Minimal time issues for the observability of Grushin-type equations

Beauchard, Karine; Dardé, Jérémi; Ervedoza, Sylvain

Annales de l'Institut Fourier, Tome 70 (2020) no. 1, pp. 247-312.

Le but de cet article est de fournir plusieurs estimées optimales sur le temps minimal nécessaire pour avoir l’observabilité d’équations de type Grushin. En effet, il est désormais bien connu que les équations de type Grushin sont des équations paraboliques dégénérées pour lesquelles des conditions géométriques sont nécessaires pour satisfaire des propriétés d’observabilité, contrairement aux équations paraboliques usuelles. Nos résultats concernent l’opérateur de Grushin ${\partial }_t-{\Delta }_x-{\left|x\right|}^2{\Delta }_y$ observé de tout le bord dans le cas multi-dimensionnel (dans le sens où $x\in {\Omega }_x$, où ${\Omega }_x$ est un ouvert de ${ℝ}^{d_x}$, avec $d_x\ge 1$, $y\in {\Omega }_y$ est un ouvert de ${ℝ}^{d_y}$ avec $d_y\ge 1$, et l’observation est faite sur $\Gamma =\partial {\Omega }_x\times {\Omega }_y$), d’un bord latéral dans le cas uni-dimensionnel (i.e. $d_x=1$), incluant certaines généralisations de la forme ${\partial }_t-{\partial }_x^2-{\left(q\left(x\right)\right)}^2{\partial }_y^2$ pour des fonctions $q$ convenables, et l’opérateur de Heisenberg ${\partial }_t-{\partial }_x^2-{(x{\partial }_z+{\partial }_y)}^2$ observé d’un bord latéral. Dans tous ces cas, notre approche repose fortement sur l’analyse de la famille d’équations obtenues en développant la solution en Fourier dans la variable $y$ (ou $(y,z)$), et en particulier sur l’asymptotique du coût de l’observabilité en fonction du paramètre de Fourier. En combinant ces estimées avec les résultats sur le taux de dissipation de chacune de ces équations, nous obtenons des inégalités d’observabilité en temps suffisamment grand. Nous montrons ensuite que les temps que nous avons obtenus pour l’observabilité sont optimaux dans plusieurs cas, en utilisant des estimées de Agmon.

The goal of this article is to provide several sharp results on the minimal time required for observability of several Grushin-type equations. Namely, it is by now well-known that Grushin-type equations are degenerate parabolic equations for which some geometric conditions are needed to get observability properties, contrarily to the usual parabolic equations. Our results concern the Grushin operator ${\partial }_t-{\Delta }_x-{\left|x\right|}^2{\Delta }_y$ observed from the whole boundary in the multi-dimensional setting (meaning that $x\in {\Omega }_x$, where ${\Omega }_x$ is a subset of ${ℝ}^{d_x}$ with $d_x\ge 1$, $y\in {\Omega }_y$, where ${\Omega }_y$ is a subset of ${ℝ}^{d_y}$ with $d_y\ge 1$, and the observation is done on $\Gamma =\partial {\Omega }_x\times {\Omega }_y$), from one lateral boundary in the one-dimensional setting (i.e. $d_x=1$), including some generalized version of the form ${\partial }_t-{\partial }_x^2-{\left(q\left(x\right)\right)}^2{\partial }_y^2$ for suitable functions $q$, and the Heisenberg operator ${\partial }_t-{\partial }_x^2-{(x{\partial }_z+{\partial }_y)}^2$ observed from one lateral boundary. In all these cases, our approach strongly relies on the analysis of the family of equations obtained by using the Fourier expansion of the equations in the $y$ (or $(y,z)$) variables, and in particular the asymptotic of the cost of observability in the Fourier parameters. Combining these estimates with results on the rate of dissipation of each of these equations, we obtain observability estimates in suitably large times. We then show that the times we obtain to get observability are optimal in several cases using Agmon type estimates.

Classification : 35K65, 93B07
Keywords: Observability, Grushin equations, Carleman estimates
Mot clés : Observabilité, Équations de Grushin, Inégalités de Carleman

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