Comptes Rendus
Partial differential equations
Non-null-controllability of the Grushin operator in 2D
[Non-contrôlabilité à zéro de l'opérateur de Grushin en dimension 2]
Comptes Rendus. Mathématique, Volume 355 (2017) no. 12, pp. 1215-1235.

We are interested in the exact null controllability of the equation tfx2fx2y2f=1ωu, with control u supported on ω. We show that, when ω does not intersect a horizontal band, the considered equation is never null-controllable. The main idea is to interpret the associated observability inequality as an L2 estimate on polynomials, which Runge's theorem disproves. To that end, we study in particular the first eigenvalue of the operator x2+(nx)2 with Dirichlet conditions on (1,1), and we show a quite precise estimation it satisfies, even when n is in some complex domain.

Nous nous intéressons à la contrôlabilité exacte à zéro de l'équation tfx2fx2y2f=1ωu sur (1,1)×T, avec contrôle u sur ω. Nous démontrons que si ω est le complémentaire d'une bande horizontale, l'équation considérée n'est contrôlable pour aucun temps. L'idée principale est d'interpréter l'inégalité d'observabilité comme une estimation sur les fonctions entières, que nous nions grâce au théorème de Runge. Pour réaliser cette interprétation, nous étudions en particulier la première valeur propre de x2+(nx)2 avec conditions de Dirichlet sur ]1,1[, et en obtenons une estimation assez précise, y compris pour certains n complexes.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.10.021

Armand Koenig 1

1 Laboratoire de mathématiques Jean-Alexandre-Dieudonné, UMR 7351 CNRS UNS, Université de Nice – Sophia Antipolis, 06108 Nice cedex 02, France
@article{CRMATH_2017__355_12_1215_0,
     author = {Armand Koenig},
     title = {Non-null-controllability of the {Grushin} operator in {2D}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1215--1235},
     publisher = {Elsevier},
     volume = {355},
     number = {12},
     year = {2017},
     doi = {10.1016/j.crma.2017.10.021},
     language = {en},
}
TY  - JOUR
AU  - Armand Koenig
TI  - Non-null-controllability of the Grushin operator in 2D
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 1215
EP  - 1235
VL  - 355
IS  - 12
PB  - Elsevier
DO  - 10.1016/j.crma.2017.10.021
LA  - en
ID  - CRMATH_2017__355_12_1215_0
ER  - 
%0 Journal Article
%A Armand Koenig
%T Non-null-controllability of the Grushin operator in 2D
%J Comptes Rendus. Mathématique
%D 2017
%P 1215-1235
%V 355
%N 12
%I Elsevier
%R 10.1016/j.crma.2017.10.021
%G en
%F CRMATH_2017__355_12_1215_0
Armand Koenig. Non-null-controllability of the Grushin operator in 2D. Comptes Rendus. Mathématique, Volume 355 (2017) no. 12, pp. 1215-1235. doi : 10.1016/j.crma.2017.10.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.10.021/

[1] S. Agmon Lectures on Exponential Decay of Solution of Second-Order Elliptic Equations, Mathematical Notes, vol. 29, Princeton University Press, Princeton, NJ, USA, 1982

[2] N.U. Arakelyan On efficient analytic continuation of power series, Math. USSR Sb., Volume 52 (1985) no. 1, pp. 21-39

[3] K. Beauchard Null controllability of Kolmogorov-type equations, Math. Control Signals Syst., Volume 26 (2014) no. 1, pp. 145-176

[4] K. Beauchard; P. Cannarsa Heat equation on the Heisenberg group: observability and applications, J. Differ. Equ., Volume 262 (2017) no. 8, pp. 4475-4521

[5] K. Beauchard; P. Cannarsa; R. Guglielmi Null controllability of Grushin-type operators in dimension two, J. Eur. Math. Soc., Volume 16 (2014) no. 1, pp. 67-101

[6] K. Beauchard; B. Helffer; R. Henry; L. Robbiano Degenerate parabolic operators of Kolmogorov type with a geometric control condition, ESAIM Control Optim. Calc. Var., Volume 21 (2015) no. 2, pp. 487-512

[7] K. Beauchard; L. Miller; M. Morancey 2d Grushin-type equations: minimal time and null controllable data, J. Differ. Equ., Volume 259 (2015) no. 11, pp. 5813-5845

[8] K. Beauchard; K. Pravda-Starov Null-controllability of hypoelliptic quadratic differential equations, 2016 | arXiv

[9] K. Beauchard; K. Pravda-Starov Null-controllability of non-autonomous Ornstein–Uhlenbeck equations, J. Math. Anal. Appl., Volume 456 (2001) no. 1, pp. 496-524

[10] P. Cannarsa; P. Martinez; J. Vancostenoble Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., Volume 47 (2008) no. 1, pp. 1-19

[11] P. Cannarsa; P. Martinez; J. Vancostenoble Global Carleman estimates for degenerate parabolic operators with applications, Mem. Amer. Math. Soc., Volume 239 (2016) no. 1133

[12] J.-M. Coron Control and Nonlinearity, Math. Surv. Monogr., vol. 143, American Mathematical Society, Boston, MA, USA, 2007

[13] T. Duyckaerts; L. Miller Resolvent conditions for the control of parabolic equations, J. Funct. Anal., Volume 263 (2012) no. 11, pp. 3641-3673

[14] A.V. Fursikov; O.Y. Imanuvilov Controllability of Evolution Equations, Lecture Note Series, vol. 34, Seoul University Press, 1996

[15] B. Helffer, F. Nier, Quantitative analysis of metastability in reversible diffusion Processes via a Witten complex approach: the case with boundary, preprint, 2004, HAL.

[16] B. Helffer; J. Sjostrand Multiples wells in the semi-classical limit I, Commun. Partial Differ. Equ., Volume 9 (1984) no. 4, pp. 337-408

[17] G. Lebeau; L. Robbiano Contrôle exact de l'équation de la chaleur, Commun. Partial Differ. Equ., Volume 20 (1995) no. 1, pp. 335-356

[18] E.L. Lindelöf Le calcul des résidus et ses applications à la théorie des fonctions, Gauthier-Villars, 1905

[19] A. Martinez An Introduction to Semiclassical and Microlocal Analysis, Universitext, Springer, New York, 2002

[20] L. Miller On the controllability of anomalous diffusions generated by the fractional laplacian, Math. Control Signals Syst., Volume 18 (2006) no. 3, pp. 260-271

[21] W. Rudin Real and Complex Analysis, McGraw Hill Education, 1986

  • Salah-Eddine Chorfi; Lahcen Maniar Stability Estimates for Initial Data in General Ornstein–Uhlenbeck Equations, Control Theory and Inverse Problems (2025), p. 137 | DOI:10.1007/978-3-031-68046-5_7
  • Pierre Lissy A non-controllability result for the half-heat equation on the whole line based on the prolate spheroidal wave functions and its application to the Grushin equation, Journal of Differential Equations, Volume 433 (2025), p. 27 (Id/No 113306) | DOI:10.1016/j.jde.2025.113306 | Zbl:8031084
  • Hoai-Minh Nguyen Local controllability of the Korteweg-de Vries equation with the right Dirichlet control, Journal of Differential Equations, Volume 435 (2025), p. 85 (Id/No 113235) | DOI:10.1016/j.jde.2025.113235 | Zbl:8035404
  • Philippe Jaming; Yunlei Wang Null-controllability of the generalized Baouendi-Grushin heat like equations, Journal of Evolution Equations, Volume 25 (2025) no. 2, p. 39 (Id/No 43) | DOI:10.1007/s00028-025-01069-7 | Zbl:8035737
  • Pei Su; Chenmin Sun; Xu Yuan Quantitative observability for one-dimensional Schrödinger equations with potentials, Journal of Functional Analysis, Volume 288 (2025) no. 2, p. 39 (Id/No 110695) | DOI:10.1016/j.jfa.2024.110695 | Zbl:1554.35278
  • Jérémi Dardé; Armand Koenig; Julien Royer Null-controllability properties of the generalized two-dimensional Baouendi-Grushin equation with non-rectangular control sets, Annales Henri Lebesgue, Volume 6 (2023), pp. 1479-1522 | DOI:10.5802/ahl.193 | Zbl:1534.35250
  • Paul Alphonse; Albrecht Seelmann Quantitative spectral inequalities for the anisotropic Shubin operators and applications to null-controllability, Comptes Rendus. Mathématique, Volume 362 (2024) no. G12, p. 1635 | DOI:10.5802/crmath.670
  • Sorin Micu; Constantin Niţă Cost evaluation of finite dimensional and approximate null-controllability for the one dimensional half-heat equation, Mathematical Control and Related Fields, Volume 14 (2024) no. 1, pp. 1-15 | DOI:10.3934/mcrf.2022054 | Zbl:1536.93086
  • Baparou Danhane; Jérôme Lohéac Ensemble controllability of parabolic type equations, Systems Control Letters, Volume 183 (2024), p. 10 (Id/No 105683) | DOI:10.1016/j.sysconle.2023.105683 | Zbl:1536.93071
  • Cyril Letrouit Exact observability properties of subelliptic wave and Schrödinger equations, Séminaire de théorie spectrale et géométrie, Volume 36 (2024), p. 51 | DOI:10.5802/tsg.373
  • Cyril Letrouit Subelliptic wave equations are never observable, Analysis PDE, Volume 16 (2023) no. 3, pp. 643-678 | DOI:10.2140/apde.2023.16.643 | Zbl:1533.35066
  • Victor Arnaiz; Chenmin Sun Sharp resolvent estimate for the damped-wave Baouendi-Grushin operator and applications, Communications in Mathematical Physics, Volume 400 (2023) no. 1, pp. 541-637 | DOI:10.1007/s00220-022-04606-4 | Zbl:1522.35159
  • Cyril Letrouit; Chenmin Sun Observability of Baouendi-Grushin-type equations through resolvent estimates, Journal of the Institute of Mathematics of Jussieu, Volume 22 (2023) no. 2, pp. 541-579 | DOI:10.1017/s1474748021000207 | Zbl:1508.93045
  • Nicolas Burq; Chenmin Sun Time optimal observability for Grushin Schrödinger equation, Analysis PDE, Volume 15 (2022) no. 6, pp. 1487-1530 | DOI:10.2140/apde.2022.15.1487 | Zbl:1501.35337
  • Lin Yan; Bin Wu; Shiping Lu; Yuchan Wang Null controllability and inverse source problem for stochastic Grushin equation with boundary degeneracy and singularity, European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations, Volume 28 (2022), p. 34 (Id/No 43) | DOI:10.1051/cocv/2022027 | Zbl:1497.93026
  • Cyprien Tamekue Null controllability of the parabolic spherical Grushin equation, European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations, Volume 28 (2022), p. 29 (Id/No 70) | DOI:10.1051/cocv/2022055 | Zbl:1511.93022
  • Camille Laurent; Matthieu Léautaud Tunneling estimates and approximate controllability for hypoelliptic equations, Memoirs of the American Mathematical Society, 1357, Providence, RI: American Mathematical Society (AMS), 2022 | DOI:10.1090/memo/1357 | Zbl:1500.35002
  • Clotilde Fermanian Kammerer; Cyril Letrouit Observability and controllability for the Schrödinger equation on quotients of groups of Heisenberg type, Journal de l'École Polytechnique – Mathématiques, Volume 8 (2021), pp. 1459-1513 | DOI:10.5802/jep.176 | Zbl:1475.35377
  • Jérémi Dardé; Julien Royer Critical time for the observability of Kolmogorov-type equations, Journal de l'École Polytechnique – Mathématiques, Volume 8 (2021), pp. 859-894 | DOI:10.5802/jep.160 | Zbl:1465.35289
  • Damien Allonsius; Franck Boyer; Morgan Morancey Analysis of the null controllability of degenerate parabolic systems of Grushin type via the moments method, Journal of Evolution Equations, Volume 21 (2021) no. 4, pp. 4799-4843 | DOI:10.1007/s00028-021-00733-y | Zbl:1485.93057
  • Karine Beauchard; Jérémi Dardé; Sylvain Ervedoza Minimal time issues for the observability of Grushin-type equations, Annales de l'Institut Fourier, Volume 70 (2020) no. 1, pp. 247-312 | DOI:10.5802/aif.3313 | Zbl:1448.35316
  • Paul Alphonse; Joackim Bernier Smoothing properties of fractional Ornstein-Uhlenbeck semigroups and null-controllability, Bulletin des Sciences Mathématiques, Volume 165 (2020), p. 52 (Id/No 102914) | DOI:10.1016/j.bulsci.2020.102914 | Zbl:1454.93024
  • Michel Duprez; Armand Koenig Control of the Grushin equation: non-rectangular control region and minimal time, European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations, Volume 26 (2020), p. 18 (Id/No 3) | DOI:10.1051/cocv/2019001 | Zbl:1447.93025
  • Karine Beauchard; Armand Koenig; Kévin Le Balc'h Null-controllability of linear parabolic-transport systems, Journal de l'École Polytechnique – Mathématiques, Volume 7 (2020), pp. 743-802 | DOI:10.5802/jep.127 | Zbl:1443.93030
  • Armand Koenig Lack of null-controllability for the fractional heat equation and related equations, SIAM Journal on Control and Optimization, Volume 58 (2020) no. 6, pp. 3130-3160 | DOI:10.1137/19m1256610 | Zbl:1453.93021
  • Jérémi Dardé; Sylvain Ervedoza Backward uniqueness results for some parabolic equations in an infinite rod, Mathematical Control and Related Fields, Volume 9 (2019) no. 4, pp. 673-696 | DOI:10.3934/mcrf.2019046 | Zbl:1442.35178

Cité par 26 documents. Sources : Crossref, zbMATH

This work was partially supported by the ERC advanced grant SCAPDE, seventh framework program, agreement No. 320845.

Commentaires - Politique