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We investigate the hard-thresholding method applied to optimal controlproblems with $L^0(\Omega)$ control cost, which penalizes the measure of thesupport 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 objectivefunctional are lower semicontinuous along the iterates. That is, the functionvalue in a weak limit point is less or equal than the lim-inf of the functionvalues along the iterates. Under a compactness assumption, we can prove thatweak limit points are strong limit points, which enables us to prove certainstationarity conditions for the limit points. Numerical experiments are carriedout, which show the performance of the method. These indicates that the methodis robust with respect to discretization. In addition, we show that solutionsobtained by the thresholding algorithm are superior to solutions of$L^1(\Omega)$-regularized problems.

Iterative hard-thresholding applied to optimal control problems with $L^0(\Omega)$ control cost