Conic nonholonomic constraints on surfaces and control systems

Abstract : This paper addresses the equivalence problem of conic submanifolds in the tangent bundle of a smooth 2-dimensional manifold. Those are treated as nonholonomic constraints whose admissible curves are trajectories of the corresponding control systems. We deal with this problem under the prism of feedback equivalence of control systems, both control-affine and fully nonlinear. The first main result of this work is a complete description of regular conic submanifolds. We also give equivalence results for a special class of conic submanifolds via the study of the Lie algebra of infinitesimal symmetries of the corresponding control systems. Then, we consider the classification problem of conic submanifolds, which is achieved via feedback classification of nonlinear control system. Our results describe and completely characterise quadratic systems, and include several normal and canonical forms.