  The Ensemble Kalman Filter (EnKF) belongs to the class of iterative particle
filtering methods and can be used for solving control--to--observable inverse
problems. In this context, the EnKF is known as Ensemble Kalman Inversion
(EKI). In recent years several continuous limits in the number of iteration and
particles have been performed in order to study properties of the method. In
particular, a one--dimensional linear stability analysis reveals possible
drawbacks in the phase space of moments provided by the continuous limits of
the EKI, but observed also in the multi--dimensional setting. In this work we
address this issue by introducing a stabilization of the dynamics which leads
to a method with globally asymptotically stable solutions. We illustrate the
performance of the stabilized version by using test inverse problems from the
literature and comparing it with the classical continuous limit formulation of
the method.
