Directed topological complexity
It provides a theoretical framework for motion planning in systems with constraints, relevant to robotics and concurrency theory.
This paper adapts the notion of topological complexity to directed topological spaces, which model systems with constrained controls, and studies its properties, calculations for certain spaces, and applications to directed homotopy equivalence.
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.