Semi-global controllability of a geometric wave equation

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 $k = 1$.

Keywords

Wave maps, semi-global controllability, quantitative stabilization

2010 Mathematics Subject Classification

Primary 35B40, 35L05. Secondary 93C20.

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Dedicated to Demetrios Christodoulou for his 70th birthday, in friendship and admiration

Received 17 February 2022

Received revised 18 August 2022

Accepted 24 September 2022

Published 18 July 2024