Pseudo-spectral optimal control of stochastic processes using Fokker Planck equation

Abstract

Motivated by the successful implementation of Pseudo-spectral (PS) methods in optimal control problems (OCP), a new technique is introduced to control the probability density function (PDF) of the state of the 1-D system described by a stochastic differential equation (SDE). In this paper, the Fokker Planck equation (FPE) is used to model the time evolution of the PDF of the stochastic process. Using FPE instead of SDE, changes the problem of stochastic optimal control to a deterministic one. FPE is a parabolic PDE. Solving an OCP with PDE constraint is computationally a difficult task. We use two strategies to efficiently solve this OCP problem: firstly, we use PS methods in order to transform the OCP to a non-linear programming (NLP) with fewer discretization points but higher order of accuracy, and secondly, we utilize Genetic algorithm (GA) to solve this large-scale NLP in a more efficient approach than gradient-based optimization methods. The simulation results based on Monte-Carlo simulations prove the performance of the proposed method.

Keywords

• pseudo-spectral optimal control
• fokker planck equation
• stochastic process
• genetic algorithm
• legendre pseudo-spectral method