Two Strings with a Dynamic Interior Mass: A Feedback Control Design with Guaranteed Exponential Decay

TL;DR

This paper presents a feedback control method for stabilizing a coupled two-string system with an interior mass, guaranteeing exponential decay without structural constraints, applicable to various hybrid systems, and validated through simulations.

Contribution

It introduces a novel feedback control design ensuring unconditional exponential stability for a coupled string system with an interior mass, improving upon previous methods.

Findings

Unconditional exponential stability achieved without wave speed constraints.

Lower-order damping can be omitted when combined with other feedbacks.

Numerical simulations confirm theoretical stability results.

Abstract

This paper investigates the exponential stabilization of a coupled two-string system joined by a dynamic interior mass. The combined effect of three feedback mechanisms, boundary damping from tip velocity, higher-order nodal damping from angular velocity, and lower-order nodal damping from mass velocity, is analyzed using a Lyapunov framework. Exponential stability is established unconditionally, without constraints on wave speeds or mass location, improving upon earlier results that lower-order nodal damping, as in {Hansen-Zuazua'95}, or boundary damping alone, as in {Lee-You'89}, does not ensure exponential decay without additional structural conditions. Moreover, the lower-order feedback can be removed without loss of exponential decay when combined with the other two mechanisms, via a compact perturbation argument. These results apply to hybrid systems with interior or tip mass interfaces, including overhead cranes, deep-sea cables, and fluid structure interaction. Theoretical findings are validated through numerical simulations.