Abstract : We investigate the long-time behavior of solutions of quasilinear hyperbolic systems with transparent boundary conditions when small source terms are
incorporated in the system. Even if the finite-time stability of the system is not preserved, it is shown here that an exponential convergence towards
the steady state still holds with a decay rate which is proportional to the logarithm
of the amplitude of the source term. The result is stated for a system with dynamical boundary conditions in order to deal with initial data that are free of any compatibility condition.
