On an example for the Uniform Tauberian theorem in abstract control systems

TL;DR

This paper investigates the asymptotic behavior of value functions in abstract control systems, establishing conditions under which long-term average and discounted average value functions converge uniformly and share the same limit.

Contribution

The paper refines the conditions required for the Uniform Tauberian Theorem to hold in abstract control systems, specifically relaxing the closure condition of feasible processes.

Findings

Established equivalence of convergence for long-time and discounted averages under refined conditions.

Proved the limits of value functions are identical when conditions are satisfied.

Extended the applicability of the Uniform Tauberian Theorem in control theory.

Abstract

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.