Further on Pinning Synchronization of Dynamical Networks with Coupling Delay

SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 1933-1952, August 2024.
Abstract. Though extensively studied, the longstanding pinning synchronization problem of dynamical networks with coupling delay has not been well solved until now. In this paper, we further investigate this problem. By proposing a system of functional differential inequalities, we derive synchronization criteria for dynamical networks with coupling delay under linear pinning control, where the threshold of the admissible delay and the control gain threshold are estimated. Since the estimated control gain threshold could be very large when the delay draws close to the delay threshold, we also use the adaptive pinning control scheme to avoid control gain estimation. Pinning synchronization criteria of networks under adaptive control are derived and the delay threshold is given. This is the first time that general coupling delay has been addressed in the field of pinning control. Finally, two numerical examples are presented to validate the theoretical results.