Adaptive exponential synchronization in mean square for Markovian jumping neutral-type coupled neural networks with time-varying delays by pinning control
Introduction
The neural networks (NNs) has been extensively investigated over the last decades for its practical applications in many areas including image processing and signal, automatic control, associative memories, combinatorial optimization, and so on (see [1], [2], [3], [4]). In NNs, one of the most interesting phenomena is synchronization, such as drive-response synchronization of neural networks, synchronization of biological neural networks, and so on. Motivated by both the basic science and the technological practice, the study of synchronization problems among an array of coupled neural networks has become an active research topic in the past few years (see [5], [6], [7]).
In the case where the coupled network cannot synchronize by itself, therefore many control techniques have been developed to drive the network to achieve synchronize, such as linear state feedback control [8], sampled data control [9], impulsive control [10] and adaptive control [11]. All of them have a common feature that the controller needs to be added to each node. But in practice, it is too difficult to add controllers to all nodes in a large-scale coupled network. To reduce the number of controlled nodes, pinning control is introduced, in which controllers are only applied to partial nodes. This case of control techniques has been earlier reported in paper [12], [13], [14], [15]. In addition, the adaptive pinning control method, which is utilized to get the appropriate control gains effectively, has received considerable research attention. An adaptive pinning control method is proposed in [16] to synchronize for a delayed complex dynamical network with free coupling matrix. Besides these, there are many literatures to study adaptive pinning control problems of networks [17], [18], [19].
On the other hand, time delays commonly exist in various systems [20], [21], [22], [23], because of the finite switching speeds of amplifiers and traffic congestions in signal transmission processes, which become one of the main sources for causing instability and poor performance of neural networks. So far, much efforts have been paid for analyzing synchronization behaviors of NNs with various types of time delays, such as constant time delays, time-varying delays, discrete and distributed delays. Neutral-type NNs are a special type of time delayed NNs, in which the time-delays occur not only in the system states but also in the derivatives of system states [24], [25], [26], [27]. Meanwhile, many NNs can happen abrupt changes in their structure and parameters, which can be described by a continuous time, finite state Markov chain [28], [29]. It is called Markovian jumping neutral neural networks when the neural network with neutral-type delays and Markovian jumping parameters. These kinds of systems are widely studied by many scholars. In [30], the stochastic stability problem of neutral-type neural networks with Markovian jumping parameters is considered. By using the adaptive control approach, the drive-response synchronization of neutral-type delayed neural networks is investigated in [31]. In [32], by using adaptive control method, the exponential synchronization of coupled neutral-type complex dynamical networks is considered, in which the adaptive controller is added to all nodes. To the best of our knowledge, the problem of adaptive pinning synchronization for neutral-type neural networks with Markovian switching parameters has received very little research attention.
In this paper, we concern with the analysis issue for adaptive pinning synchronization of neutral-type neural networks with Markovian switching parameters. By using Lyapunov stability theory and adaptive pinning control approach, several criteria are given to guarantee exponential synchronization of neutral-type coupled neural networks with Markovian switching parameters. Two numerical examples are also presented to show the effectiveness of the proposed method. The main contributions of this paper are as follows:
(1) A new class of Markovian jumping neutral-type neural networks is considered, the switching parameters are modeled as a continuous time, finite state Markov chain.
(2) A new adaptive pinning law, which depend on Markovian chain and error states, is designed.
The notations are quite standard. Throughout this paper, , Rn and denote the set of non-negative real numbers, n dimensional Euclidean space and the set of all real matrices, respectively. The superscript T denotes matrix transposition, denotes the trace of the corresponding matrix and In denotes an n dimensional identity matrix. stands for the Euclidean norm in Rn. is the largest eigenvalue of A, is the minimum eigenvalue of A; stands for the block diagonal matrix, ⊗ denotes the Kronecker product of matrix. Let be a complete probability space with a filtration satisfying the usual conditions (i.e. the filtration contains all P-null sets and is increasing and right continuous). denotes the family of all continuous functions from to Rn.
Section snippets
Model and preliminaries
Let be a right continuous Markovian chain in a complete probability space taking values in a finite state set with generator given by where and is the transition rate from i to j if , while .
In this paper, we consider the Markovian switching neutral-type coupled neural networks as follows:
Main result
Our object is to design an adaptive controller such that the neutral-type coupled neural network (6) can realize synchronization. Theorem 1 Suppose that Assumption 1, Assumption 2 hold. The controlled network (1) can be globally exponentially synchronized with the trajectory of s(t) for almost every initial data, if there exist positive constants , ρ, ξ1, ξ2, ξ3 and positive definite matrices , such that
Numerical simulation
In this section, we present two numerical simulations to illustrate the feasibility and effectiveness of our results. For given the probability transition matrix, a Markov chain can be generated. We consider the probability transition matrix as follows: and Markov chain r(t) is described in Fig. 1. The total error and the synchronization total error of the network are defined as , where .
Conclusion
In this paper, we have investigated the adaptive pining synchronization problem for an array of linearly coupled neutral-type neural networks with Markovian switching parameters by using adaptive pinning control. By utilizing Lyapunov stability theory and adaptive pinning control method, some novel conditions for synchronization are derived. Furthermore, a numerical example has verified the effectiveness of the presented method.
Acknowledgements
This work is supported by the Natural Science Foundation of China (11371125), the Natural Science Foundation of the Department of Education of Hunan Province (13A013), the Educational Department of Hunan Province of China (12B024, 13C127, 15C0243), the Youth Fund Project of the Humanities and Social Science Research for the Ministry of Education of China (14YJCZH173), the Science and Technology Research Key Program for the Education Department of Hubei Province of China (D20156001), the Science
Anding Dai received the master degree from ShenZhen University in 2011 and the Ph.D. degree in Control Theory and Control Engineering from Donghua University in 2014. Currently he is a lecturer at Hunan City University. His current research interests include the synchronization, control of neural networks and complex networks.
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Anding Dai received the master degree from ShenZhen University in 2011 and the Ph.D. degree in Control Theory and Control Engineering from Donghua University in 2014. Currently he is a lecturer at Hunan City University. His current research interests include the synchronization, control of neural networks and complex networks.
Wuneng Zhou received a first class B.S. degree from Huazhong Normal University in 1982. He obtained his Ph.D. degree from Zhejiang University in 2005. Now he is a professor in Donghua University, Shanghai. His current research interests include the stability, the synchronization, control of neural networks and complex networks.
Yuhua Xu received his Ph.D. degree in Control Theory and Control Engineering from Donghua University, PR China, in 2011. Currently he is a professor at Yunyang Teachers College. His research interests include robust control, chaos synchronization, and network control.
Cuie Xiao received her Ph.D. degree in Applied Mathematics from Central South University in 2011. Currently she is a professor at Hunan City University. Her current research interest is nonlinear differential equation.