Trivialisable control-affine systems revisited

TL;DR

This paper extends the concept of trivial control systems to control-affine systems with multiple states and controls, providing new characterizations, normal forms, and geometric insights, including a novel proof for 3D scalar control systems.

Contribution

It introduces extended definitions, geometric characterizations, and normal forms for trivial control-affine systems, enhancing understanding of their structure and symmetries.

Findings

Two novel geometric characterizations of trivial control-affine systems.

A normal form for systems with transitive almost abelian symmetry Lie algebras.

A new proof of Serres' characterization for 3D scalar control systems.

Abstract

The purpose of this paper is to explore the concept of trivial control systems, namely systems whose dynamics depends on the controls only. Trivial systems have been introduced and studied by Serres in the the context of control-nonlinear systems on the plane with a scalar control. In our work, we begin by proposing an extension of the notion of triviality to control-affine systems with arbitrary number of states and controls. Next, our first result concerns two novel characterisations of trivial control-affine systems, one of them is based on the study of infinitesimal symmetries and is thus geometric. Second, we derive a normal form of trivial control-affine systems whose Lie algebra of infinitesimal symmetries possesses a transitive almost abelian Lie subalgebra. Third, we study and propose a characterisation of trivial control-affine systems on $3$-dimensional manifolds with scalar control. In particular, we give a novel proof of the previous characterisation obtained by Serres. Our characterisation is based on the properties of two functional feedback invariants: the curvature (introduced by Agrachev) and the centro-affine curvature (used by Wilkens). Finally, we give several normal forms of control-affine systems, for which the curvature and the centro-affine curvature have special properties.