Boundary-to-Displacement Asymptotic Gains for Wave Systems With Kelvin-Voigt Damping
This work offers theoretical bounds for boundary-to-displacement gains in damped wave systems, relevant for control and vibration analysis, but the results are incremental and domain-specific.
The paper provides estimates for the asymptotic gains (L2 and sup norms) of displacement for a wave equation with Kelvin-Voigt damping and boundary disturbance, showing that the asymptotic gain property holds in L2 norm without assumptions on damping coefficients. The L2 norm gain is estimated more accurately than the sup norm gain.
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.