Stochastic L1-optimal control via forward and backward sampling

The aim of this work is to present a sampling-based algorithm designed to solve a certain class of stochastic optimal control problems, utilizing forward and backward stochastic differential equations (FBSDEs). Specifically, we address the class of problems in which the running cost of the performance index involves an L1-type minimization problem in terms of the control effort. Such problems are typically called minimum fuel problems in optimal control literature. By means of a nonlinear version of the Feynman-Kac lemma, we obtain a probabilistic representation of the solution to the nonlinear Hamilton-Jacobi-Bellman equation, expressed in the form of a system of decoupled FBSDEs. This system of FBSDEs can be solved by employing linear regression techniques.