D. E. Stewart and M. Anitescu, "Optimal Control of Systems with Discontinuous Differential Equations," Preprint ANL/MCS-P1258-0605, June 2005. [pdf]
In this paper we discuss the problem of verifying and computing optimal controls of systems whose dynamics is governed by differential systems with discontinuous right hand side. In our work, we are motivated by optimal control of mechanical systems with Coulomb friction, which exhibit such right-hand side. Notwithstanding the impressive development of nonsmooth and set-valued analysis, these systems have not been closely studied either computationally or analytically. First, we show that even when the solution crosses and does not stay on the discontinuity, differentiating the results of a simulation gives gradients that have errors of a size independent of the step-size. This means that the strategy of "optimize the discretization" will usually fail for problems of this kind. We approximate the discontinuous right-hand side for the differential equations or inclusions by a smooth right-hand side. For these smoothed approximations, we show that the resulting gradients approach the true gradients provided the start and end points of the trajectory do not lie on the discontinuity, and that using Euler's method where the step size is "sufficiently small" in comparison with the smoothing parameter. Numerical results are presented for a crude model of car racing which involves Coulomb friction and slip showing that this approach is practical and can handle problems of considerable complexity.