Optimal Control Problems for the Navier-Stokes Equations
Optimal boundary control problems and related inhomogeneous boundary value problems for the Navier-Stokes equations are considered. The control is the data in the Dirichlet boundary condition. The objective functional is the drag on a body immersed in the fluid. The size of the control is limited through the application of explicit bounds or through penalization of the drag functional. A necessary step in the analysis of both the control problems and the related boundary value problems is the characterization of traces of solenoidal vector fields. Such characterization results are given in two and three dimensions as are existence results about solutions of the boundary value problems. Results about the existence of solutions of the optimal control problem are given in the two-dimensional case, as are results concerning the numerical approximation of optimal solutions.
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