Existence of optimal boundary control for the Navier-Stokes equations with mixed boundary conditions

Abstract

Variational approaches have been used successfully as a strategy to takeadvantage from real data measurements. In several applications, this approach allowsto increase the accuracy of numerical simulations. In the particular case of fluid dy-namics, it leads to optimal control problems with non-standard cost functionals which,when subject to the Navier-Stokes equations, require a non-standard theoretical frameto ensure the existence of solution. In this work, we prove the existence of solution fora class of such type of optimal control problems. Before doing that, we ensure the ex-istence and uniqueness of solution for the 3D stationary Navier-Stokes equations, withmixed-boundary conditions, a particular type of boundary conditions very common inapplications to biomedical problems