Stochastic Galerkin Method for Constrained Optimal Control Problem Governed by an Elliptic Integro-differential Pde with Stochastic Coefficients

In this paper, a stochastic finite element approximation scheme is developed for an optimal control problem governed by an elliptic integro-differential equation with stochastic coefficients. Different from the well-studied optimal control problems governed by stochastic PDEs, our control problem has the control constraints of obstacle type, which is mostly seen in real applications. We develop the weak formulation for this control and its stochastic finite element approximation scheme. We then obtain necessary and sufficient optimality conditions for the optimal control and the state, which are the base for deriving a priori error estimates of the approximation in our work. Instead of using the infinite dimensional Lagrange multiplier theory, which is currently used in the literature but often difficult to handle inequality control constraints, we use a direct approach by applying the well-known Lions’ Lemma to the reduced optimal problem. This approach is shown to be applicable for a wide range of control constraints. Finally numerical examples are presented to illustrate our theoretical results.