  Isolated long-range interacting particle systems appear generically to relax
to non-equilibrium states ("quasi-stationary states" or QSS) which are
stationary in the thermodynamic limit. A fundamental open question concerns the
"robustness" of these states when the system is not isolated. In this paper we
explore, using both analytical and numerical approaches to a paradigmatic one
dimensional model, the effect of a simple class of perturbations. We call them
"internal local perturbations" in that the particle energies are perturbed at
collisions in a way which depends only on the local properties. Our central
finding is that the effect of the perturbations is to drive all the very
different QSS we consider towards a unique QSS. The latter is thus independent
of the initial conditions of the system, but determined instead by both the
long-range forces and the details of the perturbations applied. Thus in the
presence of such a perturbation the long-range system evolves to a unique
non-equilibrium stationary state, completely different to its state in absence
of the perturbation, and it remains in this state when the perturbation is
removed. We argue that this result may be generic for long-range interacting
systems subject to perturbations which are dependent on the local properties
(e.g. spatial density or velocity distribution) of the system itself.
