Density Functions for Mesh Refinement in Numerical Optimal Control

Abstract

needed to understand how the density/monitor functions can be used in numerical optimal control and how they can influence the accuracy and robustness of numerical optimal control algorithms. Furthermore, the choice of “good” density/monitor functions for mesh discretization of optimal control problems seems to be open. Inthispaperweattempttoprovideapartialanswertotheprevious questions. We introduce a method to distribute the mesh points efficiently using density/monitor functions. Although different monitor functions can be used for mesh generation, an appropriate choice of a monitor function can generate a better quality mesh, and can improve the accuracy of the solution along with the speed of convergence. Hence, the problem of mesh generation can be treated asaproblemof findinganappropriatedensity/monitorfunction.Two possible choices of density functions are used in the numerical examples, based on the discrete control/state histories from the previousiterationduringthemeshrefinementprocess.Theproposed method avoids the numerical integration step and the use of ODE solversforthesystemdynamicsaswasdonein[8].Yet,itgeneratesa mesh with a suitable level of adaptive discretization that provides sharp resolution around the places where the control switches or the trajectory meets/leaves state constraints, thus resulting in better accuracy of the overall final solution. Numerical examples are presented to demonstrate the advantage of the proposed method, and comparisons are provided against the industry-standard sparse optimal control software (SOCS).