Optimal control of a system of reaction‐diffusion equations modeling the wine fermentation process

The investigation of a reaction–diffusion system with new nonlinear reactions kinetics to model the fermentation of wine is presented. The reactive part extends existing ordinary differential models by taking into account oxygen availability and ethanol toxicity. The presence of spatial diffusion and the inclusion of a heat equation allow geometrical and thermal effects in the model. Existence, uniqueness, and regularity of solutions to this system of reaction–diffusion partial differential equations are discussed. A boundary optimal control problem is formulated with the purpose of steering an ideal fermentation process. This optimal control problem is theoretically investigated and numerically solved in the adjoint method framework. The existence of an optimal control is proved, and its solution is characterized by means of the corresponding optimality system. To solve this system, an implicit–explicit splitting approach is considered, and a Broyden‐Fletcher‐Goldfarb‐Shanno (BFGS) optimization scheme is implemented. Results of numerical experiments demonstrate the validity of the fermentation model and of the proposed control strategy.