Symmetry actuated closed-loop Hamiltonian systems

TL;DR

This paper generalizes the theory of controlled Hamiltonian systems with symmetries to non-abelian groups, introducing symmetry actuating forces and deriving conditions for closed-loop dynamics, with applications to fluid dynamics and satellite control.

Contribution

It extends controlled Hamiltonian systems theory to non-abelian symmetry groups and introduces symmetry actuating forces, providing new matching conditions for closed-loop system analysis.

Findings

Conservation law for Hamiltonian systems with symmetry actuating forces.

Matching conditions relating forced and unforced Hamiltonian systems.

Applications to Lie-Poisson systems, charged fluids, and satellite control.

Abstract

This paper extends the theory of controlled Hamiltonian systems with symmetries due to [9, 10, 6, 7, 11] to the case of non-abelian symmetry groups $G$. The notion of symmetry actuating forces is introduced and it is shown, that Hamiltonian systems subject to such forces permit a conservation law, which arises as a controlled perturbation of the $G$-momentum map. Necessary and sufficient matching conditions are given to relate the closed-loop dynamics, associated to the forced Hamiltonian system, to an unforced Hamiltonian system. These matching conditions are then applied to general Lie-Poisson systems, to the example of ideal charged fluids in the presence of an external magnetic field ([20]), and to the satellite with a rotor example ([9, 10]).