Sala Seminari (Dip. Matematica).
We study an optimization problem with SPDE constraints, which has the peculiarity that the control parameter $s$ is the $s$-th power of the diffusion operator in the state equation. Before moving to the SPDE case, we first describe the result of Sprekels-Valdinoci for the PDE case. Then we discuss a suitable concept of solutions of the state equation and establish pathwise differentiability properties of the stochastic process w.r.t. the fractional parameter $s$. Finally, we show that under certain conditions on the noise, optimality conditions for the control problem can be established. This is joined work with Enrico Valdinoci.