Abstract We investigate the uniform stabilization of two elastic strings in series, coupled with a dynamic mass at an interior node, under three damping schemes: classical boundary damping, lower-order nodal (tip-velocity) feedback, and a novel higher-order nodal (strain-velocity) feedback. It is shown that when higher-order nodal damping is paired with boundary damping the full system is unconditionally exponentially stable; by contrast, boundary damping alone, or boundary plus lower-order nodal feedback, admits at best the sharp t^-1 t - 1 decay first established by Littman-Taylor’02 and found in the strong stabilization result of Hansen-Zuazua’95. Remarkably, even in the absence of any boundary dissipation, higher-order nodal feedback alone enforces exponential decay provided the wave-speed ratio satisfies an explicit arithmetic condition, whereas lower-order nodal feedback remains confined to the t^-1 t - 1 rate, refining and completing earlier partial results of Chen-Coleman-West’87 and Lee-You’89. These findings are illustrated by finite-difference simulations of solution profiles, eigenvalue spectra, and energy-decay curves across varying damping configurations, speed ratios, and mesh resolutions, which confirm the decisive role of the arithmetic condition in distinguishing exponential, polynomial, or no decay.
Akil et al. (Fri,) studied this question.