Asymptotic Synchronization of Phase Oscillators with a Single Input

We consider asymptotic synchronization of multi- ple phase oscillators represented by first order reduced dynamics with a phase response curve (PRC) function. A single control input is employed to synchronize the oscillators. We design a feedback control that achieves asymptotic synchronization of the oscillators. We also show that the undesired equilibria of the closed-loop system are unstable, thereby guaranteeing almost global synchronization. We extend the design to address the scenarios where the control is symmetrically or asymmetrically bounded and where the coefficients in the PRC function are unknown constants. Simulation results are presented to validate the proposed control algorithms.