Error estimates for an unregularized optimal control problem for the stationary Navier-Stokes equations

TL;DR

This paper analyzes an unregularized optimal control problem constrained by stationary Navier-Stokes equations, deriving existence, optimality conditions, and convergence of discretization with error estimates.

Contribution

It establishes existence, optimality conditions, and convergence analysis for a novel unregularized control problem with variational discretization.

Findings

Existence of optimal solutions proven.

First- and second-order optimality conditions derived.

A priori error estimates established for certain controls.

Abstract

We consider an unregularized optimal control problem subject to the steady-state Navier-Stokes equations. We derive the existence of optimal solutions and prove first- and -- necessary and sufficient -- second-order optimality conditions. To approximate solutions to the optimal control problem, we consider the variational discretization scheme. We analyze convergence properties of the discretization and prove a priori error estimates for locally optimal controls that are nonsingular and which satisfy a growth condition which implies a bang-bang structure.