| Synchronization control of complex dynamical networks is a central topic in theinvestigation of the dynamics of complex networks and has been received much at-tention by a lot of scholars. The aim of this work is to investigate the synchronizationcontrol of several complex dynamic network models, which include the periodicallyintermittent control of neural networks, the synchronization control of undirectednetworks, the adaptive intermittent control for directed networks as well as thecluster synchronization and complete synchronization of community networks.In the first part, the synchronization of two kinds of neural networks is discussedunder periodically intermittent control. First of all, the lag synchronization of neuralnetworks with mixed delays is proposed. By introducing periodically intermitten-t control and applying analysis techniques such as mathematical induction methodand the reduction to absurdity, the criteria for exponentially lag synchronization arederived in terms of the infinite norm. Secondly, the exponential synchronization fora class of reaction-difusion neural networks with mixed delays is considered. Basedon p-norm, the conditions of exponentially complete synchronization are obtainedvia introducing multi-parameters and Lyapunov theory. Especially, a feasible syn-chronization region concerning control gain and the rate of control time is derivedunder a decentralized intermittent control. Besides, the efects of reaction difusionson synchronization are considered and we pointed out that it is beneficial to realizethe synchronization of neural networks when the difusion strengths are strengthenedor the difusion spaces are reduced. It is noted that some traditional restrictions ondelays and work time are removed in our results.The synchronization control of undirected networks is analyzed in the secondpart. First, the models of undirected network with adaptive coupling weights arestudied and the criteria of complete synchronization are given based on pinning con-trol, adaptive feedback laws and Barbalat lemma. In addition, the synchronizationof complex networks with node balance and single coupling distributed delays isdiscussed by using intermittent control. Some conditions are derived to ensure therealization of exponential synchronization by using analysis technique and inequalitymethods. Moreover, a feasible synchronization region for control gains and the rate of control time are also obtained. Diferent from traditional results, the proposedsynchronization states are un-decoupled states and the influence of inner couplingmatrix and the degree of nodes on the synchronized states is included.An open problem, that is, the problem of adaptive intermittent control forcomplex network is solved in third part. First, a model of directed network is es-tablished and the diference on model representations between directed network andun-directed network is pointed out. Some criteria are established to ensure theglobally exponential synchronization by imposing decentralized adaptive intermit-tent control on partial nodes and using inequality techniques. Besides, a feasiblesynchronization region concerning the rate of control time is given. Finally, twonumerical examples are provided to show the validity and efectiveness of the theo-retical results.In the forth part, two type of community networks are investigated. Firstly,the cluster synchronization is proposed via imposing feedback control and adaptivecontrol on partial communities and some sufcient conditions are obtained based onLyapunov theory. In all, this work answers several challenging problems in pinningcontrol of directed community networks:(a) What communities should be chosenas controlled candidates?(b) How many communities are needed to be controlled?(c) How large should the control gains be used in a given community network toachieve cluster synchronization? Unlike the previous results, each community isregarded as a whole and the informations of communities are included in the derivedcriteria. Additionally, the complete synchronization of a delayed community networkis considered in this part. By combining open loop control with feedback control anddesigning adaptive update law for coupling strength, some criteria are established toensure the community networks synchronize onto an any given smoothly dynamicalstate based on inequality techniques and Barbalat lemma. Finally, the validity andefectiveness of the theoretical results are approved by two numerical examples.
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