Convergence rate for a Gauss collocation method applied to constrained optimal control

TL;DR

This paper establishes a local convergence rate for a Gauss collocation method in solving constrained optimal control problems, showing that under certain conditions, the method converges with quantifiable accuracy.

Contribution

The paper provides the first convergence rate analysis for Gauss collocation methods applied to constrained optimal control problems with strong convexity assumptions.

Findings

Convergence rate depends on the strong convexity of the Hamiltonian.

The theory applies to problems with state and costate having two square integrable derivatives.

Numerical example confirms the tightness of the convergence bounds.

Abstract

A local convergence rate is established for a Gauss orthogonal collocation method applied to optimal control problems with control constraints. If the Hamiltonian possesses a strong convexity property, then the theory yields convergence for problems whose optimal state and costate possess two square integrable derivatives. The convergence theory is based on a stability result for the sup-norm change in the solution of a variational inequality relative to a 2-norm perturbation, and on a Sobolev space bound for the error in interpolation at the Gauss quadrature points and the additional point -1. The tightness of the convergence theory is examined using a numerical example.