Boundary-to-Displacement Asymptotic Gains for Wave Systems With Kelvin-Voigt Damping

TL;DR

This paper analyzes the asymptotic displacement gains in vibrating strings with Kelvin-Voigt damping, providing bounds and novel frequency analysis methods for wave systems with boundary disturbances.

Contribution

It introduces a new frequency analysis approach to derive lower bounds for asymptotic gains in wave systems with Kelvin-Voigt damping.

Findings

Asymptotic gain in L2 norm holds without damping coefficient assumptions.

Upper bounds are derived via eigenfunction expansion or small-gain arguments.

Lower bounds are obtained using a novel frequency analysis methodology.

Abstract

We provide estimates for the asymptotic gains of the displacement of a vibrating string with endpoint forcing, modeled by the wave equation with Kelvin-Voigt and viscous damping and a boundary disturbance. Two asymptotic gains are studied: the gain in the L2 spatial norm and the gain in the spatial sup norm. It is shown that the asymptotic gain property holds in the L2 norm of the displacement without any assumption for the damping coefficients. The derivation of the upper bounds for the asymptotic gains is performed by either employing an eigenfunction expansion methodology or by means of a small-gain argument, whereas a novel frequency analysis methodology is employed for the derivation of the lower bounds for the asymptotic gains. The graphical illustration of the upper and lower bounds for the gains shows that that the asymptotic gain in the L2 norm is estimated much more accurately than the asymptotic gain in the sup norm.