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

## TL;DR

This paper develops mathematical tools for optimal control problems governed by fractional elliptic PDEs with state constraints, including dual space characterization and well-posedness, extending classical results to fractional orders.

## Contribution

It introduces the notion of state constraints for fractional elliptic PDEs and characterizes the dual of fractional Sobolev spaces, advancing the mathematical foundation for fractional optimal control.

## Key findings

- Established well-posedness of fractional PDE control problems
- Derived first order optimality conditions for fractional PDEs
- Characterized the dual of fractional Sobolev spaces

## Abstract

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.