Quadratic obstructions to small-time local controllability for multi-input systems

TL;DR

This paper establishes a necessary condition involving Lie brackets for the small-time local controllability of multi-input control-affine systems, extending previous single-input results using a Magnus-type representation.

Contribution

It generalizes the necessary controllability condition to multi-input systems and introduces a Magnus-type approach for the analysis.

Findings

Identifies quadratic obstructions to controllability

Extends single-input methods to multi-input systems

Uses interpolation inequalities to derive obstructions

Abstract

We present a necessary condition for the small-time local controllability of multi-input control-affine systems on $R^d$ . This condition is formulated on the vectors of $R^d$ resulting from the evaluation at zero of the Lie brackets of the vector fields: it involves both their direction and their amplitude. The proof is an adaptation to the multi-input case of a general method introduced by Beauchard and Marbach in the single-input case. It relies on a Magnus-type representation formula: the state is approximated by a linear combination of the evaluation at zero of the Lie brackets of the vector fields, whose coefficients are functionals of the time and the controls. Finally, obstructions to small-time local controllability result from interpolation inequalities.