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

B. Bonnard, V. Jurdjevic, I. Kupka, G. Sallet
1982 Transactions of the American Mathematical Society  
In this paper we give necessary and sufficient conditions for a family of right (or left) invariant vector fields on a Lie group G 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 G which is a semidirect product of a compact group K and a vector space V on which K acts linearly. If 5F is a family of right-invariant vector fields, then the
more » ... fields, then the values of the elements of if at the identity define a subset T of 7.(0) the Lie algebra of G. We say that if is transitive on G if the semigroup generated by U XE¡,{exp(tX): t » 0} is equal to G. Our main result is that if is transitive if and only if Lie(F), the Lie algebra generated by T, is equal to L(G).
doi:10.1090/s0002-9947-1982-0654849-4 fatcat:ttqkzu4horfgbfduadhjn2sbxe