Null-controllability of evolution equations associated with fractional Shubin operators through quantitative Agmon estimates

Alphonse, Paul

doi:10.5802/aif.3642

Null-controllability of evolution equations associated with fractional Shubin operators through quantitative Agmon estimates
[Contrôlabilité à zéro des équations d’évolution associées aux opérateurs de Shubin fractionnaires grâce à des estimées d’Agmon]

Alphonse, Paul ¹

¹ Université de Lyon, ENSL, UMPA – UMR 5669, F-69364 Lyon (France)

Annales de l'Institut Fourier, Online first, 50 p.

Résumé
Abstract

On considère les opérateurs de Shubin anisotropes ${(- Δ)}^{m} + {| x |}^{2 k}$ agissant sur l’espace $L^{2} (ℝ^{n})$ , avec $k, m \geq 1$ des entiers strictement positifs. On démontre des inégalités quantitatives et précises dans des espaces de Gelfand–Shilov pour les fonctions propres de ces opérateurs différentiels autoadjoints, grâce à une stratégie basée sur l’approche classique pour obtenir des estimées d’Agmon en théorie spectrale. En utilisant une loi de Weyl pour les valeurs propres des opérateurs de Shubin anisotropes, on décrit également les effets régularisants des semi-groupes engendrés par les puissances fractionnaires de ces opérateurs, et on donne des estimations précises en temps courts. Cette description permet de démontrer des résultats positifs de contrôlabilité à zéro pour les équations d’évolution associées posées sur tout l’espace $ℝ^{n}$ , depuis des supports de contrôle épais par rapport à des densités, et en tout temps strictement positif. On généralise en particulier des résultats connus pour les équations d’évolution associées aux oscillateurs harmoniques fractionnaires.

We consider the anisotropic Shubin operators ${(- Δ)}^{m} + {| x |}^{2 k}$ acting on the space $L^{2} (ℝ^{n})$ , with $k, m \geq 1$ some positive integers. We provide sharp quantitative estimates in Gelfand–Shilov spaces for the eigenfunctions of these selfadjoint differential operators with a strategy based on the classical approach to obtain Agmon estimates in spectral theory. By using a Weyl law for the eigenvalues of the anisotropic Shubin operators, we also describe the smoothing properties of the semigroups generated by the fractional powers of these operators, with precise estimates in short times. This description allows us to prove positive null-controllability results for the associated evolution equations posed on the whole space $ℝ^{n}$ , from control supports which are thick with respect to densities and in any positive time. We generalize in particular results known for the evolution equations associated with fractional harmonic oscillators.

Reçu le : 2020-12-11
Révisé le : 2022-12-08
Accepté le : 2022-12-16
Première publication : 2024-05-17

DOI : 10.5802/aif.3642

Classification : 93B05, 35B65, 35P10
Keywords: Null-controllability, Gelfand–Shilov regularity, Agmon estimates, Pseudodifferential calculus, Anisotropic Shubin operators
Mot clés : Contrôlabilité à zéro, régularité Gelfand–Shilov, estimées d’Agmon, calcul pseudo-différentiel, opérateurs de Shubin anisotropes.

Affiliations des auteurs :

Alphonse, Paul ¹

¹ Université de Lyon, ENSL, UMPA – UMR 5669, F-69364 Lyon (France)

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Alphonse, Paul. Null-controllability of evolution equations associated with fractional Shubin operators through quantitative Agmon estimates. Annales de l'Institut Fourier, Online first, 50 p.

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