  We investigate the convergence of an application of a proximal gradient
method to control problems with nonsmooth and nonconvex control cost. Here, we
focus on control cost functionals that promote sparsity, which includes
functionals of $L^p$-type for $p\in [0,1)$. We prove stationarity properties of
weak limit points of the method. These properties are weaker than those
provided by Pontryagin's maximum principle and weaker than $L$-stationarity.
