Bilinear control of discrete spectrum Schr\"odinger operators

Abstract

The bilinear control problem of the Schr\"odinger equation $i\frac{\partial}{\partial t}\psi(t)$ $=(A+u(t) B)\psi(t)$, where $u(t)$ is the control function, is investigated through topological irreducibility of the set $\mathfrak{M}=\{e^{-it (A+u B)}, u\in \mathbb{R}, t>0\}$ of bounded operators. This allows to prove the approximate controllability of such systems when the uncontrolled Hamiltonian $A$ has a simple discrete spectrum and under an appropriate assumption on $B$.