Insensitizing control for linear and semi-linear heat equations with partially unknown domain

TL;DR

This paper investigates insensitizing control for semi-linear heat equations on unknown perturbed domains, establishing approximate insensitization and exact insensitization results, with methods tailored for linear and semi-linear cases.

Contribution

It introduces an approximate insensitization result for semi-linear heat equations and an exact insensitization property for linear cases, addressing unknown domain perturbations.

Findings

Approximate insensitization achieved via linearization and fixed point theorem.

Exact insensitization for certain deformation families in linear case.

Duality theory developed for the linear problem using unique continuation.

Abstract

We consider a semi-linear heat equation with Dirichlet boundary conditions and globally Lipschitz nonlinearity, posed on a bounded domain of R^N (N $\in$ N *), assumed to be an unknown perturbation of a reference domain. We are interested in an insensitizing control problem, which consists in finding a distributed control such that some functional of the state is insensitive at the first order to the perturbations of the domain. Our first result consists of an approximate insensitization property on the semi-linear heat equation. It rests upon a linearization procedure together with the use of an appropriate fixed point theorem. For the linear case, an appropriate duality theory is developed, so that the problem can be seen as a consequence of well-known unique continuation theorems. Our second result is specific to the linear case. We show a property of exact insensitization for some families of deformation given by one or two parameters. Due to the nonlinearity of the intrinsic control problem, no duality theory is available, so that our proof relies on a geometrical approach and direct computations.