# Mesh Refinement Method for Solving Bang-Bang Optimal Control Problems Using Direct Collocation

## TL;DR

This paper introduces a mesh refinement technique for solving bang-bang optimal control problems with direct collocation, accurately identifying switch times and improving computational efficiency.

## Contribution

It presents a novel mesh refinement approach that automatically estimates control discontinuities and switch times, enhancing solution accuracy and efficiency.

## Key findings

- Accurately determines control switch times using roots of switching functions.
- Improves computational efficiency over previous methods.
- Demonstrates effectiveness on three example problems.

## Abstract

A mesh refinement method is developed for solving bang-bang optimal control problems using direct collocation. The method starts by finding a solution on a coarse mesh. Using this initial solution, the method then determines automatically if the Hamiltonian is linear with respect to the control, and, if so, estimates the locations of the discontinuities in the control. The switch times are estimated by determining the roots of the switching functions, where the switching functions are determined using estimates of the state and costate obtained from the collocation method. The accuracy of the switch times is then improved on subsequent meshes by dividing the original optimal control problem into multiple domains and including variables that define the locations of the switch times. While in principle any collocation method can be used, in this research the previously developed Legendre-Gauss-Radau collocation method is employed because it provides an accurate approximation of the costate which in turn improves the approximation of the switching functions. The method of this paper is designed to be used with a previously developed mesh refinement method in order to accurately approximate the solution in segments where the solution is smooth. The method is demonstrated on three examples where it is shown to accurately determine the switching structure of a bang-bang optimal control problem. When compared with previously developed mesh refinement methods, the results demonstrate that the method developed in this paper improves computational efficiency when solving bang-bang optimal control problems.