On the Numerical Integration of the Fokker–Planck Equation Driven by a Mechanical Force and the Bismut–Elworthy–Li Formula

Optimal control theory aims to find an optimal protocol to steer a system between assigned boundary conditions while minimizing a given cost functional in finite time. Equations arising from these types of problems are often non-linear and difficult to solve numerically. In this article, we describe numerical methods of integration for two partial differential equations that commonly arise in optimal control theory: the Fokker-Planck equation driven by a mechanical potential for which we use the Girsanov theorem; and the Hamilton-Jacobi-Bellman, or dynamic programming, equation for which we find the gradient of its solution using the Bismut-Elworthy-Li formula. The computation of the gradient is necessary to specify the optimal protocol. Finally, we give an example application of the numerical techniques to solving an optimal control problem without spacial discretization using machine learning.