  The paper is devoted to the asymptotic behavior of value functions of
abstract control problem with the long-time and discounted averages. The
Uniform Tauberian Theorem for these problems states that the uniform
convergence of value functions for long-time~averages (as the horizon tends to
infinity) is equivalent to the uniform convergence of value functions for
discounted averages (as the discount tends to zero), and that the limits are
identical. According to Miquel Oliu-Barton and Guillaume Vigeral, this
assertion holds if the set of all feasible processes is closed with respect to
concatenation. In this paper, we refine this condition.
