Continuing the investigations started in the recent work (Krieger and Xiang in Semi-global controllability of a geometric wave equation, 2022. arXiv:2205.00915) 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 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 and Xiang (2022) is sharp. Lastly, the global exact controllability for \(\mathbb {S}^1\) -target within minimum time is discussed.
