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Linearization of analytic vector fields in the transitive case

Authors
Journal
Journal of Differential Equations
0022-0396
Publisher
Elsevier
Publication Date
Volume
25
Issue
3
Identifiers
DOI: 10.1016/0022-0396(77)90051-1
Disciplines
  • Mathematics

Abstract

Abstract Given a nonlinear control system, linear in the controls, all of whose terms have a common critical point, Lie algebraic conditions are established for the existence of a real-analytic transformation to coordinates in which the system is bilinear, that is, of type dx dt = ∑ u iB ix . ( ∗) The hypotheses used are analyticity, transitivity of the Lie algebra L associated with ( ∗) (i.e., controllability of ( ∗)), and isomorphism of L to the Lie algebra of vector fields associated with the original nonlinear system. That the transitivity condition can be replaced by semisimplicity or compactness of L is known from work of Sternberg and Guillemin.

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