Hardy Inequalities, Observability, and Control for the Wave and Schrödinger Equations with Singular Potentials

Abstract

We address the question of exact controllability of the wave and Schrödinger equations perturbed by a singular inverse-square potential. Exact boundary controllability is proved in the range of subcritical coefficients of the singular potential and under suitable geometric conditions. The proof relies on the method of multipliers. The key point in the proof of the observability inequality is a suitable Hardy-type inequality with sharp constants. On the contrary, in the supercritical case, we prove that exact controllability is false.