Convergent Behaviors For Several Complex Network Systems

The stability of neural networks, the synchronization of coupled networks and the consen-sus of multi-agent systems all appear convergent behaviors of the nodes in complex dynamical networks. In recent years, many researchers have always paid much attention to the study of these aspects, and simultaneously applied the obtained theoretical achievements to many areas, such as engineering technology, biology, computer science, sociology, and so on. Some further researches for these three aspects will be given in this paper based on some precious works. Since time delays are unavoidably existed in the information processing of nodes in many real networks, this dissertation focuses on the systematical study of delayed systems. The main contents in details are organized as follows:1、The global exponential stability of two types of high-order delayed BAM neural net-works is investigated. Firstly, a class of higher-order delayed stochastic BAM neural networks with reaction-diffusion term was proposed. Some global exponential stability criteria are ob-tained by using the theory of stochastic differential equation, M-matrix and some well-known inequalities. The obtained criteria are based on p-norm (p≥2) and dependent on the reaction-diffusion terms, which improve some previous results. Secondly, the exponential stability is studied for a type of Markovian jumping high-order BAM neural networks with time-varying delays. By using Lyapunov functional method and linear matrix inequality, a class of easily verified criterion is obtained. Moreover, several comparisons and a numerical example are given respectively to show the effectiveness and the feasibility of this proposed results.2、The problem of global exponential synchronization for the complex network consist of N delayed stochastic neural networks is discussed. The topology graph of the associated coupled neural networks is directed and switched. Mixed time-varying delays and impulsive effects are both taken into account when modeling the dynamical behavior of each neural network. Based on single pinning impulsive control method, the global exponential synchronization are investigated for the case of |μr|< 1 and |μr|>1, respectively. By means of multiple Lyapunov functional method of switched systems, impulsive delayed differential equation theory, Halanay inequality as well as linear matrix inequality, some global exponential synchronization criteria are given, which show theoretically that the global exponential synchronization can be achieved even if only a single impulsive controller is exerted. It is worthy to be mentioned, the way in which switching and impulse happen is assumed to satisfy some looser conditions, and the derivative of the time-varying delays need not to be existed.3、The cluster synchronization of delayed complex networks with non-identical nodes is investigated based on pinning control and adaptive feedback control. The associated com-plex networks have cooperative-competitive outer coupling mechanism and the inner coupling of each node is nonlinear. By using pinning control, adaptive feedback control, Lyapunov functional method and some inequality techniques, some sufficient conditions of cluster syn-chronization for this class of complex networks are obtained. Furthermore, an effective scheme for choosing pinned nodes is proposed based on the given synchronization criteria. It is wor-thy to be mentioned that the outer coupling matrices are not assumed to be symmetric or irreducible.4、The leader-following consensus problem for delayed multi-agent systems with nonlin-ear dynamics is investigated. Firstly, a consensus protocol with directed switched topology and symmetric communication delay is designed for a class of first-order multi-agent systems. Moreover, a pinning controller based on the information of the virtual leader is exerted on this consensus protocol. By using the multiple Lyapunov functional method, the property of irreducible matrix and linear matrix inequality, some sufficient leader-following consensus criteria are obtained, and simultaneously the optimization problem for allowable maximum communication delay is proposed. Secondly, the leader-following consensus problem for a class of second-order multi-agent systems is further studied, and some leader-following consensus criteria are given. It is worthy to be mentioned that the connectivity condition of the associ-ated topology structure is looser to some extent, and the derivative of the time-varying delay is not necessary less than 1.