Local Exact Controllability of a One-Dimensional Nonlinear Schrödinger Equation

Karine Beauchard (1, 2) , Horst Lange (3) , Holger Teismann (4)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 ENS Rennes - École normale supérieure - Rennes
3 Mathematisches Institut, Universität zu Köln
4 Department of Mathematics and Statistics

We consider a one-dimensional nonlinear Schrödinger equation, modeling a Bose--Einstein condensate in an infinite square-well potential (box). This is a nonlinear control system in which the state is the wave function of the Bose--Einstein condensate and the control is the length of the box. We prove that local exact controllability around the ground state (associated with a fixed length of the box) holds generically with respect to the chemical potential $\mu $, i.e., up to an at most countable set of $\mu $-values. The proof relies on the linearization principle and the inverse mapping theorem, as well as ideas from analytic perturbation theory.Read More: http://epubs.siam.org/doi/10.1137/140951618

Mots clés: Quantum systems; controllability of PDEs

Domaines: Equations aux dérivées partielles [math.AP]

Karine Beauchard, Horst Lange, Holger Teismann. Local Exact Controllability of a One-Dimensional Nonlinear Schrödinger Equation. SIAM Journal on Control and Optimization, 2016, 53 (5), pp.2781 - 2818. ⟨10.1137/140951618⟩. ⟨hal-01377135⟩