 
Summary: Acta ApplicandaeMathematicae14 (1989), 191237. 191
© 1989 by KluwerAcademicPublishers.
Local Invariants of Smooth Control Systems
A. A. AGRACHEV, R. V. GAMKRELIDZE, and A. V. SARYCHEV
V. A. SteklovInstituteof Mathematics,Academy of Sciencesof the U.S.S.R., ul. Vavilova42,
Moscow GPS 117966, U.S.S.R.
(Received in revised form: 12 July 1988)
Abstract. Methods are presented for locally studying smooth nonlinear control systems on the
manifold M. The technique of chronological calculus is intensively exploited. The concept of
chronological connection is introduced and is used when obtaining the invariant expressions in the
form of Lie bracket polynomials for highorder variations of a nonlinear control system. The theorem
on adduction of a family of smooth vector fields to the canonical form proved in Section 4 is then
applied to the construction of a nilpotent polynomial approximation for a control system. Finally, the
relation between the attainable sets of an original system and an approximating one is established; it
implies some conclusions on the local controllability of these systems.
AMS subject classifications (1980). Primary 49B10, 49E15.
Key words. Nonlinear control systems, geometric/Lie algebraic methods; chronological calculus,
nilpotent polynomial approximation, controllability, attainable sets.
1. In this paper we study a smooth control system of the form
q=f(q)+g(q)u, qcM, uER, q(O)=qo (1.1)
