Control of the Grushin equation: non-rectangular control region and minimal time

TL;DR

This paper investigates the internal null-controllability of the Grushin equation, identifying the minimal control time for various non-rectangular regions using novel analytical methods.

Contribution

It introduces a new approach to determine minimal control time for the Grushin equation in non-rectangular regions, combining the fictitious control method and polynomial estimates.

Findings

Positive controllability results for certain regions

Negative results based on polynomial $L^2$ estimates

Characterization of minimal control time for non-rectangular regions

Abstract

This paper is devoted to the study of the internal null-controllability of the Grushin equation. We determine the minimal time of controllability for a large class of non-rectangular control region. We establish the positive result thanks to the fictitious control method and the negative one by interpreting the associated observability inequality as an $L^2$ estimate on complex polynomials.