Fast Mesh Refinement in Pseudospectral Optimal Control

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 N2; 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 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.