Feedback stabilization of quasilinear hyperbolic systems with varying delays

We consider the feedback stabilization of quasilinear hyperbolic systems on star-shaped networks. We present boundary feedback controls with varying delays. The delays are given by C
¹
-functions with bounded derivatives. We obtain the existence of unique C
¹
-solutions on a given finite time interval. In order to measure the system evolution, we introduce an L
²
-Lyapunov function with delay terms. The feedback controls yield the exponential decay of the Lyapunov function with time. This implies the exponential stability of the system. Our results can be applied on the stabilization of the isothermal Euler equations with friction that model the gas flow in pipe networks.