Synchronization Analysis Of Two Classes Of Complex Dynamical Network With Time Delay Couplings

Complex network is usually composed of a large number of interconnected nodes,in which each node has specific features and functions.In recent years,complex network has attracted much attentation by many scholars from different research fields.Synchronization of complex network means that the states of all nodes are consistent with the state of the target along time.Since synchronization in complex network not only can well explain many natural phenomena but also has many potential applications,it has been a remarkable topic in the study field of complex network.Due to the increasing complexity of network structure,the influence of time delay on complex network is inevitable,which will affect the synchronizability of a given complex network.Nowadays,much attention has been drawn to investigate the synchronization problems of complex network with time-delayed coupling.On the basis of the analysis of a large number of domestic and foreign literatures,by using the delay decomposition approach,selecting the appropriate Lyapunov-Krasovskii functional and linear matrix inequality(LMI)technique the synchronization problems for two classes of complex networks with coupling time-delay were studied.Specifically,the main contents of this paper are as follows:(1)The synchronization stability problem for a class of general complex networks with interval time-varying coupling delay and delay in the dynamical node is investigated.By using the delay-partitioning approach,the delay interval is divided into two identical sub-intervals and adjustable sub-intervals.By constructing two appropriate Lyapunov-Krasovskii functionals,several less conservative delay-dependent synchronization stability criteria are proposed in terms of linear matrix inequalities.Finally,a numerical example is given to show the effectiveness of the proposed results.(2)The synchronization stability of neutral-type complex dynamical network with interval time-varying coupling delay is studied.Firstly,the problem of synchronization stability is transformed into the stability analysis over the zero solution.By dividing the delay interval into two equidistant subintervals and constructing two appropriate Lyapunov-Krasovskii functionals,usefuling time-varying delays information and mutual convex combination inequality technique,combined with the method of free weighting matrices,several less conservative delay-dependent synchronization stability criteria are proposed in terms of linear matrix inequalities.The validity of the results is verified by numerical examples.(3)A simple linear state feedback controller is introduced to force the network to synchronize while the whole network cannot achieve synchronization by itself.The synchronization control problem is transformed into the stability analysis of the error system.Some sufficient synchronization conditions for the controlled network are derived.The minimum feedback gain can be determined based on the proposed conditions.Numerical examples are given to demonstrate the effectiveness and less conservatism of the proposed method.