We present a necessary condition for the small-time local controllability of multi-input control-affine systems on \(\mathbb {R}^d\) . This condition is formulated on the vectors of \(\mathbb {R}^d\) 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 – see Beauchard and Marbach (2024). 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.
