Boundary controllability for the semilinear Schrödinger equation

We study the boundary exact controllability for the semilinear Schrodinger equation defined on an open, bounded, connected set Ω of a complete, n-dimensional, Riemannian manifold M with metric g. We prove the local exact controllability around the equilibria under some checkable geometrical conditions. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and equilibria of the semilinear Schrodinger equation. We then establish the globally exact controllability in such a way that the state of the semilinear Schrodinger equation moves from an equilibrium in one location to an equilibrium in another location.