Randomly Occurring Cluster Synchronization of Complex Networks via Adaptive Pinning Control

The article studies the cluster synchronization for a kind of nonlinear coupled complex network with time-varying delay. Considering the networks may subject to certain uncertainties, the model of complex networks consisting of nonidentical systems with randomly occurring disturbance which described by Bernoulli stochastic variable is established. Secondly, a kind of pinning feedback controllers under randomly occurring disturbance is proposed in order to not only synchronize the systems in the same clusters but also weaken the mutual influence among clusters, which will be imposed on the systems in current cluster which have directed connections with the systems in the other clusters. Then, sufficient conditions for the realization of the cluster synchronization are derived in terms of the QUAD function class, the NCF function class and the Lyapunov stability theorem. Furthermore, the optimal feedback control gain is obtained by designing the adaptive updating laws. Finally, a numerical experiment is presented to illustrate the effectiveness of theoretical analysis.