Fast Mesh Refinement in Pseudospectral Optimal Control
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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 √(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 polynomials of over a thousandth order to solve a low-thrust, long-duration orbit transfer problem.
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