Global approximate controllability of the Camassa-Holm equation by a finite dimensional force

In this article, we consider the Camassa-Holm equation posed on the periodic domain $\mathbb{T}.$ We show that in any positive time, the Camassa-Holm equation is globally approximately controllable in $H^s(\mathbb{T})$ for $s > \frac{3}{2}$ by a three dimensional external force. The proof is based on Agrachev-Sarychev approach in geometric control theory. In particular, our three dimensional control prevents a finite time blow up of the solution of Camassa-Holm equation in $H^s(\mathbb{T})$ for $s > \frac{3}{2}.$