Some Results on Second Order Controllability Conditions

For a symmetric system, we study the problemof crossing an hypersurface in the neighborhood of a givenpoint, when we suppose that all of the available vector fieldsare tangent to the hypersurface at the point. Classically onerequires transversality of at least one Lie bracket generatedby two available vector fields. However such condition doesnot take into account neither the geometry of the hypersurfacenor the practical fact that in order to realize the direction of a Lie bracket one needs three switches among the vector fields in a short time. We find a new sufficient condition that requires a symmetric matrix to have a negative eigenvalue.This sufficient condition, which contains either the case of a transversal Lie bracket and the case of a favorable geometry of the hypersurface, is thus weaker than the classical one, easyto check and also necessary. Moreover it is constructive and produces a trajectory with at most one switch to reach the goal.