Linearization of analytic vector fields in the transitive case

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.