We propose a mathematical model to study the water usage for the irrigation of given farmland to guarantee that the field crop is kept in a good state of preservation. This problem is formulated as an optimal control problem. The lack of analytic solution leads us to turn to numerical methods to solve the problem numerically. We then apply necessary conditions of optimality to validate the numerical solution. To deal with the high degree of unpredictability of water inflow due to weather, we further propose a replan strategy and we implement it.

Artigo em fase de publicaçãoWe address necessary conditions of optimality (NCO), in the form of a maximum principle, for optimal control problems with state constraints. In particular, we are interested in the NCO that are strengthened to avoid the degeneracy phenomenon that occurs when the trajectory hits the boundary of the state constraint. In the literature on this subject, we can distinguish two types of constraint qualifications (CQ) under which the strengthened NCO can be applied: CQ involving the optimal control and CQ not involving it. Each one of these types of CQ has its own merits. The CQs involving the optimal control are not so easy to verify, but, are typically applicable to problems with less regularity on the data. In this article, we provide conditions under which the type of CQ involving the optimal control can be reduced to the other type. In this way, we also provide nondegenerate NCO that are valid under a different set of hypotheses.Fundação para a Ciência e a Tecnologia (FCT) - Project PTDC/EEACRO/ 116014/2009 "Optimal Control in Constrained and Hybrid Nonlinear Systems

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.

In this work, we address the daily irrigation problem of minimizing water consumption. This problem has the particularity that the dynamics is described via field capacity modes which can model the fact that the soil has different dynamics when it is saturated or not. An analysis is carried out to ensure that the normal Maximum Principle for nonsoomoth problems can be applied to this problem, and, it is observed that the numerical solution fulfills the necessary conditions of optimality.