On the Rate of Convergence for the Pseudospectral Optimal Control of Feedback Linearizable Systems

Over the last decade, pseudospectral (PS) computational methods for nonlinear constrained optimal control have been applied to many industrial-strength problems, notably the recent zero-propellant-maneuvering of the International Space Station performed by NASA. In this paper, we prove a theorem on the rate of convergence for the optimal cost computed using PS methods. It is a first proved convergence rate in the literature of PS optimal control. In addition to the high-order convergence rate, two theorems are proved for the existence and convergence of the approximate solutions. This paper contains several essential differences from existing papers on PS optimal control as well as some other direct computational methods. The proofs do not use necessary conditions of optimal control. Furthermore, we do not make coercivity type of assumptions. As a result, the theory does not require the local uniqueness of optimal solutions. In addition, a restrictive assumption on the cluster points of discrete solutions made in existing convergence theorems are removed.