The asymmetric sinistral/dextral Markov-Dubins problem

We consider a variation of the classical Markov-Dubins problem dealing with curvature-constrained, shortest paths in the plane with prescribed initial and terminal positions and tangents, when the lower and upper bounds of the curvature are not necessarily equal. The motivation for this problem stems from vehicle navigation applications when the vehicle may be biased in taking turns at a particular direction due to hardware failures or environmental conditions. We employ optimal control to characterize the structure of the shortest path and we resort to geometric techniques to provide sufficient conditions for optimality of the resulting path.