Bilinear optimal control for the fractional Laplacian: analysis and discretization

TL;DR

This paper investigates an optimal control problem involving the fractional Laplacian, establishing existence, optimality conditions, regularity, and developing finite element discretizations with convergence and error analysis.

Contribution

It introduces two finite element discretization strategies for fractional Laplacian control problems and provides comprehensive convergence and error estimates.

Findings

Existence of optimal solutions proven.

First and second order optimality conditions derived.

Finite element schemes analyzed for convergence and error bounds.

Abstract

We adopt the integral definition of the fractional Laplace operator and study an optimal control problem on Lipschitz domains that involves a fractional elliptic partial differential equation (PDE) as state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are considered as well. We establish the existence of optimal solutions and analyze first and, necessary and sufficient, second order optimality conditions. Regularity estimates for optimal variables are also analyzed. We develop two finite element discretization strategies: a semidiscrete scheme in which the control variable is not discretized, and a fully discrete scheme in which the control variable is discretized with piecewise constant functions. For both schemes, we analyze the convergence properties of discretizations and derive error estimates.