Controllability of hypersurface and solvable invariant systems
type: Item
instance of: scholarly article
title: Controllability of hypersurface and solvable invariant systems
zbMATH Open document ID: 0991.93014
DOI: 10.1007/BF02259622
author: Yu L. Sachkov
published in: Journal of Dynamical and Control Systems
publication date: 2000-04-13T00:00:00Z
review text: Let \(G\) be a connected Lie group, \(L\) its Lie algebra, and \(A, B_1, \dots, B_m\in L\) considered as right invariant vector fields on \(G\). Consider the set \(\Gamma=A+ \sum^m_{j=1} \mathbb{R} B_j\) and the subsemigroup \(S\) generated by the \(\exp\mathbb{R}_+X\) with \(X\in\Gamma\). The set \(\Gamma\) is referred to as an (invariant) affine control system, which is called controllable if \(S=G\). Let \(L_0\) be the Lie subalgebra generated by \(B_1,\dots,B_m\) and \(G_0\) the corresponding analytic subgroup of \(G\). Suppose that \(\text{codim} L_0=1\). Then the author shows that \(\Gamma\) is controllable iff \(A\notin L_0\) (plus the condition \(G/G_0 \cong S^1\) in case \(G_0\) is closed). If \(G\) is simply connected and solvable such that all elements of \(\operatorname {ad}L\) have real spectrum, then \(\Gamma\) is controllable iff \(L=L_0\).
reviewed by: Joachim Hilgert
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 93B27
Mathematics Subject Classification ID: 22E99
zbMATH DE Number: 1431806
zbMATH Keywords: Lie groups
zbMATH Keywords: controllability of hypersurfaces
zbMATH Keywords: affine control system
MaRDI profile type: Publication
cites work: Control systems on Lie groups
cites work: Control systems on semi-simple Lie groups and their homogeneous spaces
cites work: Controlabilité des Systémes Bilinéaires
cites work: Controllability on classical Lie groups
cites work: Q4225129
cites work: Controllability of a System of Two Symmetric Rigid Bodies in Three Space
full work available at URL: https://doi.org/10.1007/bf02259622
OpenAlex ID: W2088878831
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description: scientific article; zbMATH DE number 1431806