A fractional calculus approach to the dynamic optimization of biological reactive systems. Part II: Numerical solution of fractional optimal control problems

• Fractional optimal control problems are formulated for biological reactive systems.
• Combines numerical/analytical strategies are proposed for solving FOCP.
• One approach combines optimality conditions for a FOCP and the gradient method.
• Second approach combines an NLP solver, shooting method and Laplace transformation.
• Resulting profiles show the effect of the fractional orders in the optimal results.

This second paper of our series is concerned with the formulation and solution strategies of fractional optimal control problems (FOCP). Given the sets of fractional differential equations representing the behavior of fermentation and thermal hydrolysis reactive systems, here we formulate the corresponding FOCP’s and describe suitable techniques for solving them. An analytical/numerical strategy that combines the optimality conditions and the gradient method for FOCP as well as the predictor–corrector fractional integrator is used to obtain optimal dilution rate profiles for the fermentation case-study. For the case of the thermal hydrolysis, the strategy involves discretization of the FOCP to formulate it as a Non-Linear Programming problem; then, the solution algorithm involves the use of an NLP solver and the shooting technique coupled to an inverse Laplace transformation subroutine. The optimal profiles show the performance of the numerical solution approaches proposed and the effect of the fractional orders in the optimal results.