Damped wave systems on networks: Exponential stability and uniform approximations

TL;DR

This paper analyzes the exponential stability of damped wave systems on pipe networks, proving convergence to equilibrium and providing uniform approximation methods with numerical validation.

Contribution

It extends stability results from single pipes to networks and introduces a variational approach applicable to Galerkin approximations and finite element methods.

Findings

Proven exponential stability and convergence to equilibrium.

Established uniform exponential stability for semi-discretizations.

Numerical tests confirm decay rate bounds and theoretical predictions.

Abstract

We consider a damped linear hyperbolic system modelling the propagation of pressure waves in a network of pipes. Well-posedness is established via semi-group theory and the existence of a unique steady state is proven in the absence of driving forces. Under mild assumptions on the network topology and the model parameters, we show exponential stability and convergence to equilibrium. This generalizes related results for single pipes and multi-dimensional domains to the network context. Our proof of the exponential stability estimate is based on a variational formulation of the problem, some graph theoretic results, and appropriate energy estimates. The main arguments are rather generic and can be applied also for the analysis of Galerkin approximations. Uniform exponential stability can be guaranteed for the resulting semi-discretizations under mild compatibility conditions on the approximation spaces. A particular realization by mixed finite elements is discussed and the theoretical results are illustrated by numerical tests in which also bounds for the decay rate are investigated.