Delay-dependent stochastic stability criteria for Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates

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Abstract

In this paper, the problem of stochastic stability criterion of Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates is considered. Some new delay-dependent stability criteria are derived by choosing a new class of Lyapunov functional. The obtained criteria are less conservative because free-weighting matrices method and a convex optimization approach are considered. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.

Introduction

In recent decades, neural networks have been investigated extensively because of their successful applications in various areas such as pattern recognition, image processing, associative memory and combinatorial optimization. However, these successful applications are greatly dependent on the dynamic behaviors of neural networks. As is well known now, stability is one of the main properties of neural networks, which is a crucial feature in the design of neural networks. On the other hand, it has been recognized that the time delays often occur in various neural networks, and may cause undesirable dynamic network behaviors such as oscillation and instability. Therefore, the stability analysis for delayed neural networks has become a topic of great theoretic and practical importance in recent years [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27].

Recently, systems with Marvokian jumps have been attracting increasing research attention. This class of systems are the hybrid systems with two components in the state. The first one refers to the mode, which is described by a continuous–time finite-state Markovian process, and the second one refers to the state which is represented by a system of differential equations. The Markovian jump systems have the advantage of modeling the dynamic systems subject to abrupt variation in their structures, such as component failures or repairs, sudden environmental disturbance, changing subsystem interconnections, and operating in different points of a nonlinear plant [28]. Recently, there has been a growing interest in the study of neural networks with Markovian jumping parameters [29], [30], [31], [32], [33], [34], [35], [36], [37], [38]. In [29], the problem of stochastic robust stability for uncertain delayed neural networks with Markovian jumping parameters is investigated. The state estimation problem for a class of Markovian neural networks with discrete and distributed time-delays is studied in [30]. Without assuming the boundedness, monotonicity and differentiability of the activation functions, some results for delay-dependent stochastic stability criteria for the Markovian jumping Hopfield neural networks with time-delay are developed in [31]. Some new delay-dependent stochastic stability criteria for BAM neural networks with Markovian jumping parameters are derived in [32] based on delay partitioning idea. To the best of our knowledge, the stochastic stability analysis for Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates has never been tackled, and such a situation motivates our present study.

In this paper, the problem of stochastic stability criterion for Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates is considered. By choosing a new class of Lyapunov functional, some new delay-dependent stochastic stability criteria are derived to guarantee the stochastic stability of Markovian jumping neural networks. The obtained criteria are less conservative because free-weighting matrices method and a convex optimization approach are considered. Finally, a numerical example is given to show the effectiveness of the derived method.

Section snippets

Problem formulation

Consider the following delayed neural network:x˙(t)=-Ax(t)+Bg(x(t))+Cg(x(t-h(t)))+Dt-d(t)tg(x(s))ds+μ,x(t)=Φ(t),t[-h¯,0],where x(t)=[x1(t),x2(t),,xn(t)]TRn is the neuron state vector, g(x(·))=[g1(x1(·)),g2(x2(·)),,gn(xn(·))]TRn denotes the neuron activation function, and μ=(μ1,μ2,,μn)TRn is a constant input vector. B,C,DRn×n are the connection weight matrix and the delayed connection weight matrix,respectively. A = diag(a1, a2,  , an) with ai > 0, i = 1, 2,  , n. h(t), d(t) are time-varying

Main results

In this section, a new Lyapunov functional is constructed to derived a delay-dependent stochastic stability criterion for system (8) when the time-varying delays are mode-dependent and the transition rates are partially known.

Theorem 1

For given scalars hi  0, di > 0, ui, the system (8) with mode-dependent time-varying delays and partially known transition rates is stochastically stable if there exist symmetric positive definite matrices Pi, Q1i, Q2i, Q3i, Q4i, R1, R2, R3, R4, R5, R6, R7, positive diagonal

A Numerical example

Consider the system (8) with the following parameters:A1=2002,B1=11-1-1,C1=0.88111,D1=0.80.40.50.6,A2=2.2001.5,B2=10.60.10.3,C2=1-0.10.10.2,D2=1.20.70.60.4,A3=2.3002.5,B3=0.30.20.40.1,C3=0.50.70.70.4,D3=0.5-O.30.21.2,Γ1=0.2000.1,Γ2=0.4000.8.The three cases of the transition rates matrices are considered as:Case(1):Π=-0.80.30.50.1-0.80.70.70.4-1.1,Case(2):Π=-0.8??0.1-0.80.70.70.4-1.1,Case(3):Π=-0.8???-0.8?0.70.4-1.1.We assume condition 1: h1 = h2 = h3, u1 = u2 = u3 = 0.1, d1 = d2 = d3 = 0.5; condition 2: h1 = h2 = h

Conclusions

In this paper, the problem of stochastic stability criterion of Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates has been proposed. By choosing a new class of Lyapunov functional, some new delay-dependent stochastic stability criteria are derived to guarantee the stochastic stability of Markovian jumping neural networks. The obtained criteria are less conservative because free-weighting matrices method and a convex optimization

Acknowledgments

The authors thank the editors and the reviewers for their valuable suggestions and comments which have led to a much improved paper. This work was supported by the demonstration project of oil and gas development for carbonate reservoirs in Tarim basin under Grant 2011ZX05049 and the National Basic Research Program of China under Grant 2010CB732501.

References (40)

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