In this survey paper, we report on recent works concerning exact observability (and, by duality, exact controllability) properties of subelliptic wave and Schrödinger-type equations. These results illustrate the slowdown of propagation in directions transverse to the horizontal distribution. The proofs combine sub-Riemannian geometry, semi-classical analysis, spectral theory and non-commutative harmonic analysis.
@article{TSG_2019-2021__36__51_0, author = {Cyril Letrouit}, title = {Exact observability properties of subelliptic wave and {Schr\"odinger} equations}, journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie}, pages = {51--102}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {36}, year = {2019-2021}, doi = {10.5802/tsg.373}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/tsg.373/} }
TY - JOUR AU - Cyril Letrouit TI - Exact observability properties of subelliptic wave and Schrödinger equations JO - Séminaire de théorie spectrale et géométrie PY - 2019-2021 SP - 51 EP - 102 VL - 36 PB - Institut Fourier PP - Grenoble UR - https://proceedings.centre-mersenne.org/articles/10.5802/tsg.373/ DO - 10.5802/tsg.373 LA - en ID - TSG_2019-2021__36__51_0 ER -
%0 Journal Article %A Cyril Letrouit %T Exact observability properties of subelliptic wave and Schrödinger equations %J Séminaire de théorie spectrale et géométrie %D 2019-2021 %P 51-102 %V 36 %I Institut Fourier %C Grenoble %U https://proceedings.centre-mersenne.org/articles/10.5802/tsg.373/ %R 10.5802/tsg.373 %G en %F TSG_2019-2021__36__51_0
Cyril Letrouit. Exact observability properties of subelliptic wave and Schrödinger equations. Séminaire de théorie spectrale et géométrie, Tome 36 (2019-2021), pp. 51-102. doi : 10.5802/tsg.373. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.373/
[1] Andrei Agrachev; Davide Barilari; Ugo Boscain A comprehensive introduction to sub-Riemannian geometry. From the Hamiltonian viewpoint, Cambridge Studies in Advanced Mathematics, 181, Cambridge University Press, 2020 | Zbl
[2] Paolo Albano; Antonio Bove; Marco Mughetti Analytic hypoellipticity for sums of squares and the Treves conjecture, J. Funct. Anal., Volume 274 (2018) no. 10, pp. 2725-2753 | DOI | MR | Zbl
[3] Nalini Anantharaman; Matthieu Léautaud; Fabricio Macià Wigner measures and observability for the Schrödinger equation on the disk, Invent. Math., Volume 206 (2016) no. 2, pp. 485-599 | DOI | Zbl
[4] Nalini Anantharaman; Fabricio Macià Semiclassical measures for the Schrödinger equation on the torus, J. Eur. Math. Soc., Volume 16 (2014) no. 6, pp. 1253-1288 | DOI | Zbl
[5] Hajer Bahouri; Clotilde Fermanian Kammerer; Isabelle Gallagher Dispersive estimates for the Schrödinger operator on step-2 stratified Lie groups, Anal. PDE, Volume 9 (2016) no. 3, pp. 545-574 | DOI | Zbl
[6] Hajer Bahouri; Patrick Gérard; Chao-Jiang Xu Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg, J. Anal. Math., Volume 82 (2000) no. 1, pp. 93-118 | DOI | Zbl
[7] Claude Bardos; Gilles Lebeau; Jeffrey Rauch Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., Volume 30 (1992) no. 5, pp. 1024-1065 | DOI | MR | Zbl
[8] Karine Beauchard; Piermarco Cannarsa Heat equation on the Heisenberg group: Observability and applications, J. Differ. Equations, Volume 262 (2017) no. 8, pp. 4475-4521 | DOI | MR | Zbl
[9] Karine Beauchard; Piermarco Cannarsa; Roberto Guglielmi Null controllability of Grushin-type operators in dimension two, J. Eur. Math. Soc., Volume 16 (2014) no. 1, pp. 67-101 | MR | Zbl
[10] Karine Beauchard; Jérémi Dardé; Sylvain Ervedoza Minimal time issues for the observability of Grushin-type equations, Ann. Inst. Fourier, Volume 70 (2020) no. 1, pp. 247-312 | DOI | Numdam | MR | Zbl
[11] Karine Beauchard; Luc Miller; Morgan Morancey 2D Grushin-type equations: minimal time and null controllable data, J. Differ. Equations, Volume 259 (2015) no. 11, pp. 5813-5845 | DOI | MR | Zbl
[12] Karine Beauchard; Karel Pravda-Starov Null-controllability of hypoelliptic quadratic differential equations, Journal de l’École polytechnique – Mathématiques, Volume 5 (2018), pp. 1-43 | DOI | Numdam | MR | Zbl
[13] Ugo Boscain; Camille Laurent The Laplace-Beltrami operator in almost-Riemannian geometry, Ann. Inst. Fourier, Volume 63 (2013) no. 5, pp. 1739-1770 | DOI | Numdam | MR | Zbl
[14] Ugo Boscain; Dario Prandi Self-adjoint extensions and stochastic completeness of the Laplace–Beltrami operator on conic and anticonic surfaces, J. Differ. Equations, Volume 260 (2016) no. 4, pp. 3234-3269 | DOI | MR | Zbl
[15] Nicolas Burq Mesures semi-classiques et mesures de défaut (Astérisque), Volume 245, Société Mathématique de France, 1997, pp. 167-195 | Numdam | Zbl
[16] Nicolas Burq; Patrick Gérard Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Math. Acad. Sci. Paris, Volume 325 (1997) no. 7, pp. 749-752 | DOI | Zbl
[17] Nicolas Burq; Chenmin Sun Time optimal observability for Grushin Schrödinger equation (2021) (ArXiv preprint, to appear in Analysis & PDEs, arXiv:1910.03691)
[18] Nicolas Burq; Maciej Zworski Geometric control in the presence of a black box, J. Am. Math. Soc., Volume 17 (2004) no. 2, pp. 443-471 | DOI | MR | Zbl
[19] Nicolas Burq; Maciej Zworski Control for Schrödinger operators on tori, Math. Res. Lett., Volume 19 (2012) no. 2, pp. 309-324 | DOI | Zbl
[20] Yves Colin de Verdière; Luc Hillairet; Emmanuel Trélat Spectral asymptotics for sub-Riemannian Laplacians, I: Quantum ergodicity and quantum limits in the 3-dimensional contact case, Duke Math. J., Volume 167 (2018) no. 1, pp. 109-174 | MR | Zbl
[21] Yves Colin de Verdière; Cyril Letrouit Propagation of well-prepared states along Martinet singular geodesics, J. Spectr. Theory, Volume 12 (2022) no. 3, pp. 1235-1253 | DOI | MR | Zbl
[22] Jean-Michel Coron Control and nonlinearity, Mathematical Surveys and Monographs, American Mathematical Society, 2007 no. 136 | Zbl
[23] Laurence Corwin; Frederick P. Greenleaf Representations of nilpotent Lie groups and their applications. Part 1: Basic theory and examples, 18, Cambridge University Press, 1990 | Zbl
[24] Jérémi Dardé; Julien Royer Critical time for the observability of Kolmogorov-type equations, J. Éc. Polytech., Math., Volume 8 (2021), pp. 859-894 | DOI | Numdam | MR | Zbl
[25] Belhassen Dehman; Patrick Gérard; Gilles Lebeau Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Math. Z., Volume 254 (2006) no. 4, pp. 729-749 | DOI
[26] Michel Duprez; Armand Koenig Control of the Grushin equation: non-rectangular control region and minimal time, ESAIM, Control Optim. Calc. Var., Volume 26 (2020), 3 | MR | Zbl
[27] Thomas Duyckaerts; Luc Miller Resolvent conditions for the control of parabolic equations, J. Funct. Anal., Volume 263 (2012) no. 11, pp. 3641-3673 | DOI | MR | Zbl
[28] Semyon Dyatlov; Long Jin; Stéphane Nonnenmacher Control of eigenfunctions on surfaces of variable curvature, J. Am. Math. Soc., Volume 35 (2022) no. 2, pp. 361-465 | DOI | MR | Zbl
[29] VS Fedii On a criterion for hypoellipticity, Math. USSR, Sb., Volume 14 (1971) (1972) no. 1, pp. 15-45 | DOI | Zbl
[30] Clotilde Fermanian Kammerer; Véronique Fischer Semi-classical analysis on H-type groups, Sci. China, Math., Volume 62 (2019) no. 6, pp. 1057-1086 | MR | Zbl
[31] Clotilde Fermanian Kammerer; Véronique Fischer Quantum evolution and sub-Laplacian operators on groups of Heisenberg type, J. Spectr. Theory, Volume 11 (2021) no. 3, pp. 1313-1367 | DOI | MR | Zbl
[32] Clotilde Fermanian Kammerer; Cyril Letrouit Observability and controllability for the Schrödinger equation on quotients of groups of Heisenberg type, J. Éc. Polytech., Math., Volume 8 (2021), pp. 1459-1513 | DOI | Numdam | Zbl
[33] Valentina Franceschi; Dario Prandi; Luca Rizzi On the essential self-adjointness of singular sub-Laplacians, Potential Anal., Volume 53 (2020) no. 1, pp. 89-112 | DOI | MR | Zbl
[34] Nicola Garofalo Fractional thoughts (2017) (ArXiv preprint, arXiv:1712.03347v1)
[35] Patrick Gérard Microlocal defect measures, Commun. Partial Differ. Equations, Volume 16 (1991) no. 11, pp. 1761-1794 | DOI | MR | Zbl
[36] Patrick Gérard; Sandrine Grellier The cubic Szegö equation, Ann. Sci. Éc. Norm. Supér., Volume 43 (2010) no. 5, pp. 761-810 | DOI | Numdam | Zbl
[37] Bernard Helffer; Francis Nier Hypoelliptic estimates and spectral theory for Fokker–Planck operators and Witten Laplacians, Lecture Notes in Mathematics, 1862, Springer, 2005 | DOI | Zbl
[38] Lars Hörmander Hypoelliptic second order differential equations, Acta Math., Volume 119 (1967) no. 1, pp. 147-171 | DOI | MR | Zbl
[39] Lars Hörmander On the existence and the regularity of solutions of linear pseudodifferential equations, Enseign. Math., Volume 17 (1971) no. 2, pp. 99-103 | Zbl
[40] Lars Hörmander The analysis of linear partial differential operators III: Pseudo-differential operators, Classics in Mathematics, Springer, 2007 | DOI | Zbl
[41] Emmanuel Humbert; Yannick Privat; Emmanuel Trélat Observability properties of the homogeneous wave equation on a closed manifold, Commun. Partial Differ. Equations, Volume 44 (2019) no. 9, pp. 749-772 | DOI | MR | Zbl
[42] Stéphane Jaffard Contrôle interne exact des vibrations d’une plaque rectangulaire, Port. Math., Volume 47 (1990) no. 4, pp. 423-429 | Zbl
[43] Frédéric Jean Control of nonholonomic systems: from sub-Riemannian geometry to motion planning, Springer, 2014 | Zbl
[44] Aroldo Kaplan Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Am. Math. Soc., Volume 258 (1980) no. 1, pp. 147-153 | DOI | MR | Zbl
[45] Armand Koenig Non-null-controllability of the Grushin operator in 2D, C. R. Math. Acad. Sci. Paris, Volume 355 (2017) no. 12, pp. 1215-1235 | DOI | Numdam | MR | Zbl
[46] Andrey Kolmogoroff Zufällige Bewegungen (zur Theorie der brownschen Bewegung), Ann. Math., Volume 35 (1934), pp. 116-117 | DOI | Zbl
[47] Camille Laurent; Matthieu Léautaud Tunneling estimates and approximate controllability for hypoelliptic equations, Memoirs of the American Mathematical Society, 1357, American Mathematical Society, 2022 | DOI | Zbl
[48] Jérôme Le Rousseau; Gilles Lebeau; Peppino Terpolilli; Emmanuel Trélat Geometric control condition for the wave equation with a time-dependent observation domain, Anal. PDE, Volume 10 (2017) no. 4, pp. 983-1015 | DOI | MR | Zbl
[49] Gilles Lebeau Control for hyperbolic equations, Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1992), École Polytech., Palaiseau, 1992, pp. 1-24 | Numdam | Zbl
[50] Gilles Lebeau Contrôle de l’équation de Schrödinger, J. Math. Pures Appl., Volume 71 (1992) no. 3, pp. 267-291 | Zbl
[51] Gilles Lebeau; Luc Robbiano Contrôle exact de l’équation de la chaleur, Commun. Partial Differ. Equations, Volume 20 (1995) no. 1-2, pp. 335-356 | DOI | Numdam | Zbl
[52] Cyril Letrouit Propagation of singularities for subelliptic wave equations, Commun. Math. Phys., Volume 395 (2022) no. 1, pp. 143-178 | DOI | MR | Zbl
[53] Cyril Letrouit Quantum limits of sub-Laplacians via joint spectral calculus, Doc. Math., Volume 28 (2023) no. 1, pp. 55-104 | DOI | MR | Zbl
[54] Cyril Letrouit Subelliptic wave equations are never observable, Anal. PDE, Volume 16 (2023) no. 3, pp. 643-678 | DOI | MR | Zbl
[55] Cyril Letrouit; Chenmin Sun Observability of Baouendi–Grushin-type equations through resolvent estimates, J. Inst. Math. Jussieu, Volume 22 (2023) no. 2, pp. 541-579 | DOI | MR | Zbl
[56] Jacques-Louis Lions Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées, 8, Masson, 1988 | Zbl
[57] 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 (2020) (Hal preprint HAL Id: hal-02420212)
[58] Richard B. Melrose Propagation for the wave group of a positive subelliptic second-order differential operator, Hyperbolic equations and related topics (Katata/Kyoto, 1984) (Taniguchi Symp. HERT Katata), Academic Press Inc., 1986, pp. 181-192 | DOI | Zbl
[59] Richard B. Melrose; Johannes Sjöstrand Singularities of boundary value problems. I, Commun. Pure Appl. Math., Volume 31 (1978) no. 5, pp. 593-617 | DOI | MR | Zbl
[60] Luc Miller Resolvent conditions for the control of unitary groups and their approximations, J. Spectr. Theory, Volume 2 (2012) no. 1, pp. 1-55 | DOI | MR | Zbl
[61] Richard Montgomery Abnormal minimizers, SIAM J. Control Optim., Volume 32 (1994) no. 6, pp. 1605-1620 | DOI | MR | Zbl
[62] Richard Montgomery Hearing the zero locus of a magnetic field, Commun. Math. Phys., Volume 168 (1995) no. 3, pp. 651-675 | DOI | MR | Zbl
[63] Richard Montgomery A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, American Mathematical Society, 2002 no. 91 | Zbl
[64] Yoshinori Morimoto On the hypoellipticity for infinitely degenerate semi-elliptic operators, J. Math. Soc. Japan, Volume 30 (1978) no. 2, pp. 327-358 | MR | Zbl
[65] Dario Prandi; Luca Rizzi; Marcello Seri Quantum confinement on non-complete Riemannian manifolds, J. Spectr. Theory, Volume 8 (2018) no. 4, pp. 1221-1280 | DOI | MR | Zbl
[66] James Ralston Gaussian beams and the propagation of singularities, Studies in partial differential equations (MAA Studies in Mathematics), Volume 23, Mathematical Association of America, 1982, pp. 206-248 | MR | Zbl
[67] Linda Preiss Rothschild; Elias M. Stein Hypoelliptic differential operators and nilpotent groups, Acta Math., Volume 137 (1976) no. 1, pp. 247-320 | DOI | MR | Zbl
[68] Michael Eugene Taylor Noncommutative harmonic analysis, Mathematical Surveys and Monographs, 22, American Mathematical Society, 1986 | DOI | Zbl
[69] François Treves Symplectic geometry and analytic hypo-ellipticity, Differential equations: La Pietra 1996 (Proceedings of Symposia in Pure Mathematics), Volume 65, American Mathematical Society (1999), pp. 201-219 | MR | Zbl
Cité par Sources :