A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule

Ugo Boscain (1, 2) , Marco Caponigro (3, 4) , Thomas Chambrion (4, 3) , Mario Sigalotti (2)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 GECO - Geometric Control Design
3 IECN - Institut Élie Cartan de Nancy
4 CORIDA - Robust control of infinite dimensional systems and applications

In this paper we prove an approximate controllability result for the bilinear Schrödinger equation. This result requires less restrictive non-resonance hypotheses on the spectrum of the uncontrolled Schrödinger operator than those present in the literature. The control operator is not required to be bounded and we are able to extend the controllability result to the density matrices. The proof is based on fine controllability properties of the finite dimensional Galerkin approximations and allows to get estimates for the $L^{1}$ norm of the control. The general controllability result is applied to the problem of controlling the rotation of a bipolar rigid molecule confined on a plane by means of two orthogonal external fields.

Mots clés: Galerkin method; Control of PDEs; Quantum Control; Bilinear Schrödinger equation

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Ugo Boscain, Marco Caponigro, Thomas Chambrion, Mario Sigalotti. A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule. Communications in Mathematical Physics, 2012, 311 (2), pp.423-455. ⟨10.1007/s00220-012-1441-z⟩. ⟨hal-00557742v5⟩