Global synchronization for a class of Markovian switching complex networks with mixed time-varying delays in the delay-partition approach
© Wang et al.; licensee Springer. 2014
Received: 17 March 2014
Accepted: 10 September 2014
Published: 24 September 2014
In this paper, global synchronization problem for a class of Markovian switching complex networks (MSCNs) with mixed time-varying delays under the delay-partition approach is investigated. A novel delay-partition approach is developed to derive sufficient conditions for a new class of MSCNs with mixed time-varying delays. The proposed delay-partition approach can give global synchronization results lower conservatism. Two numerical examples are provided to illustrate the effectiveness of the theoretical results.
In recent years, complex networks have received a lot of research attention since the pioneering work of Watts and Strogatz . The main reason is two-fold: the first reason is that complex networks can be found in almost everywhere in real world, such as the Internet, WWW, the World Trade Web, genetic networks, and social networks; the second reason is that the dynamical behaviors of complex networks have found numerous applications in various fields such as physics, technology, and so on [2–4]. Complex networks are a set of inter-connected nodes, in which each node is a basic unit with specific contents or dynamics. Among all of dynamical behaviors of complex networks, synchronization is one of the most interesting topics and has been extensively investigated [5, 6]. Synchronization phenomena are very common and important in real world networks, such as synchronization phenomena on the Internet, synchronization transfer of digital or analog signals in communication networks, and synchronization related to biological neural networks. Hence, synchronization analysis in complex networks is important both in theory and application [7, 8].
On the one hand, the actual systems may experience abrupt changes in their structure and parameters caused by phenomena such as component failures or repairs, changing subsystem interconnections, and abrupt environmental disturbances. In general, these systems can be modeled by using Markov chains [9–11]. For example, in , mean-square exponential synchronization of Markovian switching stochastic complex networks with time-varying delays by pinning control was proposed and in , synchronization of Markovian jumping stochastic complex networks with distributed time delays and probabilistic interval discrete time-varying delays was considered. Besides these, the other strategies, which usually include data-driven approaches, support vector machine and multivariate statistical methods, and can be used to process these systems [14–18].
On the other hand, because of the limited speed of signals traveling through the links and the frequently delayed couplings in complex networks, gene regulatory networks, static networks and multi-agent networks, time delays often occur [19–22]. Therefore, recently, synchronization problems in neural networks with mixed-time delays have been extensively studied [23–28]. Although there is some literature [29–33] to investigate synchronization issues of complex networks, to the best of our knowledge, until now, global synchronization of MSCNs via using the delay-partition approach is still rarely paid attention to.
MSCNs model aspect. A new model for a class of MSCNs with mixed time-varying delays is proposed.
A novel delay-partition approach is developed to solve global synchronization for a new class of MSCNs with mixed time-varying delays. This causes our results to have lower conservatism.
Notations: Throughout this paper, the following mathematical notations will be used. denotes the n-dimensional Euclidean space and is the set of real matrices. The superscript T denotes the matrix transposition. means an n-dimensional identity matrix. , where , means that the matrix is real positive semi-definite. For symmetric block matrices or long matrix expressions, an asterisk ⋆ is used to represent a term that is induced by symmetry. stands for a block-diagonal matrix. The Kronecker product of matrices and is a matrix in , which is denoted as . Let be a complete probability space with a filtration satisfying the usual conditions (i.e., the filtration contains all P-null sets and is right continuous). means the expectation of the random variable x. If the dimensions of matrices are not explicitly indicated, that means they are suitable for any algebraic operations.
2 Problem formulation and preliminaries
In this section, the problem formulation and preliminaries are briefly introduced.
where , , () is the transition probability from mode i to mode j, and .
where is the continuous-time Markov process which describes the evolution of the mode at time t. and are matrices with real values in mode . , and represent node discrete time-varying delay, discrete time-varying coupling delay and distributed time-varying coupling delay, respectively. is the coupling strength in mode , where . is the inner-coupling matrix in mode . represents the outer-coupling matrix, and the diagonal elements of matrix are defined by (, ). Here, .
Remark 1 The MSCN (1), which contains Markovian switching parameters and mixed time-varying delays, in this paper is more practical than that of [29–31]. Although time delays are considered in [29–31], Markovian switching cannot be taken to describe the addressed systems. Furthermore, the MSCN (1) of this paper is clearly different from that of [32, 33]. Their primary differences are mixed time-varying delays. In , mixed time-varying delays include node discrete time-varying and distributed time-varying delays. In , mixed time-varying delays are comprised of node discrete stochastic time-varying, discrete stochastic time-varying coupling, and distributed time-varying delays.
where , is a solution of an isolated node and satisfying .
Assumption 2 (Khalil )
where , are real constant matrices with .
Lemma 1 (Langville and Stewart )
Lemma 2 (Liu et al. )
Lemma 3 (Boyd et al. )
Lemma 4 (Gu )
Lemma 5 (Boyd et al. )
3 Main results
In this section, global synchronization of the MSCN (1) is investigated by utilizing the Lyapunov-Krasovskii functional method, the stochastic analysis techniques and the delay-partition approach. Furthermore, in order to show the merits of the delay-partition approach, Corollary 1 can also be given, according to Theorem 1.
where scalar .
According to Definition 1, the MSCN (1) is global asymptotic synchronization. The proof is completed. □
Remark 2 In Theorem 1, the criterion which is the MSCN (1) with mixed time-varying delays under the delay-partition approach can achieve global asymptotic synchronization is established. In proving Theorem 1, it is clear that the time-varying delays and are divided into r and slices, respectively. In [34, 35], the delay-partition approach is used to solve state estimation and stability analysis problems of neural networks with time-varying delay. Although synchronization problems of complex network with time delays were investigated in [29–32], our results in the delay-partition approach in this paper has lower conservatism. The reason is that the integers r and become larger, and the allowable upper bounds of the time-varying delays and will be larger. This will also be analyzed in Remark 3 and be shown in numerical examples.
Corollary 1 Under Assumptions 1-2, Definition 1, for given constants , , , , , system (1) is globally asymptotically synchronized if there exist positive-definite matrices , , , , , , arbitrary matrices , , , , with appropriate dimensions and positive scalars , , , such that the LMI (4) holds for all , , and .
Remark 3 In Corollary 1, the delay-partition approach is not used to solve the synchronization problem of the MSCN (1). Therefore, the upper bounds of the time-varying delays and are and . From the analysis in Remark 2, we know that and can be divided into r and slices by using the delay-partition approach in Theorem 1. Therefore, the allowable upper bounds of the time-varying delays and of Corollary 1 are smaller than that of Theorem 1. That means conservatism of Theorem 1 is lower than that of Corollary 1.
4 Numerical example
Remark 4 From Examples 1-2, it is clear that and of Example 1 are larger than that of Example 2. That means allowable upper bounds of and of Example 1 are larger than that of Example 2. This further proves that the analysis in Remarks 2-3 is reasonable.
In this paper, we study global synchronization for a new class of MSCNs with mixed time-varying delays in the delay-partition approach. Sufficient conditions of global synchronization for the new class of MSCNs with mixed time-varying delays are derived by the new delay-partition approach. The advantage of the delay-partition approach is that the obtained results have lower conservatism. With two numerical examples, the theoretical results proposed are proved to be effective.
The work is supported by the Education Commission Scientific Research Innovation Key Project of Shanghai under Grant 13ZZ050, the Science and Technology Commission Innovation Plan Basic Research Key Project of Shanghai under Grant 12JC1400400, Education Department Scientific Project of Zhejiang Province under Grant Y201326804, and the Scientific and Technological Innovation Plan Projects of Ningbo City under Grants 2012B71011 and 2011B710038.
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