[Contrôle désensibilisant pour l’équation de la chaleur par rapport à des variations du domaine]
Cet article est dédié à l’étude de problèmes de désensibilisation par rapport à des variations du domaine, pour des fonctionnelles quadratiques dépendant de la solution de l’équation de la chaleur linéaire. Ce travail peut être vu comme la suite du travail [28], dans la mesure où nous généralisons un certain nombre de résultats qu’il contient, et où nous nous intéressons à de nouvelles propriétés en lien avec ce travail. Nous considérons des variations du domaine spatial sur lequel la solution de l’EDP est définie, et nous nous intéressons à trois questions : (i) désensibilisation approchée, (ii) désensibilisation approchée couplée avec une propriété de désensibilisation exacte sur un sous-espace vectoriel de dimension finie, (iii) désensibilisation exacte. Nous donnons des réponses positives aux points (i) et (ii), et des résultats partiels au point (iii).
This article is dedicated to desensitizing issues for a quadratic functional involving the solution of the linear heat equation with respect to domain variations. This work can be seen as a continuation of [28], insofar as we generalize several of the results it contains and investigate new related properties. In our framework, we consider variations of the spatial domain on which the solution of the PDE is defined at each time, and investigate three main issues: (i) approximate desensitizing, (ii) approximate desensitizing combined with an exact desensitizing for a finite-dimensional subspace, and (iii) exact desensitizing. We provide positive answers to questions (i) and (ii) and partial results to question (iii).
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Keywords: Heat equation, exact/approximate control, domain variations, insentizing/desensitizing properties, Brouwer fixed-point theorem
Mot clés : Équation de la chaleur, contrôle approché/exact, variations de domaines, propriétés de désensibilisation
Sylvain Ervedoza 1 ; Pierre Lissy 2 ; Yannick Privat 3
CC-BY 4.0
@article{JEP_2022__9__1397_0,
author = {Sylvain Ervedoza and Pierre Lissy and Yannick Privat},
title = {Desensitizing control for the heat equation with respect to domain variations},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1397--1429},
publisher = {\'Ecole polytechnique},
volume = {9},
year = {2022},
doi = {10.5802/jep.209},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.209/}
}
TY - JOUR AU - Sylvain Ervedoza AU - Pierre Lissy AU - Yannick Privat TI - Desensitizing control for the heat equation with respect to domain variations JO - Journal de l’École polytechnique — Mathématiques PY - 2022 SP - 1397 EP - 1429 VL - 9 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.209/ DO - 10.5802/jep.209 LA - en ID - JEP_2022__9__1397_0 ER -
%0 Journal Article %A Sylvain Ervedoza %A Pierre Lissy %A Yannick Privat %T Desensitizing control for the heat equation with respect to domain variations %J Journal de l’École polytechnique — Mathématiques %D 2022 %P 1397-1429 %V 9 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.209/ %R 10.5802/jep.209 %G en %F JEP_2022__9__1397_0
Sylvain Ervedoza; Pierre Lissy; Yannick Privat. Desensitizing control for the heat equation with respect to domain variations. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1397-1429. doi : 10.5802/jep.209. https://jep.centre-mersenne.org/articles/10.5802/jep.209/
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