Passivity analysis of Markov jump BAM neural networks with mode-dependent mixed time-delays via piecewise-constant transition rates
Introduction
The past decades have witnessed a huge and rapidly growing investigations concerned with neural network such as recurrent neural network [1], Hopfield neural network [2], cellular neural network [3], [4], and bi-directional associative memory (BAM) neural network [5], which is due to the broad applications in various fields such as medical imaging, signal processing, pattern recognition, and fixed-point computations. Among them, recurrent neural networks (RNNs), as is known to all, is a class of artificial neural network with the characteristic that internal connections form a directed cycle, such unique way of internal connecting make dynamic temporal behavior of the network possible. As a special class of RNNs, the bidirectional associative memory (BAM) neural network can store bipolar vector pairs and is composed of neurons arranged in two layers fully interconnected, respectively, with each other. According to the iterations of forward and backward information flows between the two layers, a two-way associative search for stored bipolar vector pairs was constructed. Furthermore, the Cohen–Grossberg-type BAM neural network model, as an important one in BAM neural network models and an extension of the traditional single layer neural networks model, was originally introduced by Cohen and Grossberg [6] in 1983 and investigated by considerable literatures [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18].
On one hand, time-delays caused by many practical factors do inevitably exist in network models. Moreover, time-delay systems can better describe various kinds of practical systems and have been successfully applied in a wide range of fields such as electrical circuits, biological systems, nuclear reactors, economics, etc. As a matter of course, neural networks with time-delays have also been widely studied by researchers from control communities and mathematics. More specifically, as for the stability of delayed BAM neural networks, in [19], based on the Lyapunov–Krasovskii functional and the linear matrix inequality (LMI) approach, some sufficient conditions are achieved to ensure the global asymptotic stability of stochastic Cohen–Grossberg-type BAM neural networks with mixed delays. And by employing non-smooth analysis, the global asymptotic stability of the delayed Cohen–Grossberg neural network was established [20]. Moreover, a class of BAM neural networks with time-varying delays in leakage terms were studied, and some sufficient conditions for guaranteeing the existence and exponential stability of the neural networks were obtained [21]. As for the synchronization of the BAM model, in [22], the exponential synchronization of the BAM model was derived by utilizing the linear matrix inequality (LMI) approach and impulsive control design. In addition, the asymptotic property of neutral-type BAM neural networks with time-varying and infinite distributed delays was derived based on the properties of nonnegative matrix [23] and the passivity problem for BAM neural networks with time varying delay was discussed in [24]. Taking into account of the effect of memristor and time-varying delays, in [25], the issue of the delay-dependent exponential passivity analysis of recurrent neural networks has been studied and several less conservative results in the sense of Filippov solutions have been derived.
On the other hand, there have been increasing attentions focused on the analysis and synthesis of Markov jump systems as a special class of hybrid systems, which is partly due to their well characterization of random switching structure caused by random abrupt variations [26], [27], [28], [29], [30], [31], [32]. By establishing appropriate Lyapunov–Krasovskii functional and combining with LMIs technique, mean square stabilization and mean square exponential stabilization of stochastic BAM neural networks with Markov jump parameters have been investigated in [33]. And [34] was concerned with the exponential stability for Markov jump stochastic BAM neural networks with mode-dependent probabilistic time-varying delays and impulse control. As we all know, the assumption in the above-mentioned references meant that transition probabilities or transition rates in the Markov chain or Markov process were time-invariance. Nevertheless, it cannot always be satisfied in practical systems [35], [36], and its application will inevitably be some degree of restriction even if this idealized assumption can certainly simplify the research on the Markov jump neural network. Therefore, to consider Markov jump neural with time-varying transition probabilities of transition rates is necessary and meaningful. Just as discussed in [37], [38], the passivity analysis of discrete-time stochastic neural networks with mixed time delays and time-varying transition probabilities has been investigated and a delay-dependent passivity condition has been achieved based on LMIs. Especially, the work [40] not only has involved with mode-dependent round-trip time-varying delays but also norm-bounded parametric uncertainties when coped with the problems of passivity analysis and passivity-based control for a class of uncertain stochastic jumping systems. And based on Markovian switched Lyapunov functional and linear matrix inequalities, some passivity conditions have been obtained, furthermore, output feedback controller with respect to Markovian switching systems has been presented to ensure the passivity of the corresponding closed-loop systems. In [41], by using the stochastic stability theory, the global asymptotic stability analysis for a class of Markov jump stochastic BAM neural networks was derived. In addition, the problem of passivity analysis of fuzzy BAM neural network model was considered and a set of sufficient conditions were achieved by combining the delay fractioning technique with the Lyapunov function approach in [42]. However, there are few related results available in open literature concerned with continuous-time Markov jump BAM neural networks with time-varying transition rates. Therefore, this paper is aimed at bridging the gap by engaging in the challenging study on the passivity analysis problem for a type of continuous-time neural networks with time-varying transition rates and mixed time delays.
In this paper, the subject of passivity is analyzed for Markov jump BAM neural networks with mode-dependent time-varying transition rates. The mode-dependent time-varying character of transition rates is supposed to be piecewise-constant. Time-varying transition rates are, respectively, discussed under the cases of known transition rates and partly unknown transition rates. The mode-dependent time-varying character of transition rates is supposed to be piecewise-constant. Instead of constructing relatively high conservative common Lyapunov functions or delay-independent Lyapunov–Krasovskii functions, we employ a delay-dependent Lyapunov–Krasovskii functional candidate to analyze the dynamical behavior with the advantage of less conservative functions, which in order to derive more general conditions and results. By utilizing LMIs technique and a class of Lyapunov functionals, a switching delay passivity criterion underlying known transition rates is derived, which can be easily checked by the Matlab LMI Tool Box. Furthermore, we extend the result to passivity analysis of Markov jump BAM neural networks with partly unknown transition rates. The results obtained relate on both discrete delay and distributed delay which are all switching based on Markov process r(t). Finally, a numerical example is given to illustrate the effectiveness of the results.
Notation: Throughout this paper, denote the set of real matrices. The superscript “T” stands for the transpose, and diag is a block-diagonal matrix. means the space of square-integrable vector functions over . The notation where X and Y are symmetric matrices, means that is positive semi-definite (positive definite). I is the identity matrix with compatible dimension. For is the family of continuous functions ϕ from to with the norm , where is the Euclidean norm in . denotes a complete probability space with a filtration containing all -null sets and being right continuous. Let be the family of all -measurable - valued random variables such that where represents the mathematical expectation operator with respect to the given probability measure . For an arbitrary matrix A and two symmetric matrices B and C, is a symmetric matrix, in which “*” stands for the term that is induced by symmetry. ‘?’ stands for the unknown transition rate. Denote transition rate πij is known for , and transition rate πij is unknown for , . transition rate pbd is known for , and transition rate pbd is unknown for , .
Section snippets
Preliminaries
Discuss Markov jump BAM neural networks with mode-dependent mixed time-delays of the form on a probability space as follows:where , , ,
Main results ( and )
In this section, we analyze the passivity problem of neural network (1) for and . Our objective is to present a delay-dependent passivity criterion in terms of LMIs. Before giving the main results, for the sake of presentation simplicity, we denote Theorem 1 System (1) is stochastically passive under that and , for
Extension to Markov jump neural networks with partly unknown transition probabilities ( and )
For Markov jump system, the dynamical characteristics of the system largely depend on the transition probabilities in the jumping process, it is convenient to analyze and manage the systems with complete control information of the transition probabilities, and such a large results with all known information are available. However, the cost to achieve all complete information of the transition probabilities can be very high or even beyond one׳s means, and what should be pointed out is that the
Numerical example
In this section, a numerical example is given to demonstrate the effectiveness of the proposed result. Consider a BAM neural network with mode-dependent mixed time-delays composed of two layers of neurons (i, j, respectively, denote one of the Ith layer of neurons composed of two cells and one of the Jth layer of neurons composed of four cells). Suppose, for , that B(r) and C(r) and D(r) denote the strengths of connectivity between the Ith layer of neurons and the Jth layer of neurons,
Conclusion
In this paper, stochastic passivity analysis has been discussed for Markov jump neural networks with discrete and distributed delays. We consider that the transition rates of the underlying Markov process are model-dependent time-varying piecewise-constant. Based on the LMI method and a class of stochastic Lyapunov functionals established, a passivity condition underlying known transition rates has been proposed, which not only depends upon discrete delays, but also depends upon distributed
Acknowledgments
This work was supported in part supported by both the National Natural Science Foundation of China under Grants 11401062 and 61374104. The authors would like to express sincere appreciation to the editor and anonymous reviewers for their valuable comments which have led to an improvement in the presentation of the paper.
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