Synchronization criteria for coupled stochastic neural networks with time-varying delays and leakage delay
Introduction
Complex networks have received increasing attention of researches from various fields of science and engineering such as the World Wide Web, social networks, electrical power grids, global economic markets, and so on [1], [2]. Also, in the real applications of systems, there exists naturally time-delay due to the finite information processing speed and the finite switching speed of amplifiers. It is well known that time-delay often causes undesirable dynamic behaviors such as performance degradation, and instability of the systems. Therefore, recently, the problem of synchronization of coupled neural networks with time-delay which is one of the hot research fields of complex networks has been a challenging issue due to its potential applications such as information science, biological systems and so on [3], [4].
On the other hand, in implementation of many practical systems such as aircraft, chemical and biological systems and electric circuits, there exist occasionally stochastic perturbations. It is no less important than the time-delay as a considerable factor affecting dynamics in the systems. Therefore, stochastic modeling with time-delay plays an important role in many fields of science and engineering applications. For this reason, various approaches to stability criteria for stochastic systems with time-delay have been investigated in the literature [5], [6], [7], [8]. Xu et al. [5] studied the problem of stability analysis for stochastic systems with parameter uncertainties and a class of nonlinearities. In [6], by the Lyapunov–Krasovskii's functional based on the delay fractioning approach, an exponential stability criterion for stochastic systems with time-delay was presented. Kwon [7] derived the delay-dependent stability criteria for uncertain stochastic dynamic systems with time-varying delays via the Lyapunov–Krasovskii's functional approach with two delay fraction numbers. Moreover, by choosing the Lyapunov matrices in the decomposed integral intervals, stability analysis for stochastic neural networks with time-varying delay were proposed in [8]. Above this, the study on the problems for various forms of complex networks or stochastic systems have been addressed. For more details, see the literature [9], [10], [11], [12], [13], [14], [15], [16] and references therein.
Recently, the synchronization stability problem for a class of complex dynamical networks with Markovian jumping parameters and mixed time delays had been studied in [17]. The considered model in [17] has stochastic coupling term and stochastic disturbance to reflect more realistic dynamical behaviors of the complex networks that are affected by noisy environment. Very recently, a leakage delay, which is the time delay in leakage term of the systems and a considerable factor affecting dynamics for the worse in the systems, is being put to use in the problem of stability for neural networks [18], [19]. Balasubramaniam et al. [18] investigated the problem of passivity analysis for neutral type neural networks with Markovian jumping parameters and time delay in the leakage term. By use of the topological degree theory, delay-dependent stability conditions of neural networks of neutral type with time delay in the leakage term was proposed in [19]. However, to the best of authors' knowledge, delay-dependent synchronization analysis of coupled stochastic neural networks with time-varying delay and leakage delay has not been investigated yet. Here, delay-dependent analysis has been paid more attention than delay-independent one because the sufficient conditions for delay-dependent analysis make use of the information on the size of time delay [20]. That is, the former is generally less conservative than the latter.
Motivated by the above discussions, the problem of new delay-dependent synchronization criteria for coupled stochastic neural networks with time-varying delays in network coupling and leakage delay is considered. The coupled stochastic neural networks are represented as a simple mathematical model by use of Kronecker product technique. Then, by construction of a suitable Lyapunov–Krasovskii's functional and utilization of Finsler's lemma, new synchronization criteria are derived in terms of LMIs which can be solved efficiently by standard convex optimization algorithms [21]. In order to utilize Finsler's lemma as a tool of getting less conservative synchronization criteria, it should be noted that a new zero equality from the constructed mathematical model is devised. The concept of scaling transformation matrix will be utilized in deriving zero equality of the method. Finally, three numerical examples are included to show the effectiveness of the proposed method.
Notation: is the n-dimensional Euclidean space, and denotes the set of m×n real matrix. For symmetric matrices X and Y, (respectively, ) means that the matrix is positive definite (respectively, nonnegative). In, and denote n×n identity matrix, n×n and zero matrices, respectively. En denotes the n×n matrix which all elements are 1. refers to the Euclidean vector norm and the induced matrix norm. means the maximum eigenvalue of a given square matrix. denotes the block diagonal matrix. represents the elements below the main diagonal of a symmetric matrix. For a given matrix , such that , we define as the right orthogonal complement of X; i.e., . Let be complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all P-pull sets). stands for the mathematical expectation operator with respect to the given probability measure P.
Section snippets
Problem statements
Consider the following neural networks with time-varying delays and leakage delaywhere is the neuron state vector, n denotes the number of neurons in a neural network, denotes the neuron activation function vector with , means the constant external input vector, is the self-feedback matrix, are the connection weight matrices, and h(t
Main results
In this section, new synchronization criteria for network (8) will be proposed. For the sake of simplicity on matrix representation, and are defined as block entry matrices (For example, ) and the (i,j)th entry of a matrix , respectively. The notations of several matrices are defined as:
Numerical examples
In this section, we provide three numerical examples to illustrate the effectiveness of the proposed synchronization criteria in this paper. Example 1 Consider the following coupled stochastic neural networks with three nodes:where with
Conclusions
In this paper, the delay-dependent synchronization criteria for the coupled stochastic neural networks with time-varying delays in network coupling and leakage delay have been proposed. To do this, a suitable Lyapunov–Krasovskii's functional was used to investigate the feasible region of stability criteria. By establishment of a new zero equality and utilization of Finsler's lemma, sufficient conditions for guaranteeing asymptotic synchronization for the concerned networks have been derived in
Acknowledgments
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0001045), and by a grant of the Korea Healthcare Technology R&D Project, Ministry of Health & Welfare, Republic of Korea (A100054).
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