Controllability on classical Lie groups
type: Item
instance of: scholarly article
title: Controllability on classical Lie groups
zbMATH Open document ID: 0658.93013
DOI: 10.1007/BF02551234
published in: MCSS. Mathematics of Control, Signals, and Systems
publication date: 1988-01-01T00:00:00Z
review text: This paper investigates conditions for the global controllability of bilinear systems evolving on a connected Lie group \(G\subset Gl(n,{\mathbb{R}})\) whose Lie algebra L is the normal real form of one of the classical complex Lie algebras g of type An, Bn, Cn or Dn. The systems are described by equations of the form \[ \dot x(t)=(Au+Bv)x(t),\quad u,v\in {\mathbb{R}},\quad x\in G;\quad or\quad \dot x(t)=(A+uB)x(t),\quad u\in {\mathbb{R}},\quad x\in G \] where A, \(B\in L\) and B is supposed to be diagonal. Let \(\Phi\) be the set of nonzero roots of g with respect to a Cartan subalgebra h. B is said to be A-strongly regular if the elements \(\alpha\) (B), \(\alpha \in \Phi_ A\) are nonzero and distinct where \(\Phi_ A\) is the set of roots occuring in the decomposition of A. Under the assumption that B is A-strongly regular it is shown that A, B generate the Lie algebra L if and only if A is P-irreducible. This yields necessary and sufficient conditions for global controllability in situations where this property is equivalent to the controllability rank conditions. The paper extends previous results of Gauthier, Kupka and Sallet obtained under the assumption that B is strongly regular.
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 93B25
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 22E99
Mathematics Subject Classification ID: 93C10
zbMATH DE Number: 4077042
zbMATH Keywords: semisimple Lie algebras
zbMATH Keywords: global controllability
zbMATH Keywords: bilinear systems
zbMATH Keywords: connected Lie group
Wikidata QID: Q115392075
author: Fátima Silva Leite
author: Peter E. Crouch
reviewed by: D. Hinrichsen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4077042