Rough controls for Schroedinger operators on 2-tori

TL;DR

This paper demonstrates that on rectangular 2-tori, the Schrödinger equation can be controlled and observed using rough, measurable sets of positive measure, extending previous results to less regular control functions.

Contribution

It extends controllability and observability results for the Schrödinger equation on 2-tori to rough, measurable sets, using methods from Bourgain and previous work.

Findings

Any positive measure Lebesgue set can be used for observability.

Controllability is achievable with rough localization and control functions.

Results apply to square tori over sufficiently long times.

Abstract

The purpose of this note is to use the results and methods of our previous work with Bourgain to obtain control and observability by rough functions and sets on rectangular 2-tori. We show that any Lebesgue measurable set of positive measure can be used for observability for the Schroedinger equation. This leads to controllability with rough localization and control functions. For non-empty open sets this follows from the results of Haraux '89 and Jaffard '89 while for square tori and sufficiently long times this can be deduced from the results of Jakobson '97.