# Injection-suction control for Navier-Stokes equations with slippage

## TL;DR

This paper addresses boundary control of 2D Navier-Stokes flows with slip conditions, establishing well-posedness, optimality conditions, and existence of solutions for a velocity tracking problem.

## Contribution

It introduces a novel boundary injection-suction control method for Navier-Stokes equations with slip boundary conditions, including analysis of optimal control solutions.

## Key findings

- Proved well-posedness of the controlled Navier-Stokes system.
- Established existence of an optimal control solution.
- Derived first order optimality conditions for the control problem.

## Abstract

We consider a velocity tracking problem for the Navier-Stokes equations in a 2D-bounded domain. The control acts on the boundary through a injection-suction device and the flow is allowed to slip against the surface wall. We study the well-posedness of the state equations, linearized state equations and adjoint equations. In addition, we show the existence of an optimal solution and establish the first order optimality condition.