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arXiv · 2307.08329

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

Jean-Michel Coron·Joachim Krieger·Shengquan Xiang
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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.

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Jean-Michel Coron, Joachim Krieger, Shengquan Xiang. 2023-07-17. Global controllability and stabilization of the wave maps equation from a circle to a sphere. https://arxiv.org/abs/2307.08329

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