Conic nonholonomic constraints on surfaces and control systems

TL;DR

This paper classifies conic submanifolds in the tangent bundle of surfaces as nonholonomic constraints, providing a normal form and feedback classification for quadratic control systems, advancing understanding of their geometric and control properties.

Contribution

It offers a complete description and normal form for non-degenerate conic submanifolds via feedback transformations of quadratic control-affine systems, and classifies regular conic submanifolds through feedback classification.

Findings

Normal forms for quadratic control-affine systems

Classification of regular conic submanifolds (ellipses, hyperbolas, parabolas)

Explicit criteria for feedback equivalence of conic submanifolds

Abstract

This paper addresses the equivalence problem of conic submanifolds in the tangent bundle of a smooth 2-dimensional manifold. Those are given by a quadratic relation between the velocities and are treated as nonholonomic constraints whose admissible curves are trajectories of the corresponding control systems, called quadratic systems. We deal with the problem of characterising and classifying conic submanifolds 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 non-degenerate conic submanifolds via a characterisation under feedback transformations of the novel class of quadratic control-affine systems. This characterisation can explicitly be tested on structure functions defined for any control-affine system and gives a normal form of quadratizable systems and of conic submanifolds. Then, we consider the classification problem of regular conic submanifolds (ellipses, hyperbolas, and parabolas), which is treated via feedback classification of quadratic control-nonlinear systems. Our classification includes several normal forms of quadratic systems (in particular, normal forms not containing functional parameters as well as those containing neither functional nor real parameters), and, as a consequence, gives a classification of regular conic submanifolds.