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.

中文翻译：

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Hunter-Saxton控制系统的全局渐近稳定

本文解决了周期设定下Hunter-Saxton方程的渐近稳定问题。首先设计反馈控制律，将控制状态区分为有利状态和不利状态。然后，我们通过Schauder型定点方案建立了闭环控制系统的局部适定性。我们还针对两种情况提出了一些能量估算。最后，通过引入Lyapunov泛函，针对Hunter-Saxton方程的控制问题，获得了全局渐近稳定化结果。