Insensitizing controls for the heat equation with respect to boundary variations

TL;DR

This paper investigates insensitization of a quadratic functional related to the heat equation under boundary domain variations, extending previous work and exploring approximate and exact insensitization methods.

Contribution

It generalizes prior results on insensitization for the heat equation and introduces new properties for boundary variation scenarios, including approximate and partial exact insensitization.

Findings

Positive results for approximate insensitization

Positive results for combined approximate and finite-dimensional exact insensitization

Partial results for full exact insensitization

Abstract

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).