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Observer design and boundary output feedback stabilization for semilinear parabolic system over general multidimensional domain

Kai Liu, Hua-Cheng Zhou, Zhong-Jie Han, Xiangyang Peng

TL;DR

The paper addresses the stabilization of a semilinear parabolic PDE with Lipschitz nonlinearity on general multidimensional domains using boundary output feedback. It develops a novel nonlinear observer aided by a Dirichlet lifting and leverages spectral geometry, notably the Berezin-Li-Yau inequality, to achieve prescribed observer decay and guide sensor placement; it then designs a finite-dimensional state-feedback controller that stabilizes the linear part, integrating it with the observer to yield robust boundary output feedback. The main theoretical contributions include exponential stabilization guarantees for the observer error and the closed-loop system in spatial dimensions $d\in\{1,2,3\}$ for arbitrary Lipschitz constants (via sufficiently large $N$ and suitable $m$), along with explicit sensor-placement criteria on general domains. Numerical simulations on a 2-D domain validate both the observer performance and the rapid stabilization of the closed-loop system, confirming the practicality of the proposed strategy in complex geometries.

Abstract

This paper investigates the output feedback stabilization of parabolic equation with Lipschitz nonlinearity over general multidimensional domain using spectral geometry theories. First, a novel nonlinear observer is designed, and the error system is shown to achieve any prescribed decay rate by leveraging the Berezin-Li-Yau inequality from spectral geometry, which also provides effective guidance for sensor placement. Subsequently, a finite-dimensional state feedback controller is proposed, which ensures the quantitative rapid stabilization of the linear part. By integrating this control law with the observer, an efficient boundary output feedback control strategy is developed. The feasibility of the proposed control design is rigorously verified for arbitrary Lipschitz constants, thereby resolving a persistent theoretical challenge. Finally, a numerical case study confirms the effectiveness of the approach.

Observer design and boundary output feedback stabilization for semilinear parabolic system over general multidimensional domain

TL;DR

The paper addresses the stabilization of a semilinear parabolic PDE with Lipschitz nonlinearity on general multidimensional domains using boundary output feedback. It develops a novel nonlinear observer aided by a Dirichlet lifting and leverages spectral geometry, notably the Berezin-Li-Yau inequality, to achieve prescribed observer decay and guide sensor placement; it then designs a finite-dimensional state-feedback controller that stabilizes the linear part, integrating it with the observer to yield robust boundary output feedback. The main theoretical contributions include exponential stabilization guarantees for the observer error and the closed-loop system in spatial dimensions for arbitrary Lipschitz constants (via sufficiently large and suitable ), along with explicit sensor-placement criteria on general domains. Numerical simulations on a 2-D domain validate both the observer performance and the rapid stabilization of the closed-loop system, confirming the practicality of the proposed strategy in complex geometries.

Abstract

This paper investigates the output feedback stabilization of parabolic equation with Lipschitz nonlinearity over general multidimensional domain using spectral geometry theories. First, a novel nonlinear observer is designed, and the error system is shown to achieve any prescribed decay rate by leveraging the Berezin-Li-Yau inequality from spectral geometry, which also provides effective guidance for sensor placement. Subsequently, a finite-dimensional state feedback controller is proposed, which ensures the quantitative rapid stabilization of the linear part. By integrating this control law with the observer, an efficient boundary output feedback control strategy is developed. The feasibility of the proposed control design is rigorously verified for arbitrary Lipschitz constants, thereby resolving a persistent theoretical challenge. Finally, a numerical case study confirms the effectiveness of the approach.
Paper Structure (18 sections, 7 theorems, 96 equations, 4 figures)

This paper contains 18 sections, 7 theorems, 96 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Problem formulation
  4. Novelties and Contributions
  5. Organization
  6. Notation
  7. State Estimation
  8. Preliminaries
  9. Observer Design
  10. Sensor Placement Strategy on Rectangular Domains
  11. Controller Design
  12. Preliminaries
  13. State Feedback Control
  14. Output feedback control
  15. Illustrative Example
  16. ...and 3 more sections

Key Result

Proposition 1

The eigenvalue-eigenfunction pairs $(\lambda_k, \phi_k)_{k=1}^{\infty}$ possess the following spectral properties: (1) Weyl's Law weyl1912asymptotische: The eigenvalues of the operator $\mathscr{A}$ satisfy the asymptotic behavior where $\sim$ denotes asymptotic equivalence as $k \to \infty$. (2) Berezin-Li-Yau Inequalityberezin1972convexli1983schrodinger: The $k$-th eigenvalue of the operator $\

Figures (4)

  • Figure 1: Illustrative example
  • Figure 2: Domain decomposition
  • Figure 3: Numerical verification of the asymptotic scaling for \ref{['3-term']}
  • Figure 4: Numerical results for $L=100$: (a) open-loop, (b) closed-loop, (c) estimation error.

Theorems & Definitions (15)

  • Proposition 1
  • Theorem 2
  • proof
  • Remark 1
  • Theorem 3
  • proof
  • Remark 2
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 5 more