Optimal control of steer-braking systems: Non-existence of minimizing trajectories

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 over the admissible input trajectories. 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 trajectory optimization problem provide evidence of convergence issues and of an apparently unmotivated ripple in the steer angle. Characterizing this ripple behavior is important to fully understand and exploit minimizing vehicle trajectories. We are able to isolate the key features of this chattering behavior in a very simple dynamics/objective setting. We show that the cost functional has an infimum, but an admissible minimizing input trajectory does not exist. We also show that the infimum can be arbitrarily approximated by bang‐bang inputs with a sufficiently large number of switches. We reproduce this phenomenon in numerical computations and characterize it by means of non‐existence of admissible minimizing trajectories.