LMI conditions for global asymptotic stability results for neutral-type neural networks with distributed time delays

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Abstract

In this paper, the global asymptotic stability of neutral-type neural networks with unbounded distributed delays is analyzed by utilizing the Lyapunov–Krasovskii functional and the linear matrix inequality (LMI) approach. A new sufficient condition ensuring the global asymptotic stability for neutral-type neural networks is obtained by using the powerful MATLAB LMI toolbox. Two examples are provided to illustrate the applicability of the stability results.

Introduction

The key features of neural networks are asynchronous parallel processing, continuous time dynamics and global interaction of network elements. Neural networks and their various generalizations have attracted the attention of the scientific community due to their promising potential for tasks of classification, associative memory, and parallel computation and their ability to solve difficult optimization problems [1], [7], [8]. In such applications, the stability of the network is prerequisite. Many researchers have a lot of contributions to these subjects. Stability is a basic knowledge for dynamical systems and is useful in the application to the real systems. Because of this, the stability of neural networks has been deeply investigated in the literature. Delay systems have been largely used to describe propagation and transport phenomena or population dynamics. In economic systems, delays appear in a natural way since decisions and effect (investment policy, commodity market evolution, etc.) are separated by some time interval. The existence of time delays may degrade system performance and cause oscillation in a network, leading to instability. Therefore, it is important to investigate the stability of delayed neural networks. The global asymptotic stability results for different classes of delayed neural networks were proposed in [2], [3], [4], [9].

Most works on delayed neural networks have dealt with the stability analysis problems for neural networks with discrete delays. Neural network usually has a spatial nature due to the presence of various parallel pathways with a variety of axon sizes and lengths, so it is desirable to model them by introducing unbounded delays. In these circumstances the signal propagation is not instantaneous and cannot be modelled with discrete delays. A more appropriate way is to incorporate continuously distributed delays in neural network models. In recent years there has been a growing research interest in the study of neural networks with distributed delays [15], [16], [17], [18], [20], [21]. Another type of time-delays, namely, neutral-type time-delays, has recently drawn much research attention. So far there are only a few papers that have taken neutral-type phenomenon into account in delayed neural networks, see for example [5], [6], [10], [11], [12], [13], [19]. Practically, such phenomenon always appears in the study of automatic control, population dynamics and vibrating masses attached to an elastic bar, etc. Recently, Rakkiyappan and Balasubramaniam [14] studied the global exponential stability results for neutral-type neural networks with distributed delays by using LMI approach.

Based on the above discussions, we consider a class of neutral-type neural networks with unbounded distributed time delays described by a neutral integro differential equations. The main purpose of this paper is to study the global asymptotic stability for neutral-type neural networks with unbounded distributed delays. To the best of the authors knowledge, there were no results for global asymptotic stability analysis problem for neutral-type neural networks with distributed delays. It is therefore, our intension in this paper is to investigate the global asymptotic stability analysis problem for a class of neutral-type neural networks with distributed delays. By using the Lyapunov–Krasovskii functional technique, global asymptotic stability conditions for the considered delayed neural networks are given in terms of LMIs, which can be easily calculated by MATLAB LMI toolbox. We also provide a numerical example to demonstrate the effectiveness of the proposed stability results.

Section snippets

Global stability results

Throughout the manuscript we will use the notation A>0(orA<0) to denote that A is a symmetric and positive definite (or negative definite) matrix. The notation AT and A-1 means that the transpose of A and the inverse of a square matrix, respectively. If A, B are symmetric matrices A>B (AB), then A-B is positive definite (positive semi-definite). z denotes the Euclidean norm of a vector z and A denotes the induced norm of the matrix A, that is A=λM(ATA).

In this paper, we consider a class

Main results

We establish the following stability criteria by using the Lyapunov method in terms of LMIs:

Theorem 3.1

The equilibrium point of system(4) is globally asymptotically stable if there exist positive definite matrices P,Q,R and a positive diagonal matrices M,N such thatΞ=Ψ11Ψ12Ψ13Ψ14Ψ15Ψ22Ψ23Ψ24Ψ25Ψ33Ψ34Ψ35Ψ44Ψ45Ψ55<0,whereΨ11=-PA-ATP+ATRA+ATNA+LQL+LML+PBL+ATRBL+ATNBL+LBTRBL+LBTNBL,Ψ12=PC-ATRC-ATNC+LBTRC+LBTNC,Ψ13=PD-ATRD-ATND+LBTRD+LBTND,Ψ14=PE-ATRE-ATNE+LBTRE+LBTNE,Ψ15=PF-ATRF-ATNF+LBTRF+LBTNF,Ψ22=

Example

Consider the following coefficient matrices of two-neuron neural networks of neutral-type with distributed delays satisfying (2)A=3.6003.6,B=1.1980.10.11.198,C=0.10.160.050.1,D=0.2000.2,E=0.4-0.20.30.2,F=0.3-0.150.5-0.2,where τ(t)=h(t)=0.3+0.15sin2t. Solving the LMI in Theorem 3.1 using MATLAB LMI toolbox, we obtain the solution asP=59.8780-6.3974-6.397430.4082,Q=13.0598-3.0734-3.073416.0019,R=7.8943-3.6823-3.68234.9155,M=34.3347009.1941,N=10.3356004.0831.The above results shows that all the

Conclusion

A new sufficient condition is derived to guarantee the global asymptotic stability of the equilibrium point for neutral-type neural networks with unbounded distributed delays. To the best of our knowledge, the results presented here have been not appeared in the related literature. The stability criteria is expressed in terms of LMIs, which are less conservative and less restrictive and can be easily verified by using MATLAB LMI toolbox.

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The work of the authors was supported by UGC-SAP(DRS), New Delhi India under the sanctioned No. F510/6/DRS/2004 (SAP-1).

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