A unified approach of obstructions to small-time local controllability for scalar-input systems

TL;DR

This paper introduces a unified method to identify obstructions to small-time local controllability in scalar-input systems, proving known conditions, a 1986 conjecture, and discovering new obstructions using advanced algebraic and analytical tools.

Contribution

It provides a unified framework for analyzing controllability obstructions, proving all known conditions, confirming a longstanding conjecture, and deriving new necessary conditions.

Findings

Proves all known necessary conditions for controllability.

Confirms a 1986 conjecture by Kawski.

Derives new obstructions, including a novel condition for the third quadratic bracket.

Abstract

We present a unified approach for determining and proving obstructions to small-time local controllability of scalar-input control systems. Our approach views obstructions to controllability as resulting from interpolation inequalities between the functionals associated with the formal Lie brackets of the system. Using this approach, we give compact unified proofs of all known necessary conditions, we prove a conjecture of 1986 due to Kawski, and we derive entirely new obstructions. Our work doubles the number of previously-known necessary conditions, all established in the 1980s. In particular, for the third quadratic bracket, we derive a new necessary condition which is complementary to the Agrachev-Gamkrelidze sufficient ones. We rely on a recent Magnus-type representation formula for the state, a new Hall basis of the free Lie algebra over two generators, an appropriate use of Sussmann's infinite product to compute the Magnus expansion, and Gagliardo-Nirenberg interpolation inequalities.