Global Approximate Controllability of the Camassa-Holm Equation by a Finite Dimensional Force

In this paper, we consider the Camassa-Holm equation posed on the periodic domain $\mathbb{T}$. We show that Camassa-Holm equation is globally approximately controllable by three dimensional external force in $H^s(\mathbb{T})$ for $s>\frac{3}{2}$ . The proof is based on Agrachev-Sarychev approach in geometric control theory.