Null-forms of conic systems in R3 are determined by their symmetries

Abstract

We address the problem of characterisation of null-forms of conic 3-dimensional systems, that is, control-affine systems whose field of admissible velocities forms a conic (without parameters) in the tangent space. Those systems have been previously identified as the simplest control systems under a conic nonholonomic constraint or as systems of zero curvature. In this work, we propose a direct characterisation of null-forms of conic systems among all control-affine systems by studying the Lie algebra of infinitesimal symmetries. Namely, we show that the Lie algebra of infinitesimal symmetries characterises uniquely null-forms of conic systems.