  We discuss homotopy properties of endpoint maps for affine control systems.
We prove that these maps are Hurewicz fibrations with respect to some $W^{1,p}$
topology on the space of trajectories, for a certain $p>1$. We study critical
points of geometric costs for these affine control systems, proving that if the
base manifold is compact then the number of their critical points is infinite
(we use Lusternik-Schnirelmann category combined with the Hurewicz property).
In the special case where the control system is subriemannian this result can
be read as the corresponding version of Serre's theorem, on the existence of
infinitely many geodesics between two points on a compact riemannian manifold.
In the subriemannian case we show that the Hurewicz property holds for all
$p\geq1$ and the horizontal-loop space with the $W^{1,2}$ topology has the
homotopy type of a CW-complex (as long as the endpoint map has at least one
regular value); in particular the inclusion of the horizontal-loop space in the
ordinary one is a homotopy equivalence.
