Analysis of a toy model for optimal crop protection

TL;DR

This paper studies an optimal control problem for crop protection using a toy model, proving the existence and structure of optimal interventions, and illustrating results with numerical simulations.

Contribution

It introduces a relaxation method to prove the existence of bang-bang controls and analyzes the shape of optimal intervention zones under geometric assumptions.

Findings

Existence of optimal bang-bang controls.

Shape results for intervention zones under geometric assumptions.

Numerical simulations illustrating theoretical results.

Abstract

In this paper we investigate an optimal control problem involving a toy model for the protection on a crop field. Precisely, we consider a protection on a crop field and we want to place intervention zones represented by a control, in order to maximise the protection on the field during a given period. Using a relaxation method, we prove that there exists a control which maximises the protection and, moreover, it must be a bang-bang control. Furthermore, with additional assumptions on the crop field geometry, some results on the shape of the optimal intervention are proved using comparison results for elliptic equations via Schwarz and Steiner symmetrizations. Finally, some numerical simulations are performed in order to illustrate those results.