Global controllability and stabilization of the wave maps equation from a circle to a sphere

TL;DR

This paper proves that wave maps from a circle to a sphere are globally controllable and stabilizable under certain energy bounds, extending previous semi-global results to full global results and analyzing minimal time control.

Contribution

It establishes the first global exact controllability results for wave maps from a circle to a sphere for dimensions greater than one, and characterizes the energy threshold for stabilization.

Findings

Global exact controllability for wave maps with target spheres of dimension > 1.

Energy threshold of 2π for stabilization via feedback laws.

Minimal time controllability for wave maps from a circle.

Abstract

Continuing the investigations started in the recent work [Krieger-Xiang, 2022] on semi-global controllability and stabilization of the $(1+1)$-dimensional wave maps equation with spatial domain $\mathbb{S}^1$ and target $\mathbb{S}^k$, where {\it semi-global} refers to the $2\pi$-energy bound, we prove global exact controllability of the same system for $k>1$ and show that the $2\pi$-energy bound is a strict threshold for uniform asymptotic stabilization via continuous time-varying feedback laws indicating that the damping stabilization in [Krieger-Xiang, 2022] is sharp. Lastly, the global exact controllability for $\mathbb{S}^1$-target within minimum time is discussed.