Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control

Karine Beauchard (1, 2) , Camille Laurent (3)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 ENS Rennes - École normale supérieure - Rennes
3 LJLL - Laboratoire Jacques-Louis Lions

We consider a linear Schrödinger equation, on a bounded interval, with bilinear control, that represents a quantum particle in an electric field (the control). We prove the controllability of this system, in any positive time, locally around the ground state. Similar results were proved for particular models (by the first author and with J.M. Coron), in non optimal spaces, in long time and the proof relied on the Nash-Moser implicit function theorem in order to deal with an a priori loss of regularity. In this article, the model is more general, the spaces are optimal, there is no restriction on the time and the proof relies on the classical inverse mapping theorem. A hidden regularizing effect is emphasized, showing there is actually no loss of regularity. Then, the same strategy is applied to nonlinear Schrödinger equations and nonlinear wave equations, showing that the method works for a wide range of bilinear control systems.

Mots clés: control of partial differential equations; bilinear control; Schrödinger equation; quantum systems; wave equation; inverse mapping theorem

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

Karine Beauchard, Camille Laurent. Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control. Journal de Mathématiques Pures et Appliquées, 2010, 94 (5), pp.520 - 554. ⟨10.1016/j.matpur.2010.04.001⟩. ⟨hal-00448504⟩