Reduced Control Systems on Symmetric Lie Algebras

2023/07/25 by Malvetti, Emanuel, Dirr, Gunther, Ende, Frederik vom +1
#37N20 (Primary) 93B03 #93B05 #93C10 (Secondary) #93D99 #FOS: Mathematics #Optimization and Control (math.OC) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2307.13664

Abstract

For a symmetric Lie algebra \mathfrak g=\mathfrak k⊕\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 ad_ki for k_i∈\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⊆\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.

Related

• Equivalence of Control Systems with Linear Systems on Lie Groups and Homogeneous Spaces
• Affine and bilinear systems on Lie groups
• The General Solution for Affine Control Systems on Lie Groups
• Solution Curve for Linear Control Systems on Lie Groups
• Control sets of linear systems on Lie groups