Hessians in Birkhoff-Theoretic Trajectory Optimization

TL;DR

This paper analyzes Hessians in Birkhoff-theoretic trajectory optimization, revealing eigenvalue distributions and discussing computational challenges for large-scale problems.

Contribution

It derives Hessians for Birkhoff methods and proves eigenvalue bounds, advancing understanding of their spectral properties and computational complexity.

Findings

Approximately 80% of eigenvalues lie in [-2, 4]

Eigenvalue distribution is narrow across problems

Discusses challenges in large-scale trajectory optimization

Abstract

This paper derives various Hessians associated with Birkhoff-theoretic methods for trajectory optimization. According to a theorem proved in this paper, approximately 80% of the eigenvalues are contained in the narrow interval [-2, 4] for all Birkhoff-discretized optimal control problems. A preliminary analysis of computational complexity is also presented with further discussions on the grand challenge of solving a million point trajectory optimization problem.