On Discrete-Time Optimality Conditions for Pseudospectral Methods

Abstract

One of the most efficient families of techniques for solving space trajectory optimization problems are pseudospectral (PS) methods. Among the rich variety of PS methods, the class of Legendre PS methods are most thoroughly studied for optimal control and trajectory optimization applications. In particular, the Legendre-Gauss-Lobatto PS method is widely used for boundary-value type problems while the Legendre-Gauss-Radau PS method was recently proposed for solving innate-horizon optimal control problems as a means to manage conditions at inanity. Both methods satisfy the Convector Mapping Principle, the mathematical principle associated with the consistency of approximations that allows one to generate dual maps (such as Hamiltonians, adjoins etc) without resorting to solving di'cult two-point boundary-value problems. In this paper we prove that a combination of weighted interpolants, their duals, and a proper definition of orthogonality allows us to formulate a generalized Covector Mapping Theorem that applies to all such PS methods. The consequences of this theorem are that it clarifies the connections between theory and computation, the impact of these connections on solving trajectory optimization problems, and the selection of the correct PS method for solving problems quickly and efficiently. A classical benchmark continuous-thrust orbit transfer problem is used to illustrate the concepts.