Random adaptive control for cluster synchronization of complex networks with distinct communities

In this paper, we investigate the cluster synchronization for complex networks with time-varying delayed couplings, stochastic disturbance, and non-identical nodes in different clusters. Based on randomly occurring controllers, some Bernoulli stochastic variables are introduced to describe the controllers, then, a fraction of nodes in clusters, which have direct connections to the other clusters, is controlled, and the states of the whole dynamical networks can be globally forced to the objective cluster states. Sufficient conditions are derived to guarantee the realization of the mean square cluster synchronization pattern for all initial values by means of Lyapunov stability theory, Itô differential formula, and LMI approach. Besides, by designing the randomly occurring adaptive update law, some suitable control gains are obtained. Finally, numerical simulations are also given to demonstrate the effectiveness and validity of the main result.