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arXiv · 1904.12992

Fast Mesh Refinement in Pseudospectral Optimal Control

N. Koeppen·I. M. Ross·L. C. Wilcox·R. J. Proulx
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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.

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N. Koeppen, I. M. Ross, L. C. Wilcox, R. J. Proulx. 2019-04-29. Fast Mesh Refinement in Pseudospectral Optimal Control. https://arxiv.org/abs/1904.12992

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