Conic Nonholonomic Constraints on Surfaces and Control Systems

Timothée Schmoderer (1, 2) , Witold Respondek (1, 3)

1 LMI - Laboratoire de Mathématiques de l'INSA de Rouen Normandie
2 PRISME - Laboratoire pluridisciplinaire de recherche en ingénierie des systèmes, mécanique et énergétique
3 Institute of Automatic Control

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 quadratisable 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.

Mots clés: nonlinear control system; feedback equivalence; conic submanifolds; nonholonomic constraint; normal forms; pseudo-Riemannian geometry

Domaines: Géométrie différentielle [math.DG]; Automatique; Systèmes dynamiques [math.DS]; Optimisation et contrôle [math.OC]

Timothée Schmoderer, Witold Respondek. Conic Nonholonomic Constraints on Surfaces and Control Systems. Journal of Dynamical and Control Systems, In press, ⟨10.1007/s10883-023-09659-9⟩. ⟨hal-03634034v1⟩