On small-time local controllability

TL;DR

This paper proves that small-time local controllability is preserved under certain polynomial perturbations of real analytic control-affine systems, connecting two open conjectures in the field.

Contribution

It establishes a general theorem showing the robustness of small-time local controllability under polynomial perturbations of vector fields.

Findings

Controllability is preserved under polynomial perturbations of order higher than the original system.

Connects two long-standing open conjectures in control theory.

Provides a framework for analyzing perturbations in real analytic systems.

Abstract

In this paper, we study small-time local controllability of real analytic control-affine systems under small perturbations of their vector fields. Consider a real analytic control system $\mathcal{X}$ which is small-time locally controllable and whose reachable sets shrink with the polynomial rate of order $N$ with respect to time. We will prove a general theorem which states that any real analytic control-affine system whose vector fields are perturbations of the vector fields of $\mathcal{X}$ with polynomials of order higher than $N$ is again small-time locally controllable. In particular, we show that this result connects two long-standing open conjectures about small-time local controllability of systems.