On étudie la contrôlabilité approchée d’une équation de Grushin avec potentiel singulier sur le rectangle . 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 .
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 with an inverse square potential. This model corresponds to the heat equation for the Laplace-Beltrami operator associated to the Grushin metric on , studied by Boscain and Laurent. The operator is both degenerate and singular on the line .
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.
Keywords: unique continuation, degenerate parabolic equation, singular parabolic equation, Grushin operator, self-adjoint extensions, singular Sturm-Liouville operators, Carleman estimate.
Mot clés : contrôlabilité approchée, équation parabolique dégénérée, opérateur de Grushin, extensions autoadjointes, opérateurs de Sturm-Liouville singuliers, inégalité de Carleman.
@article{AIF_2015__65_4_1525_0, author = {Morancey, Morgan}, title = {Approximate controllability for a {2D} {Grushin} equation with potential having an internal singularity}, journal = {Annales de l'Institut Fourier}, pages = {1525--1556}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {65}, number = {4}, year = {2015}, doi = {10.5802/aif.2966}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2966/} }
TY - JOUR AU - Morancey, Morgan TI - Approximate controllability for a 2D Grushin equation with potential having an internal singularity JO - Annales de l'Institut Fourier PY - 2015 SP - 1525 EP - 1556 VL - 65 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2966/ DO - 10.5802/aif.2966 LA - en ID - AIF_2015__65_4_1525_0 ER -
%0 Journal Article %A Morancey, Morgan %T Approximate controllability for a 2D Grushin equation with potential having an internal singularity %J Annales de l'Institut Fourier %D 2015 %P 1525-1556 %V 65 %N 4 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2966/ %R 10.5802/aif.2966 %G en %F AIF_2015__65_4_1525_0
Morancey, Morgan. Approximate controllability for a 2D Grushin equation with potential having an internal singularity. Annales de l'Institut Fourier, Tome 65 (2015) no. 4, pp. 1525-1556. doi : 10.5802/aif.2966. http://www.numdam.org/articles/10.5802/aif.2966/
[1] On extensions of the Bessel operator on a finite interval and the half-line, Ukr. Mat. Visn., Volume 9 (2012) no. 2, p. 147-156, 297 | MR
[2] The heat equation with a singular potential, Trans. Amer. Math. Soc., Volume 284 (1984) no. 1, pp. 121-139 | DOI | MR | Zbl
[3] Null controllability of Grushin-type operators in dimension two, J. Eur. Math. Soc. (JEMS), Volume 16 (2014) no. 1, pp. 67-101 | DOI | MR | Zbl
[4] The Laplace-Beltrami operator in almost-Riemannian geometry, Ann. Inst. Fourier (Grenoble), Volume 63 (2013) no. 5, pp. 1739-1770 | DOI | Numdam | MR
[5] The Laplace-Beltrami operator on conic and anticonic-type surfaces (http://arxiv.org/abs/1305.5271v1)
[6] Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier, C. R. Acad. Sci. Paris Sér. I Math., Volume 329 (1999) no. 11, pp. 973-978 | DOI | MR | Zbl
[7] Null controllability of degenerate heat equations, Adv. Differential Equations, Volume 10 (2005) no. 2, pp. 153-190 | MR | Zbl
[8] Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., Volume 47 (2008) no. 1, pp. 1-19 | DOI | MR | Zbl
[9] Null controllability in large time for the parabolic Grushin operator with singular potential, Geometric control theory and sub-Riemannian geometry (Springer INdAM Ser.), Volume 5, Springer, Cham, 2014, pp. 87-102 | DOI | MR | Zbl
[10] Carleman estimates and null controllability for boundary-degenerate parabolic operators, C. R. Math. Acad. Sci. Paris, Volume 347 (2009) no. 3-4, pp. 147-152 | DOI | MR | Zbl
[11] Unique continuation and approximate controllability for a degenerate parabolic equation, Appl. Anal., Volume 91 (2012) no. 8, pp. 1409-1425 | DOI | MR | Zbl
[12] Introduction aux problèmes d’évolution semi-linéaires, Mathématiques & Applications (Paris) [Mathematics and Applications], 1, Ellipses, Paris, 1990, pp. 142 | MR | Zbl
[13] Control and nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007, pp. xiv+426 | MR | Zbl
[14] Boundary controllability of parabolic equations, Uspekhi Mat. Nauk, Volume 48 (1993) no. 3(291), pp. 211-212 | MR | Zbl
[15] Control and stabilization properties for a singular heat equation with an inverse-square potential, Comm. Partial Differential Equations, Volume 33 (2008) no. 10-12, pp. 1996-2019 | DOI | MR | Zbl
[16] Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim., Volume 52 (2014) no. 4, pp. 2037-2054 | DOI | MR | Zbl
[17] Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ., Volume 6 (2006) no. 2, pp. 325-362 | DOI | MR | Zbl
[18] Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983, pp. viii+279 | MR | Zbl
[19] Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975, pp. xv+361 | MR | Zbl
[20] Unique continuation for some evolution equations, J. Differential Equations, Volume 66 (1987) no. 1, pp. 118-139 | DOI | MR | Zbl
[21] Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal., Volume 254 (2008) no. 7, pp. 1864-1902 | DOI | MR | Zbl
[22] Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems, Discrete Contin. Dyn. Syst. Ser. S, Volume 4 (2011) no. 3, pp. 761-790 | DOI | MR | Zbl
[23] The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., Volume 173 (2000) no. 1, pp. 103-153 | DOI | MR | Zbl
[24] Sturm-Liouville theory, Mathematical Surveys and Monographs, 121, American Mathematical Society, Providence, RI, 2005, pp. xii+328 | MR | Zbl
Cité par Sources :