Convergence Rate for a Radau hp Collocation Method Applied to Constrained Optimal Control

TL;DR

This paper establishes the local convergence rate of an hp-collocation method based on Radau points for unconstrained optimal control problems, showing exponential convergence under smoothness and convexity assumptions.

Contribution

It provides the first convergence rate analysis for Radau hp-collocation methods applied to constrained optimal control problems, including exponential convergence results.

Findings

Convergence is exponential with respect to polynomial degree.

Discrete solutions converge to continuous solutions as collocation points or mesh intervals increase.

The hp-scheme guarantees convergence with sufficiently small mesh, unlike global polynomial methods.

Abstract

For unconstrained control problems, a local convergence rate is established for an $hp$-method based on collocation at the Radau quadrature points in each mesh interval of the discretization. If the continuous problem has a sufficiently smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a local minimizer in a neighborhood of the continuous solution, and as either the number of collocation points or the number of mesh intervals increase, the discrete solution convergences to the continuous solution in the sup-norm. The convergence is exponentially fast with respect to the degree of the polynomials on each mesh interval, while the error is bounded by a polynomial in the mesh spacing. An advantage of the $hp$-scheme over global polynomials is that there is a convergence guarantee when the mesh is sufficiently small, while the convergence result for global polynomials requires that a norm of the linearized dynamics is sufficiently small. Numerical examples explore the convergence theory.