  A classical problem in ergodic continuous time control consists of studying
the limit behavior of the optimal value of a discounted cost functional with
infinite horizon as the discount factor $\lambda$ tends to zero. In the
literature, this problem has been addressed under various controllability or
ergodicity conditions ensuring that the rescaled value function converges
uniformly to a constant limit. In this case the limit can be characterized as
the unique constant such that a suitable Hamilton-Jacobi equation has at least
one continuous viscosity solution. In this paper, we study this problem without
such conditions, so that the aforementioned limit needs not be constant. Our
main result characterizes the uniform limit (when it exists) as the maximal
subsolution of a system of Hamilton-Jacobi equations. Moreover, when such a
subsolution is a viscosity solution, we obtain the convergence of optimal
values as well as a rate of convergence. This mirrors the analysis of the
discrete time case, where we characterize the uniform limit as the supremum
over a set of sub-invariant half-lines of the dynamic programming operator. The
emerging structure in both discrete and continuous time models shows that the
supremum over sub-invariato half-lines with respect to the Lax-Oleinik
semigroup/dynamic programming operator, captures the behavior of the limit cost
as discount vanishes.
