Water management in agriculture is a key issue due to the increasing problem of water scarcity worldwide. Based on the recent progress in the dynamic modeling of plant growth in interaction with the water resource, our objective is to study the optimal control problem of crop irrigation. For this purpose, we first describe the LNAS model for sugar beet growth, driving the dynamics of both plant biomass and soil water reserve. We then introduce the utility function corresponding to the farmer's profit and derive the value function from the Hamilton-Jacobi-Bellman (HJB) equation. Then a backward finite-difference scheme is implemented to solve the HJB equation. It is proved to converge under a proper Courant-Friedrichs-Lewy condition for the discretization step. A few numerical simulations are provided to illustrate the resolution.