Insensitizing controls for the heat equation with respect to boundary variations

Sylvain Ervedoza (1) , Pierre Lissy (2) , Yannick Privat (3, 4)

1 IMB - Institut de Mathématiques de Bordeaux
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 TONUS - TOkamaks and NUmerical Simulations
4 IRMA - Institut de Recherche Mathématique Avancée

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).

Mots clés: heat equation; exact/approximate control; domain variations; insensitization properties; Brouwer fixed-point theorem

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Sylvain Ervedoza, Pierre Lissy, Yannick Privat. Insensitizing controls for the heat equation with respect to boundary variations. 2020. ⟨hal-03083177v1⟩