Asymptotic analysis of an optimal control problem for a viscous incompressible fluid with Navier slip boundary conditions

TL;DR

This paper investigates the asymptotic behavior of an optimal control problem for the Navier-Stokes equations with Navier slip boundary conditions as the friction coefficient approaches infinity, showing convergence to a Dirichlet boundary condition problem.

Contribution

It provides a rigorous analysis of how optimal controls and states for Navier-Stokes with slip conditions converge to those with Dirichlet conditions as friction increases.

Findings

Optimal controls converge to Dirichlet boundary control as friction coefficient tends to infinity.

States and adjoint states also converge in the asymptotic limit.

The analysis bridges slip and no-slip boundary condition control problems.

Abstract

We consider an optimal control problem for the Navier-Stokes system with Navier slip boundary conditions. We denote by $\alpha$ the friction coefficient and we analyze the asymptotic behavior of such a problem as $\alpha\to \infty$. More precisely, we prove that if we take an optimal control for each $\alpha$, then there exists a sequence of optimal controls converging to an optimal control of the same optimal control problem for the Navier-Stokes system with the Dirichlet boundary condition. We also show the convergence of the corresponding direct and adjoint states.