Optimal Control of Fractional Elliptic PDEs with State Constraints and Characterization of the dual of Fractional Order Sobolev Spaces

This paper introduces the notion of state constraints for optimal control problems governed by fractional elliptic PDEs of order $s \in (0,1)$. There are several mathematical tools that are developed during the process to study this problem, for instance, the characterization of the dual of the fractional order Sobolev spaces and well-posedness of fractional PDEs with measure-valued datum. These tools are widely applicable. We show well-posedness of the optimal control problem and derive the first order optimality conditions. Notice that the adjoint equation is a fractional PDE with measure as the right-hand-side datum. We use the characterization of the fractional order dual spaces to study the regularity of the state and adjoint equations. We emphasize that the classical case ($s=1$) was considered by E. Casas in \cite{ECasas_1986a} but almost none of the existing results are applicable to our fractional case.