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.
Categories: math.AP math.OC