Optimal control of linear non-local parabolic problems with an integral kernel

TL;DR

This paper studies optimal control of a linear non-local heat equation with an integral kernel, addressing theoretical and numerical aspects, and demonstrating effectiveness through examples.

Contribution

It introduces a theoretical and computational framework for controlling non-local heat equations with integral kernels, inspired by biological applications.

Findings

Effective control strategies demonstrated through numerical examples

Implementation of Lions' low-regret approach for incomplete data

Theoretical insights into non-local parabolic control problems

Abstract

We consider a linear non-local heat equation in a bounded domain $\Omega\subset\mathbb{R}^d$, $d\geq 1$, with Dirichlet boundary conditions, where the non-locality is given by the presence of an integral kernel. Motivated by several applications in biological systems, in the present paper we study some optimal control problems from a theoretical and numerical point of view. In particular, we will employ the classical low-regret approach of J.-L. Lions for treating the problem of incomplete data and provide a simple computational implementation of the method. The effectiveness of the results are illustrated by several examples.