On the bilinear control of the Gross-Pitaevskii equation

TL;DR

This paper investigates the controllability of the Schrödinger equation with harmonic potential, demonstrating how smoothing effects can impose limitations on control capabilities in both linear and nonlinear cases.

Contribution

It provides new insights into how space-time smoothing effects influence the controllability of the Schrödinger equation with harmonic potential.

Findings

Smoothing effects can create obstructions to controllability.

Examples illustrate limitations in controlling the Schrödinger equation.

Analysis applies to both linear and nonlinear cases.

Abstract

In this paper we study the bilinear-control problem for the linear and non-linear Schr{\"o}dinger equation with harmonic potential. By the means of different examples, we show how space-time smoothing effects (Strichartz estimates, Kato smoothing effect) enjoyed by the linear flow, can help to prove obstructions to controllability.