Global controllability of a nonlinear Korteweg-de Vries equation

We are interested in both the global exact controllability to the trajectories and in the global exact controllability of a nonlinear Korteweg-de Vries equation in a bounded interval. The local exact controllability to the trajectories by means of one boundary control, namely the boundary value at the left endpoint, has already been proved independently by L. Rosier and by O. Glass and S. Guerrero. We first introduce here two more controls: the boundary value at the right endpoint and the right member of the equation, assumed to be x-independent. Then we prove that, thanks to these three controls, one has the global exact controllability to the trajectories, for any positive time T . Finally, introducing a fourth control on the first derivative at the right endpoint, we get the global exact controllability, for any positive time T .