Global Controllability of the Kawahara Equation at Any Time

In this article, we prove that the nonlinear Kawahara equation on the periodic domain $\mathbb{T}$ (the unit circle in the plane) is globally approximately controllable in $H^s(\mathbb{T})$ for $s \in \mathbb{N}$, at any time $T > 0$, using a two-dimensional control force. The proof is based on the Agrachev-Sarychev approach in geometric control theory.