Time optimal observability for Grushin Schrödinger equation

Burq, Nicolas; Sun, Chenmin

doi:10.2140/apde.2022.15.1487

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Time optimal observability for Grushin Schrödinger equation

Nicolas Burq and Chenmin Sun

Vol. 15 (2022), No. 6, 1487–1530

DOI: 10.2140/apde.2022.15.1487

Abstract

We consider the two-dimensional Grushin Schrödinger equation posed on a finite cylinder $Ω = {(- 1, 1)}_{x} \times 𝕋_{y}$ with Dirichlet boundary condition. We obtain sharp observability by any horizontal strip, with the optimal time $T_{*} > 0$ depending on the size of the strip. Consequently, we prove the exact controllability for the Grushin Schrödinger equation. By exploiting the concentration of eigenfunctions of a harmonic oscillator at $x = 0$ , we also show that the observability fails for any $T \leq T_{*}$ .

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Keywords

controllability, observability, subelliptic, Schrödinger equation, semiclassical analysis

Mathematical Subject Classification 2010

Primary: 35Q41, 35Q93, 93B07

Milestones

Received: 9 October 2019

Revised: 2 February 2021

Accepted: 19 March 2021

Published: 10 November 2022

Authors

Nicolas Burq
Université Paris-Sud, Université Paris-Saclay and Institut Universitaire de France, Laboratoire de Mathématiques d’Orsay, UMR 8628 du CNRS Orsay France

Chenmin Sun
Université de Cergy-Pontoise, Laboratoire de Mathématiques AGM, UMR 8088 du CNRS Cergy-Pontoise France

Vol. 15, No. 6, 2022