Control and numerical approximation of fractional diffusion equations

TL;DR

This paper surveys control properties and numerical methods for fractional diffusion equations, focusing on finite element approximations, control scenarios, and the potential of stochastic optimization for simultaneous control.

Contribution

It provides a comprehensive overview of numerical control techniques for fractional diffusive models, including error estimates, convergence, and innovative control strategies.

Findings

Finite element methods effectively approximate fractional Laplacian models.

Interior and exterior control scenarios are feasible despite non-locality.

Stochastic optimization can reduce computational costs in simultaneous control problems.

Abstract

The aim of this work is to give a broad panorama of the control properties of fractional diffusive models from a numerical analysis and simulation perspective. We do this by surveying several research results we obtained in the last years, focusing in particular on the numerical computation of controls. Our reference model will be a non-local diffusive dynamics driven by the fractional Laplacian on a bounded domain $\Omega$. The starting point of our analysis will be a Finite Element approximation for the associated elliptic model in one and two space-dimensions, for which we also present error estimates and convergence rates in the $L^2$ and energy norm. Secondly, we will address two specific control scenarios: firstly, we consider the standard interior control problem, in which the control is acting from a small subset $\omega\subset\Omega$. Secondly, we move our attention to the exterior control problem, in which the control region $\mathcal O\subset\Omega^c$ is located outside $\Omega$. This exterior control notion extends boundary control to the fractional framework, in which the non-local nature of the models does not allow for controls supported on $\partial\Omega$. We will conclude by discussing the interesting problem of simultaneous control, in which we consider families of parameter-dependent fractional heat equations and we aim at designing a unique control function capable of steering all the different realizations of the model to the same target configuration. In this framework, we will see how the employment of stochastic optimization techniques may help in alleviating the computational burden for the approximation of simultaneous controls. Our discussion is complemented by several open problems related with fractional models which are currently unsolved and may be of interest for future investigation.