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We prove the continuity and the H\"older equivalence w.r.t.\ an Euclideandistance of the value function associated with the $L^1$ cost of thecontrol-affine system $\dot q = \drift(q)+\sum_{j=1}^m u_j f_j(q)$, satisfyingthe strong H\"ormander condition. This is done by proving a result in the samespirit as the Ball-Box theorem for driftless (or sub-Riemannian) systems. Thetechniques used are based on a reduction of the control-affine system to alinear but time-dependent one, for which we are able to define a generalizationof the nilpotent approximation and through which we derive estimates for theshape of the reachable sets. Finally, we also prove the continuity of the valuefunction associated with the $L^1$ cost of time-dependent systems of the form$\dot q = \sum_{j=1}^m u_j f_j^t(q)$.

H\"older equivalence of the value function for control-affine systems