Stochastic perturbation method for optimal control problem governed by parabolic 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 parabolic PDEs with small uncertainties. First, we propose the finite dimensional noise assumption for the random coefficient, and expand the state and co-state variables up to a certain order with respect to a small parameter by using the perturbation technique. Then, we insert all the expansions into the known deterministic parametric optimality system, and obtain the first-order and second-order optimality systems. These two systems are discretized by the finite element method in space and the backward Euler method in time to establish the first-order and second-order approximate schemes. Further, a priori error estimates are derived for the state, co-state and control variables of the first-order and second-order approximation, respectively. Finally, some numerical experiments are presented to verify the theoretical results. • Use the perturbation technique to expand the state and co-state variables. • Obtain the deterministic first-order and second-order optimality systems. • Establish the first-order and second-order approximate schemes. • Derive a priori error estimates for the state, co-state and control variables. [ABSTRACT FROM AUTHOR]