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

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.