Critical time for the observability of Kolmogorov-type equations

Dardé, Jérémi; Royer, Julien

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 859-894.

Nous nous intéressons à l’observabilité d’équations de type Kolmogorov bi-dimensionnelles présentant une dégénérescence quadratique. Nous donnons un majorant et un minorant du temps critique. Dans une configuration symétrique, ces bornes coïncident et donnent alors précisément le temps critique d’observabilité. La preuve est basée sur des estimées de Carleman et sur l’étude des propriétés spectrales d’une famille d’opérateurs de Schrödinger non auto-adjoints, en particulier la localisation de la première valeur propre et des estimées de type Agmon pour les fonctions propres correspondantes.

This paper is devoted to the observability of a class of two-dimensional Kolmogorov-type equations presenting a quadratic degeneracy. We give lower and upper bounds for the critical time. These bounds coincide in symmetric settings, giving a sharp result in these cases. The proof is based on Carleman estimates and on the spectral properties of a family of non-selfadjoint Schrödinger operators, in particular the localization of the first eigenvalue and Agmon type estimates for the corresponding eigenfunctions.

Classification : 35K65, 93B07, 47B28, 47A10, 47D06
Keywords: Observability, Kolmogorov equations, Carleman estimates, non-selfadjoint operators, resolvent estimates
Mot clés : Observabilité, équations de Kolmogorov, estimées de Carleman, opérateurs non auto-adjoints, estimées de résolvante

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