On uniform observability of gradient flows in the vanishing viscosity limit

Laurent, Camille; Léautaud, Matthieu

Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 439-506.

We consider a transport equation by a gradient vector field with a small viscous perturbation $-\epsilon {\Delta }_g$. We study uniform observability (resp. controllability) properties in the (singular) vanishing viscosity limit $\epsilon \to 0^{+}$, that is, the possibility of having a uniformly bounded observation constant (resp. control cost). We prove with a series of examples that in general, the minimal time for uniform observability may be much larger than the minimal time needed for the observability of the limit equation $\epsilon =0$. We also prove that the two minimal times coincide for positive solutions. The proofs rely on a semiclassical reformulation of the problem together with (a) Agmon estimates concerning the decay of eigenfunctions in the classically forbidden region [HS84] (b) fine estimates of the kernel of the semiclassical heat equation [LY84].

Nous considérons l’équation de transport par un champ de gradient avec une petite perturbation visqueuse $-\epsilon {\Delta }_g$. Nous étudions la propriété d’observabilité (resp. de contrôlabilité) uniforme dans la limite (singulière) de viscosité évanescente $\epsilon \to 0^{+}$, c’est-à-dire la possibilité d’avoir une constante d’observabilité (resp. un coût du contrôle) uniforme. Nous prouvons avec une série d’exemples que le temps minimal pour l’observabilité uniforme peut être bien plus grand que le temps minimal pour l’équation limite $\epsilon =0$. Nous montrons aussi que les deux temps minimaux coïncident pour les solutions positives. Les preuves reposent sur une reformulation semiclassique du problème ainsi que (a) des estimées d’Agmon de décroissance des fonctions propres dans la zone classiquement interdite [HS84] (b) des estimées fines du noyaux de la chaleur semiclassique [LY84].

Classification: 93B07, 93B05, 35B25, 35F05, 35K05, 93C73
Keywords: Transport equation, gradient flow, vanishing viscosity limit, parabolic equation, minimal control time, semiclassical Schrödinger operator
Mot clés : Équation de transport, flot gradient, limite de viscosité évanescente, équation parabolique, temps minimal de contrôle, opérateur de Schrödinger semiclassique

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