Optimal feedback control of Dubins' car tracking circular reference paths

The problem of driving a car-like vehicle moving forward in the plane with a lower bounded turning radius is considered. We analyse the motion along circular reference paths with radius greater than or equal to the car's minimal turning radius. From Pontryagin's maximum principle we deduce necessary conditions minimal-length paths that have to verify to reach tangentially the reference circle. We then construct the shortest paths synthesis in the two dimensional reduced state space defined by the lateral distance from the car to the circle and the heading angle error. Finally, a variable structure feedback control on the angular velocity of the vehicle is derived.