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 continuous limits in the number of iteration and particles have been performed in order to study properties and reduce the complexity of the method. These limits lead to a high–dimensional conservation law which provides a statistical description of the solution of an inverse problem. Linear stability analysis reveals a possible instability of the solution provided by the continuous limits of the EKI. In this talk, we discuss how to address this issue by introducing a stabilization of the dynamics at the continuous level which leads to a method with globally asymptotically stable solutions. We illustrate theoretical results and the performance of the stabilized version by using test inverse problems from the literature.