On the Optimal Control of Propagation Fronts

TL;DR

This paper investigates optimal control strategies for reaction-diffusion fronts, deriving simplified models and explicit solutions for 1D traveling waves, with applications to pest eradication and moving set optimization.

Contribution

It introduces a new approach to control reaction-diffusion fronts by deriving measure-valued controls and a sharp interface limit for simplified modeling.

Findings

Explicit solutions for 1D traveling wave control problems

Measure-valued optimal controls in certain cases

Derivation of simplified models via sharp interface limits

Abstract

We consider a controlled reaction-diffusion equation, motivated by a pest eradication problem. Our goal is to derive a simpler model, describing the controlled evolution of a contaminated set. In this direction, the first part of the paper studies the optimal control of 1-dimensional traveling wave profiles. Using Stokes' formula, explicit solutions are obtained, which in some cases require measure-valued optimal controls. In the last section we introduce a family of optimization problems for a moving set. We show how these can be derived from the original parabolic problems, by taking a sharp interface limit.