| In this thesis, we discuss the synchronization behaviors of some complex dynam-ical networks.As far as the research contents are concerned, the thesis contains two parts. In the former, it mainly focuses on the studies of the synchronization stability problems for complex dynamical networks with various time delays, including constant time delays, interval time delays, time delays with known probability and mixed delays. By utilizing Lyapunov stability theory and linear matrix inequalities(LMIs) and so on, the corresponding synchronization stability criteria are achieved. In the latter, the adaptive synchronization problems for delayed complex networks are considered.As far as the research subjects are concerned, we develop three new models in this thesis.(1) Establish a probability-distribution-dependent model for a class of delayed complex networks and provide corresponding cluster synchronization sta-bility criteria. In the existing references, the time delays in complex networks are deterministic, the corresponding criteria were derived based only the information of variation range of the time delays, but when it comes to the case that some values of the time delays are very large but the probabilities of the delays taking such large values are very small, the existing methods may lead to more conservative results. Considering probability distributions of time delays, a new model is introduced by translating the distribution probabilities of the time delays into parameter matri-ces of the translating systems. Consequently, by combining the Lyapunov method and LMI technique, some delay-distribution-dependent stability criteria of delayed complex networks are achieved.(2) Establish a class of stochastic complex networks with Markovian jump and probabilistic interval time-varying delays, all matrices contain Markovian jumping parameters in our model, the stochastic coupling term and stochastic disturbance are investigated in order to reflect more realistic dynam-ical behaviors of complex networks that are affected by noisy environment.(3) The adaptive synchronization control problem of complex dynamical networks with non-delayed and delayed nonlinearly coupled nodes is investigated, and each node is a general Lur’e system, the network model considered can represent both the di-rected and undirected weighted networks. Based on the Lyapunov stability theory and property of the coupling matrix, the novel delay-independent nonlinear feedback controllers are designed to globally synchronize the networks under the assumption that coupling coefficients are known. When coupling coefficients are unavailable, the delay-independent adaptive controllers are introduced to guarantee the global syn-chronization of the uncertain networks. Moreover, the nonlinearly coupled functions may not satisfy the Lipschitz conditions.As far as the analysis methods are concerned, in chapter2, we investigate the synchronization stability of general continuous(discrete) complex dynamical networks with constant time delays, in order to reduce the conservatism, the delay is divided into several segments, by employing a novel delay-segment-dependent Lyapunov func-tional and the free-weighting matrix technique, some new-dependent synchronization stability conditions are derived in terms of linear matrix inequalities, which can be solved efficiently by the LMI toolbox. This new criteria based on a delay fractioning approach can reduce the conservation as the delay fractioning number increases. In chapter3, the synchronization stability of general continuous(discrete) complex dy-tamical networks with interval time delays. Basing on a piecewise analysis method, the variation interval of the time delay firstly divided into several subintervals, by checking the variation of derivative of a Lyapunov functional in every subinterval, in order to reduce the conservatism, we have used the free-weighting matrix technique and the convexity in the matrix inequality, when the number of the divided subinter-vals d=1, d=2and d=3, several new delay-dependent synchronization stability criteria are derived respectively. More, the corresponding criteria can provide much less conservative results with the increasing of the number of the divided subinter-vals. The free-weighting matrix technique and the convexity in the matrix inequality are also used in other chapters.More specifically, the main contents of this thesis are as follows:(1) The synchronization stability problem is investigated for a class of contin-uous/discrete complex dynamical network with time delays. In order to reduce the conservatism, the delay is divided into several segments, by employing a novel delay-segment-dependent Lyapunov functional, some new delay-dependent synchronization stability conditions are derived in terms of linear matrix inequalities, which can be solved efficiently by the LMI toolbox. This new criteria based on a delay fractioning approach can reduce the conservation as the delay fractioning number increases.(2) The synchronization stability problem of continuous/discrete complex dy-namical networks with time-varying delays is investigated based on a piecewise anal-ysis method, the variation interval of the time delay is firstly divided into several subintervals, by checking the variation of derivative of a Lyapunov functional in ev-ery subinterval, several new delay-dependent synchronization stability conditions are derived in the form of linear matrix inequalities, which are easy to be verified by the LMI toolbox. Some numerical examples show that, when the number of the divided subintervals increases, the corresponding criteria can provide much less conservative results.(3) We investigate the cluster synchronization stability problem for a class of complex dynamical networks with stochastic nonlinearities and probabilistic inter-val time-varying delays. The stochastic nonlinearities are considered here to reflect more realistic dynamical behaviors of complex networks, the delay in this paper is assumed to be random and its probability distribution is known a prior. Based on the stochastic analysis techniques and the properties of the Kronecker product, delay-dependent cluster synchronization stability criteria are derived in the form of linear matrix inequalities which can be readily solved by using the LMI toolbox in Matlab, the solvability of derived conditions depends on not only the size of the delay, but also the probability of the delay taking values in some intervals. At last, a numerical example is given to illustrate the feasibility and effectiveness of the proposed method.(4) The paper investigates the synchronization stability problem for a class of complex dynamical networks with Marvovian jumping parameters and mixed time delays. The complex networks consist of m modes and the networks switch from one mode to another according to a markovian chain with known transition probability. The mixed time-delays are composed of discrete and distributed delays, the discrete time delay is assumed to be random and its probability distribution is known a prior. In terms of the probability distribution of the delays, the new type of system model with probability-distribution-dependent parameter matrices is proposed. Based on the stochastic analysis techniques and the properties of the Kronecker product, delay-dependent synchronization stability criteria in the mean square are derived in the form of linear matrix inequalities which can be readily solved by using the LMI toolbox in Matlab, the solvability of derived conditions depends on not only the size of the delay, but also the probability of the delay taking values in some intervals.(5)This paper investigates the global synchronization of general complex dy-namical networks with non-delayed and delayed nonlinearly coupled nodes, and each node is a general Lur’e system, the networks model considered can represent both the directed and undirected weighted networks. Based on the Lyapunov stability theory and property of the coupling matrix, the novel delay-independent nonlinear feedback controllers are designed to globally synchronize the networks under the assumption that coupling coefficients are known. When coupling coefficients are unavailable, the delay-independent adaptive controllers are introduced to guarantee the global synchronization of the uncertain networks. Two numerical examples of directed and undirected delayed dynamical networks are given, respectively, using Chua’s circuits system as the nodes of the networks, which demonstrate the effectiveness of proposed results. |
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