Cluster Synchronization for Multi-Weighted and Directed Complex Networks via Pinning Control

In this brief, we investigate the cluster synchronization of multi-weighted complex networks with pinning control. We assume that the network contains multiple coupling matrices, and they can be all asymmetric (including the symmetric case), which is more practical in the real world. For multiple asymmetric coupling matrices, the Lyapunov function is hard to design. We firstly define a candidate region for every matrix. Then if these regions are not too far, we can find (at least) a common vector lying in each region. This method has geometrical meaning, so it is different from previous LMI method. That is to say, we can design a new Lyapunov function with a vector combining all normalized left eigenvectors (NLEVec) corresponding to the zero eigenvalue of coupling matrices. With some sufficient conditions presented accordingly, the realization of the cluster synchronization is guaranteed. Especially, we carefully discuss the case with three coupling matrices. The adaptive rule for cluster synchronization is also designed. Finally, numerical simulations are given to demonstrate our results' effectiveness.