Stabilization and control for the nonlinear Schrödinger equation on a compact surface

In this paper, we study the stabilization property and the exact controllability for the nonlinear Schrödinger equation on a two dimensional compact Riemannian manifold, without boundary. We use a pseudo-differential dissipation. The proofs are based on a result of propagation of singularities and on recent dispersion estimates (Strichartz type inequalities) due to N. Burq, P. Gérard and N. Tzvetkov. © 2006 Springer-Verlag.