The Bellman Pseudospectral Method

Abstract

Based on the discoveries of a recently proposed algorithm for low-thrust trajectory optimization, we present the Bellman Pseudospectral (PS) method for a generic optimal control problem. In our original algorithm, we combined the properties of PS methods with Bellman's principle to provide an optimal solution to multi-scale and long horizon trajectory optimization problems. In this paper, we generalize this concept to provide a low cost solution to generate feasible solutions to optimal control problems. In the limit, this algorithm converges to our original concept; hence, our current proposal may also be considered as a cheap mesh-refinement technique for trajectory optimization in contrast to the more expensive PS knotting method. To facilitate the generalizations, we replace the convergence requirements in our original algorithm to controllability arguments. An application of the Bellman PS algorithm to an attitude control problem shows that the algorithm compares favorably to the PS knotting method.