A parametric optimal control problem applied to daily irrigation

¹ Center for Research and Development in Mathematics and Applications (CIDMA), University of Aveiro, 3810-193 Aveiro, Portugal
² Physics Center of Minho and Porto Universities (CF-UM-UP), 4710-057 Braga, Portugal
³ Faculty of Engineering, DEEC, SYSTEC, ISR, University of Porto, 4200-465 Porto, Portugal

^* Corresponding author: anapaiao@ua.pt

Received: 13 May 2024
Accepted: 9 November 2024

Abstract

We conduct sensitivity analysis of an optimal control problem applied to agricultural irrigation. The aim is to minimize the square of the amount of irrigation water while ensuring the health grow of the crop. A crucial parameter of our model is the percentage of water loss due to deep percolation, β, a parameter hard to estimate and subject to perturbations. Thus, our problem is a parametric state constrained optimal control problem with a L² cost. Our goal is to study how perturbations of β affect the optimal solutions of our problem using sensitivity analysis of our problem using sensitivity analysis.

To solve numerically our optimal control problem we use the direct method, transcribing the problem into a non-linear programming problem. We show how sensitivity analysis applied to the non-linear programming problem provides information on the variation of optimal solutions of the original problem in terms of β. Valid approximations of optimal solution and cost, provided by sensitivity analysis, are computed for values of β within of a certain neighbourhood. Remarkably, we show that for all β in such neighbourhood, the irrigation period is kept constant. Only the flow rate of irrigation water changes.