Spectral Algorithm for Pseudospectral Methods in Optimal Control

Recent convergence results with pseudospectral methods are exploited to design a robust, multigrid, spectral algorithm for computing optimal controls. The design of the algorithm is based on using the pseudospectral differentiation matrix to locate switches, kinks, corners, and other discontinuities that are typical when solving practical optimal control problems. The concept of pseudospectral knots and Gaussian quadrature rules are used to generate a natural spectral mesh that is dense near the points of interest. Several stopping criteria are developed based on new error-estimation formulas and Jackson’s theorem. The sequence is terminated when all of the convergence criteria are satisfied. Numerical examples demonstrate the key concepts proposed in the design of the spectral algorithm. Although a vast number of theoretical and algorithmic issues still remain open, this paper advances pseudospectral methods along several new directions and outlines the current theoretical pitfalls in computation and control.