A priori error estimate of perturbation method for optimal control problem governed by elliptic PDEs with small uncertainties.

In this paper, we investigate the first-order and second-order perturbation approximation schemes for an optimal control problem governed by elliptic PDEs with small uncertainties. The optimal control minimizes the expectation of a cost functional with a deterministic constrained control. First, using a perturbation method, we expand the state and co-state variables up to a certain order with respect to a parameter that controls the magnitude of uncertainty in the input. Then we take the expansions into the known deterministic parametric optimality system to derive the first-order and second-order optimality systems which are both deterministic problems. After that, the two systems are discretized by finite element method directly. The strong and weak error estimates are derived for the state, co-state and control variables, respectively. We finally illustrate the theoretical results by two numerical examples. [ABSTRACT FROM AUTHOR]