Dynamic stabilization of serially connected strings with a dynamical interior mass: unveiling the role of higher-order nodal damping

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 $$t^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> 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 $$t^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> 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.