Reduced Control Systems on Symmetric Lie Algebras

TL;DR

This paper studies control systems on symmetric Lie algebras, showing they can be reduced to simpler systems on maximal Abelian subspaces, with applications to simulation and control theory.

Contribution

It establishes a reduction method for control systems on symmetric Lie algebras to systems on Abelian subspaces, simplifying analysis and control design.

Findings

Control systems on symmetric Lie algebras can be reduced to systems on maximal Abelian subspaces.

The reduction preserves controllability properties under general conditions.

A simulation result with respect to Weyl group-induced preorder is proved.

Abstract

For a symmetric Lie algebra $\mathfrak g=\mathfrak k\oplus\mathfrak p$ we consider a class of bilinear or more general control-affine systems on $\mathfrak p$ defined by a drift vector field $X$ and control vector fields $\mathrm{ad}_{k_i}$ for $k_i\in\mathfrak k$ such that one has fast and full control on the corresponding compact group $\mathbf K$. We show that under quite general assumptions on $X$ such a control system is essentially equivalent to a natural reduced system on a maximal Abelian subspace $\mathfrak a\subseteq\mathfrak p$, and likewise to related differential inclusions defined on $\mathfrak a$. We derive a number of general results for such systems and as an application we prove a simulation result with respect to the preorder induced by the Weyl group action.