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

Ping Lin, Hatem Zaag

TL;DR

This work studies the heat equation under feedback control to achieve approximate controllability of blowup points. The authors develop a two-tier approach: first, reconstruct a blowup solution with a prescribed profile using nonlinear and linear control steps, and second, prove stability of the blowup profile with respect to initial data via a spectral and topological finite-dimensional reduction. They show that a bounded feedback operator $P(t)$ can steer the system so that blowup occurs near a chosen time and location, with the profile robust to small initial perturbations (an anti-interference property). The results extend the understanding of blowup controllability by enabling precise placement and timing of blowup through feedback, with implications for controlled blowup phenomena in nonlinear parabolic equations.

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.

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

TL;DR

This work studies the heat equation under feedback control to achieve approximate controllability of blowup points. The authors develop a two-tier approach: first, reconstruct a blowup solution with a prescribed profile using nonlinear and linear control steps, and second, prove stability of the blowup profile with respect to initial data via a spectral and topological finite-dimensional reduction. They show that a bounded feedback operator can steer the system so that blowup occurs near a chosen time and location, with the profile robust to small initial perturbations (an anti-interference property). The results extend the understanding of blowup controllability by enabling precise placement and timing of blowup through feedback, with implications for controlled blowup phenomena in nonlinear parabolic equations.

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.
Paper Structure (8 sections, 17 theorems, 280 equations)

This paper contains 8 sections, 17 theorems, 280 equations.

Table of Contents

  1. Introduction
  2. Reconstructing a blowup solution with a prescribed profile
  3. Formulation of the problem and preliminary lemmas
  4. Proof of Proposition \ref{['lemma1.3']}
  5. Stability of blowup profile for auxiliary system
  6. Formulation of the problem
  7. Behavior of "initial data" $\overline{\psi}$ for $(T,a)$ near $(\widehat{T},\widehat{a})$
  8. Proof of Theorem \ref{['Main Tho']}

Key Result

Proposition 1.2

For any $a\in \omega$ and any $T>0$, there exist $T_1\in (0, T/2)$ and $\widetilde{y}_0\in C_0^\infty(\omega)$ such that for any $y_0\in H_0^1(\Omega)$, the solution $y$ to (xe1.1) with the following feedback control exists on $[0,T)$, $T$ is the blowup time of $y$ and $y$ has a unique blowup point $a$, where $p>1$. Here, $P\in C_S([0,T-T_1); \Sigma^+(H))$ is the unique mild solution to the follo

Theorems & Definitions (23)

  • Definition 1.1: Notions of blowup time and point
  • Proposition 1.2
  • Definition 1.3: Notion of feedback approximate controllability of blowup points
  • Theorem 1
  • Proposition 1.4
  • Theorem 2
  • Definition 2.1: Shrinking set
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 13 more