Irrigation Perturbed Optimal Linear-Control Problem

We analyse the irrigation problem as an optimal control problem within the context of agricultural planning. By formulating irrigation as an optimal control system, we aim to optimize the water usage, while ensuring efficient agricultural productivity. A key parameter in our model is the percentage of water loss due to deep percolation, denoted by $\beta$, which is difficult to estimate and sensitive to environmental perturbations. As a result, the problem becomes a parametric optimal linear-control one with an inequality state constraint. The main goal of this work consists in investigating how perturbations of $\beta$ affect the optimal solution and the optimal cost, with the help of parameter sensitivity analysis. To achieve this, we reformulate the parametric optimal linear-control problem as a finite-dimensional non-linear programming one for which we derive necessary and second-order sufficient optimality conditions. Additionally, we design a parameter sensitivity analysis. Numerical results are provided to illustrate the designed parameter sensitivity analysis which shows that as $\beta$ increases, the irrigation period of time and the irrigation water flow rate also increase.