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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 $\mathrm{\Omega }={\left(-1,1\right)}_{x}×{\mathbb{𝕋}}_{y}$ with Dirichlet boundary condition. We obtain sharp observability by any horizontal strip, with the optimal time ${T}_{\ast }>0$ 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 $T\le {T}_{\ast }$.

##### Keywords
controllability, observability, subelliptic, Schrödinger equation, semiclassical analysis
##### Mathematical Subject Classification 2010
Primary: 35Q41, 35Q93, 93B07