Non-Existence of Minimizing Trajectories for Steer-Braking Systems

In this paper we investigate an optimal control problem in which the objective is to decelerate a simplified vehicle model, subject to input constraints, from a given initial velocity down to zero by minimizing a quadratic cost functional. The problem is of interest because, although it involves apparently simple drift-less dynamics, a minimizing trajectory does not exist. This problem is motivated by a minimum-time problem for a fairly complex car vehicle model on a race track. Numerical computations run on the car problem provide evidence of non-existence of a minimizing trajectory and of an apparently unmotivated ripple in the steer angle. We flit Abstract this situation to a very simple dynamics/objective setting, show that no minimizing trajectory exists, and reproduce the oscillating behavior on the steer angle as a mean to reduce the cost functional.