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Time optimal observability for Grushin Schrödinger equation

Nicolas Burq and Chenmin Sun

Vol. 15 (2022), No. 6, 1487–1530
Abstract

We consider the two-dimensional Grushin Schrödinger equation posed on a finite cylinder Ω = (1,1)x × 𝕋y 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 T.

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