How many symmetries does admit a nonlinear single-input control system around an equilibrium?

Abstract

We describe all symmetries of a single-input nonlinear control system that is not feedback linearizable and whose first-order approximation is controllable, around an equilibrium point. For a system such that a feedback transformation bringing it into the canonical form is analytic, we prove that the set of all local symmetries of the system is exhausted by exactly two 1-parameter families of symmetries if the system is odd, and by exactly one 1-parameter family otherwise. We also prove that the form of the set of symmetries is completely described by the canonical form of the system: possessing a nonstationary symmetry, a 1-parameter family of symmetries, or being odd corresponds, respectively, to the fact that the drift vector field of the canonical form is periodic, does not depend on the first variable, or is odd. If the feedback transformation bringing the system to its canonical form is formal, we show an analogous result for an infinitesimal symmetry: its existence is equivalent to the fact that the drift vector field of the formal canonical form does not depend on the first variable. We illustrate our results by studying the symmetries of a variable-length pendulum.