Transitivity of families of invariant vector fields on the semidirect products of Lie groups

Authors:
B. Bonnard, V. Jurdjevic, I. Kupka and G. Sallet

Journal:
Trans. Amer. Math. Soc. **271** (1982), 525-535

MSC:
Primary 49E15; Secondary 22E15, 58F40

MathSciNet review:
654849

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Abstract: In this paper we give necessary and sufficient conditions for a family of right (or left) invariant vector fields on a Lie group to be transitive. The concept of transitivity is essentially that of controllability in the literature on control systems. We consider families of right (resp. left) invariant vector fields on a Lie group which is a semidirect product of a compact group and a vector space on which acts linearly. If is a family of right-invariant vector fields, then the values of the elements of at the identity define a subset of the Lie algebra of . We say that is transitive on if the semigroup generated by is equal to . Our main result is that is transitive if and only if , the Lie algebra generated by , is equal to .

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1982-0654849-4

Article copyright:
© Copyright 1982
American Mathematical Society