  It has been observed that the very important motion planning problem of
robotics mathematically speaking boils down to the problem of finding a section
to the path-space fibration, raising the notion of topological complexity, as
introduced by M. Farber. The above notion fits the motion planning problem of
robotics when there are no constraints on the actual control that can be
applied to the physical apparatus. In many applications, however, a physical
apparatus may have constrained controls, leading to constraints on its
potential future dynamics. In this paper we adapt the notion of topological
complexity to the case of directed topological spaces, which encompass such
controlled systems, and also systems which appear in concurrency theory. We
study its first properties, make calculations for some interesting classes of
spaces, and show applications to a form of directed homotopy equivalence.
