  We consider optimal control problems for partial differential equations where
the controls take binary values but vary over the time horizon, they can thus
be seen as dynamic switches. The switching patterns may be subject to
combinatorial constraints such as, e.g., an upper bound on the total number of
switchings or a lower bound on the time between two switchings. While such
combinatorial constraints are often seen as an additional complication that is
treated in a heuristic postprocessing, the core of our approach is to
investigate the convex hull of all feasible switching patterns in order to
define a tight convex relaxation of the control problem. The convex relaxation
is built by cutting planes derived from finite-dimensional projections, which
can be studied by means of polyhedral combinatorics. A numerical example for
the case of a bounded number of switchings shows that our approach can
significantly improve the dual bounds given by the straightforward continuous
relaxation, which is obtained by relaxing binarity constraints.
