  We consider the question of asymptotic stability of quantum trajectories
undergoing quantum non-demolition imperfect measurement, that is to say the
convergence of the estimated trajectory towards the true trajectory whose
parameters and initial state are not necessarily known. We give conditions on
the estimated initial state and regions of validity for the estimated
parameters so that this convergence is ensured. We illustrate these results
through numerical simulations on the physical example [1] and discuss the
asymptotic stability for a more realistic general case where decoherence acts
on the system. In this case, the evolution is described by new Kraus operators
which do not satisfy the quantum non-demolition property.
