© 1989 by Kluwer Academic Publishers. Local Invariants of Smooth Control Systems

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 high-order 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) where M is a C=-manifo ld and f (q) , g(q) are complete smooth vector fields. For t>0, we define a mapping F,: Lo~[0, t]-->M, which maps control u (.