Singular controls with non-trivial independent trajectory of generic driftless control-affine systems (Theory of singularities of smooth mappings and around it)

Abstract

We consider singular controls with non-trivial independent trajectory and corresponding bi-extremals for generic smooth driftless control-affine systems on a finite dimensional smooth manifold in the sense of Whitney topology. Then we have the following theorem: for generic driftless control-affine systems of two or more smooth vector fields on a finite dimensional manifold, any bi-extremal with the non-trivial independent trajectory is of minimal order and any singular control with the non-trivial independent trajectory is of corank one.