Directed topological complexity

It has been observed that the motion planning problem of robotics reduces mathematically to the problem of finding a section of the path-space fibration, leading to the notion of topological complexity, as introduced by M. Farber. In this approach one imposes no limitations on motion of the system assuming that any continuous motion is admissible. In many applications, however, a physical apparatus may have constrained controls, leading to constraints on its potential dynamics. In the present 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 properties of this new notion and make calculations for some interesting classes of examples.