Some limit theorems are derived for a class of controled Markov systems with small noises. The aim is to understand the effects of strategies of actions on the limiting behaviors of the systems, so that optimization for associated utilities can be considered. In deriving the limits, we apply a large deviation approach with a somewhat new technique of proof. An almost sure convergence theorem is then derived. To meet more realistic situations, the noises need not to be smooth or Lipschitz continuous even are allowed to be discontinuous in the states. Dependence of the limits on the strategies can be found, which involves a certain differential equation. Some illustrative examples are provided.

对于一类具有小噪声的受控马尔可夫系统，导出了一些极限定理。目的是了解行动策略对系统限制行为的影响，以便可以考虑对相关效用进行优化。在推导极限时，我们应用了一种大偏差方法和一种新的证明技术。然后推导出几乎肯定的收敛定理。为了满足更现实的情况，噪声不需要是平滑的或 Lipschitz 连续的，甚至可以在状态中不连续。可以找到限制对策略的依赖性，这涉及到一定的微分方程。提供了一些说明性示例。