  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.
