Feedback stabilization of a linear fluid–membrane system with time delay

Abstract

A coupled parabolic–hyperbolic system of partial differential equations modelling the interaction of a fluid and a membrane is considered. The model is reformulated as an abstract Cauchy problem and thereby constructing a semigroup for the evolution. This is done by eliminating the pressure. The system is stabilized through a feedback force applied to the membrane incorporating a time delay. The spectral properties and stability are considered under suitable conditions on the fluid viscosity, damping coefficient and delay coefficient.