Desensitizing control for the heat equation with respect to domain variations

Ervedoza, Sylvain; Lissy, Pierre; Privat, Yannick

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1397-1429.

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).

Classification : 35K05, 93C20, 49K20
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

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