Optimizing the Fractional Power in a Model with Stochastic PDE Constraints

We study an optimization problem with SPDE constraints, which has the peculiarity that the control parameter is the -th power of the diffusion operator in the state equation. Well-posedness of the state equation and differentiability properties with respect to the fractional parameter are established. We show that under certain conditions on the noise, optimality conditions for the control problem can be established.