Mixed-integer optimal control problems with switching costs: a shortest path approach

We investigate an extension of Mixed-Integer Optimal Control Problems by adding switching costs, which enables the penalization of chattering and extends current modeling capabilities. The decomposition approach, consisting of solving a partial outer convexification to obtain a relaxed solution and using rounding schemes to obtain a discrete-valued control can still be applied, but the rounding turns out to be difficult in the presence of switching costs or switching constraints as the underlying problem is an Integer Program. We therefore reformulate the rounding problem into a shortest path problem on a parameterized family of directed acyclic graphs (DAGs). Solving the shortest path problem then allows to minimize switching costs and still maintain approximability with respect to the tunable DAG parameter θ . We provide a proof of a runtime bound on equidistant rounding grids, where the bound is linear in time discretization granularity and polynomial in θ . The efficacy of our approach is demonstrated by a comparison with an integer programming approach on a benchmark problem.