Insensitizing controls for the heat equation with respect to boundary variations

This article is dedicated to insensitization issues of a quadratic functional involving the solution of the linear heat equation with respect to domains variations. This work can be seen as a continuation of [P. Lissy, Y. Privat, and Y. Simpor\'e. Insensitizing control for linear and semi-linea...

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Published inJournal de l'École polytechnique. Mathématiques Vol. Tome 9; pp. 1397 - 1429
Main Authors Ervedoza, Sylvain, Lissy, Pierre, Privat, Yannick
Format Journal Article
LanguageEnglish
Published École polytechnique 2022
Subjects
Analysis of PDEs
Mathematics
Optimization and Control
Brouwer fixed-point theorem
domain variations
insensitization properties
heat equation
exact/approximate control
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Summary:This article is dedicated to insensitization issues of a quadratic functional involving the solution of the linear heat equation with respect to domains variations. This work can be seen as a continuation of [P. Lissy, Y. Privat, and Y. Simpor\'e. Insensitizing control for linear and semi-linear heat equations with partially unknown domain. ESAIM Control Optim. Calc. Var., 25:Art. 50, 21, 2019], insofar as we generalize several of the results it contains and investigate new related properties. In our framework, we consider boundary variations of the spatial domain on which the solution of the PDE is defined at each time, and investigate three main issues: (i) approximate insensitization, (ii) approximate insensitization combined with an exact insensitization for a finite-dimensional subspace, and (iii) exact insensitization. We provide positive answers to questions (i) and (ii) and partial results to question (iii).
ISSN:2429-7100
2270-518X