Semi-global weak stabilization of bilinear Schrödinger equations

Abstract

We consider a linear Schrödinger equation, on a bounded domain, with bilinear control, representing a quantum particle in an electric field (the control). Recently, Nersesyan proposed explicit feedback laws and proved the existence of a sequence of times $(t_n{)}_{n\in \mathbb{N}}$ for which the values of the solution of the closed loop system converge weakly in $H^2$ to the ground state. Here, we prove the convergence of the whole solution, as $t\to +\infty$. The proof relies on control Lyapunov functions and an adaptation of the LaSalle invariance principle to PDEs.