Controllability and stabilization of a degenerate/singular Schrödinger equation

Abstract

The aim of this paper is to prove controllability and stabilization properties for a degenerate and singular Schrödinger equation with degeneracy and singularity occurring at the boundary of the spatial domain. We first address the boundary control problem. In particular, by combining multiplier techniques and compactness-uniqueness argument, we prove direct and inverse inequalities for the associated adjoint system. Consequently, via the Hilbert Uniqueness Method, we deduce exact boundary controllability for the control system under consideration in any time T>0. Moreover, we investigate the stabilization problem for this class of equations in the range of subcritical coefficients of the singular potential. By introducing a suitable linear boundary feedback, we prove that the solution decays exponentially in an appropriate energy space.