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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.

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