Synchronizing a Weighted and Weakly-Connected Kuramoto-Oscillator Digraph With a Pacemaker

This paper is devoted to the stability conditions of synchronizing a network of Kuramoto oscillators with a pacemaker, where the coupling topology is a weighted and weakly connected digraph. Sufficient conditions to achieve phase agreement and frequency synchronization are addressed for identical and non-identical Kuramoto oscillators, respectively, which are related to the initial relative phase (with respect to the phase of the pacemaker) configurations and the communication topologies. Additionally, the lower bounds on exponential synchronization and phase-locking rates are obtained. Numerical examples illustrate the effectiveness of the derived sufficient conditions and the specified control strategies.