[Temps minimal de commutation de la magnétisation dans de petites structures ellipsoïdales ferromagnétiques]
Dans cet article, nous considérons un matériau ferromagnétique de forme ellipsoïdale. Le moment magnétique associé possède alors deux équilibres asymptotiquement stables opposés, de la forme $\pm \overline{m}$. Pour utiliser ces matériaux à des fins de stockage de mémoire, il est nécessaire de savoir comment contrôler le moment magnétique. Nous utilisons comme variable de contrôle un champ magnétique externe spatialement uniforme et nous considérons la question du renversement du moment magnétique, c’est-à-dire le passage de la configuration $+\overline{m}$ à la configuration $-\overline{m}$, en un temps minimal. Bien entendu, il est nécessaire d’imposer des restrictions sur le champ magnétique externe utilisé. Nous incluons donc une contrainte sur la norme $L^\infty $ des contrôles, supposée inférieure à une valeur seuil $U$. Nous montrons que, de manière générique par rapport aux dimensions de l’ellipsoïde, il existe une valeur minimale de $U$ pour que ce problème ait une solution. Nous la caractérisons alors précisément. Enfin, nous étudions certaines configurations particulières associées à des géométries présentant des propriétés de symétrie et montrons que, dans ce cas, le moment magnétique peut être contrôlé en temps minimal sans imposer de condition de seuil sur $U$.
In this paper, we consider a ferromagnetic material of ellipsoidal shape. The associated magnetic moment has then two asymptotically stable opposite equilibria, of the form $\pm \overline{m}$. In order to use these materials for memory storage purposes, it is necessary to know how to control the magnetic moment. We use as a control variable a spatially uniform external magnetic field and consider the question of flipping the magnetic moment, i.e., changing it from the $+\overline{m}$ configuration to the $-\overline{m}$ one, in minimal time. Of course, it is necessary to impose restrictions on the external magnetic field used. We therefore include a constraint on the $L^\infty $ norm of the control, assumed to be less than a threshold value $U $. We show that, generically with respect to the dimensions of the ellipsoid, there is a minimal value of $U $ for this problem to have a solution. We then characterize it precisely. Finally, we investigate some particular configurations associated to geometries enjoying symmetry properties and show that in this case the magnetic moment can be controlled in minimal time without imposing a threshold condition on $U $.
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Keywords: Ferromagnetic materials, Landau-Lifshitz equation, optimal control, minimal time
Mots-clés : Matériaux ferromagnétiques, équation de Landau-Lifshitz, contrôle optimal, temps minimal
Raphaël Côte 1 ; Clémentine Courtès 2 ; Guillaume Ferrière 2 ; Yannick Privat 3
CC-BY 4.0
@article{JEP_2025__12__147_0,
author = {Rapha\"el C\^ote and Cl\'ementine Court\`es and Guillaume Ferri\`ere and Yannick Privat},
title = {Minimal time of magnetization switching in small ferromagnetic ellipsoidal samples},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {147--184},
publisher = {\'Ecole polytechnique},
volume = {12},
year = {2025},
doi = {10.5802/jep.287},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.287/}
}
TY - JOUR AU - Raphaël Côte AU - Clémentine Courtès AU - Guillaume Ferrière AU - Yannick Privat TI - Minimal time of magnetization switching in small ferromagnetic ellipsoidal samples JO - Journal de l’École polytechnique — Mathématiques PY - 2025 SP - 147 EP - 184 VL - 12 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.287/ DO - 10.5802/jep.287 LA - en ID - JEP_2025__12__147_0 ER -
%0 Journal Article %A Raphaël Côte %A Clémentine Courtès %A Guillaume Ferrière %A Yannick Privat %T Minimal time of magnetization switching in small ferromagnetic ellipsoidal samples %J Journal de l’École polytechnique — Mathématiques %D 2025 %P 147-184 %V 12 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.287/ %R 10.5802/jep.287 %G en %F JEP_2025__12__147_0
Raphaël Côte; Clémentine Courtès; Guillaume Ferrière; Yannick Privat. Minimal time of magnetization switching in small ferromagnetic ellipsoidal samples. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 147-184. doi : 10.5802/jep.287. https://jep.centre-mersenne.org/articles/10.5802/jep.287/
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