Approximate controllability for a 2D Grushin equation with potential having an internal singularity

Morancey, Morgan

Annales de l'Institut Fourier, Tome 65 (2015) no. 4, pp. 1525-1556.

On étudie la contrôlabilité approchée d’une équation de Grushin avec potentiel singulier sur le rectangle $(-1,1)\times (0,1)$. Ce modèle est inspiré de l’équation de la chaleur pour l’opérateur de Laplace-Beltrami associé à la métrique de Grushin. Cet opérateur parabolique est à la fois dégénéré et singulier sur la droite $\{x=0\}$.

L’étude de la contrôlabilité approchée repose sur une propriété de prolongement unique du système adjoint.

Le potentiel est dégénéré à l’intérieur du domaine d’étude ce qui fait de l’étude du caractère bien posé le point central de cet article. Une extension autoadjointe de l’opérateur singulier est construite en imposant des conditions de transmission adéquate à travers la singularité.

Enfin, la propriété de prolongement unique repose sur la décomposition de Fourier de la solution du problème 2D suivant l’une des variables et sur la preuve d’une inégalité de Carleman pour le système 1D vérifié par les coefficients de Fourier. Cette inégalité de Carleman utilise l’inégalité de Hardy.

This paper is dedicated to approximate controllability for Grushin equation on the rectangle $(x,y)\in (-1,1)\times (0,1)$ with an inverse square potential. This model corresponds to the heat equation for the Laplace-Beltrami operator associated to the Grushin metric on ${ℝ}^2$, studied by Boscain and Laurent. The operator is both degenerate and singular on the line $\{x=0\}$.

The approximate controllability is studied through unique continuation of the adjoint system. For the range of singularity under study, approximate controllability is proved to hold whatever the degeneracy is.

Due to the internal inverse square singularity, a key point in this work is the study of well-posedness. An extension of the singular operator is designed imposing suitable transmission conditions through the singularity.

Then, unique continuation relies on the Fourier decomposition of the 2d solution in one variable and Carleman estimates for the 1d heat equation solved by the Fourier components. The Carleman estimate uses a suitable Hardy inequality.

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