# External optimal control of fractional parabolic PDEs

## TL;DR

This paper introduces a novel optimal control concept for fractional parabolic PDEs, allowing control placement outside the domain, and demonstrates theoretical analysis and numerical benefits over classical local models.

## Contribution

It develops a new framework for external control in fractional parabolic PDEs, extending classical models and providing convergence analysis and numerical validation.

## Key findings

- External control placement is feasible and advantageous in fractional PDEs.
- The approach includes convergence rates for approximating Dirichlet solutions.
- Numerical examples confirm theoretical benefits of nonlocal models.

## Abstract

In this paper we introduce a new notion of optimal control, or source identification in inverse, problems with fractional parabolic PDEs as constraints. This new notion allows a source/control placement outside the domain where the PDE is fulfilled. We tackle the Dirichlet, the Neumann and the Robin cases. For the fractional elliptic PDEs this has been recently investigated by the authors in \cite{HAntil_RKhatri_MWarma_2018a}. The need for these novel optimal control concepts stems from the fact that the classical PDE models only allow placing the source/control either on the boundary or in the interior where the PDE is satisfied. However, the nonlocal behavior of the fractional operator now allows placing the control in the exterior. We introduce the notions of weak and very-weak solutions to the parabolic Dirichlet problem. We present an approach on how to approximate the parabolic Dirichlet solutions by the parabolic Robin solutions (with convergence rates). A complete analysis for the Dirichlet and Robin optimal control problems has been discussed. The numerical examples confirm our theoretical findings and further illustrate the potential benefits of nonlocal models over the local ones.