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

Abstract

We propose a unified approach to determine and prove obstructions to small-time local controllability of scalar-input control systems. Our approach relies on a recent Magnus-type representation formula of the state, a new Hall basis of the free Lie algebra over two generators and an appropriate use of Sussmann's infinite product to compute the Magnus expansion.First, we recover the necessary conditions, due to Sussmann [19] and Stefani [18], concerning the strongest obstruction at each even order of the control. We also recover our classification of quadratic obstructions of [3], involving the regularity of the control, and Kawski's tight necessary condition of [11] concerning the second quadratic drift.Then, we prove and generalize a conjecture of 1986 due to Kawski [10] on a new family of loose necessary conditions, linked with quadratic drifts. In the particular case of the third quadratic drift, we state and prove a tight necessary condition, which is new.Eventually, as a further illustration of the approach, we derive an entirely new obstruction, linked with a bracket of the sixth order with respect to the control and for which the functional measuring the amplitude of the drift is not directly a Sobolev norm of the control.