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

TL;DR

This paper proves the existence of solutions for a class of optimal boundary control problems involving the 3D stationary Navier-Stokes equations with mixed boundary conditions, relevant for biomedical applications.

Contribution

It establishes the existence of solutions for complex optimal control problems with non-standard cost functionals constrained by Navier-Stokes equations with mixed boundary conditions.

Findings

Existence and uniqueness of solutions for 3D stationary Navier-Stokes with mixed boundary conditions.

Proof of existence for a class of optimal control problems with non-standard cost functionals.

Application relevance to biomedical fluid dynamics.

Abstract

Variational approaches have been used successfully as a strategy to take advantage from real data measurements. In several applications, this approach gives a means to increase the accuracy of numerical simulations. In the particular case of fluid dynamics, it leads to optimal control problems with non standard cost functionals which, when constraint to the Navier-Stokes equations, require a non-standard theoretical frame to ensure the existence of solution. In this work, we prove the existence of solution for a class of such type of optimal control problems. Before doing that, we ensure the existence and uniqueness of solution for the 3D stationary Navier-Stokes equations, with mixed-boundary conditions, a particular type of boundary conditions very common in applications to biomedical problems.