Aperiodically intermittent event-triggered pinning control on cluster synchronization of directed complex networks

This paper is dedicated to investigating the exponential cluster synchronization in a class of nonlinearly coupled complex networks with non-identical nodes and an asymmetrical coupling matrix. A novel aperiodically intermittent pinning control (APIPC) protocol is presented, which takes full account of the cluster-tree topology structure of the networks and pins only the nodes in the current cluster that have directional links to neighboring clusters. Since it is difficult to precisely determine the intermittent control instants and rest instants of APIPC in advance, the event-triggered mechanism (ETM) is thus proposed. Based on the concept of the minimal control ratio and the segmentation analysis method, sufficient requirements for realizing the exponential cluster synchronization are derived. Moreover, the Zeno behavior of ETM is excluded by rigorous analysis. Eventually, the effectiveness and advantages of the established theorems and control strategies are demonstrated by two numerical simulations.