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 T. We show that in any positive time, the Camassa–Holm equation is globally approximately controllable in Hs(T) for s > 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 Hs(T) for s > 3/2.