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Rapid stabilization for a wave equation with boundary disturbance

Patricio Guzmán, Agustín Huerta, Hugo Parada

TL;DR

This work addresses the rapid stabilization of an unstable wave equation subject to an unknown boundary disturbance. It combines backstepping with Lyapunov methods and a sign-based multivalued operator to design a boundary feedback $U(t)$ located at the disturbance site, achieving exponential decay of the energy $E(t)$ at any prescribed rate $d$ for both Dirichlet-Dirichlet and Dirichlet-Neumann configurations. The closed-loop models are differential inclusions due to the sign operator, and well-posedness is established via maximal monotone operator theory. Overall, the paper extends disturbed PDE stabilization to enable rapid stabilization with rigorous stability guarantees and explicit decay bounds.

Abstract

In this paper, we study the rapid stabilization of an unstable wave equation, in which an unknown disturbance is located at the boundary condition. We address two different boundary conditions: Dirichlet- Dirichlet and Dirichlet-Neumann. In both cases, we design a feedback law, located at the same place as the unknown disturbance, that forces the exponential decay of the energy for any desired decay rate while suppressing the effects of the unknown disturbance. For the feedback design, we employ the backstepping method, Lyapunov techniques and the sign multivalued operator. The well-posedness of the closed-loop system, which is a differential inclusion, is shown with the maximal monotone operator theory.

Rapid stabilization for a wave equation with boundary disturbance

TL;DR

This work addresses the rapid stabilization of an unstable wave equation subject to an unknown boundary disturbance. It combines backstepping with Lyapunov methods and a sign-based multivalued operator to design a boundary feedback located at the disturbance site, achieving exponential decay of the energy at any prescribed rate for both Dirichlet-Dirichlet and Dirichlet-Neumann configurations. The closed-loop models are differential inclusions due to the sign operator, and well-posedness is established via maximal monotone operator theory. Overall, the paper extends disturbed PDE stabilization to enable rapid stabilization with rigorous stability guarantees and explicit decay bounds.

Abstract

In this paper, we study the rapid stabilization of an unstable wave equation, in which an unknown disturbance is located at the boundary condition. We address two different boundary conditions: Dirichlet- Dirichlet and Dirichlet-Neumann. In both cases, we design a feedback law, located at the same place as the unknown disturbance, that forces the exponential decay of the energy for any desired decay rate while suppressing the effects of the unknown disturbance. For the feedback design, we employ the backstepping method, Lyapunov techniques and the sign multivalued operator. The well-posedness of the closed-loop system, which is a differential inclusion, is shown with the maximal monotone operator theory.

Paper Structure

This paper contains 11 sections, 12 theorems, 95 equations, 1 table.

Table of Contents

  1. Introduction
  2. Dirichlet-Dirichlet case
  3. Feedback design
  4. Well-posedness
  5. Conclusion
  6. Dirichlet-Neumann case
  7. Feedback design
  8. Well-posedness
  9. Conclusion
  10. Concluding remarks
  11. Acknowledgments

Key Result

Theorem 2.1

Let $\lambda\in C^{2}([0,L])$ and $\beta\in C([0,L])$. Then, there exist a unique $k=k(x,y)$ in $C^{2}(\mathcal{T})$ and a unique $s=s(x,y)$ in $C^{2}(\mathcal{T})$ such that

Theorems & Definitions (24)

  • Remark 1.1
  • Remark 2.1
  • Theorem 2.1: Theorem 3.1 in sck2010
  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.5
  • ...and 14 more