Optimal Control of the Unsteady Navier-Stokes Equations

1 2 A general method based on adjoint formulation is discussed for the optimal control of distributed parameter systems (including boundary parameter) which is especially suitable for large dimensional control problems. Strategies for efficient and robust implementation of the method are described. A set of consistency checks are proposed for the validation of the adjoint code. The method is applied to the problem of controlling vor-tex shedding behind a cylinder (through suction/ blowing on the cylinder surface) governed by the unsteady two-dimensional incompressible Navier-Stokes equations with time-dependent boundary conditions. Three types of objective functions are considered, with regularization to circumvent ill-posedness. These objective functions involve integration over a space-time domain. A complete control of vortex shedding is demonstrated for Reynolds numbers up to 110. The optimal values of the suction/blowing parameters are found to be insensitive to initial conditions of the model when the time window of control is larger than the vortex shedding period, the inverse of the Strouhal frequency. Although this condition is necessary for robust control, it is observed that a shorter window of control may suffice to suppress vortex shedding.