Delay-dependent stochastic stability criteria for Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates
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:where is the neuron state vector, denotes the neuron activation function, and is a constant input vector. 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:The three cases of the transition rates matrices are considered as: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)
- et al.
Global asymptotic stability analysis of bidirectional associative memory neural networks with constant time delays
Neurocomputing
(2005) - et al.
Improved asymptotic stability criteria for neural networks with interval time-varying delay
Expert Syst. Appl.
(2010) - et al.
Exponential stability of recurrent neural networks with time-varying discrete and distributed delays
Nonlinear Anal. Real World Appl.
(2009) - et al.
Exponential stability analysis for uncertain neural networks with interval time-varying delays
Appl. Math. Comput.
(2009) Exponential stability of recurrent neural networks with both time-varying delays and general activation functions via LMI approach
Neurocomputing
(2008)- et al.
New results for global stability of a class of neutral-type neural systems with time delays
Appl. Math. Comput.
(2009) - et al.
Further results on state estimation for neural networks of neutral-type with time-varying delay
Appl. Math. Comput.
(2009) - et al.
Improved delay-dependent stability criterion for neural networks with time-varying delays
Phys. Lett. A
(2009) - et al.
Improved stability criteria for neural networks with time-varying delay
Phys. Lett. A
(2009) - et al.
Global exponential stability of generalized recurrent neural networks with discrete and distributed delays
Neural Netw.
(2006)
Global asymptotic stability of stochastic fuzzy cellular neural networks with multiple time-varying delays
Expert Syst. Appl.
Improved delay-dependent stability criterion for neural networks with time-varying delay
Appl. Math. Comput.
New delay-dependent exponential stability criteria for neural networks with discrete and distributed time-varying delays
Neurocomputing
New LMI based delay-dependent criterion for global asymptotic stability of cellular neural networks
Neurocomputing
A new approach to exponential stability analysis of neural networks with time-varying delays
Neural Netw.
Global exponential stability of impulsive Cohen–Grossberg neural network with time-varying delays
Nonlinear Anal. Real World Appl.
On improved delay-dependent criterion for global stability of bidirectional associative memory neural networks with timevarying delays
Appl. Math. Comput.
New asymptotic stability criteria for neural networks with time-varying delay
Phys. Lett. A
Mean-square exponential stability of stochastic Hopfield neural networks with time-varying discrete and distributed delays
Phys. Lett. A
Improved delay-dependent exponential stability for uncertain stochastic neural networks with time-varying delays
Phys. Lett. A
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