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, Birkhoff interpolation concepts are advanced over an arbitrary grid to generate well-conditioned PS optimal control discretizations. It is shown that the condition number increases only as $\sqrt{N}$ in general, but it 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 polynomials of over one-thousandth order to solve a low-thrust long-duration orbit transfer problem.

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We take this opportunity to thank the anonymous reviewers for providing many detailed comments. Their comments greatly improved the quality of this paper.

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Received14 June 2018
Accepted19 September 2018
Published online20 December 2018

Fast Mesh Refinement in Pseudospectral Optimal Control

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