An Optimal Control Problem for Stochastic Linear PDE’s Driven by a Gaussian White Noise

A computationally efficient technique for the numerical solution of constrained optimal control problems governed by linear stochastic partial differential equations (SPDEs) is considered in this paper. Using the Wiener-Ito chaos expansion of the solution and the control, the stochastic problem is reformulated to a set of deterministic equations. To obtain these chaos coefficients, we use the usual Galerkin finite element method using standard techniques. Once this representation is computed, the statistics of the numerical solution can be easily evaluated. To illustrate our ideas we consider an optimal control problem of a linear elliptic equation with a quadratic cost functional and a distributed stochastic control which lies in the Hida distribution spaces.