A Uniform Tauberian Theorem in Optimal Control

Abstract

In an optimal control framework, we consider the value V T (x) of the problem starting from state x with finite horizon T , as well as the value V λ (x) of the λ-discounted problem starting from x. We prove that uniform convergence (on the set of states) of the values V T (·) as T tends to infinity is equivalent to uniform convergence of the values V λ (·) as λ tends to 0, and that the limits are identical. An example is also provided to show that the result does not hold for pointwise convergence. This work is an extension, using similar techniques, of a related result in a discrete-time framework [15].

On representation formulas for long run averaging optimal control problem

“…also [19,18]). Our main aim is to provide a representation formula of the cluster points in the uniform convergence topology of v T , as T → +∞, and of u λ , as λ → 0 + .…”

“…The existence of the limit of v T , as T → +∞, and of u λ , as λ → 0 + , is a crucial problem studied in a huge literature: We refer the reader to [1][2][3]10,[17][18][19] and the references therein. The works in ergodic control deal with sufficient conditions proving the existence of a limit -in the uniform convergence topology -which is constant, i.e., which does not depend on the initial condition x 0 .…”

Uniform Assymptotics in the Average Continuous Control of Piecewise Deterministic Markov Processes : Vanishing Approach

“…We partially extend the results of [19] to continuous control of piecewise deterministic Markov process. In our main result (Theorem 4.1), we prove that, whenever lim δ→0 v δ exists uniformly on the state space, the limit value lim t→∞ V t also exists (which gives a sup/inf long time average value function).…”

“…A further generalization (cf. [19]) allows the limit value function with respect to a system governed by controlled deterministic dynamics to depend on the initial data. In the Brownian diffusion setting, similar results have been obtained in [4].…”

“…Following the idea of [19], we propose a sufficient criterion for the existence of a (uniform) limit value function for continuous control problems with long run average costs using a uniform vanishing technique.…”