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.

We give necessary and sufficient conditions for the controllability of a Schrödinger equation involving the sub-Laplacian of a nilmanifold obtained by taking the quotient of a group of Heisenberg type by one of its discrete sub-groups. This class of nilpotent Lie groups is a major example of stratified Lie groups of step 2. The sub-Laplacian involved in these Schrödinger equations is subelliptic, and, contrary to what happens for the usual elliptic Schrödinger equation for example on flat tori or on negatively curved manifolds, there exists a minimal time of controllability. The main tools used in the proofs are (operator-valued) semi-classical measures constructed by use of representation theory and a notion of semi-classical wave packets that we introduce here in the context of groups of Heisenberg type.

Dans cet article, nous donnons des conditions nécessaires et des conditions suffisantes pour la contrôlabilité d’une équation de Schrödinger impliquant un opérateur sous-elliptique sur une variété compacte. Cet opérateur est le sous-laplacien d’une variété obtenue en quotientant un groupe de type Heisenberg par l’un de ses sous-groupes discrets. Cette classe de groupes nilpotents est un exemple important de groupes de Lie de pas 2. Le sous-laplacien est alors un opérateur sous-elliptique et nous montrons qu’à la différence de ce qui se passe pour le cas elliptique sur le tore ou sur des surfaces à courbures négatives, il existe un temps minimal de contrôlabilité pour l’équation de Schrödinger associée à ce sous-laplacien. Les principaux outils que nous utilisons sont des mesures semi-classiques à valeurs opérateurs construites via la théorie des représentations et une notion de paquets d’ondes semi-classiques que nous introduisons ici dans le contexte des groupes de type Heisenberg.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.176
Classification: 35R03, 35H20, 35J10, 93B07, 35Q93
Keywords: Sub-elliptic operator, control theory, observability, nilmanifold, H-type group
Mot clés : Opérateur sous-elliptique, théorie du contrôle, nilvariété, groupe de type Heisenberg
Fermanian Kammerer, Clotilde 1, 2; Letrouit, Cyril 3, 4

1 Univ Paris Est Creteil, CNRS, LAMA F-94010 Creteil, France
2 Univ Gustave Eiffel, LAMA F-77447 Marne-la-Vallée, France
3 Sorbonne Université, Université Paris-Diderot, CNRS, Inria, LJLL 75005 Paris, France
4 DMA, École normale supérieure, CNRS, PSL Research University 75005 Paris, France
@article{JEP_2021__8__1459_0,
     author = {Fermanian Kammerer, Clotilde and Letrouit, Cyril},
     title = {Observability and controllability for {the~Schr\"odinger} equation on quotients of groups of {Heisenberg} type},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1459--1513},
     publisher = {Ecole polytechnique},
     volume = {8},
     year = {2021},
     doi = {10.5802/jep.176},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jep.176/}
}
TY  - JOUR
AU  - Fermanian Kammerer, Clotilde
AU  - Letrouit, Cyril
TI  - Observability and controllability for the Schrödinger equation on quotients of groups of Heisenberg type
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2021
SP  - 1459
EP  - 1513
VL  - 8
PB  - Ecole polytechnique
UR  - http://www.numdam.org/articles/10.5802/jep.176/
DO  - 10.5802/jep.176
LA  - en
ID  - JEP_2021__8__1459_0
ER  - 
%0 Journal Article
%A Fermanian Kammerer, Clotilde
%A Letrouit, Cyril
%T Observability and controllability for the Schrödinger equation on quotients of groups of Heisenberg type
%J Journal de l’École polytechnique — Mathématiques
%D 2021
%P 1459-1513
%V 8
%I Ecole polytechnique
%U http://www.numdam.org/articles/10.5802/jep.176/
%R 10.5802/jep.176
%G en
%F JEP_2021__8__1459_0
Fermanian Kammerer, Clotilde; Letrouit, Cyril. 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. http://www.numdam.org/articles/10.5802/jep.176/

[1] Anantharaman, Nalini; Fermanian Kammerer, Clotilde; Macià, Fabricio Semiclassical completely integrable systems: long-time dynamics and observability via two-microlocal Wigner measures, Amer. J. Math., Volume 137 (2015) no. 3, pp. 577-638 | DOI | MR | Zbl

[2] Anantharaman, Nalini; Macià, Fabricio Semiclassical measures for the Schrödinger equation on the torus, J. Eur. Math. Soc. (JEMS), Volume 16 (2014) no. 6, pp. 1253-1288 | DOI | Zbl

[3] Bahouri, Hajer Non prolongement unique des solutions d’opérateurs ‘somme de carrés’, Ann. Inst. Fourier (Grenoble), Volume 36 (1986) no. 4, pp. 137-155 | DOI | MR | Zbl

[4] Bahouri, Hajer; Fermanian Kammerer, Clotilde; Gallagher, Isabelle Dispersive estimates for the Schrödinger operator on step-2 stratified Lie groups, Anal. PDE, Volume 9 (2016) no. 3, pp. 545-574 | DOI | Zbl

[5] Bahouri, Hajer; Gérard, Patrick; Xu, Chao-Jiang Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg, J. Anal. Math., Volume 82 (2000), pp. 93-118 | DOI | Zbl

[6] Beauchard, K.; Cannarsa, P. Heat equation on the Heisenberg group: observability and applications, J. Differential Equations, Volume 262 (2017) no. 8, pp. 4475-4521 | DOI | MR | Zbl

[7] Beauchard, Karine; Dardé, Jérémi; Ervedoza, Sylvain Minimal time issues for the observability of Grushin-type equations, Ann. Inst. Fourier (Grenoble), Volume 70 (2020) no. 1, pp. 247-312 http://aif.cedram.org/item?id=AIF_2020__70_1_247_0 | DOI | MR | Zbl

[8] Bonfiglioli, A.; Lanconelli, E.; Uguzzoni, F. Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monographs in Math., Springer, Berlin, 2007 | Zbl

[9] Bony, Jean-Michel Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble), Volume 19 (1969) no. 1, pp. 277-304 | DOI | Numdam | Zbl

[10] Burq, Nicolas; Sun, Chenmin Time optimal observability for Grushin Schrödinger equation, 2019 (to appear in Anal. PDE) | arXiv

[11] Burq, Nicolas; Zworski, Maciej Control for Schrödinger operators on tori, Math. Res. Lett., Volume 19 (2012) no. 2, pp. 309-324 | DOI | Zbl

[12] Chabu, Victor; Fermanian Kammerer, Clotilde; Macià, Fabricio Semiclassical analysis of dispersion phenomena, Analysis and partial differential equations: perspectives from developing countries (Springer Proc. Math. Stat.), Volume 275, Springer, Cham, 2019, pp. 84-108 | DOI | MR | Zbl

[13] Chabu, Victor; Fermanian Kammerer, Clotilde; Macià, Fabricio Wigner measures and effective mass theorems, Ann. H. Lebesgue, Volume 3 (2020), pp. 1049-1089 | DOI | MR | Zbl

[14] Combescure, Monique; Robert, Didier Coherent states and applications in mathematical physics, Theoretical and Math. Physics, Springer, Dordrecht, 2012 | DOI | Zbl

[15] Corwin, Lawrence J.; Greenleaf, Frederick P. Representations of nilpotent Lie groups and their applications. Part I. Basic theory and examples, Cambridge Studies in Advanced Math., 18, Cambridge University Press, Cambridge, 1990 | Zbl

[16] Duprez, Michel; Koenig, Armand Control of the Grushin equation: non-rectangular control region and minimal time, ESAIM Control Optim. Calc. Var., Volume 26 (2020), 3, 18 pages | DOI | MR | Zbl

[17] Fermanian Kammerer, Clotilde Mesures semi-classiques 2-microlocales, C. R. Acad. Sci. Paris Sér. I Math., Volume 331 (2000) no. 7, pp. 515-518 | DOI | MR | Zbl

[18] Fermanian Kammerer, Clotilde Analyse à deux échelles d’une suite bornée de L 2 sur une sous-variété du cotangent, Comptes Rendus Mathématique, Volume 340 (2005) no. 4, pp. 269-274 | DOI | MR | Zbl

[19] Fermanian Kammerer, Clotilde; Fischer, Véronique Quantum evolution and sub-Laplacian operators on groups of Heisenberg type, 2019 (to appear in J. Spectral Theory) | arXiv

[20] Fermanian Kammerer, Clotilde; Fischer, Véronique Semi-classical analysis on H-type groups, Sci. China Math., Volume 62 (2019) no. 6, pp. 1057-1086 | DOI | MR | Zbl

[21] Fermanian Kammerer, Clotilde; Fischer, Véronique Defect measures on graded Lie groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), Volume 21 (2020), pp. 207-291 | DOI | MR | Zbl

[22] Fermanian Kammerer, Clotilde; Gérard, Patrick Mesures semi-classiques et croisement de modes, Bull. Soc. math. France, Volume 130 (2002) no. 1, pp. 123-168 | DOI | Numdam | MR | Zbl

[23] Fermanian Kammerer, Clotilde; Lasser, Caroline Propagation through generic level crossings: a surface hopping semigroup, SIAM J. Math. Anal., Volume 40 (2008) no. 1, pp. 103-133 | DOI | MR | Zbl

[24] Fischer, Veronique; Ruzhansky, Michael Quantization on nilpotent Lie groups, Progress in Math., 314, Birkhäuser/Springer, Cham, 2016 | DOI | MR | Zbl

[25] Gérard, Patrick Mesures semi-classiques et ondes de Bloch, Séminaire sur les Équations aux Dérivées Partielles, 1990–1991, École Polytechnique, Palaiseau, 1991 (Exp. No. XVI, 19 p.) | Numdam | Zbl

[26] Gérard, Patrick Microlocal defect measures, Comm. Partial Differential Equations, Volume 16 (1991) no. 11, pp. 1761-1794 | DOI | MR | Zbl

[27] Gérard, Patrick; Leichtnam, Éric Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J., Volume 71 (1993) no. 2, pp. 559-607 | DOI | MR | Zbl

[28] Gérard, Patrick; Markowich, Peter A.; Mauser, Norbert J.; Poupaud, Frédéric Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., Volume 50 (1997) no. 4, pp. 323-379 Erratum: Ibid. 53 (2000), no. 2, p. 280–281 | DOI | MR | Zbl

[29] Hagedorn, G. A. Semiclassical quantum mechanics. I. The 0 limit for coherent states, Comm. Math. Phys., Volume 71 (1980) no. 1, pp. 77-93 http://projecteuclid.org/euclid.cmp/1103907396 | DOI | MR

[30] Helffer, B.; Martinez, A.; Robert, D. Ergodicité et limite semi-classique, Comm. Math. Phys., Volume 109 (1987) no. 2, pp. 313-326 http://projecteuclid.org/euclid.cmp/1104116844 | DOI | Zbl

[31] Hörmander, Lars Hypoelliptic second order differential equations, Acta Math., Volume 119 (1967), pp. 147-171 | DOI | MR | Zbl

[32] Jaffard, S. Contrôle interne exact des vibrations d’une plaque rectangulaire, Portugal. Math., Volume 47 (1990) no. 4, pp. 423-429 | Zbl

[33] Kaplan, Aroldo Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc., Volume 258 (1980) no. 1, pp. 147-153 | DOI | MR | Zbl

[34] Koenig, Armand Non-null-controllability of the Grushin operator in 2D, Comptes Rendus Mathématique, Volume 355 (2017) no. 12, pp. 1215-1235 | DOI | MR | Zbl

[35] Lasser, Caroline; Teufel, Stefan Propagation through conical crossings: an asymptotic semigroup, Comm. Pure Appl. Math., Volume 58 (2005) no. 9, pp. 1188-1230 | DOI | MR | Zbl

[36] Laurent, Camille; Léautaud, Matthieu Tunneling estimates and approximate controllability for hypoelliptic equations, 2017 (to appear in Mem. Amer. Math. Soc.) | arXiv

[37] Lebeau, Gilles Control for hyperbolic equations, Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1992), École Polytechnique, Palaiseau, 1992, p. 24 | Numdam | MR | Zbl

[38] Lebeau, Gilles Contrôle de l’équation de Schrödinger, J. Math. Pures Appl. (9), Volume 71 (1992) no. 3, pp. 267-291 | Zbl

[39] Lebeau, Gilles Équation des ondes amorties, Algebraic and geometric methods in mathematical physics (Kaciveli, 1993) (Math. Phys. Stud.), Volume 19, Kluwer Acad. Publ., Dordrecht, 1996, pp. 73-109 | DOI | Zbl

[40] Letrouit, Cyril Subelliptic wave equations are never observable, 2020 | arXiv

[41] Letrouit, Cyril; Sun, Chenmin Observability of Baouendi-Grushin-type equations through resolvent estimates, 2020 (to appear in J. Inst. Math. Jussieu) | arXiv

[42] Lions, J.-L. Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1. Contrôlabilité exacte, Recherches en Mathématiques Appliquées, 8, Masson, Paris, 1988 | Zbl

[43] Lions, Pierre-Louis; Paul, Thierry Sur les mesures de Wigner, Rev. Mat. Iberoamericana, Volume 9 (1993) no. 3, pp. 553-618 | DOI | MR | Zbl

[44] Macià, Fabricio High-frequency propagation for the Schrödinger equation on the torus, J. Funct. Anal., Volume 258 (2010) no. 3, pp. 933-955 | DOI | MR | Zbl

[45] Macià, Fabricio The Schrödinger flow in a compact manifold: high-frequency dynamics and dispersion, Modern aspects of the theory of partial differential equations (Oper. Theory Adv. Appl.), Volume 216, Birkhäuser/Springer Basel AG, Basel, 2011, pp. 275-289 | DOI | MR

[46] Macià, Fabricio High-frequency dynamics for the Schrödinger equation, with applications to dispersion and observability, Nonlinear optical and atomic systems (Lect. Notes in Math.), Volume 2146, Springer, Cham, 2015, pp. 275-335 | DOI | MR | Zbl

[47] Macià, Fabricio; Rivière, Gabriel Two-microlocal regularity of quasimodes on the torus, Anal. PDE, Volume 11 (2018) no. 8, pp. 2111-2136 | DOI | MR | Zbl

[48] Macià, Fabricio; Rivière, Gabriel Observability and quantum limits for the Schrödinger equation on 𝕊 d , Probabilistic methods in geometry, topology and spectral theory, American Mathematical Society, Providence, RI, 2019, pp. 139-153 | DOI | Zbl

[49] Miller, Luc Propagation d’ondes semi-classiques à travers une interface et mesures 2-microlocales, Ph. D. Thesis, École Polytechnique, Palaiseau (1996)

[50] Nier, Francis A semi-classical picture of quantum scattering, Ann. Sci. École Norm. Sup. (4), Volume 29 (1996) no. 2, pp. 149-183 | DOI | Numdam | MR | Zbl

[51] Pedersen, Niels Vigand Matrix coefficients and a Weyl correspondence for nilpotent Lie groups, Invent. Math., Volume 118 (1994) no. 1, pp. 1-36 | DOI | MR | Zbl

[52] Tartar, Luc H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, Volume 115 (1990) no. 3-4, pp. 193-230 | DOI | MR | Zbl

[53] Taylor, Michael E. Noncommutative harmonic analysis, Math. Surveys and Monographs, 22, American Mathematical Society, Providence, RI, 1986 | DOI | MR | Zbl

Cited by Sources: