 
Summary: Acta Applicandae Mathematicae 32: 157, 1993. 1
9 1993 KluwerAcademic Publishers. Printed in the Netherlands.
Local Controllability and Semigroups
of Diffeomorphisms
A.A. AGRACHEV and R.V. GAMKRELIDZE
SteMov Mathematical lnstitute, ul. Vavilova 42, l17966, Moscow GSP1, Russia
(Received: 15 November 1992)
Abstract. The local structure of orbits of semigroups, generated by families of diffeomorphisms, is
studied by Lie theory methods. New sufficient conditions for local controllability are obtained which
take into account ordinary, as well as fastswitching variations.
Mathematics Subject Classification (1991). 93BXX.
Key words. Local controllability, vector field, flow, Lie bracket.
1. Introduction
1. Let M be a realanalytic manifold, Vect M the Lie algebra of analytic vector
fields on M. We consider a vector field f E Vect M as a differential operator of first
order on the algebra of smooth functions C ~ (M) which satisfies the differentiation
rule
f(W1W2) " (fw1)W2 + wI(fW2) VWI,W2 E Coo(M).
A point z E M will be identified with the corresponding homomorphism
z: Coo(M) ~ I~, ~ ~ ~(z).
