最优控制中的一个一致Tauberian定理
A uniform Tauberian theorem in optimal control
摘要：
在一个最优控制框架中，我们考虑了从状态$x$开始的具有有限时域$T$的问题的值$V_T(x)$，以及从$x$开始的$λ$ -折扣问题的值$V_λ(x)$.我们证明了当$T$趋于无穷大时，值$V_T(\cdot)$的一致收敛性（在状态集上）等价于当$λ$趋于0时，值$V_λ(\cdot)$的一致收敛性，并且极限是相同的。最后给出了一个例子，证明了该结果不适用于逐点收敛。这项工作是一个扩展，使用类似的技术，在离散时间框架\cite{LehSys}的相关结果。
英文摘要：
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 $V_T(\cdot)$ as $T$ tends to infinity is equivalent to uniform convergence of the values $V_λ(\cdot)$ 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 \cite{LehSys}.
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