  We investigate the hard-thresholding method applied to optimal control
problems with $L^0(\Omega)$ control cost, which penalizes the measure of the
support of the control. As the underlying measure space is non-atomic,
arguments of convergence proofs in $l^2$ or $\mathbb R^n$ cannot be applied.
Nevertheless, we prove the surprising property that the values of the objective
functional are lower semicontinuous along the iterates. That is, the function
value in a weak limit point is less or equal than the lim-inf of the function
values along the iterates. Under a compactness assumption, we can prove that
weak limit points are strong limit points, which enables us to prove certain
stationarity conditions for the limit points. Numerical experiments are carried
out, which show the performance of the method. These indicates that the method
is robust with respect to discretization. In addition, we show that solutions
obtained by the thresholding algorithm are superior to solutions of
$L^1(\Omega)$-regularized problems.
