Fast Mesh Refinement in Pseudospectral Optimal Control
For practitioners of pseudospectral optimal control, this work provides a practical solution to maintain spectral accuracy at high polynomial orders, enabling efficient mesh refinement for challenging problems.
This paper addresses the ill-conditioning of pseudospectral optimal control discretizations as polynomial order increases, proposing a Birkhoff interpolation-based method that reduces condition number growth from N^2 to sqrt(N) (or constant for fixed boundary). The method enables mesh refinement with polynomials over 1000th order, demonstrated on a low-thrust orbit transfer problem.
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.