Optimal control problems with L0(Ω) constraints: maximum principle and proximal gradient method

We investigate optimal control problems with L0constraints, which restrict the measure of the support of the controls. We prove necessary optimality conditions of Pontryagin maximum principle type. Here, a special control perturbation is used that respects the L0constraint. First, the maximum principle is obtained in integral form, which is then turned into a pointwise form. In addition, an optimization algorithm of proximal gradient type is analyzed. Under some assumptions, the sequence of iterates contains strongly converging subsequences, whose limits are feasible and satisfy a subset of the necessary optimality conditions.