  We consider a state estimation problem for gas pipeline flow modeled by the
one-dimensional barotropic Euler equations. In order to reconstruct the system
state, we construct an observer system of Luenberger type based on distributed
measurements of one state variable. First, we show the existence of
Lipschitz-continuous semi-global solutions of the observer system and of the
original system for initial and boundary data satisfying smallness and
compatibility conditions for a single pipe and for general networks. Second,
based on an extension of the relative energy method we prove that the state of
the observer system converges exponentially in the long time limit towards the
original system state. We show this for a single pipe and for star-shaped
networks.
