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Lyapunov functions for linear damped wave equations in one-dimensional space with dynamic boundary conditions

Yacine Chitour, Hoai-Minh Nguyen, Christophe Roman

Abstract

We establish the exponential decay of the solutions of the damped wave equations in one-dimensional space where the damping coefficient is a nowhere-vanishing function of space. The considered PDE is associated with several dynamic boundary conditions, also referred to as Wentzell/Ventzel boundary conditions in the literature. The analysis is based on the determination of appropriate Lyapunov functions and some further analysis. This result is associated with a regulation problem inspired by a real experiment with a proportional-integral control. Some numerical simulations and additional results on closed wave equations are also provided.

Lyapunov functions for linear damped wave equations in one-dimensional space with dynamic boundary conditions

Abstract

We establish the exponential decay of the solutions of the damped wave equations in one-dimensional space where the damping coefficient is a nowhere-vanishing function of space. The considered PDE is associated with several dynamic boundary conditions, also referred to as Wentzell/Ventzel boundary conditions in the literature. The analysis is based on the determination of appropriate Lyapunov functions and some further analysis. This result is associated with a regulation problem inspired by a real experiment with a proportional-integral control. Some numerical simulations and additional results on closed wave equations are also provided.
Paper Structure (10 sections, 9 theorems, 109 equations, 6 figures, 1 table)

This paper contains 10 sections, 9 theorems, 109 equations, 6 figures, 1 table.

Table of Contents

  1. Problem statement.
  2. Main result
  3. Proof of Theorem \ref{['th:main1']}
  4. Discussion on the proof of the Theorem \ref{['th:main1']}
  5. Proof on Theorem \ref{['theo_robust']}
  6. Exponential stability for the zero input system with no disturbance.
  7. Numerical schemes and simulations.
  8. Conclusion
  9. Proof of Theorem \ref{['th:wellposeness']}
  10. Additional materials

Key Result

Theorem 1

Considering assumption item1 and item2, the abstract problem sys:abs is well-posed. In order words for any initial data $\mathcal{X}_0\in \text{Dom}(\mathcal{A})$, there exists a unique solution to the abstract problem sys:abs, such that for any $t\geq 0$, $\mathcal{X}(t)\in \text{Dom}(\mathcal{A})\ $X_w$ is the state space of weak solutions and the Hilbert space considered and is defined in def:X

Figures (6)

  • Figure 1: The boundary velocities times responses.
  • Figure 2: The objectives times responses.
  • Figure 3: The control law time response.
  • Figure 4: The distributed velocity $\dot{v}(t,x)$ time response.
  • Figure 5: The distributed position $v(t,x)$ time response.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Proposition 1
  • Remark 2
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Remark 3
  • ...and 2 more