Limits of MPCC Formulations in Direct Optimal Control with Nonsmooth Differential Equations

In this work we discuss the limits of direct methods for Optimal Control Problems (OCPs) for some classes of nonsmooth dynamic systems. We highlight the equivalence between Filippov systems and a subclass of Differential Complementarity Systems (DCSs). Direct methods for optimal control with DCSs yield Mathematical Programs with Complementarity Constraints (MPCC), which are often solved with relaxation or smoothing methods. Due to the equivalence, results from the first class transfer to the DCSs. Therefore, to get the right numerical sensitivities one has to have a step-size of h = o(σ), where σ is the relaxation or smoothing parameter in the MPCC. A possible consequence of wrong numerical sensitivities is the appearance of spurious local solutions of the discretized OCP. We demonstrate and highlight the limits of MPCC approaches in direct optimal control on a simple counterexample.