝑉(𝑘)==𝜃𝜃^𝑘+1_𝑘+1𝑉(𝑘+1)−𝜃^𝑘+1𝑉(𝑘)+𝜃^𝑘+1𝑉(𝑘)−𝜃^𝑘𝑉(𝑘)≤𝛽₃Δ𝑉(𝑘)+𝜃^𝑘(𝜃−1)𝑉(𝑘)𝜃^𝑘‖𝑢(𝑘)‖²+‖𝑣(𝑘)‖²+𝛽₄𝜃^𝑘⎛⎜⎜⎝^𝑘−1_𝑖=_{𝑘−𝜎𝑀}‖𝑢(𝑖)‖²+^𝑘−1_𝑖=_{𝑘−𝜏𝑀}‖𝑣(𝑖)‖²⎞⎟⎟⎠,(3.26) where 𝛽₃=−𝛽𝜃+(𝜃−1)(𝜆_max(𝑃₁)+𝜆_max(𝑃₂)) and 𝛽₄=(𝜃−1)(𝛽₁+𝛽₂).
Therefore, for any integer 𝑁≥𝜏_𝑀+1 and 𝑁≥𝜎_𝑀+1, summing up both sides of (3.26) from 0 to 𝑁−1 with respect to 𝑘, we have𝜃^𝑁𝑉(𝑁)−𝑉(0)≤𝛽_3𝑁−1_𝑘=0+𝛽𝜃^𝑘_4𝑁−1‖𝑢(𝑘)‖²+‖𝑣(𝑘)‖²_𝑘=0𝜃^𝑘⎛⎜⎜⎝^𝑘−1_𝑖=_{𝑘−𝜎𝑀}‖𝑢(𝑖)‖²+^𝑘−1_𝑖=_{𝑘−𝜏𝑀}‖𝑣(𝑖)‖²⎞⎟⎟⎠.(3.27) Here, we note that for 𝜏_𝑀≥1, 𝜎_𝑀≥1, we have ^𝑁⁻¹_𝑘=0𝜃^𝑘⎛⎜⎜⎝^𝑘−1_𝑖=_{𝑘−𝜎𝑀}‖𝑢(𝑖)‖²+^𝑘−1_𝑖=_{𝑘−𝜏𝑀}‖𝑣(𝑖)‖²⎞⎟⎟⎠=𝜎_𝑀𝜎_𝑀+1𝜃^𝜎𝑀sup_{−𝜎𝑀≤𝑖≤0}‖𝑢(𝑖)‖²+𝜏_𝑀𝜏_𝑀+1𝜃^𝜏𝑀×sup_{−𝜏𝑀≤𝑖≤0}‖𝑣(𝑖)‖²+𝜎_𝑀𝜃^{𝜎𝑀𝑁−1}_𝑘=0𝜃^𝑘‖𝑢(𝑘)‖²+𝜏_𝑀𝜃^{𝜏𝑀𝑁−1}_𝑘=0𝜃^𝑘‖𝑣(𝑘)‖².(3.28) Substituting (3.28) in (3.27) gives 𝜃^𝑁𝑉(𝑁)≤𝛽₃+𝛽₄𝜎_𝑀𝜃^𝜎𝑀+𝜏_𝑀𝜃^𝜏𝑀^𝑇−1_𝑘=0+𝛽₄𝜃^𝑘‖𝑢(𝑘)‖²+‖𝑣(𝑘)‖²𝜎_𝑀𝜎_𝑀+1𝜃^𝜎𝑀+𝜏_𝑀𝜏_𝑀+1𝜃^𝜏𝑀sup_{−𝜎𝑀≤𝑖≤0}‖𝑢(𝑖)‖²+sup_{−𝜏𝑀≤𝑖≤0}‖𝑣(𝑖)‖²+𝑉(0()3..29) We can observe that 𝑉(𝑁)≥𝜆_min𝑃₁,𝜆_min𝑃₂‖𝑢(𝑁)‖²+‖𝑣(𝑁)‖².(3.30) It follows easily from (3.24) that 𝑉(0)≤𝛽₁𝜎_𝑀sup_{−𝜎𝑀≤𝑖≤0}‖𝑢(𝑖)‖²+𝛽₂𝜏_𝑀sup_{−𝜏𝑀≤𝑖≤0}‖𝑣(𝑖)‖².(3.31) Then, it follows from (3.25), (3.29), and (3.31) that ‖𝑢(𝑁)‖+‖𝑣(𝑁)‖≤𝜈𝒢^𝑇sup_{−𝜎𝑀≤𝑖≤0}‖𝑢(𝑖)‖+sup_{−𝜏𝑀≤𝑖≤0}‖𝑣(𝑖)‖,(3.32) where 𝒢=𝜔^−1/2 and 𝜈=⎷𝛽₁𝜎_𝑀+𝛽₂𝜏_𝑀+𝛽₄𝜎_𝑀𝜎_𝑀+1𝜃^𝜎𝑀+𝜏_𝑀𝜏_𝑀+1𝜃^𝜏𝑀𝜆_min𝑃₁,𝜆_min𝑃₂.(3.33) This indicates that the discrete-time stochastic BAM neural network (2.7) is said to be globally exponentially stable. This completes the proof of this theorem.

For a deterministic BAM neural network, we have the following system of equations:𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐶𝑓(𝑦(𝑘))+𝑊𝑔(𝑦(𝑘−𝜏(𝑘)))+𝑀^+∞_ℳ₌₁𝑦(𝑘+1)=𝐵𝑦(𝑘)+𝐷𝑓(𝑥(𝑘))+𝑉𝑔̂(𝑥(𝑘−𝜎(𝑘)))+𝑁_+∞𝜇_ℳℎ(𝑦(𝑘−ℳ))+𝐼,_𝒩₌₁𝜌_𝒩ℎ(𝑥(𝑘−𝒩))+𝐽.(3.34)

Then, by Theorem 3.1, it is very easy to obtain the following theorem.

Theorem 3.2. Under Assumptions 1–4, the discrete-time BAM neural network (3.34) is globally exponentially stable, if there exist diagonal matrices Λ₁=diag{𝜆¹⁽¹⁾,𝜆²⁽¹⁾,…,𝜆^𝑛(1)}>0, Λ₂=diag{𝜆¹⁽²⁾,𝜆²⁽²⁾,…,𝜆^𝑛(2)}>0, Γ₁=diag{𝛾¹⁽¹⁾,𝛾²⁽¹⁾,…,𝛾^𝑛(1)}>0, Γ₂=diag{𝛾¹⁽²⁾,𝛾²⁽²⁾,…,𝛾^𝑛(2)}>0, Ω₁=diag{𝜔¹⁽¹⁾,𝜔²⁽¹⁾,…,𝜔^𝑛(1)}>0 and Ω₂=diag{𝜔¹⁽²⁾,𝜔²⁽²⁾,…,𝜔^𝑛(2)}>0 and positive definite matrices 𝑃₁>0,𝑃₂>0,𝑄₁>0,𝑄₂>0𝑅₁>0, and 𝑅₂>0, such that the following LMIs hold: Ξ₃=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣Ψ¹¹∗−0𝑄₂Λ₂𝑈₂Γ₂𝑉₂0Ω₂𝑊₂0∗∗Π0₃₃0000∗∗∗−0Γ₂000∗∗∗∗Π0₅₅00∗∗∗∗∗−0Ω₂0∗∗∗∗∗∗Π0₇₇Ξ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,₄=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣Φ¹¹∗−0𝑄₁Λ₁𝐿₂Γ₁𝑀₂0Ω₁𝑁₂0∗∗Θ0₃₃0000∗∗∗−0Γ₁000∗∗∗∗Θ0₅₅00∗∗∗∗∗−0Ω₁0∗∗∗∗∗∗Θ0₇₇⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.35) where ΦΨ₁₁₁₁=𝐴^𝑇𝑃₁𝐴−2𝑃₁−Λ₂𝑈₁−Γ₂𝑉₁−Ω₂𝑊₁+1+𝜎_𝑀−𝜎_𝑚𝑄₂+𝜌𝑅₂,=𝐵^𝑇𝑃₂𝐵−2𝑃₂−Λ₁𝐿₁−Γ₁𝑀₁−Ω₁𝑁₁+1+𝜏_𝑀−𝜏_𝑚𝑄₁+𝜇𝑅₁,(3.36) and Π₃₃,Π₅₅,Π₇₇,Θ₃₃,Θ₅₅,andΘ₇₇ are defined in Theorem 3.1.

Proof. Similar to the proof of Theorem 3.1, we can derive the stability result. The proof is straightforward and hence omitted.

If we neglect the distributed delay term in (2.2), it can be reduced to𝑥𝑦((𝑘𝑘++11))==[𝐴𝐵𝑥𝑦((𝑘𝑘))++𝐶𝐷𝑓𝑓((𝑦𝑥((𝑘𝑘))))++𝑊𝑉𝑔𝑔̂((𝑦𝑥((𝑘𝑘−−𝜏𝜎((𝑘𝑘))))))++𝐼𝐽]++𝛿𝜒(𝑥(𝑦(𝑘()𝑘,)𝑦,𝑥(𝑘(𝑘−−𝜏𝜎(𝑘()𝑘)),)𝑘,)𝑘𝑤)𝑤₁(₂𝑘),(𝑘).(3.37)

For system (3.37), we have the following stability result.

Corollary 3.3. Under Assumptions 1–5, the discrete-time BAM neural network (3.37) is globally exponentially stable, if there exist diagonal matrices Λ₁=diag{𝜆¹⁽¹⁾,𝜆²⁽¹⁾,…,𝜆^𝑛(1)}>0, Λ₂=diag{𝜆¹⁽²⁾,𝜆²⁽²⁾,…,𝜆^𝑛(2)}>0, Γ₁=diag{𝛾¹⁽¹⁾,𝛾²⁽¹⁾,…,𝛾^𝑛(1)}>0, and Γ₂=diag{𝛾¹⁽²⁾,𝛾²⁽²⁾,…,𝛾^𝑛(2)}>0, and positive definite matrices 𝑃₁>0,𝑃₂>0,𝑄₁>0,and 𝑄₂>0, such that the following LMI holds: Ξ₅=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣Υ¹¹∗−0𝑄₂Λ₂𝑈₂Γ₂𝑉₂0∗∗Π0₃₃00∗∗∗−0Γ₂0∗∗∗∗Π0₅₅Ξ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,₆=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣Σ¹¹∗−0𝑄₁Λ₁𝐿₂Γ₁𝑀₂0∗∗Θ0₃₃00∗∗∗−0Γ₁0∗∗∗∗Θ0₅₅⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.38) where ΥΣ₁₁₁₁=𝐴^𝑇𝑃₁𝐴−2𝑃₁−Λ₂𝑈₁−Γ₂𝑉₁+1+𝜎_𝑀−𝜎_𝑚𝑄₂,=𝐵^𝑇𝑃₂𝐵−2𝑃₂−Λ₁𝐿₁−Γ₁𝑀₁+1+𝜏_𝑀−𝜏_𝑚𝑄₁,(3.39) and Π₃₃,Π₅₅,Θ₃₃,andΘ₅₅ are defined in Theorem 3.1.

4. Numerical Example

To illustrate the effectiveness of our stability criterion, we give the following numerical example.

Example 4.1. Consider the SBAM neural networks (2.2) with the following parameters: 𝐷𝑀=𝐾𝐴=⎡⎢⎢⎢⎢⎣=−=⎡⎢⎢⎢⎢⎣0⎡⎢⎢⎢⎢⎣00−.002⎡⎢⎢⎢⎢⎣.02000.0.41.00.23−0000−00...000132.00.02..336.070000.002..00.−22⎤⎥⎥⎥⎥⎦40⎤⎥⎥⎥⎥⎦0⎤⎥⎥⎥⎥⎦0,.1,.,5⎤⎥⎥⎥⎥⎦,𝜏𝑊(𝐵𝑘=)𝑁==⎡⎢⎢⎢⎢⎣⎡⎢⎢⎢⎢⎣=−3000000..⎡⎢⎢⎢⎢⎣+33.−20s000.i−004n..00240..21.0002𝑘000...215𝜋−⎤⎥⎥⎥⎥⎦0−00,.006.05..43.⎤⎥⎥⎥⎥⎦3,⎤⎥⎥⎥⎥⎦𝐶,=𝑉⎡⎢⎢⎢⎢⎣−𝐺=000.=4.⎡⎢⎢⎢⎢⎣1−0⎡⎢⎢⎢⎢⎣00000.40.00...322200−0000..00.4103..23.1−0⎤⎥⎥⎥⎥⎦0000.0..3,32.⎤⎥⎥⎥⎥⎦2⎤⎥⎥⎥⎥⎦,,𝐼=−3sin𝑘𝜋2,𝜎(𝑘)=2−cos𝑘2𝜋,2,𝐽=2cos𝑘2𝜋,𝜇^𝑘𝑓(𝑦(𝑘))=𝑔(𝑦(𝑘))=ℎ(𝑦(𝑘))=⎡⎢⎢⎢⎢⎣tanh−4𝑦=₁𝜌_𝑘=𝑒^−4𝑘,tanh−4𝑦₂(𝑘)tanh−𝑦₃(𝑘)𝑓(𝑦(𝑘))=𝑔̂(𝑦(𝑘))=ℎ(𝑦(𝑘))=⎡⎢⎢⎢⎢⎣tanh−𝑥₁(𝑘)⎤⎥⎥⎥⎥⎦,tanh−4𝑥₂(𝑘)tanh−𝑥₃(𝑘)(𝑘)⎤⎥⎥⎥⎥⎦.(4.1) It can be verified that 𝜎_𝑚=𝜏_𝑚=3, 𝜎_𝑀=𝜏_𝑀=4, 𝑙⁺¹=𝑚⁺¹=𝑛⁺¹=2, 𝑙⁻¹=𝑚⁻¹=𝑛⁻¹=−2, 𝑙⁺²=𝑚⁺²=𝑛⁺²=2, 𝑙⁻²=𝑚⁻²=𝑛⁻²=−2, 𝑙⁺³=𝑚⁺³=𝑛⁺³=1, 𝑙⁻³=𝑚⁻³=𝑛⁻³=−1, 𝑢⁺¹=𝑣⁺¹=𝑤⁺¹=1, 𝑢⁻¹=𝑣⁻¹=𝑤⁻¹=−1, 𝑢⁺²=𝑣⁺²=𝑤⁺²=2, 𝑢⁻²=𝑣⁻²=𝑤⁻²=−2, 𝑢⁺³=𝑣⁺³=𝑤⁺³=1 and 𝑢⁻³=𝑣⁻³=𝑤⁻³=−1 with 𝐿𝑈₁=𝑀₁=𝑁₁=⎡⎢⎢⎢⎢⎣−004−004−001⎤⎥⎥⎥⎥⎦,𝐿₂=𝑀₂=𝑁₂=⎡⎢⎢⎢⎢⎣000000000₁⎤⎥⎥⎥⎥⎦,=𝑉₁=𝑊₁=⎡⎢⎢⎢⎢⎣−001−004−001⎤⎥⎥⎥⎥⎦,𝑈₂=𝑉₂=𝑊₂=⎡⎢⎢⎢⎢⎣000000000⎤⎥⎥⎥⎥⎦.(4.2) By using Matlab LMI toolbox, we solve the LMIs (3.1) in Theorem 3.1 and obtain the feasible solutions as follows: 𝑄𝑃₁=⎡⎢⎢⎢⎢⎣−−030.5..72145278076−−070.5..56155251075−−−090...206525861645⎤⎥⎥⎥⎥⎦,𝑃₂=⎡⎢⎢⎢⎢⎣−007..20.6517154244−−060.2..20500124206−−008.0..04102540467₁⎤⎥⎥⎥⎥⎦,𝑅=⎡⎢⎢⎢⎢⎣−200..80.2462183107−030..06.2647880002−−200.1..22727830872⎤⎥⎥⎥⎥⎦,𝑄₂=⎡⎢⎢⎢⎢⎣−200..55.6332784598−030..56.3337848931−−200.1..634279487₁81⎤⎥⎥⎥⎥⎦,Λ=⎡⎢⎢⎢⎢⎣101...862701360890021...623023610923112...231134001034⎤⎥⎥⎥⎥⎦,𝑅₂=⎡⎢⎢⎢⎢⎣−−200.2..57060313466−−000.8..47270173806−−201.2..42071377404₁⎤⎥⎥⎥⎥⎦,Γ=⎡⎢⎢⎢⎢⎣−6.500450−10.004004−16.006900⎤⎥⎥⎥⎥⎦,Λ₂=⎡⎢⎢⎢⎢⎣−12.000795−9.800357−15.005096₁⎤⎥⎥⎥⎥⎦,Ω=⎡⎢⎢⎢⎢⎣1.7000503.6007652.100403⎤⎥⎥⎥⎥⎦,Γ₂=⎡⎢⎢⎢⎢⎣2.3001763.2009882.000030₁⎤⎥⎥⎥⎥⎦,𝜆=⎡⎢⎢⎢⎢⎣1.7000503.6007652.100403⎤⎥⎥⎥⎥⎦,Ω₂=⎡⎢⎢⎢⎢⎣2.3001763.2009882.000030₀⎤⎥⎥⎥⎥⎦,=1.7593,𝜖₀=3.6570.(4.3) Then, it follows from Theorem 3.1 that the SBAMNN (2.7) with given parameters is globally exponentially stable in the mean square. Our main purpose in this example is to estimate the maximum allowable upper bound delay 𝜎_𝑀 and 𝜏_𝑀for given lower bound 𝜎_𝑚 and 𝜏_𝑚 (Table 1). For instance, if we set 𝜎_𝑚=𝜏_𝑚=2, the allowable time delay upper bound obtained by Gao and Cui [19] is 4. However, in our paper, we obtained that for any time delay satisfying 0<𝜏(𝑡)≤𝜏_𝑀=foranylargefinitevalue, 0<𝜎(𝑡)≤𝜎_𝑀=foranylargefinitevalue. This is much larger than that in [19], which shows the less conservativeness of our developed method (Figure 1).

Table 1 

Allowable upper bound for 𝜎_𝑀 and 𝜏_𝑀 for given 𝜎_𝑚 and 𝜏_𝑚.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1 

State trajectories of for Example 4.1.

5. Conclusion

In this paper, we have considered the stability analysis problem for a class of discrete-time stochastic BAM neural networks with both discrete and distributed delays. Employing a Lyapunov-Krasovskii functional and a Linear matrix inequality approach has been developed to establish sufficient conditions for the SBAMNNs to be globally exponentially stable. It has been shown that the delayed SBAMNNs are globally exponentially stable if some LMIs are solvable and the feasibility of such LMIs can be easily checked by using the numerically efficient LMI toolbox in Matlab. A numerical example has been given to demonstrate the effectiveness of the obtained stability conditions.

Acknowledgments

The work of the first author was supported by UGC Rajiv Gandhi National Fellowship. The work of the second author was supported by the Korean Research Foundation Grant funded by the Korean Government with Grant no. KRF 2010-0003495 and the work of the third author was supported by the CSIR, New Delhi. The authors are very much thankful to the reviewers and editors for their valuable comments and suggestions for improving this work.