Feedback approximate controllability of blowup points for the heat equation with anti-interference blowup profile

TL;DR

This paper demonstrates that feedback controls can approximately steer blowup points in the heat equation, with stable blowup profiles under small initial perturbations, indicating an anti-interference control strategy.

Contribution

It establishes the approximate controllability of blowup points using bounded feedback controls and proves the stability of blowup profiles against small initial data perturbations.

Findings

Feedback controls achieve approximate controllability of blowup points.

Blowup profiles are stable under small initial data perturbations.

The feedback operator remains bounded before blowup.

Abstract

This paper is concerned with a feedback approximate controllability problem of blowup points for the heat equation. We show that the system is approximately controllable for blowup points with feedback controls and the feedback operator is bounded at any time before blowup. It is also proved that the blowup profile for feedback controllability of blowup points is stable with respect to initial data. That is, suppose that the initial data has a very small perturbation, the blowup profiles also have tiny changes. More precisely, it just undergoes a tiny translation in space and time. This means that our feedback strategy is anti-interference.