Optimal trajectory approximation by cubic splines on fed-batch control problems

Optimal control problems appear in several engineering fields and in particular on the control of fed-batch fermentation processes. These problems are often described by sets of nonlinear differential and algebraic equations, usually subject to constraints in the state and control variables. Tradicional approaches to the optimal feed trajectory computation consists in getting a linear spline that approximates the trajectory, which optimizes a given performance of the fed-batch fermentation process. This approach leads to non-differentiable trajectories that can pose some problems to implement in practice, resulting in a possible discrepancy of the simulated and real performances. In this paper we develop a technique to obtain a cubic spline for the approximate trajectory, leading to a smooth approximation function. We provide numerical results for a set of case studies where the AMPL modeling language, CVODE ordinary differential equations solver and a particle swarm algorithm were used.