Critical time for the observability of Kolmogorov-type equations

Jérémi Dardé; Julien Royer

doi:10.5802/jep.160

Critical time for the observability of Kolmogorov-type equations
[Temps critique pour l’observabilité d’équations de type Kolmogorov]

Jérémi Dardé ¹ ; Julien Royer ¹

¹ Institut de Mathématiques de Toulouse, UMR5219, Univ. de Toulouse, CNRS, UPS F-31062 Toulouse Cedex 9, France

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 859-894.

Résumé
Abstract

Nous nous intéressons à l’observabilité d’équations de type Kolmogorov bi-dimensionnelles présentant une dégénérescence quadratique. Nous donnons un majorant et un minorant du temps critique. Dans une configuration symétrique, ces bornes coïncident et donnent alors précisément le temps critique d’observabilité. La preuve est basée sur des estimées de Carleman et sur l’étude des propriétés spectrales d’une famille d’opérateurs de Schrödinger non auto-adjoints, en particulier la localisation de la première valeur propre et des estimées de type Agmon pour les fonctions propres correspondantes.

This paper is devoted to the observability of a class of two-dimensional Kolmogorov-type equations presenting a quadratic degeneracy. We give lower and upper bounds for the critical time. These bounds coincide in symmetric settings, giving a sharp result in these cases. The proof is based on Carleman estimates and on the spectral properties of a family of non-selfadjoint Schrödinger operators, in particular the localization of the first eigenvalue and Agmon type estimates for the corresponding eigenfunctions.

Reçu le : 2020-06-17
Accepté le : 2021-03-02
Publié le : 2021-03-30

DOI : 10.5802/jep.160

Classification : 35K65, 93B07, 47B28, 47A10, 47D06
Keywords: Observability, Kolmogorov equations, Carleman estimates, non-selfadjoint operators, resolvent estimates
Mot clés : Observabilité, équations de Kolmogorov, estimées de Carleman, opérateurs non auto-adjoints, estimées de résolvante

Affiliations des auteurs :

Jérémi Dardé ¹ ; Julien Royer ¹

¹ Institut de Mathématiques de Toulouse, UMR5219, Univ. de Toulouse, CNRS, UPS F-31062 Toulouse Cedex 9, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JEP_2021__8__859_0,
     author = {J\'er\'emi Dard\'e and Julien Royer},
     title = {Critical time for the observability of {Kolmogorov-type} equations},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {859--894},
     publisher = {\'Ecole polytechnique},
     volume = {8},
     year = {2021},
     doi = {10.5802/jep.160},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.160/}
}

TY  - JOUR
AU  - Jérémi Dardé
AU  - Julien Royer
TI  - Critical time for the observability of Kolmogorov-type equations
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2021
SP  - 859
EP  - 894
VL  - 8
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.160/
DO  - 10.5802/jep.160
LA  - en
ID  - JEP_2021__8__859_0
ER  -

%0 Journal Article
%A Jérémi Dardé
%A Julien Royer
%T Critical time for the observability of Kolmogorov-type equations
%J Journal de l’École polytechnique — Mathématiques
%D 2021
%P 859-894
%V 8
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.160/
%R 10.5802/jep.160
%G en
%F JEP_2021__8__859_0

Jérémi Dardé; Julien Royer. Critical time for the observability of Kolmogorov-type equations. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 859-894. doi : 10.5802/jep.160. https://jep.centre-mersenne.org/articles/10.5802/jep.160/

Bibliographie
Cité par

[ABM20] D. Allonsius, F. Boyer & M. Morancey - “Analysis of the null-controllability of degenerate parabolic systems of Grushin type via the moments method”, 2020, to appear in J. Evol. Equ. | HAL

[Agm85] S. Agmon - “Bounds on exponential decay of eigenfunctions of Schrödinger operators”, in Schrödinger operators (Como, 1984), Lect. Notes in Math., vol. 1159, Springer, Berlin, 1985, p. 1-38 | DOI

[AKBGBdT16] F. Ammar Khodja, A. Benabdallah, M. González-Burgos & L. de Teresa - “New phenomena for the null controllability of parabolic systems: minimal time and geometrical dependence”, J. Math. Anal. Appl. 444 (2016) no. 2, p. 1071-1113 | DOI | MR

[Bak12] L. Bakri - “Quantitative uniqueness for Schrödinger operator”, Indiana Univ. Math. J. 61 (2012) no. 4, p. 1565-1580 | DOI | MR

[BBM20] A. Benabdallah, F. Boyer & M. Morancey - “A block moment method to handle spectral condensation phenomenon in parabolic control problems”, Ann. H. Lebesgue 3 (2020), p. 717-793 | DOI | MR

[BC17] K. Beauchard & P. Cannarsa - “Heat equation on the Heisenberg group: observability and applications”, J. Differential Equations 262 (2017) no. 8, p. 4475-4521 | DOI | MR | Zbl

[BCG14] K. Beauchard, P. Cannarsa & R. Guglielmi - “Null controllability of Grushin-type operators in dimension two”, J. Eur. Math. Soc. (JEMS) 16 (2014) no. 1, p. 67–101 | DOI | MR | Zbl

[BDE20] K. Beauchard, J. Dardé & S. Ervedoza - “Minimal time issues for the observability of Grushin-type equations”, Ann. Inst. Fourier (Grenoble) 70 (2020) no. 1, p. 247-312 | DOI | MR

[Bea14] K. Beauchard - “Null controllability of Kolmogorov-type equations”, Math. Control Signals Systems 26 (2014) no. 1, p. 145-176 | DOI | MR

[BHHR15] K. Beauchard, B. Helffer, R. Henry & L. Robbiano - “Degenerate parabolic operators of Kolmogorov type with a geometric control condition”, ESAIM Control Optim. Calc. Var. 21 (2015) no. 2, p. 487-512 | DOI | MR | Zbl

[BLR92] C. Bardos, G. Lebeau & J. Rauch - “Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary”, SIAM J. Control Optim. 30 (1992) no. 5, p. 1024-1065 | DOI | MR | Zbl

[BMM15] K. Beauchard, L. Miller & M. Morancey - “2D Grushin-type equations: minimal time and null controllable data”, J. Differential Equations 259 (2015) no. 11, p. 5813-5845 | DOI | MR | Zbl

[BS19] N. Burq & C. Sun - “Time optimal observability for Grushin Schrödinger equation”, 2019 | HAL

[BZ09] K. Beauchard & E. Zuazua - “Some controllability results for the 2D Kolmogorov equation”, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009) no. 5, p. 1793-1815 | DOI | Numdam | MR

[CMV16] P. Cannarsa, P. Martinez & J. Vancostenoble - Global Carleman estimates for degenerate parabolic operators with applications, Mem. Amer. Math. Soc., vol. 239, no. 1133, American Mathematical Society, Providence, RI, 2016 | DOI

[DK20] M. Duprez & A. Koenig - “Control of the Grushin equation: non-rectangular control region and minimal time”, ESAIM Control Optim. Calc. Var. 26 (2020), article ID 3, 18 pages | DOI | MR

[Dup17] M. Duprez - “Controllability of a $2 \times 2$ parabolic system by one force with space-dependent coupling term of order one”, ESAIM Control Optim. Calc. Var. 23 (2017) no. 4, p. 1473-1498 | DOI | MR

[Ego63] J. V. Egorov - “Some problems in the theory of optimal control”, Ž. Vyčisl. Mat. i Mat. Fiz. 3 (1963), p. 887-904 | MR

[EN00] K.-J. Engel & R. Nagel - One-parameter semigroups for linear evolution equations, Graduate Texts in Math., vol. 194, Springer-Verlag, New York, 2000 | MR

[FI96] A. V. Fursikov & O. Y. Imanuvilov - Controllability of evolution equations, Lect. Notes Series, vol. 34, Seoul National University, Seoul, 1996 | MR

[FR71] H. O. Fattorini & D. L. Russell - “Exact controllability theorems for linear parabolic equations in one space dimension”, Arch. Rational Mech. Anal. 43 (1971), p. 272-292 | DOI | MR | Zbl

[Hel88] B. Helffer - Semi-classical analysis for the Schrödinger operator and applications, Lect. Notes in Math., vol. 1336, Springer-Verlag, Berlin, 1988 | DOI | Zbl

[Hel11] B. Helffer - “On pseudo-spectral problems related to a time-dependent model in superconductivity with electric current”, Confluentes Math. 3 (2011) no. 2, p. 237-251 | DOI | MR

[Hel13] B. Helffer - Spectral theory and its applications, Cambridge Studies in Advanced Math., vol. 139, Cambridge University Press, Cambridge, 2013 | MR

[Hen14] R. Henry - “On the semi-classical analysis of Schrödinger operators with purely imaginary electric potentials in a bounded domain”, 2014 | arXiv

[HS10] B. Helffer & J. Sjöstrand - “From resolvent bounds to semi-group bounds”, 2010, Actes du colloque d’Évian 2009 | arXiv

[HSV13] M. Hitrik, J. Sjöstrand & J. Viola - “Resolvent estimates for elliptic quadratic differential operators”, Anal. PDE 6 (2013) no. 1, p. 181-196 | DOI | MR

[Kat80] T. Kato - Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1980

[Koe17] A. Koenig - “Non-null-controllability of the Grushin operator in 2D”, Comptes Rendus Mathématique 355 (2017) no. 12, p. 1215-1235 | DOI | MR | Zbl

[Koe20] A. Koenig - “Lack of null-controllability for the fractional heat equation and related equations”, SIAM J. Control Optim. 58 (2020) no. 6, p. 3130-3160 | DOI | MR | Zbl

[KRRS17] D. Krejčiřík, N. Raymond, J. Royer & P. Siegl - “Non-accretive Schrödinger operators and exponential decay of their eigenfunctions”, Israel J. Math. 221 (2017) no. 2, p. 779-802 | DOI | Zbl

[KS15] D. Krejčiřík & P. Siegl - “Elements of spectral theory without the spectral theorem”, in Non-selfadjoint operators in quantum physics, Wiley, Hoboken, NJ, 2015, p. 241-291 | DOI | Zbl

[KSTV15] D. Krejčiřík, P. Siegl, M. Tater & J. Viola - “Pseudospectra in non-Hermitian quantum mechanics”, J. Math. Phys. 56 (2015) no. 10, article ID 103513, 32 pages | DOI | MR

[Lis19] P. 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”, 2019 | HAL

[LL17] C. Laurent & M. Léautaud - “Tunneling estimates and approximate controllability for hypoelliptic equations”, 2017, to appear in Mem. Amer. Math. Soc. | HAL

[LR95] G. Lebeau & L. Robbiano - “Contrôle exact de l’équation de la chaleur”, Comm. Partial Differential Equations 20 (1995) no. 1-2, p. 335-356 | DOI

[LS20] C. Letrouit & C. Sun - “Observability of Baouendi-Grushin-type equations through resolvent estimates”, 2020 | HAL

[RT74] J. Rauch & M. Taylor - “Exponential decay of solutions to hyperbolic equations in bounded domains”, Indiana Univ. Math. J. 24 (1974), p. 79-86 | DOI | MR

[TW09] M. Tucsnak & G. Weiss - Observation and control for operator semigroups, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel, 2009 | DOI

[VS10] O. Vallée & M. Soares - Airy functions and applications to physics, Imperial College Press, London, 2010 | DOI

Cité par Sources :