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Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with distributed and Neumann controls. (English) Zbl 0747.76063

Optimal control problems for the stationary incompressible Navier-Stokes equations are considered. Two controls on the right-hand side of the Navier-Stokes equations and the Neumann type boundary conditions, respectively, have to be chosen such a way that an optimality criterion depending on the velocity field for the solution and the controls themselves takes a minimum. Two cases of criterions are proposed, either a sum of 4th power of \(L^ 4\)-distance of the velocity field to a desired flow and the squares of \(L^ 2\)-norms of the controls, or the viscous drag have to be minimized. In Section 2 the existence of (local) minimizers of the first mentioned criterion on a suitable admissibility set is shown. The reason for the choice of the \(L^ 4\)-term in the criterion is explained. In Section 3 the constrained minimization problem is turned into an unconstrained one by use of Lagrange multipliers. Hence, the optimal velocity and pressure fields together with the Lagrange multipliers are determined by an optimality system of partial differential equations. The optimal controls can easily be derived from the Lagrange multipliers. A regularity result is given. In Section 4 finite element discretizations for the optimality system are considered. By fitting the problem into the framework of a suitable class of nonlinear problems error estimates are given. In Section 5 four variations of the problem are considered: the drag functional as criterion, lack of boundary control, distributed control only on part of the domain and Neumann control only on part of the boundary. Questions about efficient solution methods and implementation are excluded from the presentation.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
49J20 Existence theories for optimal control problems involving partial differential equations
49K20 Optimality conditions for problems involving partial differential equations
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