# Fast Mesh Refinement in Pseudospectral Optimal Control

## TL;DR

This paper introduces a novel Birkhoff interpolation-based mesh refinement method for pseudospectral optimal control that maintains low condition numbers and spectral accuracy even at very high polynomial orders, enabling efficient solutions to complex problems.

## Contribution

It develops a Birkhoff interpolation approach to produce well-conditioned pseudospectral discretizations with condition numbers growing only as -20 words)

## Key findings

- Condition number grows as -20 words)
- Achieves spectral accuracy at high polynomial orders
- Successfully solves a 1000th order orbit transfer problem

## Abstract

Mesh refinement in pseudospectral (PS) optimal control is embarrassingly easy --- simply increase the order $N$ of the Lagrange interpolating polynomial and the mathematics of convergence automates the distribution of the grid points. Unfortunately, as $N$ increases, the condition number of the resulting linear algebra increases as $N^2$; hence, spectral efficiency and accuracy are lost in practice. In this paper, we advance Birkhoff interpolation concepts over an arbitrary grid to generate well-conditioned PS optimal control discretizations. We show that the condition number increases only as $\sqrt{N}$ in general, but is independent of $N$ for the special case of one of the boundary points being fixed. Hence, spectral accuracy and efficiency are maintained as $N$ increases. The effectiveness of the resulting fast mesh refinement strategy is demonstrated by using \underline{polynomials of over a thousandth order} to solve a low-thrust, long-duration orbit transfer problem.