The flow of gas through a pipeline network
can be modelled by a coupled system of 1-d quasilinear hyperbolic equations.
Often for the solution of
control problems it is convenient to replace the quasilinear model
by a simpler semilinear model.
We analyze the behavior of such a semilinear model on a star-shaped network.
The model is derived from the diagonal form of the quasilinear model by replacing the eigenvalues by the sound speed multiplied by
1 or -1 respectively, thus neglecting the influence of the gas velocity
which is justified in the applications since it is much smaller than the sound speed.
For a star-shaped network of pipes
we present boundary feedback laws
that stabilize the system state
exponentially fast
to a position of rest
for sufficiently small initial data.
We show the exponential decay of
the $L^2$-norm
for arbitrarily long pipes.
This is remarkable since in general
even for linear systems, for certain source terms
the system can become exponentially unstable
if the space interval is too long.
Our proofs are based upon
an observability inequality and suitably chosen Lyapunov functions.
Numerical examples including a comparison of
the semilinear and the quasilinear model are presented.
Boundary feedback stabilization of a semilinear model for the flow in star-shaped gas networks