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

Abstract

This paper introduces the notion of state constraints for optimal control problems governed by fractional elliptic partial differential equations. Several mathematical tools are developed during the process to study these problems, for instance, the characterization of the dual of fractional-order Sobolev spaces and the well-posedness of fractional elliptic equations with measure-valued data. 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 partial differential equation with a measure as the right-hand-side datum. We use the characterization of the fractional- order dual spaces to study the regularity of solutions of the state and adjoint equations. We emphasize that the classical case was considered by E. Casas, but almost none of the existing results are applicable to our fractional case. As an application of the regularity result of the adjoint equation, we establish the Sobolev regularity of the optimal control. In addition, under this setup, even weaker controls can be used.