A uniform Tauberian theorem in optimal control

TL;DR

This paper establishes a uniform convergence equivalence between finite horizon and discounted value functions in optimal control, extending previous discrete-time results and highlighting the importance of uniform convergence.

Contribution

It proves a uniform Tauberian theorem linking finite horizon and discounted optimal control values, extending known discrete-time results to continuous-time frameworks.

Findings

Uniform convergence of $V_T$ implies uniform convergence of $V_$ as $ o 0$

Limits of $V_T$ and $V_$ coincide under uniform convergence

Pointwise convergence does not guarantee the same result

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 $V_\lambda(x)$ of the $\lambda$-discounted problem starting from $x$. We prove that uniform convergence (on the set of states) of the values $V_T(\cdot)$ as $T$ tends to infinity is equivalent to uniform convergence of the values $V_\lambda(\cdot)$ as $\lambda$ 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 \cite{LehSys}.