Optimizing the fractional power in a model with stochastic PDE constraints

TL;DR

This paper investigates an optimization problem constrained by stochastic partial differential equations where the control parameter is the fractional power of the diffusion operator, establishing well-posedness, differentiability, and optimality conditions.

Contribution

It introduces a novel control problem involving the fractional power of the diffusion operator in SPDEs, with new results on well-posedness and differentiability w.r.t. the fractional parameter.

Findings

Proved well-posedness of the SPDE with fractional diffusion

Established differentiability of the state with respect to the fractional parameter

Derived optimality conditions for the control problem

Abstract

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. Well-posedness of the state equation and differentiability properties with respect to the fractional parameter $s$ are established. We show that under certain conditions on the noise, optimality conditions for the control problem can be established.