  We consider the p-system in Eulerian coordinates on a star-shaped network.
Under suitable transmission conditions at the junction and dissipative boundary
conditions in the exterior vertices, we show that the entropy solutions of the
system are exponentially stabilizable. Our proof extends the strategy by Coron
et al. (2017) and is based on a front-tracking algorithm used to construct
approximate piecewise constant solutions whose BV norms are controlled through
a suitable exponentially-weighted Glimm-type Lyapunov functional.
