Complete entrainment of Kuramoto oscillators with inertia on networks via gradient-like flow

We study the asymptotic complete entrainment of Kuramoto oscillators with inertia on symmetric and connected network. We provide several sufficient conditions for the asymptotic complete entrainment in terms of initial phase-frequency configurations, strengths of inertia and coupling, and natural frequency distributions. For this purpose, we reinterpret the Kuramoto oscillators with inertia as a second-order gradient-like flow, and adopt analytical methods based on several Lyapunov functions to apply the convergence estimate studied by Haraux and Jendoubi [21]. Our approach does not require any spectral information of the graph associated with the given network structure.