Local controllability of 1D Schr\"odinger equations with bilinear control and minimal time

TL;DR

This paper investigates the local controllability of 1D Schrödinger equations with bilinear control, establishing conditions under which controllability is possible in large time but not in small time, linked to the second order term behavior.

Contribution

It provides a general framework explaining when local controllability is achievable in large time but not in small time for bilinear Schrödinger equations.

Findings

Controllability holds in large time under certain conditions.

A positive minimal time is necessary for controllability in some cases.

Second order term behavior determines the minimal controllability time.

Abstract

We consider a linear Schr\"odinger equation, on a bounded interval, with bilinear control. Beauchard and Laurent proved that, under an appropriate non degeneracy assumption, this system is controllable, locally around the ground state, in arbitrary time. Coron proved that a positive minimal time is required for this controllability, on a particular degenerate example. In this article, we propose a general context for the local controllability to hold in large time, but not in small time. The existence of a positive minimal time is closely related to the behaviour of the second order term, in the power series expansion of the solution.