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

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.


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 discretetime 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.

Problem Description and Preliminaries
Consider the following discrete-time stochastic BAM neural networks with both discrete and distributed delays of the following form: or, in an equivalent form, where τ m , τ M , σ m , and σ M are known positive integer; M, N denotes the distributed timevarying delays.Then 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 where

2.8
Assumption 3. Obviously, the activation functions f j , f i , g j , g i , h j , and h i i, j ∈ N satisfy the following condition:

2.10
Remark 2.2.Assumption 4 ensures that the terms which is significant for the subsequent analysis.
Assumption 5.There exist constant matrices G and K such that

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 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:

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:

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.

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 λ 0 > 0 and 0 > 0 if there exist diagonal matrices and R 2 > 0, such that the following LMIs hold:

3.2
Proof.Let us choose the Lyapunov-Krasovskii functional as

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 have where

3.5
By using Lemma 2.4, we have

3.8
It is clear from 2.9 that where e k denotes the unit column vector having "1" element on its kth row and zeros elsewhere.Consequently,

3.21
Then from 3.5 -3.8 and 3.16 -3.21 , we obtain 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 that

3.23
From 3.3 , it can be verified that

3.25
Then by 3.23 and 3.24 , we have

ISRN Discrete Mathematics
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 have

3.29
We can observe that

3.30
It follows easily from 3.24 that

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:

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 and R 2 > 0, such that the following LMIs hold:
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 to

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

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: 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 M for 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 for any large finite value, 0 < σ t ≤ σ M for any largefinite value.This is much larger than that in 19 , which shows the less conservativeness of our developed method Figure 1 .

Conclusion
In this paper, we have considered the stability analysis problem for a class of discretetime 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.
• 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.
and w i are some constants.

Table 1 :
Allowable upper bound for σ M and τ M for given σ m and τ m .