The characterization and numerical solution of two non-smooth optimal control problems governed by a Fokker–Planck (FP) equation are investigated in the framework of the Pontryagin maximum principle (PMP). The two FP control problems are related to the problem of determining open- and closed-loop controls for a stochastic process whose probability density function is modelled by the FP equation. In both cases, existence and PMP characterisation of optimal controls are proved, and PMP-based numerical optimization schemes are implemented that solve the PMP optimality conditions to determine the controls sought. Results of experiments are presented that successfully validate the proposed computational framework and allow to compare the two control strategies.

中文翻译：

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解决福克-普朗克最优控制问题的庞特里亚金极大原理

在庞特里亚金极大值原理（PMP）的框架下，研究了两个由Fokker-Planck（FP）方程控制的非光滑最优控制问题的特征和数值解。这两个FP控制问题与确定随机过程的开环和闭环控制的问题有关，该随机过程的概率密度函数由FP方程建模。在这两种情况下，都证明了最优控制的存在性和PMP的特性，并实施了基于PMP的数值优化方案，以解决PMP最优性条件以确定所寻求的控制。提出的实验结果成功验证了所提出的计算框架并允许比较这两种控制策略。