Remarks on the nonlinear stability of the Kuramoto model with inertia

In this short note, we present an a priori nonlinear stability estimate for the Kuramoto model with finite inertia in $\ell ^{\infty }$-norm under some a priori condition on the size of the phase diameter. As a direct corollary of our nonlinear stability estimate, we show that phase-locked states obtained in Choi, Ha, and Yun (2011) are orbital-stable in $\ell ^{\infty }$-norm, which means that the perturbed phase-locked state approaches the phase-shift of the given phase-locked state. The phase-shift is explicitly determined by the averages of initial phase and frequency distribution and the strength of inertia $m$.