Quantitative Approximation Properties for the Fractional Heat Equation

TL;DR

This paper investigates the quantitative approximation capabilities of solutions to the fractional heat equation, combining local and nonlocal operators, and provides bounds on controllability costs using propagation of smallness.

Contribution

It offers an alternative proof of qualitative approximation results and introduces quantitative bounds on controllability costs for fractional heat equations.

Findings

Established bounds on the cost of approximate controllability.

Provided an alternative proof for qualitative approximation results.

Extended results to a broader class of local and nonlocal operators.

Abstract

In this note we analyse \emph{quantitative} approximation properties of a certain class of \emph{nonlocal} equations: Viewing the fractional heat equation as a model problem, which involves both \emph{local} and \emph{nonlocal} pseudodifferential operators, we study quantitative approximation properties of solutions to it. First, relying on Runge type arguments, we give an alternative proof of certain \emph{qualitative} approximation results from \cite{DSV16}. Using propagation of smallness arguments, we then provide bounds on the \emph{cost} of approximate controllability and thus quantify the approximation properties of solutions to the fractional heat equation. Finally, we discuss generalizations of these results to a larger class of operators involving both local and nonlocal contributions.