ISRN Discrete MathematicsVolume 2011 (2011), Article ID 153409, 23 pagesdoi:10.5402/2011/153409

Research Article

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

R. Raja,¹ R. Sakthivel,² and S. Marshal Anthoni³

¹Department of Mathematics, Periyar University, Salem 636 011, India²Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea³Department of Mathematics, Anna University of Technology, Coimbatore 641 047, India

Received 13 September 2011; Accepted 23 October 2011

Academic Editors: C.-K. Lin and N. I. Trinajstić

Copyright © 2011 R. Raja et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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, and denote, respectively, the -dimensional Euclidean space and the set of all real matrices. denotes the identity matrix with appropriate dimensions and denotes the diagonal matrix. For real symmetric matrices and , the notation (resp., ) means that the matrix is positive semidefinite (resp., positive definite). and stands for the Euclidean norm in . (resp., ) stands for the maximum (resp., minimum) eigenvalue of the matrix . 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: or, in an equivalent form, for , where and are the neural state vector; and are the state feedback coefficient matrices; , and are, respectively, the connection weight matrices, the discretely delayed connection weight matrices, and distributed delayed connection weight matrices; and denote the discrete time-varying delays satisfying where , and are known positive integer; denotes the distributed time-varying delays.Then denote the neuron activation functions. The constant vectors and are the external inputs from outside the system; and are scalar constants, where and are scalar Wiener process (Brownian motion) on the probability space with with being the mathematical expectation operator; and are the nonlinear vector function representing the disturbance intensities.

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

Assumption 1. For , the neuron activation functions , , , , , and in (2.2) are continuous as well as bounded on .

Assumption 2. For , the neuron activation functions in (2.2) satisfy where , and are some constants.

Remark 2.1. Assumption 2 was first introduced by Liu et al. [25]. The constants , and 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 and of system (2.2) to the origin. Let and ; then system (2.2) can be transformed to where

Assumption 3. Obviously, the activation functions satisfy the following condition:

Assumption 4. The constants satisfy the following convergent conditions:

Remark 2.2. Assumption 4 ensures that the terms and are convergent, which is significant for the subsequent analysis.

Assumption 5. There exist constant matrices and such that

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 and such that

Lemma 2.4. Let and be any n-dimensional real vectors and let be an positive semidefinite matrix. Then, the following matrix inequality holds:

Lemma 2.5. Let be a positive semidefinite matrix, , and . If the series concerned are convergent, the following inequality holds:

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 and if there exist diagonal matrices , , , , , and and positive definite matrices , such that the following LMIs hold: where

Proof. Let us choose the Lyapunov-Krasovskii functional as In order to analyze the global exponential stability of the stochastic BAM neural network, we calculate differences of the Lyapunov function , along with the trajectories of the BAM neural network (2.7); then we have where By using Lemma 2.4, we have It is clear from (2.9) that which is equivalent to where denotes the unit column vector having “1” element on its th row and zeros elsewhere.Consequently, Similarly, from (3.10)–(3.14), we have Then from (3.5)–(3.8) and (3.16)–(3.21), we obtain where .Therefore, if the LMIs (3.1) hold, it can be concluded that . It follows that . 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 such that From (3.3), it can be verified that where and .Choose a scalar , satisfying Then by (3.23) and (3.24), we have where and .Therefore, for any integer and , summing up both sides of (3.26) from 0 to with respect to , we have Here, we note that for , , we have Substituting (3.28) in (3.27) gives We can observe that It follows easily from (3.24) that Then, it follows from (3.25), (3.29), and (3.31) that where and 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:

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 and positive definite matrices , and , such that the following LMIs hold: where and 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 to

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 , , , and , and positive definite matrices and , such that the following LMI holds: where and 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: It can be verified that , , , , , , , , , , , , and with By using Matlab LMI toolbox, we solve the LMIs (3.1) in Theorem 3.1 and obtain the feasible solutions as follows: 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 and for given lower bound and (Table 1). For instance, if we set , 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 , . 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 and for given and .

In [19]

In this paper

(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.

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