An optimal boundary control problem related to the time dependent Navier-Stokes equations

TL;DR

This paper investigates a boundary control problem for the 2D Navier-Stokes equations with mixed boundary conditions, deriving optimality conditions and proposing solution algorithms with simulations.

Contribution

It provides a comprehensive theoretical framework for boundary control with mixed conditions and analyzes the Dirichlet control via the reduced functional approach.

Findings

Derivation of first-order optimality conditions

Development of descent algorithms for control

Numerical simulations demonstrating effectiveness

Abstract

In this work, we study a boundary control problem for the evolutionary Navier-Stokes equations, under mixed boundary conditions, in two dimensions. The cost functional here considered is of quadratic type, depending on both state and control variables. We provide a comprehensive theoretical framework to address the analysis and the derivation of a system of first-order optimality conditions that characterizes the solution of the control problem. We take advantage of an adequate treatment of the Dirichlet control through the study of the reduced functional. Despite the fact that this approach is quite common, a detailed analysis for the case of mixed boundary conditions with is still lacking. Finally, solution-finding algorithms of descent type are proposed and illustrated with several simulations.