  Extended formulations are an important tool in polyhedral combinatorics. Many
combinatorial optimization problems require an exponential number of
inequalities when modeled as a linear program in the natural space of
variables. However, by adding artificial variables, one can often find a small
linear formulation, i.e., one containing a polynomial number of variables and
constraints, such that the projection to the original space of variables yields
a perfect linear formulation. Motivated by binary optimal control problems with
switching constraints, we show that a similar approach can be useful also for
optimization problems in function space, in order to model the closed convex
hull of feasible controls in a compact way. More specifically, we present small
extended formulations for switches with bounded variation and for dwell-time
constraints. For general linear switching point constraints, we devise an
extended model linearizing the problem, but show that a small extended
formulation that is compatible with discretization cannot exist unless P=NP.
