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A classical problem in ergodic continuous time control consists of studyingthe limit behavior of the optimal value of a discounted cost functional withinfinite horizon as the discount factor $\lambda$ tends to zero. In theliterature, this problem has been addressed under various controllability orergodicity conditions ensuring that the rescaled value function convergesuniformly to a constant limit. In this case the limit can be characterized asthe unique constant such that a suitable Hamilton-Jacobi equation has at leastone continuous viscosity solution. In this paper, we study this problem withoutsuch conditions, so that the aforementioned limit needs not be constant. Ourmain result characterizes the uniform limit (when it exists) as the maximalsubsolution of a system of Hamilton-Jacobi equations. Moreover, when such asubsolution is a viscosity solution, we obtain the convergence of optimalvalues as well as a rate of convergence. This mirrors the analysis of thediscrete time case, where we characterize the uniform limit as the supremumover a set of sub-invariant half-lines of the dynamic programming operator. Theemerging structure in both discrete and continuous time models shows that thesupremum over sub-invariato half-lines with respect to the Lax-Oleiniksemigroup/dynamic programming operator, captures the behavior of the limit costas discount vanishes.

Analysis of the vanishing discount limit for optimal control problems in continuous and discrete time