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

TL;DR

This paper establishes the first proven convergence rate for pseudospectral methods in optimal control of feedback linearizable systems, demonstrating high-order convergence without requiring local uniqueness or coercivity assumptions.

Contribution

It introduces the first proven convergence rate for pseudospectral optimal control and removes restrictive assumptions present in prior convergence theorems.

Findings

Proved a high-order convergence rate for PS methods.

Established existence and convergence of approximate solutions.

Removed restrictive assumptions from previous convergence theorems.

Abstract

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.