Feedback controllability for blowup points of heat equation

TL;DR

This paper investigates how feedback control can be used to induce a heat equation solution to blow up at a specific point within a domain at a predetermined time, highlighting the conditions for controllability of blowup points.

Contribution

It demonstrates the possibility of controlling the blowup point of the heat equation using feedback control within a subset of the domain, establishing conditions for uniqueness.

Findings

A feedback control can make a chosen point the unique blowup point.

If the point is outside the control region, it cannot be the unique blowup point.

The control region must contain the blowup point for controllability.

Abstract

This paper concerns a controllability problem for blowup points on heat equation. It can be described as follows: In the absence of control, the solution to the linear heat system globally exists in a bounded domain $\Omega$. While, for a given time $T>0$ and a point $a$ in this domain, we find a feedback control, which is acted on an internal subset $\omega$ of this domain, such that the corresponding solution to this system blows up at time $T$ and holds unique point $a$. We show that $a\in \omega$ can be the unique blowup point of the corresponding solution with a certain feedback control, and for any feedback control, $a\in \Omega\setminus \overline{\omega}$ could not be the unique blowup point.