The study of a stochastic optimal control problem with feedback control leads us to a deterministic optimal control problem governed by a Kolmogorov equation with open-loop control. We analyse the relationship between the stochastic and the deterministic optimal control problems. We show the existence of an optimal control in a particular case, derive the optimality conditions for the deterministic problem and translate these results to the stochastic problem. Additional comments and further extensions are discussed. Some auxiliary results are also given.

中文翻译：

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带有反馈输入的随机最优控制问题

对带有反馈控制的随机最优控制问题的研究将我们引向了一个由具有开环控制的 Kolmogorov 方程控制的确定性最优控制问题。我们分析了随机和确定性最优控制问题之间的关系。我们展示了在特定情况下存在最优控制，推导出确定性问题的最优条件，并将这些结果转化为随机问题。讨论了其他评论和进一步的扩展。还给出了一些辅助结果。