Global asymptotic stabilization of the Hunter-Saxton control system

Abstract

In this paper, the asymptotic stabilization problem of the Hunter-Saxton equation in the periodic setting is solved. We first design feedback control law, and distinguish the controlled state into favorable situation and unfavourable situation. Then, we establish the local well-posedness for the closed-loop control system by the means of a fixed-point scheme of Schauder’s type. We also present some energy estimates with respect to two situations. Finally, by introducing a Lyapunov functional, we obtain global asymptotic stabilization result for the control problem of Hunter-Saxton equation.