Reduced Approach for Stochastic Optimal Control Problems

Abstract

In this paper, we develop and analyze the reduced approach for solving optimal control problems constrained by stochastic partial differential equations (SPDEs). Compared to the classical approaches based on Monte Carlo method to the solution of stochastic optimal control and optimization problems, e.g. Lagrange multiplier method, optimization methods based on sensitivity equations or adjoint equations, our strategy can take best advantage of all sorts of gradient descent algorithms used to deal with the unconstrained optimization problems but with less computational cost. Specifically, we represent the sample solutions for the constrained SPDEs or the state equations by their associated inverse-operators and plug them into the objective functional to explicitly eliminate the constrains, the constrained optimal control problems are then converted into the equivalent unconstrained ones, which implies the computational cost for solving the adjoint equations of the derived Lagrange system is avoided and faster convergent rate is expected. The stochastic Burgers’ equation with additive white noise is used to illustrate the performance of our reduced approach. It no doubt has great potential application in stochastic optimization problems.