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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(t^{-1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>t</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> decay first established by &#xa0;[Littman-Taylor’02] and found in the strong stabilization result of &#xa0;[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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(t^{-1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>t</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> rate, refining and completing earlier partial results of &#xa0;[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.</p>

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

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  • 发表日期：

    2026年2月27日

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  • Mohammad Akil,
  • Zoe Brown,
  • Ibtissam Issa,
  • Ahmet Özkan Özer

摘要

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.