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Rapid stabilization of the heat equation with localized disturbance

Patricio Guzmán, Hugo Parada, Christian Calle-Cárdenas

TL;DR

This work tackles exponential stabilization of a multidimensional heat equation subject to a spatially localized, unknown disturbance. It develops a novel multivalued feedback that combines the Frequency Lyapunov method with a sign operator, enabling robust stabilization with actuation in a small subdomain. The analysis relies on a weak spectral inequality for Laplacian eigenfunctions and a finite-dimensional modal truncation, leading to a differential inclusion whose well-posedness is established via maximal monotone operator theory. An explicit feedback law is derived that achieves exponential decay at any prescribed rate λ without requiring disturbance modeling, highlighting robust stabilization in distributed PDEs with localized actuation.

Abstract

This paper studies the rapid stabilization of a multidimensional heat equation in the presence of an unknown spatially localized disturbance. A novel multivalued feedback control strategy is proposed, which synthesizes the frequency Lyapunov method (introduced by Xiang [41]) with the sign multivalued operator. This methodology connects Lyapunov-based stability analysis with spectral inequalities, while the inclusion of the sign operator ensures robustness against the disturbance. The closed-loop system is governed by a differential inclusion, for which well-posedness is proved via the theory of maximal monotone operators. This approach not only guarantees exponential stabilization but also circumvents the need for explicit disturbance modeling or estimation.

Rapid stabilization of the heat equation with localized disturbance

TL;DR

This work tackles exponential stabilization of a multidimensional heat equation subject to a spatially localized, unknown disturbance. It develops a novel multivalued feedback that combines the Frequency Lyapunov method with a sign operator, enabling robust stabilization with actuation in a small subdomain. The analysis relies on a weak spectral inequality for Laplacian eigenfunctions and a finite-dimensional modal truncation, leading to a differential inclusion whose well-posedness is established via maximal monotone operator theory. An explicit feedback law is derived that achieves exponential decay at any prescribed rate λ without requiring disturbance modeling, highlighting robust stabilization in distributed PDEs with localized actuation.

Abstract

This paper studies the rapid stabilization of a multidimensional heat equation in the presence of an unknown spatially localized disturbance. A novel multivalued feedback control strategy is proposed, which synthesizes the frequency Lyapunov method (introduced by Xiang [41]) with the sign multivalued operator. This methodology connects Lyapunov-based stability analysis with spectral inequalities, while the inclusion of the sign operator ensures robustness against the disturbance. The closed-loop system is governed by a differential inclusion, for which well-posedness is proved via the theory of maximal monotone operators. This approach not only guarantees exponential stabilization but also circumvents the need for explicit disturbance modeling or estimation.
Paper Structure (12 sections, 8 theorems, 121 equations, 1 table)

This paper contains 12 sections, 8 theorems, 121 equations, 1 table.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Spectral Inequality
  4. Feedback Design
  5. Design of u
  6. Design of u
  7. Feedback law
  8. Well-Posedness
  9. Concluding Remarks and Perspectives
  10. Weak Spectral Inequality Proof
  11. Inner product
  12. Isomorphism of $\mathscr{C}$

Key Result

Theorem 2.1

Let us assume A_1 and A_2. Let $y_{0}$ in $L^{2}(\Omega)$ be the initial condition. Let $\lambda$ in $(0, \infty)$ be the desired decay rate. Then, there exists a feedback law $\mathscr{G_\lambda}: L^{2}(\Omega) \rightarrow$$L^{2}(\Omega)$ such that P is exponentially stable in $L^{2}(\Omega)$, with

Theorems & Definitions (18)

  • Theorem 2.1
  • Remark 2.1
  • Proposition 3.1
  • Remark 3.1
  • Remark 3.2
  • Proposition 5.1
  • proof
  • Remark 5.1
  • Proposition 5.2
  • proof
  • ...and 8 more