Reduced Control Systems on Symmetric Lie Algebras

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.

27 pages, 5 figures