Elsevier

Neurocomputing

Volume 121, 9 December 2013, Pages 365-378
Neurocomputing

Effects of leakage time-varying delays in Markovian jump neural networks with impulse control

https://doi.org/10.1016/j.neucom.2013.05.018Get rights and content

Abstract

In this paper, the stability analysis problem is investigated for delayed neural networks with mixed time-varying delays, impulsive control and Markovian jumping parameters. The mixed time-varying delays include leakage, discrete and distributed time-varying delays. Sufficient conditions for the global exponential stability in the mean square are derived by using Lyapunov–Krasovskii functional having triple integral terms and model transformation technique. The stability criterion that depends on the upper bounds of the leakage time-varying delay and its derivative is given in terms of linear matrix inequalities (LMIs), which can be efficiently solved via standard numerical softwares. Finally, three numerical examples and simulations are given to demonstrate the usefulness and effectiveness of the presented results.

Introduction

During the last two decades, neural networks (NNs) are widely studied, because of their massive potentials of application in modern society of science and technology. Its potential applications are signal processing, pattern recognition, static image processing, associative memory, and combinatorial optimization [1], [2], [3], [4], [5]. In such applications, it is major importance to ensure that the designed neural network is stable. Further, when the neural networks are implemented with the help of very large-scale integrated electronic circuits, the finite switching speed of amplifiers and the communication time of neurons may induce time delays in the interaction between the neurons. It has been shown that time delays may cause undesirable dynamic network behaviors such as oscillation and instability. Thus, it is important to investigate the stability analysis of delayed neural networks in the recent years [6], [7], [8], [9], [10], [11], [12], [13], [14].

It is well known that the stochastic modeling plays an important role in the branches of science and technology. A particular area of interest is neural networks with Markovian jumping parameters. Markovian jumping systems can be considered as a special class of hybrid systems, and it is sometimes the case that a neural network has finite state representations (also called modes, patterns, or clusters), and the modes that may switch (or jump) from one to another at different times. In [15], [16], it has been shown that the network states may switch (or jump) between different neural networks modes according to a Markovian chain. Further, these structures are subjected to random abrupt changes due to some unexpected factors such as component failures or repairs, sudden environmental disturbance and failures that occurred in components or interconnections and executor faults, etc. Hence, the stability analysis problem for neural networks with Markovian jumping becomes more and more significant. Moreover, many relevant results have been reported in the literature [17], [18], [19], [20], [21], [22], [23], [24], [25], [26].

Recently, the effect of leakage delay in dynamical neural networks is one of the research topics and it has been studied by many researchers in the literature. As correctly pointed out in the literature [27], [28], the time delay in stabilizing negative feedback term has a tendency to destabilize the system. Furthermore, sometimes it has more significant effect on dynamics of neural networks than other kinds of delays. Hence, the leakage term also has great impact on the dynamical behavior of neural networks. Therefore, there are many authors considering the problem of stability analysis of neural networks and system involving time delay in the leakage term, see for example [28], [29], [30], [31], [32]. In addition, the authors in [33], [34] discussed the problem of global stability analysis of equilibrium of neural networks with time delay in the leakage term under impulsive perturbations. The problem of synchronization of chaotic neural networks with time delay in the leakage term and parametric uncertainties based on sampled-data control has been presented in [35]. Also, it is well known that impulsive effects are likely to exist in the neural network systems [36], [37], [38], [39], [40]. For instance, in the implementation of electronic networks, the state is subjected to instantaneous perturbations and also it experiences abrupt change at certain moments. These abrupt changes might be caused by switching phenomenon, frequency changes or other sudden noises, which exhibit impulsive effects [41], [42]. Hence, it is necessary to consider the impulsive control to the stability problem of neural networks with delays to reflect a more realistic dynamics. In the meanwhile, effect of leakage time-varying delay on stability of nonlinear differential systems has been investigated in [43]. To the best of authors’ knowledge, so far, no results on the effects of leakage time-varying delays in Markovian jump neural networks with impulse control are available in the existing literature. This motivates our current research.

In this paper, we consider a class of Markovian jumping neural networks with leakage time-varying delays and impulsive effects. Then, by using a suitable Lyapunov Krasovskii's functional, model transformation and some analysis techniques, some delay-dependent criteria ensuring the global exponential stability in the mean square of the unique equilibrium point are obtained, which depend on the upper bounds of the leakage time-varying delay and its derivative. The criteria are given in terms of LMIs and can be efficiently solved via standard numerical softwares such as the MATLAB LMI toolbox [44]. It should be mentioned that Zhu and Cao [45] obtained some interesting and important results about the stability of Markovian jump system based on similar methods. However, we focus on the discussion of the leakage time-varying delays, which was not investigated in [45]. Thus, our development result is more general than the recent work [45].

Notations Let Rn denote the n-dimensional Euclidean space and the superscript ”T” denote the transpose of a matrix or vector. I denotes the identity matrix with compatible dimensions. diag() denotes a block diagonal matrix. For square matrices, M1 and M2, the notation M1>(,<,)M2 denotes M1M2 is a positive-definite (positive-semi-definite, negative, negative-semi-definite) matrix. λmin(·) and λmax(·) stand for the minimum and maximum eigenvalues of a given matrix, respectively. Let (Ω,F,P) be a complete probability space with a natural filtration {Ft}t0 and E[·] stands for the correspondent expectation operator with respect to the given probability measure P. Also, let τ>0 and C([τ,0];Rn) denote the family of continuously differentiable function ϕ from [τ,0] to Rn with the uniform norm ϕ=max{maxτθ0|ϕ(θ)|,maxτθ0|ϕ(θ)|}. Denote by CF02([τ,0];Rn) the family of bounded F0-measurable, C([τ,0];Rn)-valued stochastic variables ξ={ξ(θ):τθ0} such that τ0E|ξ(θ)|2ds<.

Section snippets

Model description and preliminaries

Let {r(t),t0} is a right-continuous Markov chain on the probability space (Ω,F,P) taking values in a finite state space S={1,2,,N} with generator Q=(qij)N×N given by P{r(t+Δt)=j|r(t)=i}={qijΔt+o(Δt),ij,1+qiiΔt+o(Δt),i=j,where Δt>0 and limΔt0o(Δt)/Δt=0, qij0 is the transition rate from i to j, if ij while qii=jiqij.

Consider the following delayed recurrent neural networks with Markovian jumping parameters and leakage time-varying delay:{ẋ(t)=C(r(t))x(tσ(t))+A(r(t))f(1)(x(t))+B(r(t))f(2

Main results

In this section, under Assumptions 1 and 2, we will investigate the exponential stability in the mean square of the equilibrium point for the system (2).

Theorem 3.1

Let α0 be a fixed positive constant and assume that Assumptions 1 and 2 hold. Then the equilibrium point of Eq. (2) is exponentially stable in the mean square, if there exist positive definite matrices X,Q1,Q2,Q3,Q4,Q5,R1,R2,R3,T1,T2,T3,Z1,Pi,Mi,{i=1,2,,N}, positive diagonal matrices G1,G2 and any appropriate dimensions matrices Y,N1,N2,N3

Numerical examples and simulation

In this section, three numerical examples are given to illustrate the effectiveness and usefulness of the theoretical results.

Example 4.1

Consider a two-dimensional Markovian jump neural network with impulse control and time varying delays{ẋ(t)=C(r(t))x(tσ(t))+A(r(t))f(1)(x(t))+B(r(t))f(2)(x(tτ(t)))+V,ttk,x(tk)x(tk)=Dk(r(t)){x(tk)C(r(t))tkσ(tk)tkx(s)ds},kZ+where x(t)=(x1(t),x2(t))T, V=(0,0)T, τ(t)=0.99cost+0.9, and the Markov process {r(t),t0} taking values in S={1,2} with generator Q=[2233]

Conclusions

In this paper, the stability analysis problem for Markovian jump recurrent neural networks with impulse control and leakage time-varying delays has been studied. By constructing an appropriate Lyapunov–Krasovskii functional and model transformation technique, some stability criteria that depend on the upper bounds of the leakage time-varying delay and its derivative have been presented in terms of LMIs, which can be efficiently solved via standard numerical softwares. The numerical examples and

R. Rakkiyappan undergraduated in the field of Mathematics during 1999–2002 from Sri Ramakrishna Mission Vidyalaya College of Arts and Science. He postgraduated in Mathematics from PSG College of Arts and Science affiliated to Bharathiar University, Coimbatore, Tamilnadu, India during 2002–2004. He was awarded the Doctor of Philosophy in 2011 from the Department of Mathematics, Gandhigram Rural University, Gandhigram, Tamil Nadu, India. His research interests is in the field of qualitative

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      Over the past decade, much attention has also been paid to the passivity problem of delayed neural networks, please refer to [14–16] for the continuous-time case, and [17,18] for the discrete-time case. Recently, a special type of time delay, namely, leakage delay (or forgetting delay), has been identified and investigated due to its existence in many real systems such as neural networks, population dynamics and some fuzzy systems [19–29]. In [19,20], the stability problem has been considered for a class of nonlinear time-delay systems with leakage delay, and in [21–29], the problems of stability, passivity and state estimation have been investigated for several kinds of delayed neural networks with leakage delay.

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    R. Rakkiyappan undergraduated in the field of Mathematics during 1999–2002 from Sri Ramakrishna Mission Vidyalaya College of Arts and Science. He postgraduated in Mathematics from PSG College of Arts and Science affiliated to Bharathiar University, Coimbatore, Tamilnadu, India during 2002–2004. He was awarded the Doctor of Philosophy in 2011 from the Department of Mathematics, Gandhigram Rural University, Gandhigram, Tamil Nadu, India. His research interests is in the field of qualitative theory of stochastic and impulsive systems, Neural Networks and delay differential systems. He has published more than 45 papers in international journals. Now he is working as an Assistant Professor in Department of Mathematics, Bharathiyar University, Coimbatore.

    A. Chandrasekar was born in 1989. He received the B.Sc. degree in Mathematics from Thiruvalluvar Government Arts college affiliated to Periyar University, Salem in 2009 and the M.Sc. degree in Mathematics from Bharathiar University, Coimbatore, Tamilnadu, India, in 2011. He is pursuing M.Phil. Mathematics in Bharathiar University, Coimbatore, Tamilnadu, India. His research interests include Neural Networks, stochastic and impulsive systems.

    S. Lakshmanan undergraduated in the field of Mathematics during 2002–2005 from Government Arts College, Salem-7. He postgraduated in Mathematics from Sri Ramakrishna Mission Vidyalaya College of Arts and Science affiliated to Bharathiar University, Coimbatore, Tamilnadu, India during 2005–2007. He received the Master of Philosophy and Doctor of Philosophy from department of Mathematics, Gandhigram Rural University, Gandhigram, Tamil Nadu, India, in 2008 and 2012, respectively. His research interests are in the field of qualitative theory of stochastic systems and Neural Networks. Now he is working as a Post-Doctoral Research Associate in the Department of Electrical Engineering/Information and Communication Engineering, Yeungnam University, Kyongsan, Republic of Korea.

    Ju H. Park received the Ph.D. degree in Electronics and Electrical Engineering from POSTECH, Pohang, Republic of Korea, in 1997. From May 1997 to February 2000, he was a Research Associate in ERC-ARC, POSTECH. In March 2000, he joined Yeungnam University, Kyongsan, Republic of Korea, where he is currently the Chuma Chair Professor. From December 2006 to December 2007, he was a Visiting Professor in the Department of Mechanical Engineering, Georgia Institute of Technology. His research interests include robust control and filtering, neural networks, complex networks, and chaotic systems. He has published a number of papers in these areas. He serves as Editor of International Journal of Control, Automation and Systems. He is also an Associate Editor/Editorial Board member for several international journals, including Applied Mathematics and Computation, Journal of The Franklin Institute, Journal of Applied Mathematics and Computing, IET Control Theory and Applications, etc.

    H.Y. Jung received the Ph.D. degree in electronics engineering from the Institut National des Sciences Appliquées de Lyon (INSA de Lyon), France, in 1988. He is currently a Professor in the Department of Information and Communications Engineering, Yeungnam University, Korea. Both teaching and research interests include digital signal/image processing, intelligent vehicles, and nonlinear systems.

    The work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2010-0009373).

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