Planar length-optimal paths under acceleration constraints

This paper considers the problem of finding minimum length trajectories for a particle moving in a two-dimensional plane from a given initial position and velocity to a specified terminal heading under a magnitude constraint on the acceleration. Unlike previous work on related problems, variations in the magnitude of the velocity vector are allowed. Pontryagin´s maximum principle is used to show that the length-optimal paths possess a special property whereby the angle bisector between the acceleration and velocity vectors is a constant. This property is used to obtain the optimal acceleration vector and to show that the length-optimal paths are arcs of alysoids. A numerical example is presented and the solutions of the length-optimal problem are compared with those of the corresponding time-optimal problem