FEEDBACK CLASSIFICATION OF NONLINEAR CONTROL-SYSTEMS ON 3-MANIFOLDS

We consider nonlinear control-affine systems with two inputs evolving on three-dimensional manifolds. We study their local classification un der static state feedback. Under the assumption that the control vecto r fields are independent we give complete classification of generic sy stems. We prove that out of a ''singular'' smooth curve a generic cont rol system is either structurally stable and thus equivalent to one of six canonical forms (models) or finitely determined and thus equivale nt to one of two canonical forms with real parameters. Moreover, we sh ow that at points of the ''singular'' curve the system is not finitely determined and we give normal forms containing functional moduli. We also study geometry of singularities, i.e., we describe surfaces, curv es, and isolated points where the system admits its canonical forms.