Numerical approximation of optimal flow control problems by a penalty method : Error estimates and numerical results

The purpose of this paper is to present numerically convenient approaches to solve optimal Dirichlet control problems governed by the steady Navier--Stokes equations. We will examine a penalized Neumann control approach for solving Dirichlet control problems from numerical and computational points of view. The control is affected by the suction or injection of fluid through the boundary or by boundary surface movements in the tangential direction. The control objective is to minimize the vorticity in the flow or to drive the velocity field to a desired one. We develop sequential quadratic programming methods to solve these optimal control problems. The effectiveness of the optimal control techniques in flow controls and the feasibility of the proposed penalized Neumann control approaches for flow control problems are demonstrated by numerical experiments for a viscous, incompressible fluid flow in a two-dimensional channel and in a cavity geometry.