Global Controllability and Stabilization for the Nonlinear Schrödinger Equation on Some Compact Manifolds of Dimension 3

We prove global internal controllability in large time for the nonlinear Schrödinger equation on some compact manifolds of dimension 3. The result is proved under some geometrical assumptions: geometric control and unique continuation. We give some examples where they are fulfilled on T³, S³, and S²× S¹. We prove this by two different methods, both inherently interesting. The first one combines stabilization and local controllability near 0. The second one uses successive controls near some trajectories. We also get a regularity result about the control if the data are assumed smoother. If the H¹ norm is bounded, it gives a local control in H¹ with a smallness assumption only in L². We use Bourgain spaces to solve the equation in H¹.