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In this paper, we discuss optimality conditions for optimization problemsinvolving random state constraints, which are modeled in probabilistic oralmost sure form. While the latter can be understood as the limiting case ofthe former, the derivation of optimality conditions requires substantiallydifferent approaches. We apply them to a linear elliptic partial differentialequation (PDE) with random inputs. In the probabilistic case, we rely on thespherical-radial decomposition of Gaussian random vectors in order to formulatefully explicit optimality conditions involving a spherical integral. In thealmost sure case, we derive optimality conditions and compare them to a modelbased on robust constraints with respect to the (compact) support of the givendistribution.

Optimality conditions in control problems with random state constraints in probabilistic or almost-sure form