Advances in Pseudospectral Methods for Optimal Control

Recently, the Legendre pseudospectral (PS) method migrated from theory to flight application onboard the International Space Station for performing a finite-horizon, zeropropellant maneuver. A small technical modification to the Legendre PS method is necessary to manage the limiting conditions at infinity for infinite-horizon optimal control problems. Motivated by these technicalities, the concept of primal-dual weighted interpolation, introduced earlier by the authors, is used to articulate a unified theory for all PS methods for optimal control. This theory illuminates the previously hidden fact of the unit weight function implicit in the the Legendre PS method based on Legendre-Gauss-Lobatto points. The unified framework also reveals why this Legendre PS method is the most appropriate method for solving finite-horizon optimal control problems with arbitrary boundary conditions. This conclusion is borne by a proper definition of orthogonality needed to generate convergent approximations in Hilbert spaces. Special boundary conditions are needed to ensure the convergence of the Legendre PS method based on the Legendre-Gauss-Radau (LGR) and the Legendre-Gauss (LG) points. These facts are illustrated by simple examples and counter examples which reveal when and why PS methods based on LGR and LG points fail. A new kind of consistency in the primal-dual weight functions allows us to generate dual maps (such as Hamiltonians, adjoints etc) without resorting to solving difficult two-point boundary-value problems. These concepts are encapsulated in a unified Covector Mapping Theorem.