Minimal Water Consumption for a Crop Fertirrigation Model

In this work, we consider a simplified model of crop fertirrigation as a non-autonomous controlled system, with soil moisture, nitrogen content, and biomass as state variables and the delivered water flow rate as input. We study the problem of minimizing the total water quantity delivered during the agricultural season under the constraints that the crops are not suffering from water or nitrogen stress at any time. We establish sufficient conditions for the feasibility of the problem and depict several control strategies depending on the initial nitrogen content. In particular, we show that this problem can exhibit an infinity of singular trajectories of the same cost.