A decomposition theorem for singular control systems on Lie groups.
type: Item
instance of: scholarly article
title: A decomposition theorem for singular control systems on Lie groups.
zbMATH Open document ID: 1037.22020
DOI: 10.1016/S0898-1221(03)00023-2
publication date: 2003-12-14T00:00:00Z
review text: The purpose of the paper under review is twofold. In the first place, the concept of a singular control system \({\mathcal S}_G\) on a connected finite-dimensional Lie group \(G\) with Lie algebra \(\text{ Lie}(G)\) has been introduced. This concept depends upon a pair of derivations \((E,D)\) where \(E\) plays on \(\text{ Lie}(G)\) the analogous role as the singular matrix of order \(n \geq 1\) on the simply connected abelian Lie group which is formed by the real vector space \({\mathbb R}^n\), and \(D\) induces the vector field on \(G\) corresponding to the drift vector field of the control system \({\mathcal S}_{{\mathbb R}^n}\). The second aim is to decompose the system into two subsystems: The linear control system and the differential-algebraic control system. In order to obtain the desired decomposition, the authors outline the construction of a principal fibre bundle and a canonical connection associated to any non-invertible derivation on \(\text{ Lie}(G)\) like \(E\). The authors consider the Jordan decomposition of \(E\), denote by \(\text{ Lie}(V)\) its generalized 0-eigenspace and assume that the connected Lie group \(V\) is closed. They also assume that the drift vector field is projectable onto the homogeneous manifold \(G/V\). In particular they extend to the singular system \({\mathcal S}_G\) the decomposition result for the class of singular control systems \({\mathcal S}_{{\mathbb R}^n}\). Finally, an example of decomposition for a singular control system \({\mathcal S}_N\) on the simply connected three-dimensional Heisenberg nilpotent Lie group \(N\) with real Heisenberg Lie algebra \(\text{ Lie}(N)\) of depth or step 2 is studied [\textit{W. Schempp}, Proc. Am. Math. Soc. 92, 103--110 (1984; Zbl 0525.43007); \textit{W. Schempp}, Harmonic analysis on the Heisenberg nilpotent Lie group, with applications to signal theory (1986; Zbl 0632.43001); \textit{A. Bellaïche}, The tangent space in sub-Riemannian geometry, Prog. Math. 144, 1--78 (1996; Zbl 0…
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 34H05
Mathematics Subject Classification ID: 93C15
zbMATH DE Number: 2017214
Wikidata QID: Q115338571
author: Víctor Ayala
author: Wolfgang Kliemann
reviewed by: Walter Schempp
MaRDI profile type: Publication
cites work: Q4225129
cites work: Q3661950
cites work: Controllability properties of a class of control systems on lie groups
cites work: Observable linear pairs
cites work: Solvability, controllability, and observability of continuous descriptor systems
cites work: Radar Ambiguity Functions, the Heisenberg Group, and Holomorphic Theta Series
full work available at URL: s0898-1221(03)00023-2
OpenAlex ID: W2015716776
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 2017214