Insensitizing controls for the heat equation with respect to boundary variations

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. Simporé. 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).

An insensitizing control problem for a linear Stabilized Kuramoto-Sivashinsky system

“…For insensitizing problems of equations in fluid mechanics we refer to the works [9,12,13,22] and for a phase field system to [7]. Most recently, the insensitizing control problem has been addressed from a numerical point of view in [6], for fourth-order parabolic equations in [25] and with respect to shape variations in [18,28].…”