Asymptotic behaviour of a system modelling rigid structures floating in a viscous fluid

Abstract

The PDE system introduced in Maity et al. (2019) describes the interaction of surface water waves with a floating solid, and takes into account the viscosity µ of the fluid. In this work, we study the Cummins type integro-differential equation for unbounded domains, that arises when the system is linearized around equilibrium conditions. A proof of the input-output stability of the system is given, thanks to a diffusive representation of the generalized fractional operator $\sqrt{1 + \mu s}$. Moreover, relying on Matignon (1996) stability result for fractional systems, explicit solutions are established both in the frequency and the time domains, leading to an explicit knowledge of the decay rate of the solution. Finally, numerical evidence is provided of the transition between different decay rates as a function of the viscosity $\mu$.