Extendibility and path components of admissible paths for the Dubins problem

An admissible path for the Dubins problem is a curvature constrained path connecting two directed points in the plane. In this work we characterize the behavior of homotopies of admissible paths in terms of boundedness or unboundedness of their path length. The connected path components are studied for the possible topologies when given boundary conditions. Moreover, for a type of configuration of directed points in the plane we prove the existence of a compact region in which no admissible path lying on it can be arbitrarily extended by admissible paths.