Exact Controllability for the Schrödinger Equation

The exact controllability of Schrödinger equation in bounded domains with Dirichlet boundary condition is studied. Both the boundary controllability and the internal controllability problems are considered. Concerning the boundary controllability, the paper proves the exact controllability in $H^{ - 1} (\Omega )$ with $L^2 $-boundary control. On the other hand, the exact controllability in $L^2 (\Omega )$ is proved with $L^2 $-controls supported in a neigh-borhood of the boundary. Both results hold for an arbitrarily small time. The method of proof combines both the HUM (Hilbert Uniqueness Method) and multiplier techniques.