A pointwise tracking optimal control problem for a fractional, semilinear PDE

TL;DR

This paper studies an optimal control problem involving a fractional semilinear PDE with pointwise state tracking, establishing existence and optimality conditions despite the challenges posed by fractional operators and singular adjoint equations.

Contribution

It introduces a framework for analyzing pointwise tracking control problems for fractional PDEs, including existence and optimality conditions, addressing nonconvexity and singular adjoint issues.

Findings

Existence of optimal solutions is proven.

First-order optimality conditions are derived.

Second-order conditions are established for the control problem.

Abstract

We analyze an optimal control problem with pointwise tracking for a fractional semilinear elliptic partial differential equation. The diffusion is characterized by the spectral fractional Laplacian $(-\Delta)^s$ with $s \in (1/2,1)$, a range that guarantees the well-posedness of point evaluations of the state. In addition to the nonconvexity of the control problem, the main difficulty is that the adjoint equation is a fractional partial differential equation with a singular right-hand side: a linear combination of Dirac measures. We establish the existence of optimal solutions and derive first-order as well as necessary and sufficient second-order optimality conditions.