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

[Contrôlabilité approchée d’une équation de Grushin 2D avec potentiel singulier]

Morancey, Morgan

Annales de l'Institut Fourier, Tome 65 (2015), p. 1525-1556 / Harvested from Numdam

Résumé

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 $\lbrace x=0\rbrace $.

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.

Abstract

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 $\mathbb{R}^2$, studied by Boscain and Laurent. The operator is both degenerate and singular on the line $\lbrace x=0 \rbrace $.

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.

Détail

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/aif.2966
Classification:  93B05,  35K65,  34B24
Mots clés: contrôlabilité approchée, équation parabolique dégénérée, opérateur de Grushin, extensions autoadjointes, opérateurs de Sturm-Liouville singuliers, inégalité de Carleman.

Bibliographie

[1] Alekseeva, Viktoriya S.; AnanʼEva, Aleksandra Yu. On extensions of the Bessel operator on a finite interval and the half-line, Ukr. Mat. Visn., Tome 9 (2012) no. 2, p. 147-156, 297 | MR 2986603

[2] Baras, Pierre; Goldstein, Jerome A. The heat equation with a singular potential, Trans. Amer. Math. Soc., Tome 284 (1984) no. 1, pp. 121-139 | Article | MR 742415 | Zbl 0556.35063

[3] Beauchard, K.; Cannarsa, P.; Guglielmi, R. Null controllability of Grushin-type operators in dimension two, J. Eur. Math. Soc. (JEMS), Tome 16 (2014) no. 1, pp. 67-101 | Article | MR 3141729 | Zbl 1293.35148

[4] Boscain, Ugo; Laurent, Camille The Laplace-Beltrami operator in almost-Riemannian geometry, Ann. Inst. Fourier (Grenoble), Tome 63 (2013) no. 5, pp. 1739-1770 | Article | Numdam | MR 3186507

[5] Boscain, Ugo; Prandi, Dario The Laplace-Beltrami operator on conic and anticonic-type surfaces (http://arxiv.org/abs/1305.5271v1)

[6] Cabré, Xavier; Martel, Yvan Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier, C. R. Acad. Sci. Paris Sér. I Math., Tome 329 (1999) no. 11, pp. 973-978 | Article | MR 1733904 | Zbl 0940.35105

[7] Cannarsa, P.; Martinez, P.; Vancostenoble, J. Null controllability of degenerate heat equations, Adv. Differential Equations, Tome 10 (2005) no. 2, pp. 153-190 | MR 2106129 | Zbl 1145.35408

[8] Cannarsa, P.; Martinez, P.; Vancostenoble, J. Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., Tome 47 (2008) no. 1, pp. 1-19 | Article | MR 2373460 | Zbl 1168.35025

[9] Cannarsa, Piermarco; Guglielmi, Roberto Null controllability in large time for the parabolic Grushin operator with singular potential, Geometric control theory and sub-Riemannian geometry, Springer, Cham (Springer INdAM Ser.) Tome 5 (2014), pp. 87-102 | Article | MR 3205097 | Zbl 1291.93036

[10] Cannarsa, Piermarco; Martinez, Partick; Vancostenoble, Judith Carleman estimates and null controllability for boundary-degenerate parabolic operators, C. R. Math. Acad. Sci. Paris, Tome 347 (2009) no. 3-4, pp. 147-152 | Article | MR 2538102 | Zbl 1162.35330

[11] Cannarsa, Piermarco; Tort, Jacques; Yamamoto, Masahiro Unique continuation and approximate controllability for a degenerate parabolic equation, Appl. Anal., Tome 91 (2012) no. 8, pp. 1409-1425 | Article | MR 2959541 | Zbl 1248.35034

[12] Cazenave, Thierry; Haraux, Alain Introduction aux problèmes d’évolution semi-linéaires, Ellipses, Paris, Mathématiques & Applications (Paris) [Mathematics and Applications], Tome 1 (1990), pp. 142 | MR 1299976 | Zbl 0786.35070

[13] Coron, Jean-Michel Control and nonlinearity, American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs, Tome 136 (2007), pp. xiv+426 | MR 2302744 | Zbl 1140.93002

[14] Èmanuilov, O. Yu. Boundary controllability of parabolic equations, Uspekhi Mat. Nauk, Tome 48 (1993) no. 3(291), p. 211-212 | MR 1243631 | Zbl 0800.93172

[15] Ervedoza, Sylvain Control and stabilization properties for a singular heat equation with an inverse-square potential, Comm. Partial Differential Equations, Tome 33 (2008) no. 10-12, pp. 1996-2019 | Article | MR 2475327 | Zbl 1170.35331

[16] Gueye, Mamadou Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim., Tome 52 (2014) no. 4, pp. 2037-2054 | Article | MR 3227458 | Zbl 1327.35211

[17] Martinez, P.; Vancostenoble, J. Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ., Tome 6 (2006) no. 2, pp. 325-362 | Article | MR 2227700 | Zbl 1179.93043

[18] Pazy, A. Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, Applied Mathematical Sciences, Tome 44 (1983), pp. viii+279 | MR 710486 | Zbl 0516.47023

[19] Reed, Michael; Simon, Barry Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London (1975), pp. xv+361 | MR 493420 | Zbl 0242.46001

[20] Saut, Jean-Claude; Scheurer, Bruno Unique continuation for some evolution equations, J. Differential Equations, Tome 66 (1987) no. 1, pp. 118-139 | Article | MR 871574 | Zbl 0631.35044

[21] Vancostenoble, J.; Zuazua, E. Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal., Tome 254 (2008) no. 7, pp. 1864-1902 | Article | MR 2397877 | Zbl 1145.93009

[22] Vancostenoble, Judith Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems, Discrete Contin. Dyn. Syst. Ser. S, Tome 4 (2011) no. 3, pp. 761-790 | Article | MR 2746432 | Zbl 1213.93018

[23] Vazquez, Juan Luis; Zuazua, Enrike The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., Tome 173 (2000) no. 1, pp. 103-153 | Article | MR 1760280 | Zbl 0953.35053

[24] Zettl, Anton Sturm-Liouville theory, American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs, Tome 121 (2005), pp. xii+328 | MR 2170950 | Zbl 1103.34001