Time optimal observability for Grushin Schr\"odinger equation

TL;DR

This paper establishes the precise conditions and minimal time needed for observing and controlling the two-dimensional Grushin Schrödinger equation on a finite cylinder, revealing the role of eigenfunction concentration.

Contribution

It provides the sharp observability results with the optimal time for the Grushin Schrödinger equation and demonstrates exact controllability under these conditions.

Findings

Sharp observability with optimal time T_* depending on strip size

Exact controllability of the Grushin Schrödinger equation

Observability fails for any T ≤ T_* due to eigenfunction concentration

Abstract

We consider two dimensional Grushin Schr\"odinger equation posed on a finite cylinder $\Omega=(-1,1)_x\times \T_y$ with Dirichlet boundary condition. We obtain the 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 Grushin Schr\"odinger equation. By exploiting the concentration of eigenfunctions of harmonic oscillator at $x=0$, we also show that the observability fails for any $T\leq T_*$.