Ergodic control of stochastic Navier-Stokes equation with Lévy noise

In this work we consider the controlled stochastic Navier-Stokes equations perturbed by Lévy noise in a two dimensional bounded domain and a semimartingale formulation is used to characterize the probability law defining the space-time statistical solution. The existence and uniqueness of invariant measures or stationary measures is examined under suitable assumptions on the noise coefficient. Then we establish the existence of an ergodic control which is optimal in the class of all stationary measures for the system for a suitable class of cost functions. Thus for this system, it is possible to choose a stationary control that corresponds to a statistically stationary turbulent state with certain prescribed cost functional attaining a minimum.