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 𝕊1 and target $𝕊k$. First we show that damping stabilizes the system when the energy is strictly below the threshold $2π$, 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 $k=1$.