Controllability properties from the exterior under positivity constraints for a 1-D fractional heat equation

TL;DR

This paper investigates the controllability of a 1-D fractional heat equation with positivity constraints, revealing a minimal time for control and characterizing the control measures needed, supported by numerical validation.

Contribution

It establishes the existence of a minimal control time for positive controllability of fractional heat equations with exterior controls, specifically for 1/2<s<1, and characterizes the control measures at this time.

Findings

Controllability is possible only if 1/2<s<1.

A minimal positive control time T_min exists.

Controls at T_min are Radon measures.

Abstract

We study the controllability to trajectories, under positivity constraints on the control or the state, of a one-dimensional heat equation involving the fractional Laplace operator $ (-\partial_x^2)^s$ (with $0<s<1$) on the interval $(-1,1)$. Our control function is localized in an open set $\mathcal O$ in the exterior of $(-1,1)$, that is, $\mathcal O\subset (\mathbb{R} \setminus (-1,1))$. We show that there exists a minimal (strictly positive) time $T_{\rm min}$ such that the fractional heat dynamics can be controlled from any initial datum in $L^2(-1,1)$ to a positive trajectory through the action of an exterior positive control, if and only if $\frac 12<s<1$. In addition, we prove that at this minimal controllability time, the constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. Finally, we provide several numerical illustrations that confirm our theoretical results.