On the stability of solutions of generalised pipe equation under nonhomogeneous dirichlet boundary control

Abstract

We consider the initial boundary value problem for a linear fourth order wave equation which is a generalization of a mathematical model describing dynamic of pipes conveying fluids, under non-homogeneous boundary conditions. We show that under some restriction on the parameters of the equation all solutions of the problem tend to zero as t -> infinity. We also show that the zero solution of corresponding homogeneous equation under homogeneous Dirichlet boundary conditions is globally asymptotically stable, and all solutions tend to zero with an exponential rate as t -> infinity.