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

TL;DR

This paper proves that a Gauss collocation method for unconstrained optimal control problems converges exponentially fast under certain smoothness and convexity conditions, providing the first such convergence rate result for this class of methods.

Contribution

It establishes the first convergence rate for an orthogonal collocation method based on Gauss quadrature applied to unconstrained optimal control problems.

Findings

Discrete solutions converge exponentially fast to the continuous solution.

Convergence rate depends on smoothness and convexity conditions.

First such result for global polynomial collocation methods in optimal control.

Abstract

A local convergence rate is established for an orthogonal collocation method based on Gauss quadrature applied to an unconstrained optimal control problem. 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 the number of collocation points increases, the discrete solution convergences exponentially fast in the sup-norm to the continuous solution. This is the first convergence rate result for an orthogonal collocation method based on global polynomials applied to an optimal control problem.