Approximate controllability for a 2D Grushin equation with potential having an internal singularity
Annales de l'Institut Fourier, Volume 65 (2015) no. 4, pp. 1525-1556.

This paper is dedicated to approximate controllability for Grushin equation on the rectangle (x,y)(-1,1)×(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 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.

On étudie la contrôlabilité approchée d’une équation de Grushin avec potentiel singulier sur le rectangle (-1,1)×(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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.2966
Classification: 93B05,  35K65,  34B24
Keywords: unique continuation, degenerate parabolic equation, singular parabolic equation, Grushin operator, self-adjoint extensions, singular Sturm-Liouville operators, Carleman estimate.
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Morancey, Morgan. Approximate controllability for a 2D Grushin equation with potential having an internal singularity. Annales de l'Institut Fourier, Volume 65 (2015) no. 4, pp. 1525-1556. doi : 10.5802/aif.2966. https://aif.centre-mersenne.org/articles/10.5802/aif.2966/

[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 INdAM Ser.) Tome 5, Springer, Cham, 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, Mathématiques & Applications (Paris) [Mathematics and Applications], Tome 1, Ellipses, Paris, 1990, 142 pages | MR: 1299976 | Zbl: 0786.35070

[13] Coron, Jean-Michel Control and nonlinearity, Mathematical Surveys and Monographs, Tome 136, American Mathematical Society, Providence, RI, 2007, xiv+426 pages | 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, Applied Mathematical Sciences, Tome 44, Springer-Verlag, New York, 1983, viii+279 pages | 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, xv+361 pages | 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, Mathematical Surveys and Monographs, Tome 121, American Mathematical Society, Providence, RI, 2005, xii+328 pages | MR: 2170950 | Zbl: 1103.34001

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