  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.
