FEEDBACK-CONTROL FOR UNSTEADY-FLOW AND ITS APPLICATION TO THE STOCHASTIC BURGERS-EQUATION

Mathematical methods of control theory are applied to the problem of c ontrol of fluid flow with the long-range objective of developing effec tive methods for the control of turbulent flows. The procedure of how to cast the problem of controlling turbulence into a problem in optima l control theory is presented using model problems through the formali sm and language of control theory. Then we present a suboptimal contro l and feedback procedure for general stationary and time-dependent pro blems using methods of calculus of variations through the adjoint stat e and gradient algorithms. This suboptimal feedback control procedure is applied to the stochastic Burgers equation. Two types of controls a re investigated: distributed and boundary controls. The control inputs are the momentum forcing for the distributed control and the boundary velocity for the boundary control. Costs to be minimized are defined as the sum of the mean-square velocity gradient inside the domain for the distributed control or the square velocity gradient at the wall fo r the boundary control; and in both cases a term was added to account for the implementation cost. Several cases of both controls have been numerically simulated to investigate the performances of the control a lgorithm. Most cases considered show significant reductions of the cos ts. Another version of the feedback procedure more effective for pract ical implementation has been considered and implemented, and the appli cation of this algorithm also shows significant reductions of the cost s. Finally, dependence of the control algorithm on the time-discretiza tion method is discussed.