  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.
