A switching cost aware rounding method for relaxations of mixed-integer optimal control problems

This article investigates a class of Mixed-Integer Optimal Control Problems (MIOCPs) with switching costs. We introduce the problem class of Minimal-Switching-Cost Optimal Control Problems (MSCP) with an objective function that consists of two summands, a continuous term depending on the state vector and an encoding of the discrete switching costs. State vectors of Mixed-Integer Optimal Control problems can be approximated by means of sequences of roundings of appropriate relaxations, which often result in a switching cost blow-up. We reformulate the problem such that trading convergence of the state vector against increasing switching costs is possible, which then allows to conserve known convergence properties of previous approaches for Mixed-Integer Optimal Control approximations. To demonstrate the findings and applicability, we present validating numerical results and the trade-off capability of our approach for a benchmark problem.