Rapid stabilization of a linearized bilinear 1-D Schrodinger equation

We consider the one dimensional Schrodinger equation with a bilinear control and prove the rapid stabilization of the linearized equation around the ground state. The feedback law ensuring the rapid stabilization is obtained using a transformation mapping the solution to the linearized equation on the solution to an exponentially stable target linear equation. A suitable condition is imposed on the transformation in order to cancel the non-local terms arising in the kernel system. The continuity and invertibility of the transformation follows from exact controllability of the linearized system. (C) 2017 Published by Elsevier Masson SAS.