Analysis of a toy model for optimal crop protection

TL;DR

The paper formulates an optimal-control problem for crop-field protection where a fast-responding protective state $p_u$ evolves under a linear elliptic-diffusion relation and the objective $\mathcal{J}(u) = \int_0^T \int_Ω p_u dx dt$ is maximized under a budget constraint $\int_Ω u ≤ L$. Employing a relaxation approach and convexity arguments, it proves the existence and uniqueness of a bang-bang optimizer, and, for special geometries, derives explicit or structured shape results for the intervention region using Schwarz and Steiner symmetrizations. Theoretical results are complemented by 2D numerical simulations that validate the bang-bang nature and symmetry properties of the optimal interventions across various domain geometries, including disks and rectangles. Overall, the work provides both rigorous structural insights and practical guidance on spatially allocating protective resources in crop protection while respecting environmental and budget constraints.

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.

Figures (5)

• Figure 1: (1a) The optimal control on $\Omega_1$. (1b) The protection for the optimal control at the final time $T$. The values of the cost functional (1c) and the surface of the intervention zone (1d) with respect to the iterations.
• Figure 2: (2a) The optimal control on $\Omega_2$. (2b) The protection for the optimal control at the final time $T$. The values of the cost functional (2c) and the surface of the intervention zone (2d) with respect to the iterations.
• Figure 3: (3a) The optimal control on the circular domain $\Omega_3$. (3b) The protection for the optimal control at the final time $T$. The values of the cost functional (3c) and the surface of the intervention zone (3d) with respect to the iterations.
• Figure 4: (4a) The optimal control on the rectangular domain $\Omega_4$. (4b) The protection for the optimal control at the final time $T$. The values of the cost functional (4c) and the surface of the intervention zone (4d) with respect to the iterations.
• Figure 5: (5a) The value of $\mathcal{J}$ at the optimal control and the surface of the intervention zone on the different domains considered. (5b) A circular domain with an annular intervention zone and the value of $\mathcal{J}$ for this control.

Theorems & Definitions (32)

• Theorem 1.1
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• proof : Proof of the existence and uniqueness of a bang-bang solution
• Remark 3.1
• proof : Proof of item \ref{['firstitem']}