Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Alberto Domínguez Corella, Nicolai Jork, Šarká Nečasová, John Sebastian H. Simon

TL;DR

This work addresses stability of bang-bang velocity-tracking controls for the 2D Navier–Stokes equations in the absence of explicit regularization, examining how solutions respond to perturbations in tracking data, initial conditions, and regularization terms. It combines a rigorous functional-analytic setup with a thorough treatment of the nonlinear NS dynamics, its linearization (Oseen equations) and adjoint systems, to derive first- and second-order optimality conditions and to establish stability via strong Hölder subregularity of the optimality mapping. A central contribution is a Hölder-type stability estimate with exponent $1/μ$ for perturbations, under growth assumptions that guarantee bang-bang structure; this is complemented by an abstract appendix that makes the results broadly applicable to PDE-constrained optimization. The findings provide theoretical guarantees for the robustness of bang-bang velocity tracking under data and model perturbations and extend the toolkit for analyzing stability in non-smooth optimal control problems governed by fluid dynamics.

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 Hölder subregularity of the optimality mapping, which stems from the necessary conditions of the problem.

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

TL;DR

This work addresses stability of bang-bang velocity-tracking controls for the 2D Navier–Stokes equations in the absence of explicit regularization, examining how solutions respond to perturbations in tracking data, initial conditions, and regularization terms. It combines a rigorous functional-analytic setup with a thorough treatment of the nonlinear NS dynamics, its linearization (Oseen equations) and adjoint systems, to derive first- and second-order optimality conditions and to establish stability via strong Hölder subregularity of the optimality mapping. A central contribution is a Hölder-type stability estimate with exponent for perturbations, under growth assumptions that guarantee bang-bang structure; this is complemented by an abstract appendix that makes the results broadly applicable to PDE-constrained optimization. The findings provide theoretical guarantees for the robustness of bang-bang velocity tracking under data and model perturbations and extend the toolkit for analyzing stability in non-smooth optimal control problems governed by fluid dynamics.

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 Hölder subregularity of the optimality mapping, which stems from the necessary conditions of the problem.
Paper Structure (10 sections, 30 theorems, 122 equations)

This paper contains 10 sections, 30 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Analysis of the governing equations
  4. The velocity tracking problem
  5. Stability under perturbations
  6. Conclusion
  7. Appendix
  8. A first-order variant of the Ekeland principle
  9. Strong Hölder subregularity of the optimality mapping
  10. Hölder growth of real-valued functions

Key Result

Theorem 3.1

Suppose that $u\in L^2(I;V^*)$ and $y_0\in H$, then there exists a unique $y \in W(I)$ such that weakvar:NS holds. Furthermore, the energy estimate holds for some constant $c>0$ independent of $u$ and $y_0$.

Theorems & Definitions (51)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4: hinze2002
  • Theorem 3.5: dwachsmuth2006
  • Theorem 3.6: hinze2002
  • Theorem 3.7: dwachsmuth2006
  • Lemma 3.8
  • proof
  • ...and 41 more