A FEM for an optimal control problem subject to the fractional Laplace equation

TL;DR

This paper develops finite element methods for solving linear-quadratic optimal control problems constrained by the fractional Laplace equation, providing error estimates and numerical validation.

Contribution

It introduces a discretization approach using the Balakrishnan formula and compares variational and fully discrete schemes with error analysis.

Findings

Finite element error estimates are derived for both schemes.

Numerical experiments confirm the theoretical error bounds.

A tailored linear system solution method is proposed.

Abstract

We study the numerical approximation of linear-quadratic optimal control problems subject to the fractional Laplace equation with its spectral definition. We compute an approximation of the state equation using a discretization of the Balakrishnan formula that is based on a finite element discretization in space and a sinc quadrature approximation of the additionally involved integral. A tailored approach for the numerical solution of the resulting linear systems is proposed. Concerning the discretization of the optimal control problem we consider two schemes. The first one is the variational approach, where the control set is not discretized, and the second one is the fully discrete scheme where the control is discretized by piecewise constant functions. We derive finite element error estimates for both methods and illustrate our results by numerical experiments.