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We consider the optimal control of a PDE with random source term subject toprobabilistic or almost sure state constraints. In the main theoretical result,we provide an exact formula for the Clarke subdifferential of the probabilityfunction without a restrictive assumption made in an earlier paper. The focusof the paper is on numerical solution algorithms. As for probabilisticconstraints, we apply the method of spherical radial decomposition. Almost sureconstraints are dealt with a Moreau--Yosida smoothing of the constraintfunction accompanied by Monte Carlo sampling of the given distribution or itssupport or even just the boundary of its support. Moreover, one can understandthe almost sure constraint as a probabilistic constraint with safety level onewhich offers yet another perspective. Finally, robust optimization can beapplied efficiently when the support is sufficiently simple. A comparativestudy of these five different methodologies is carried out and illustrated.

Numerical solution of an optimal control problem with probabilistic and almost sure state constraints