Bilinear control of discrete spectrum Schrödinger operators

Zied Ammari (1) , Kais Ammari (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 FSM - Faculté des Sciences de Monastir

The bilinear control problem of the Schrödinger 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. Under an appropriate assumption on $B$, this allows to prove the approximate controllability of such systems when the uncontrolled Hamiltonian $A$ has a simple discrete spectrum.

Mots clés: Bilinear control; Approximate controllability; Schrödinger operators; Invariant subspaces; Topological irreducibility

Domaines: Equations aux dérivées partielles [math.AP]

Zied Ammari, Kais Ammari. Bilinear control of discrete spectrum Schrödinger operators. 2010. ⟨hal-00479584⟩