Analysis and pinning control for generalized synchronization of delayed coupled neural networks with different dimensional nodes

Abstract

This paper investigate the generalized synchronization and pinning adaptive generalized synchronization for delayed coupled different dimensional neural networks with hybrid coupling, respectively. First, some sufficient conditions for reaching the generalized synchronization and pinning generalized synchronization of the considered network are acquired by using some inequality techniques and Lyapunov functional method. Second, because the precise parameter values of network cannot be obtained in some situations, we also purse the study on the generalized synchronization analysis and pinning control for the case of coupled different dimensional neural networks with parameter uncertainties. Third, two numerical examples are provided for substantiating the effectiveness of the derived results.

Introduction

In the past several decades, a great deal of attention has been paid to complex networks because of their potential applications to model and study many systems, e.g., global economic markets, metabolic systems, communication networks, and so on [1], [2], [3], [4], and many meaningful results about the dynamical behaviors including stability, synchronization and passivity for complex networks have been obtained [5], [6], [7], [8], [9], [10], [11]. Coupled neural networks (CNNs), a special class of complex networks, have been applied into many scientific and engineering fields, such as harmonic oscillation generation, pattern recognition, secure communication and image encryption, thus have attracted considerable attention from lots of researchers recently [12], [13], [14], [15], [16], [17], [18]. In [16], Tang et al. examined the mean square exponential synchronization of delayed impulsive CNNs, several criteria have been established based on the comparison principle and Lyapunov stability theory. The problem of synchronization for randomly delayed CNNs with Markovian jumping was solved by Yang et al. [18], and several synchronization criteria were established by making use of some inequalities and Lyapunov functional method. As is well known, the exact values of parameters in CNNs usually cannot be obtained in many cases because of environmental noises and model errors. For better analysis of dynamical behaviors of CNNs, it is very necessary and meaningful to investigate the robust dynamical behaviors of CNNs with parameter uncertainties, and some results on this topic have been emerged in [19], [20], [21], [22]. With the help of stability theory and impulsive control method, Li et al. [21] gained some robust synchronization conditions for CNNs with parameter uncertainties.

To the best of our knowledge, many CNNs cannot realize synchronization by themselves in practical situations, so suitable controllers should be applied for achieving synchronization in these networks. So far, a large number of controllers have been developed such that CNNs achieve synchronization, for instance, pinning control [23], intermittent control [24], impulsive control [25], fuzzy control [26] and so on. Generally speaking, it is unrealistic to apply control schemes to all nodes in CNNs when the size of the considered network is extremely huge. Therefore, pinning control, in which a small fraction of nodes are chosen to be controlled, has provoked a rapid growing attention recently. From the engineering point of view, pinning control scheme is a more effective way comparing with other control strategies due to greatly reducing the number of controlled nodes and consumption. To date, there have been some results about CNNs by exploiting pinning control methods [27], [28], [29], [30], [31]. In [29], Wang et al. proposed two kinds of CNNs with state coupling and spatial diffusion coupling, respectively, and acquired some synchronization conditions via pinning control technique and Lyapunov functional method. However, very few researchers considered CNNs with parameter uncertainties by utilizing pinning control strategies [32]. Zheng and Cao [32] focused on robust synchronization for CNNs with mixed delays, and derived some synchronization conditions by means of intermittent pinning control and Halanay inequality.

However, in the most existing results [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], there is a common assumption that the nodes composing the networks are identical. Nevertheless, the assumption that the nodes are completely identical is not realistic in many real-world networks [33], such as in the neural networks, where the internal neurons in the nervous system are impossible to be totally identical due to the differences of the parameters. As is well known, many researchers have considered the networks with non-identical nodes of the same state dimension [34], [35], [36], [37], [38]. By designing some controllers, the finite-time synchronization problem of complex networks with non-identical nodes that have discontinuous functions was concerned by Yang et al. in [37]. Unfortunately, the non-identical nodes in the above-mentioned literatures [33], [34], [35], [36], [37], [38] have the same state dimension. However, a network with different dimensional non-identical nodes can reflect more real complex networks than those with non-identical nodes of the same state dimension. Therefore, it is worthwhile to study the dynamical behaviors of complex networks with different dimensional non-identical nodes, and some results on this topic have been reported [39], [40], [41], [42], [43], [44]. In [43], the authors devoted to studying the complex networks with nodes of different dimensions, and established several synchronization criteria by designing suitable controllers. Regrettably, there exists very little research on CNNs with different dimensional nodes [45]. By designing decentralized controllers and utilizing matrix inequalities, the stabilization condition for CNNs with different dimensional nodes was derived by Tan in [45]. Therefore, it is necessary and important to devote more efforts on the CNNs with different dimensional nodes. As far as we know, the model of CNNs with different dimensional nodes and parameter uncertainties and the synchronization problem of such network via pinning control method have never been studied.

From above discussions, we deal with the generalized synchronization and pinning generalized synchronization problems for delayed CNNs (DCNNs) with different dimensional nodes in this paper. There are three main novelties in our work. First, the models of DCNNs consisting of different dimensional nodes with and without parameter uncertainties are presented, respectively. Second, several criteria are established to guarantee the generalized synchronization of DCNNs with and without parameter uncertainties by virtue of Lyapunov functional method and some inequality techniques. Third, by designing appropriate pinning adaptive controllers, we derive some sufficient conditions to ensure pinning generalized synchronization for these two types of networks.