ABSTRACT
In studying any class of objects one would like to have classifications of these objects divided into subclasses subject to natural equivalence relations and, hopefully, a canonical object in each subclass. This chapter introduces several collections of invariant functions as complete invariants for equivalence of arbitrary transitive families of smooth vector fields. The invariants in the form of functions have two advantages over the Lie brackets invariants: they work in the nonanalytic case, and they give the coordinate change realizing the equivalence. The chapter considers the state equivalence problem for smooth systems. The most natural equivalence relation for nonlinear control systems seems to be equivalence up to coordinate changes in the state space. For analytic, transitive systems, and local equivalence, one complete set of such invariants has been known in control theory since the papers of A. Krener and H. J. Sussmann.
