A classical problem in ergodic control consists of studying the limit behaviour of the optimal value of a discounted cost functional with infinite horizon as the discount fact tends to zero. In this talk we present some recente results obtained in collaboration with P. Cannarsa, S. Gaubert, M. Quincampoix on the study of this problem without the classical controllability assumption. Our main result characterises the uniform limit (when it exists) as the maximal subsolution of a system of Hamilton-Jacobi equations. Finally we will discuss some positive and negative results on the existence of such a uniform limit.
