Dynamic Optimization via Functional Control Vector Parametrization and Simple Gradient-Based Algorithms

This work introduces a novel control vector parametrization strategy based on basis function expansions. Considering that the control signals belong to Hilbert space L2[0, tf], where tf denotes the final reaction time, they are expressed as linear combinations of a selected function basis. The original problem, formulated as a nonlinear dynamic optimization problem, is thereby transformed into a static nonlinear optimization problem, which is subsequently solved using a direct gradient-based method. The proposed approach avoids the use of adjoint variables or complex numerical techniques, demonstrating that even a simple gradient-based algorithm can achieve competitive performance without the need for more sophisticated optimization tools. The methodology is applied to two benchmark chemical processes: (i) biodiesel production in a batch reactor, aimed at maximizing the concentration of ethyl esters at the final time tf, and (ii) a fed-batch fermentation process involving two control actions, where the objective is to maximize the concentration of extracellular xylitol at the final time tf. Results demonstrate that high-quality control trajectories can be obtained with a small number of parameters, yielding better or comparable performance indices than previously reported methods while ensuring continuous and implementable control profiles, without requiring any smoothing stage.