A Type II Hamiltonian Variational Principle and Adjoint Systems for Lie Groups

TL;DR

This paper introduces a new variational principle on cotangent bundles of Lie groups with fixed initial and final conditions, facilitating adjoint sensitivity analysis and optimization on Lie groups.

Contribution

It develops a globally defined Type II variational principle on Lie groups and explores adjoint systems for optimization and sensitivity analysis.

Findings

Defined a global Type II variational principle on cotangent bundles of Lie groups

Analyzed properties of adjoint systems on Lie groups

Demonstrated applications to optimization problems on Lie groups

Abstract

We present a novel Type II variational principle on the cotangent bundle of a Lie group which enforces Type II boundary conditions, i.e., fixed initial position and final momentum. In general, such Type II variational principles are only globally defined on vector spaces or locally defined on general manifolds; however, by left translation, we are able to define this variational principle globally on cotangent bundles of Lie groups. Type II boundary conditions are particularly important for adjoint sensitivity analysis, which is our motivating application. As such, we additionally discuss adjoint systems on Lie groups, their properties, and how they can be used to solve optimization problems subject to dynamics on Lie groups.