We investigate the convergence of the proximal gradient method applied to control problems with non-smooth and non-convex control cost. Here, we focus on control cost functionals that promote sparsity, which includes functionals of $$L^p$$ L p -type for $$p\in [0,1)$$ p ∈ [ 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.

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 ∈ [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.