Prescribed-Time Synchronization of Second-Order Kuramoto Oscillators

This paper addresses the problem of frequency and phase synchronization in a network of second-order Kuramoto oscillators with inertia. Distinct from traditional asymptotic synchronization, which theoretically requires infinite time, a prescribed-time control scheme is proposed to force the system states to synchronize within a user-defined duration. Specifically, a coordinate transformation is introduced to eliminate the drift induced by heterogeneous natural power injections, and a time-varying scaling function is incorporated into the controller design to regulate the convergence rate. Based on a constructed Lyapunov function containing cross-terms, rigorous theoretical analysis ensures that the synchronization error converges to zero at the prescribed time regardless of initial conditions. Finally, numerical simulations are provided to demonstrate the effectiveness and robustness of the proposed approach.