Finite-time And Fixed-time Synchronization Of Discontinuous Complex Dynamical Networks

Complex networks are ubiquitous in real life and the synchronization of complex networks is a widespread phenomenon in nature that cannot be ignored.Recently,the asymptotic synchronization and control of complex networks have attracted much attention from scholars.However,in the real complex networks of physics,engineering and other fields,the finite-time and fixed-time synchronization and control of networks are more realistic and applied values although the related work is just beginning and the related results are very few.In view of this,some related theories,including nonsmooth analysis,differential inclusion,differential equations with discontinuous right-hand sides and impulsive differential equations,are comprehensively used to investigate the finite-time and fixed-time synchronization of complex networks with discontinuous right-hand sides as well as the finite-time and fixed-time synchronization of complex networks with impulsive effects.In the first part,the finite-time and fixed-time synchronization of complex networks with discontinuous right-hand sides are discussed under a unified control framework design.First of all,in the sense of Filippov solutions,nonsmooth analysis,differential inclusion and the method of contradiction are utilized to analyze the finite-time and fixed-time synchronization of dynamical systems with discontinuous right-hand sides and some criteria are established.Additionally,a new control strategy is designed for a class of complex networks with discontinuous node dynamics.Under the unified control law,some criteria are derived to realize the finite-time and fixed-time synchronization of the networks by means of the established stability theorem and the theory of differential equations with discontinuous right-hand sides.It is pointed out from the criteria that whether the networks are finite-time synchronized or fixed-time synchronized depends on the value of a key parameter in the control strategy under the unified controller and the same conditions.Furthermore,the asymptotical,exponential and finite-time synchronization of the networks are discussed by designing another kind of unified control strategy.Finally,some numerical simulations are given to show the effectiveness of the theoretical results.Considering the impulsive phenomenon in the real networks,the finite-time synchronization of complex dynamical networks with impulsive effects is studied in the second part.Firstly,an impulsive differential inequality of finite-time stability is established and an upper estimation of the settling time is obtained via using the comparison principle and mathematical induction.In addition,the continuous control strategy is designed for a class of impulsive complex networks.By applying the established differential inequality and the theory of impulsive differential equations,the criteria of finite-time synchronization and the estimation of the settling time are derived in sense of the 1-norm and 2-norm.The validity and effectiveness of the theoretical results are eventually verified by numerical examples and simulations.In the third part,the fixed-time synchronization is investigated for complex dynamical networks with impulsive effects.Firstly,by means of the comparison principle,the theory of average impulsive interval and the classified discussion,a theorem of fixed-time stability is derived for impulsive nonlinear systems,and an optimal estimation is obtained for the settling-time independent of the initial values of systems.Moreover,a discontinuous control strategy is designed for the impulsive complex networks.By applying the established stability theorem and the theory of impulsive differential equations,the criteria of fixed-time synchronization for the networks and the estimation of the settling time are derived.To show the feasibility of the obtained results,some numerical examples with simulations are provided.