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 $\Tot$, $S^3$ and $S^2\times S^1$. 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 $H^1$ norm is bounded, it gives a local control in $H^1$ with a smallness assumption only in $L^2$. We use Bourgain spaces