TIME OPTIMAL OBSERVABILITY FOR GRUSHIN SCHRÖDINGER EQUATION

Nicolas Burq (1) , Chenmin Sun (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

We consider two dimensional Grushin Schrödinger equation posed on a finite cylinder Ω = (−1, 1) x × 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 Schrödinger equation. By exploiting the concentration of eigenfunctions of harmonic oscillator at x = 0, we also show that the observability fails for any T ≤ T * .

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Nicolas Burq, Chenmin Sun. TIME OPTIMAL OBSERVABILITY FOR GRUSHIN SCHRÖDINGER EQUATION. Analysis & PDE, 2021, 15 (6), pp.1487--1530. ⟨10.2140/apde.2022.15.1487⟩. ⟨hal-02309020v2⟩