[Stabilisation globale de l’équation de Schrödinger non linéaire cubique défocalisante sur le tore]
In this article, we prove the (uniform) global exponential stabilization of the cubic defocusing nonlinear Schrödinger equation on the torus $(\mathbb{R}/2 \pi \mathbb{Z})^d$, for $d=1$, $2$ or $3$, with a linear damping localized in a subset of the torus satisfying some geometrical assumptions. In particular, this answers an open question of Dehman, Gérard and Lebeau from 2006. Our approach is based on three ingredients. First, we prove the well-posedness of the closed-loop system in Bourgain spaces. Secondly, we derive new Carleman estimates for the nonlinear equation by directly including the cubic term in the conjugated operator. Thirdly, by conjugating with energy estimates and Morawetz multipliers method, we then deduce quantitative observability estimates leading to the uniform exponential decay of the total energy of the system. As a corollary of the global stabilization result, we obtain an upper bound of the minimal time of the global null-controllability of the nonlinear equation by using a stabilization procedure and a local null-controllability result.
Dans cet article, nous prouvons la stabilisation exponentielle globale (uniforme) de l’équation de Schrödinger non linéaire défocalisationte cubique sur le tore $(\mathbb{R}/2 \pi \mathbb{Z})^d$, pour $d=1$, $2$ ou $3$, avec un amortissement linéaire localisé dans un sous-ensemble du tore satisfaisant certaines hypothèses géométriques. Cela répond notamment à une question ouverte de Dehman, Gérard et Lebeau de 2006. Notre approche repose sur trois ingrédients. Premièrement, nous prouvons le caractère bien posé du système en boucle fermée dans les espaces de Bourgain. Deuxièmement, nous obtenons de nouvelles estimations de Carleman pour l’équation non linéaire en incluant directement le terme cubique dans l’opérateur conjugué. Troisièmement, en conjuguant les estimations d’énergie et la méthode des multiplicateurs de Morawetz, nous en déduisons ensuite des estimations d’observabilité quantitative conduisant à la décroissance exponentielle uniforme de l’énergie totale du système. Comme corollaire du résultat de stabilisation globale, nous obtenons une borne supérieure sur le temps minimal de contrôlabilité globale à zéro de l’équation non linéaire en utilisant une procédure de stabilisation et un résultat de contrôlabilité locale à zéro.
Accepté le :
Publié le :
Keywords: Stabilization, Controllability, Nonlinear Schrödinger equation, Bourgain spaces, Carleman estimates
Mots-clés : Stabilisation, Contrôlabilité, Equation de Schrödinger non linéaire, Espaces de Bourgain, Estimations de Carleman
Kévin Le Balc’h 1 ; Jérémy Martin 1
CC-BY 4.0
@article{AFST_2025_6_34_3_539_0,
author = {K\'evin Le Balc{\textquoteright}h and J\'er\'emy Martin},
title = {Global stabilization of the cubic defocusing nonlinear {Schr\"odinger} equation on the torus},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {539--579},
publisher = {Universit\'e de Toulouse, Toulouse},
volume = {Ser. 6, 34},
number = {3},
year = {2025},
doi = {10.5802/afst.1819},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1819/}
}
TY - JOUR AU - Kévin Le Balc’h AU - Jérémy Martin TI - Global stabilization of the cubic defocusing nonlinear Schrödinger equation on the torus JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2025 SP - 539 EP - 579 VL - 34 IS - 3 PB - Université de Toulouse, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1819/ DO - 10.5802/afst.1819 LA - en ID - AFST_2025_6_34_3_539_0 ER -
%0 Journal Article %A Kévin Le Balc’h %A Jérémy Martin %T Global stabilization of the cubic defocusing nonlinear Schrödinger equation on the torus %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2025 %P 539-579 %V 34 %N 3 %I Université de Toulouse, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1819/ %R 10.5802/afst.1819 %G en %F AFST_2025_6_34_3_539_0
Kévin Le Balc’h; Jérémy Martin. Global stabilization of the cubic defocusing nonlinear Schrödinger equation on the torus. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 3, pp. 539-579. doi : 10.5802/afst.1819. https://afst.centre-mersenne.org/articles/10.5802/afst.1819/
[1] Global uniqueness theorems for linear and nonlinear waves, J. Funct. Anal., Volume 269 (2015) no. 11, pp. 3458-3499 | DOI | MR | Zbl
[2] Wigner measures and observability for the Schrödinger equation on the disk, Invent. Math., Volume 206 (2016) no. 2, pp. 485-599 | DOI | MR | Zbl
[3] Semiclassical measures for the Schrödinger equation on the torus, J. Eur. Math. Soc., Volume 16 (2014) no. 6, pp. 1253-1288 | DOI | MR | Zbl
[4] Some inverse stability results for the bistable reaction-diffusion equation using Carleman inequalities, C. R. Math., Volume 347 (2009) no. 11-12, pp. 619-622 | DOI | Numdam | MR | Zbl
[5] Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal., Volume 3 (1993) no. 2, pp. 107-156 | DOI | MR | Zbl
[6] Global solutions of nonlinear Schrödinger equations, Colloquium Publications, 46, American Mathematical Society, 1999, viii+182 pages | DOI | MR | Zbl
[7] Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Am. J. Math., Volume 126 (2004) no. 3, pp. 569-605 http://muse.jhu.edu/... | DOI | MR | Zbl
[8] Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math., Volume 159 (2005) no. 1, pp. 187-223 | DOI | MR | Zbl
[9] Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations, Ann. Sci. Éc. Norm. Supér. (4), Volume 38 (2005) no. 2, pp. 255-301 | DOI | Numdam | MR | Zbl
[10] Rough controls for Schrödinger operators on 2-tori, Ann. Henri Lebesgue, Volume 2 (2019), pp. 331-347 | DOI | Numdam | MR | Zbl
[11] Asymptotic behavior of cubic defocusing Schrödinger equations on compact surfaces, Z. Angew. Math. Phys., Volume 69 (2018) no. 4, 100, 27 pages | DOI | MR | Zbl
[12] Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10, Courant Institute; American Mathematical Society, 2003, xiv+323 pages | DOI | Numdam | MR | Zbl
[13] Control and nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, 2007, xiv+426 pages | DOI | MR | Zbl
[14] Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Math. Z., Volume 254 (2006) no. 4, pp. 729-749 | DOI | MR | Zbl
[15] Stabilization and control for the subcritical semilinear wave equation, Ann. Sci. Éc. Norm. Supér. (4), Volume 36 (2003) no. 4, pp. 525-551 | DOI | Numdam | MR | Zbl
[16] Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d’espace (d’après Bourgain), Séminaire Bourbaki, Vol. 1994/95 (Astérisque), Volume 237, Société Mathématique de France, 1996, pp. 163-187 (Exp. No. 796) | MR | Zbl
[17] A note on the exact internal control of nonlinear Schrödinger equations, Quantum control: mathematical and numerical challenges (CRM Proceedings & Lecture Notes), Volume 33, American Mathematical Society, 2003, pp. 127-137 | DOI | MR
[18] The energy-critical defocusing NLS on , Duke Math. J., Volume 161 (2012) no. 8, pp. 1581-1612 | DOI | MR | Zbl
[19] Contrôle interne exact des vibrations d’une plaque rectangulaire, Port. Math., Volume 47 (1990) no. 4, pp. 423-429 | MR | Zbl
[20] Fourier series in control theory, Springer Monographs in Mathematics, Springer, 2005, x+226 pages | DOI | MR | Zbl
[21] Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. II. -estimates, J. Inverse Ill-Posed Probl., Volume 12 (2004) no. 2, pp. 183-231 | DOI | MR | Zbl
[22] Global controllability and stabilization for the nonlinear Schrödinger equation on an interval, ESAIM, Control Optim. Calc. Var., Volume 16 (2010) no. 2, pp. 356-379 | DOI | Numdam | MR | Zbl
[23] Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifolds of dimension 3, SIAM J. Math. Anal., Volume 42 (2010) no. 2, pp. 785-832 | DOI | MR | Zbl
[24] Internal control of the Schrödinger equation, Math. Control Relat. Fields, Volume 4 (2014) no. 2, pp. 161-186 | DOI | MR | Zbl
[25] Contrôle de l’équation de Schrödinger, J. Math. Pures Appl. (9), Volume 71 (1992) no. 3, pp. 267-291 | MR | Zbl
[26] Exact controllability for the Schrödinger equation, SIAM J. Control Optim., Volume 32 (1994) no. 1, pp. 24-34 | DOI | MR | Zbl
[27] Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights, Inverse Probl., Volume 24 (2008) no. 1, 015017, 18 pages | DOI | MR | Zbl
[28] Observability and control of Schrödinger equations, SIAM J. Control Optim., Volume 40 (2001) no. 1, pp. 211-230 | DOI | MR | Zbl
[29] Local exact controllability and stabilizability of the nonlinear Schrödinger equation on a bounded interval, SIAM J. Control Optim., Volume 48 (2009) no. 2, pp. 972-992 | DOI | MR | Zbl
[30] Control and stabilization of the nonlinear Schrödinger equation on rectangles, Math. Models Methods Appl. Sci., Volume 20 (2010) no. 12, pp. 2293-2347 | DOI | MR | Zbl
[31] Exponential stability of the nonlinear Schrödinger equation with locally distributed damping on compact Riemannian manifold, Adv. Nonlinear Anal., Volume 10 (2021) no. 1, pp. 569-583 | DOI | MR | Zbl
[32] Remarks on the controllability of the Schrödinger equation, Quantum control: mathematical and numerical challenges (CRM Proceedings & Lecture Notes), Volume 33, American Mathematical Society, 2003, pp. 193-211 | DOI | MR
Cité par Sources :