Bilinear control of high frequencies for a 1D Schrödinger equation

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

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

In this article, we consider a 1D linear Schrödinger equation with potential V and bilinear control. Under appropriate assumptions on V , we prove the exact controllability of high frequencies, in H 3 , locally around any H 3-trajectory of the free system. In particular, any initial state in H 3 can be steered to a regular state, for instance a nite sum of eigen-functions of (−∆ + V). This fact, coupled with a previous result due to Nersesyan, proves the global exact controllability of the system in H 3 , with smooth controls, under appropriate assumptions.

Mots clés: Schrödinger equation; Quantum control; Control of PDE; bilinear control

Domaines: Mathématiques [math]

Karine Beauchard, Camille Laurent. Bilinear control of high frequencies for a 1D Schrödinger equation. Mathematics of Control, Signals, and Systems, 2017, 29 (2), pp.article 11. ⟨10.1007/s00498-017-0187-8⟩. ⟨hal-01333625⟩