Semi-global controllability of a geometric wave equation

TL;DR

This paper establishes semi-global controllability and stabilization of a 1+1 dimensional wave maps equation on a circle, identifying energy thresholds and adapting control methods to achieve low-energy exact controllability.

Contribution

It proves semi-global controllability and stabilization for the wave maps equation, addressing energy thresholds and adapting iterative control procedures for low-energy cases.

Findings

Damping stabilizes the system below energy threshold 2π.

Harmonic maps obstruct global stabilization.

Achieved low-energy exact controllability, optimal for k=1.

Abstract

We prove the semi-global controllability and stabilization of the $(1+1)$-dimensional wave maps equation with spatial domain $\mathbb{S}^1$ and target $\mathbb{S}^k$. First we show that damping stabilizes the system when the energy is strictly below the threshold $2\pi$, where harmonic maps appear as obstruction for global stabilization. Then, we adapt an iterative control procedure to get low-energy exact controllability of the wave maps equation. This result is optimal in the case $k=1$.