# Controllability of the one-dimensional fractional heat equation under positivity constraints

## TL;DR

This paper investigates the controllability of a one-dimensional fractional heat equation with positivity constraints, establishing minimal control time and the nature of controls needed, supported by numerical simulations.

## Contribution

It proves the existence of a minimal positive control time for fractional heat equations with constraints and characterizes the control functions needed.

## Key findings

- Existence of a minimal control time T_min for s > 1/2
- Controllability achieved with Radon measure controls at T_min
- Numerical simulations confirm theoretical results

## Abstract

In this paper, we analyze the controllability properties under positivity constraints on the control or the state of a one-dimensional heat equation involving the fractional Laplacian $(-\Delta)^s$ ($0<s<1$) on the interval $(-1,1)$. We prove the existence of 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 a positive control, when $s>1/2$. Moreover, we show that in this minimal time constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. We also give some numerical simulations that confirm our theoretical results.