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Annals of the International Society of Dynamic Games vol 12 : Advances in Dynamic Games, P.Cardaliaguet and R.Cressman (Ed.) (2013) 199-215
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A uniform Tauberian theorem in optimal control
Miquel Oliu-Barton1, Guillaume Vigeral2

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 $W_\lambda(x)$ 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 $W_\lambda(\cdot)$ 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 by Lehrer and Sorin in a discrete-time framework.
1 :  C&O - Equipe combinatoire et optimisation
2 :  CEREMADE - CEntre de REcherches en MAthématiques de la DEcision