Synchronization Of Complex Dynamic Networks With Distributed Delayed Impulsive Control

In recent years,with the development of information technology,complex dynamic networks have attracted more and more researchers’ attention.It is of great significance not only in theory but also in engineering field.Synchronization is an interesting dynamic behavior of complex dynamic networks,which is also very common in real life,such as forest firefly glow,applause,and beating of the heart.In this paper,the problem of synchronization of complex dynamic networks with various time delays is studied via the distributed delayed impulsive controller and pinning controller.Firstly,we mainly investigate the synchronization of complex dynamic networks with both system delay and coupled delay through distributed delayed impulsive control.Instead of constraining the impulsive weight and impulsive delay separately,a new concept of limiting average delayed impulsive weight is proposed to obtain more relaxed conditions.At the meantime,based on the Lyapunov theory and linear matrix inequality method,some certain flexible criteria of global exponential synchronization are given and the corresponding convergence rate is estimated.It is interesting to see that though impulsive weights work negatively,the networks can still achieve synchronization under some conditions.Namely,the delays in impulsive control are able to promote synchronization potentially.Then,we focus on the synchronization of complex dynamic networks with distributed delay.We extends the concept of limiting average delay weight to a more general form and expands the comparison principle by taking delayed impulses into account,where the constraint that system delays must exist between two consecutive impulsive instants is released.Without any restriction on system delays,sufficient conditions of global exponential synchronization for complex dynamic networks are obtained.Additionally,the corresponding convergence rate is also estimated.Finally,we study pinning impulsive synchronization for complex dynamic networks from a large delay perspective.Using the average impulsive interval method,we have derived some flexible synchronization criteria.It has shown that network synchronization can be achieved via controlling a small fraction of nodes.Particularly,by applying a strict comparison principle for impulsive delayed neural networks,the effects of the impulsive elasticity coefficient on the synchronization are revealed to be different from the case of delay-free neural networks.To be specific,one can verify that the increase of elasticity coefficient may bring desynchronizing impact to the original synchronized networks.More ulteriorly,the results will still hold if the delay is sufficiently large,which implies that the delay is independent of the size of impulsive interval.In addition,the synchronization region of different impulsive weight,impulsive interval and impulsive control proportion are described fully and the criterion that can be easily verified is given.