Hölder equivalence of the value function for control-affine systems

We prove the continuity and we give a Holder estimate for the value function associated with the L cost of the control-affine system q̇ = f0(q) + ∑m j=1 ujfj(q), satisfying the strong Hörmander condition. This is done by proving a result in the same spirit as the Ball-Box theorem for driftless (or sub-Riemannian) systems. The techniques used are based on a reduction of the control-affine system to a linear but time-dependent one, for which we are able to define a generalization of the nilpotent approximation. Finally, we also prove the continuity of the value function associated with the L cost of time-dependent systems of the form q̇ = ∑m j=1 ujf t j (q).