Exponential synchronization of the Kuramoto model with inertia and frustration under locally coupled network

We study the collective synchronized behavior of the Kuramoto model with inertia and frustration effects on a connected and symmetric network. We aim to establish sufficient frameworks for achieving complete frequency synchronization, taking into account initial configuration, small inertia and frustration, and large coupling strength. More precisely, we first demonstrate that the phase diameter will be uniformly bounded by a small value after a finite time. Then we prove that the frequency diameter exhibits exponential decay to zero. Our approach relies on a careful construction of energy functionals, which effectively control the dissipation of phase and frequency diameters.