Weak classification of the full free Lie algebra

Determine whether there exists a subset of the basis B of the free Lie algebra generated by X_0 and X_1 that satisfies both the stated sufficient and necessary conditions for smooth small-time local controllability at every homogeneous level.

Background

The paper introduces a basis B whose quartic components weakly classify the relevant homogeneous layers through a complementarity between brackets that generate controllability and brackets whose evaluations must be compensated. The proposed extension asks for a subset B_bad of the entire basis B that would provide a global weak classification: vanishing evaluations on B_bad would suffice for smoothly-STLC under the Lie algebra rank condition, while smooth controllability would force every bracket in B_bad to be linearly compensated by the remaining basis brackets.

The authors explain that the main difficulty in proving sufficiency is establishing motions in both the positive and negative directions associated with even a single putatively good bracket. For necessity, the difficulty is that some drift functionals are weak and are not directly controlled by Sobolev norms, so more advanced interpolation inequalities may be required. This problem is therefore a proposed global extension of the paper’s quartic classification results.