Semi-global weak stabilization of bilinear Schr'odinger equations

摘要

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