Adaptively Refined Mesh for Collocation-Based Dynamic Optimization

This thesis deals with the topic of efficient numerical solutions to dynamic optimization problems using a direct collocation approach with adaptive mesh refinement. A novel h-method is proposed and implemented in a newly developed dynamic optimization framework that utilizes direct collocation with flipped Legendre-Gauss-Radau points. The mesh refinement algorithm aims to uniformize control trajectories by successive bisection based on slope and curvature analysis in each interval. For smooth problems and under suitable convergence, a termination guarantee is established. The algorithm has significant advantages over traditional direct collocation approaches without mesh refinement in terms of accuracy and computation time. The dynamic optimization framework allows for accessible and expressive modeling and proves to be very effective when applied to a variety of example problems, including academic and real-world physical applications. However, the proposed $h$-method offers room for improvement since no direct error estimates are incorporated, and it is shown that the problem class studied in this thesis allows for an extension to the broad and advanced class of pseudospectral mesh refinement algorithms. Moreover, it is demonstrated that embedding the method in a modern modeling and simulation environment would greatly benefit in terms of performance, symbolic handling, and for error analysis as well as validation of optimal solutions.