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<p>In this article, we investigate the stability of a high-density wave oscillating on a ring equipped with an internal dynamic controller. At the ring boundaries, the wave vanishes on the lower boundary and releases significant kinetic energy on the upper one. The analysis begins with a proof of the existence and uniqueness of the solution using the semigroup method, along with a demonstration of the total energy decay over time. Then, using Nakao’s method, we establish that this energy decay is exponential under a specific condition on the internal controller parameter. Finally, numerical experiments based on the finite difference method are conducted to illustrate and confirm the exponential stability.</p>

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Theoretical and numerical stabilization of the wave equation subject to internal and dynamic controllers

  • 期刊论文

  • 发表日期：

    2025年7月14日

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  • Kalifa Lassana Barry,
  • Karima Laoubi

摘要

In this article, we investigate the stability of a high-density wave oscillating on a ring equipped with an internal dynamic controller. At the ring boundaries, the wave vanishes on the lower boundary and releases significant kinetic energy on the upper one. The analysis begins with a proof of the existence and uniqueness of the solution using the semigroup method, along with a demonstration of the total energy decay over time. Then, using Nakao’s method, we establish that this energy decay is exponential under a specific condition on the internal controller parameter. Finally, numerical experiments based on the finite difference method are conducted to illustrate and confirm the exponential stability.