Local controllability of 1D linear and nonlinear Schr\"odinger equations with bilinear control

TL;DR

This paper proves local controllability of 1D linear and nonlinear Schr"odinger equations with bilinear control in optimal spaces and arbitrary positive time, using classical inverse mapping techniques.

Contribution

It extends controllability results to more general models and spaces, removing previous restrictions on time and regularity, and demonstrates the method's applicability to nonlinear equations.

Findings

Controllability holds in any positive time around the ground state.

The proof uses the classical inverse mapping theorem without loss of regularity.

The approach applies to nonlinear Schr"odinger and wave equations.

Abstract

We consider a linear Schr\"odinger 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 Schr\"odinger equations and nonlinear wave equations, showing that the method works for a wide range of bilinear control systems.