The problem of driving a vehicle along a given path is considered. The vehicle is supposed to move forward only, with a given velocity profile, and to have a bounded turning radius. Such a model, also known as 'Dubins' vehicle', is relevant to the kinematics of road vehicles as well as aircraft cruising at constant altitude, or sea vessels. We consider the optimal control problem consisting of minimizing the length travelled by the vehicle starting from a generic configuration to connect to a specified route. A feedback law is proposed, such that straight routes can be approached optimally, while the system is asymptotically stabilized. Experimental results are reported showing real-time feasibility of the proposed approach.