Delayed Pinning Control for Synchronization of Dynamical Networks

Though pinning control of dynamical networks with or without delay has been well explored in the past two decades, little attention has been paid to the delay originated from pinning control itself. This paper investigates the delayed pinning control scheme for lag synchronization and lag quasi-synchronization of dynamical networks. Two pinning control schemes with uniform delay and nonuniform delays are proposed, which are more practical than the widely-used delay-free pinning control scheme. For the two pinning control schemes, criteria for lag synchronization and lag quasi-synchronization of dynamical networks are derived by proposing a system of functional differential inequalities, and in the meantime, an upper bound of tolerable delays that allows synchronization is given. In particular, the ultimate bound of the synchronization error is estimated for lag quasi-synchronization. Finally, the theoretical results are demonstrated by two numerical examples.