Prescribing the best decay rate of the wave equation using internal delayed feedback damping

Kaïs Ammari; Islam Boussaada; Silviu-Iulian Niculescu; Sami Tliba; Kaïs Ammari; Islam Boussaada; Silviu-Iulian Niculescu; Sami Tliba

doi:10.3934/cam.2026001

Communications in Analysis and Mechanics

2026, Volume 18, Issue 1: 1-36. doi: 10.3934/cam.2026001

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Prescribing the best decay rate of the wave equation using internal delayed feedback damping

1.
LR Analyse et Contrôle des EDPs, LR 22ES03, Département de Mathématiques, Faculté des Sciences de Monastir, Université de Monastir, 5019 Monastir, Tunisie
2.
Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes (L2S), Inria Saclay Île-de-France, DISCO Team, 3 Rue Joliot-Curie, 91190, Gif-sur-Yvette, France
3.
Institut Polytechnique des Sciences Avancées (IPSA), 63 Boulevard de Brandebourg, 94200 Ivry-sur-Seine, France
4.
Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes (L2S), 3 Rue Joliot-Curie, 91190 Gif-sur-Yvette, France

Received: 30 April 2025 Revised: 10 October 2025 Accepted: 07 November 2025 Published: 04 January 2026
35B05, 93D15, 93D20

In this paper, we investigated the stabilization of the damped wave equation through the use of an internal feedback mechanism incorporating time delay. This work builds upon the partial pole placement paradigm, a recent theoretical framework originally developed for functional differential equations, which enables the selective assignment of eigenvalues within a prescribed region of the complex plane. Using this approach, we designed an internal delayed feedback law that guarantees the exponential stabilization of the resulting closed-loop system. A distinctive feature of our control strategy lies in its ability to prescribe the optimal exponential decay rate for each modal cluster, thereby achieving the fastest possible stabilization consistent with the system's spectral limitations. This can be achieved regardless of the stabilizability domain being delay-independent or delay-dependent. This allows for a highly efficient control mechanism tailored to the specific dynamical behavior of the wave equation. To illustrate the practical relevance of our theoretical findings, we applied the proposed method to the control of transverse vibrations in a taut string. Numerical simulations confirmed the robustness and effectiveness of the feedback design, underscoring its potential for broader applications in the control of distributed parameter systems with delay effects.
- internal stabilization,
- internal delay,
- wave equations
Citation: Kaïs Ammari, Islam Boussaada, Silviu-Iulian Niculescu, Sami Tliba. Prescribing the best decay rate of the wave equation using internal delayed feedback damping[J]. Communications in Analysis and Mechanics, 2026, 18(1): 1-36. doi: 10.3934/cam.2026001

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Abstract

In this paper, we investigated the stabilization of the damped wave equation through the use of an internal feedback mechanism incorporating time delay. This work builds upon the partial pole placement paradigm, a recent theoretical framework originally developed for functional differential equations, which enables the selective assignment of eigenvalues within a prescribed region of the complex plane. Using this approach, we designed an internal delayed feedback law that guarantees the exponential stabilization of the resulting closed-loop system. A distinctive feature of our control strategy lies in its ability to prescribe the optimal exponential decay rate for each modal cluster, thereby achieving the fastest possible stabilization consistent with the system's spectral limitations. This can be achieved regardless of the stabilizability domain being delay-independent or delay-dependent. This allows for a highly efficient control mechanism tailored to the specific dynamical behavior of the wave equation. To illustrate the practical relevance of our theoretical findings, we applied the proposed method to the control of transverse vibrations in a taut string. Numerical simulations confirmed the robustness and effectiveness of the feedback design, underscoring its potential for broader applications in the control of distributed parameter systems with delay effects.

References

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