A stochastic optimal control problem for incompressible Newtonian channel flow past a circular cylinder is used as a prototype optimal control problem for the stochastic Navier-Stokes equations. The inlet flow and the rotation speed of the cylinder are allowed to have stochastic perturbations. The control acts on the cylinder via adjustment of the rotation speed. Possible objectives of the control include, among others, tracking a desired (given) velocity field or minimizing the kinetic energy, enstrophy, or the drag of the flow over a given body. Owing to the high computational requirements, the direct application of the classical Monte Carlo methods for our problem is limited. To overcome the difficulty, we use a multi-fidelity data-driven compressive sensing based polynomial chaos expansions (MDCS-PCE). An effective gradient-based optimization for the discrete optimality systems resulted from the MDCS-PCE discretization is developed. The strategy can be applied broadly to many stochastic flow control problems. Numerical tests are performed to validate our methodology.
