  We prove the continuity and the H\"older equivalence w.r.t.\ an Euclidean
distance of the value function associated with the $L^1$ cost of the
control-affine system $\dot q = \drift(q)+\sum_{j=1}^m u_j f_j(q)$, satisfying
the strong H\"ormander 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 and through which we derive estimates for the
shape of the reachable sets. Finally, we also prove the continuity of the value
function associated with the $L^1$ cost of time-dependent systems of the form
$\dot q = \sum_{j=1}^m u_j f_j^t(q)$.
