A Priori Error Estimate of Stochastic Galerkin Method for Optimal Control Problem Governed by Stochastic Elliptic PDE with Constrained Control

In this paper, we investigate a stochastic Galerkin approximation scheme for an optimal control problem governed by an elliptic PDE with random field in its coefficients. The objective is to minimize the expectation of a cost functional with the deterministic constrained control. We represent the random elliptic PDE in term of the generalized polynomial chaos expansion and obtain the deterministic optimal problem. By applying the well-known Lions' Lemma to the reduced optimal problem, we obtain the necessary and sufficient optimality conditions. We establish a scheme to approximate the optimality system through the discretization with respect to both the spatial space and the probability space by Stochastic Galerkin method. Then a priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results.