This seminar is intended for non-specialists to whom I would first and succinctly introduce the general ideas behind the weak KAM theory in dynamical systems, the theory of homogenization in the analysis of PDEs and the theory of ergodic control. I will then and especially insist on the link between them, and which manifests itself in a particular PDE known as "ergodic stationary Hamilton-Jacobi equation". A qualitative study of these problems will also be discussed, and a link with an optimization problem in the space of measures will be mentioned. I will finally provide an example of Singular Perturbations in a control problem. 

For the sake of clarity, I will focus on the deterministic case, but much of these results are valid in a stochastic framework. A discussion will conclude the presentation.