Localization for nonlocal gradient-based optimal control problems

TL;DR

This paper investigates optimal control problems within nonlocal function spaces, analyzing how solutions approximate local problems as fractional and horizon parameters vary.

Contribution

It introduces a framework for nonlocal optimal control problems with two parameters and studies their convergence to local problems.

Findings

Analysis of convergence as fractional parameter s approaches 1

Analysis of convergence as horizon parameter δ approaches 0

Comparison between convex and non-convex energy densities

Abstract

In this paper we consider optimal control problems in the nonlocal function space framework of Bellido-2023, where there are two different parameters: a horizon parameter $\delta > 0$; and a fractional parameter $s \in (0, 1)$. The constraints are given in the form of minimizing an energy density, and we will focus on two particular cases: the well-posed case where the underlying energy density is convex and is given by the nonlocal $p$-Laplacian; and a more general poly/quasiconvex energy for which minimizers exist but may not be unique. The study is concluded by analyzing the approximation to local problems in two parallel ways, either taking the fractional parameter $s$ to $1$ or the horizon parameter $\delta$ to $0$.