Stability and Boundary Controllability of a Linearized Model of Flow in an Elastic Tube

¹ Department of Mathematics and Computer Science, University of the Philippines Baguio, Governor Pack Road, 2600 Baguio City, Philippines.
grperalta@upb.edu.ph
² Institut für Mathematik und Wissenschaftliches Rechnen, NAWI Graz, Karl-Franzens-Universität Graz, Heinrichstraße 36, 8010 Graz, Austria.
georg.propst@uni-graz.at

Received: 19 March 2013

Abstract

We consider a model describing the flow of a fluid inside an elastic tube that is connected to two tanks. We study the linearized system through semigroup theory. Controlling the pressures in the tanks renders a hyperbolic PDE with boundary control. The linearization induces a one-dimensional linear manifold of equilibria; when those are factored out, the corresponding semigroup is exponentially stable. The location of the eigenvalues in dependence on the viscosity is discussed. Exact boundary controllability of the system is achieved by the Riesz basis approach including generalized eigenvectors. A minimal time for controllability is given. The corresponding result for internal distributed control is stated.