Stability analysis of the Navier-Stokes velocity tracking problem with bang-bang controls

TL;DR

This paper analyzes the stability of velocity-tracking solutions for the 2D Navier-Stokes equations with bang-bang controls, focusing on how solutions respond to data perturbations and regularization effects.

Contribution

It provides a stability analysis based on the H"older subregularity of the optimality mapping for a velocity-tracking control problem without regularizer.

Findings

Convergence rates of solutions under Tikhonov regularization.

Stability results for bang-bang control solutions.

Insights into the effect of data perturbations on solution stability.

Abstract

This paper focuses on the stability of solutions for a velocity-tracking problem associated with the two-dimensional Navier-Stokes equations. The considered optimal control problem does not possess any regularizer in the cost, and hence bang-bang solutions can be expected. We investigate perturbations that account for uncertainty in the tracking data and the initial condition of the state, and analyze the convergence rate of solutions when the original problem is regularized by the Tikhonov term. The stability analysis relies on the H\"older subregularity of the optimality mapping, which stems from the necessary conditions of the problem.