Fast Mesh Refinement in Pseudospectral Optimal Control

Abstract

Mesh refinement in pseudospectral (PS) optimal control is embarrassingly easy: simply increase the order of the Lagrange interpolating polynomial, and the mathematics of convergence automates the distribution of the grid points. Unfortunately, as increases, the condition number of the resulting linear algebra increases as ; 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 in general, but it is independent of for the special case of one of the boundary points being fixed. Hence, spectral accuracy and efficiency are maintained as 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.