Lebesgue Constants Arising in a Class of Collocation Methods

TL;DR

This paper analyzes the growth of Lebesgue constants for specific quadrature point sets used in orthogonal collocation methods, showing they grow as the square root of the number of points, which impacts residual decay in optimal control.

Contribution

It provides new estimates for Lebesgue constants associated with Gauss and Radau quadrature points, relevant for orthogonal collocation in optimal control.

Findings

Lebesgue constants grow as O(√N) for the studied point sets.

Estimates imply exponential decay of residuals for smooth solutions.

Results inform the stability and accuracy of collocation schemes.

Abstract

Estimates are obtained for the Lebesgue constants associated with the Gauss quadrature points on $(-1, +1)$ augmented by the point $-1$ and with the Radau quadrature points on either $(-1, +1]$ or $[-1, +1)$. It is shown that the Lebesgue constants are $O(\sqrt{N})$, where $N$ is the number of quadrature points. These point sets arise in the estimation of the residual associated with recently developed orthogonal collocation schemes for optimal control problems. For problems with smooth solutions, the estimates for the Lebesgue constants can imply an exponential decay of the residual in the collocated problem as a function of the number of quadrature points.