ISS-Based Robustness to Various Neglected Damping Mechanisms for the 1-D Wave PDE

TL;DR

This paper investigates the robustness of the 1-D wave equation for an elastic string under various neglected damping mechanisms, using ISS theory and Lyapunov functionals to ensure stability against disturbances.

Contribution

It introduces a comprehensive ISS analysis for the wave equation considering multiple damping effects, including passive boundary damping, which was previously less studied.

Findings

ISS stability established for various damping mechanisms

Lyapunov functionals constructed for different norms

Robustness demonstrated against boundary and distributed disturbances

Abstract

This paper is devoted to the study of the robustness properties of the 1-D wave equation for an elastic vibrating string under four different damping mechanisms that are usually neglected in the study of the wave equation: (i) friction with the surrounding medium of the string (or viscous damping), (ii) thermoelastic phenomena (or thermal damping), (iii) internal friction of the string (or Kelvin-Voigt damping), and (iv) friction at the free end of the string (the so-called passive damper). The passive damper is also the simplest boundary feedback law that guarantees exponential stability for the string. We study robustness with respect to distributed inputs and boundary disturbances in the context of Input-to-State Stability (ISS). By constructing appropriate ISS Lyapunov functionals, we prove the ISS property expressed in various spatial norms.