  This paper formalises the concepts of weakly and weakly regularly persistent
input trajectory as well as their link to the Observability Grammian and the
existence and uniqueness of solutions of Moving Horizon Estimation (MHE)
problems. Additionally, thanks to a new time-uniform Implicit Function Theorem,
these notions are proved to imply the stability of MHE solutions with respect
to small additive perturbation in the measurements and in the dynamics, both
uniformly and non-uniformly in time. Finally, examples and counter-examples of
weakly persistent and weakly regularly persistent input trajectories are given
in the case of 2D bearing-only navigation.
