Analysis of a toy model for optimal crop protection

Overview & Synopsis

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 maximize the protection on the field during a given period. We prove that there exists a unique control which maximizes 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.