  We address second-order optimality conditions for optimal control problems
involving sparsity functionals which induce spatio-temporal sparsity patterns.
We employ the notion of (weak) second subderivatives. With this approach, we
are able to reproduce the results from Casas, Herzog, and Wachsmuth (ESAIM
COCV, 23, 2017, p. 263-295). Our analysis yields a slight improvement of one of
these results and also opens the door for the sensitivity analysis of this
class of problems.
