Today at the Institute seminar professor of M.V. Lomonosov Moscow state University Andrey V.Fursikov gave a talk on “One optimal control problem for Navier-Stokes system”.
In the talk one optimal boundary control problem for the three-dimensional, evolutionary Navier-Stokes equations in an exterior of bounded domain Ω is considered. The control objective is to minimize the drag functional, and control is effected through the Dirichlet boundary condition on dΩ.
First of all the proper space of vector-fields on boundary dΩ where we look for the control is found. Adding the norm of this space as a term of the cost functional well-posedness of considered optimal control problem is obtained. The existence of an optimal solution is proved. A strong form of an optimality system of equations is derived.
These results are based on a specially created theory of boundary value problems for Navier-Stokes equations with non zero boundary condition belonging to a non standard space of traces . Derivation of optimality system is based also on regularity results for the adjoint Oseen equations with regular initial data which do not satisfy the compatibility conditions.

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