Analytical Study for Different Extremal State Solutions of an Irrigation Optimal Control Problem with Field Capacity Modes

Abstract

In this paper we study the problem of daily irrigation of an agricultural crop using optimal control. The dynamics is a model based on field capacity modes, where the state, x, represents the water in the soil and the control variable, u, is the flow rate of water from irrigation. The variation of water in the soil depends on the field capacity of the soil, $$x_{FC}$$ x FC , weather conditions, losses due to deep percolation and irrigation. Our goal is to minimize the amount of water used for irrigation while keeping the crop in a good state of preservation. To enforce such requirement, the state constraint $$x(t) \ge x_{\min }$$ x ( t ) ≥ x min is coupled with the dynamics, where $$x_{\min }$$ x min is the hydrological need of the crop. Consequently, the problem under study is a state constrained optimal control problem. Under some mild assumptions, we consider several basic profiles for the optimal trajectories. Appealing to the Maximum Principle (MP), we characterize analytically the solution and its multipliers for each case. We then illustrate the analytical results running some computational simulations, where the analytical information is used to partially validate the computed solutions. The need to study irrigation strategies is of foremost importance nowadays since 80% of the fresh water used on our planet is used in agriculture.

Estimation of Hydraulic Parameters from the Soil Water Characteristic Curve

“…At this stage, the soil is said to be at field capacity. At FC, the water and air contents of the soil are considered to be ideal for crop growth [53][54][55]. The permanent wilting point refers to the soil moisture that means, essentially, no water is available to the plant.…”

Untitled

“…Despite some inconsistencies in the obtained research results, all the mentioned authors believe that the greatest efficiency of local irrigation, reduction of energy consumption, and high efficiency of their use are achieved at clearly defined upper and lower thresholds for moistening the local soil volume (Vozhehova et al, 2017;Lemos-Paiаo et al, 2022;Scholasch & Rienth, 2019). Reducing soil moisture, or increasing it beyond the established limits, reduces the yield of plantings, the quality of berries, and the efficiency of using energy resources (Weiler et al, 2018).…”