Synchronization of linearly coupled networks with delays via aperiodically intermittent pinning control

In this paper, we investigate the exponential synchronization problem for linearly coupled networks with delay by pinning a simple aperiodically intermittent controller. The network topology can be directed. Different from previous works, the intermittent control can be aperiodic. Two types of delay are considered. The first case is that the delay is time-varying and large, and in this case, there is no restriction imposed on the delay and the control (and/or rest) width. The other one is that the delay is small enough so that it is less than the minimum of control width. Different approaches are provided to investigate these two cases, and some criteria are given to realize exponential synchronization. Furthermore, by applying the adaptive approach to the second model, we establish a general adaptive theory for intermittent control, which can be applied not only to networks without time delay, but also to delayed networks, regardless of whether the intermittent control is periodic or aperiodic. Finally, the numerical simulations are given to verify the validness of the theoretical results.