Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stabilization of 2D Navier-Stokes equations by means of actuators with locally supported vorticity

Sérgio S. Rodrigues, Dagmawi A. Seifu

TL;DR

This work develops explicit, finite-dimensional feedback laws to globally stabilize the two-dimensional incompressible Navier–Stokes system to a prescribed time-dependent trajectory under Lions boundary conditions. By formulating the problem in vorticity and constructing actuators whose vorticity is locally supported, the authors derive an explicit oblique-projection feedback ${\mathbf K}_M^\lambda$ that yields exponential convergence at rate $\mu$ for sufficiently large actuator count $M$ and gain $\lambda$. A key contribution is the actuator construction with fixed total vorticity support volume independent of $M$, together with a rigorous analysis showing independence of the stabilization parameters $M_*$ and $\lambda_*$, and a detailed link between the velocity and vorticity formulations. The paper also connects stabilization with observer design through a Luenberger-like interpretation and validates the approach numerically on a triangular domain, demonstrating the method’s computational efficiency and potential for data assimilation and output-based control of incompressible flows.

Abstract

Exponential stabilization to time-dependent trajectories for the incompressible Navier-Stokes equations is achieved with explicit feedback controls. The fluid is contained in two-dimensional spatial domains and the control force is, at each time instant, a linear combination of a finite number of given actuators. Each actuator has its vorticity supported in a small subdomain. The velocity field is subject to Lions boundary conditions. Simulations are presented showing the stabilizing performance of the proposed feedback. The results also apply to a class of observer design problems.

Stabilization of 2D Navier-Stokes equations by means of actuators with locally supported vorticity

TL;DR

This work develops explicit, finite-dimensional feedback laws to globally stabilize the two-dimensional incompressible Navier–Stokes system to a prescribed time-dependent trajectory under Lions boundary conditions. By formulating the problem in vorticity and constructing actuators whose vorticity is locally supported, the authors derive an explicit oblique-projection feedback that yields exponential convergence at rate for sufficiently large actuator count and gain . A key contribution is the actuator construction with fixed total vorticity support volume independent of , together with a rigorous analysis showing independence of the stabilization parameters and , and a detailed link between the velocity and vorticity formulations. The paper also connects stabilization with observer design through a Luenberger-like interpretation and validates the approach numerically on a triangular domain, demonstrating the method’s computational efficiency and potential for data assimilation and output-based control of incompressible flows.

Abstract

Exponential stabilization to time-dependent trajectories for the incompressible Navier-Stokes equations is achieved with explicit feedback controls. The fluid is contained in two-dimensional spatial domains and the control force is, at each time instant, a linear combination of a finite number of given actuators. Each actuator has its vorticity supported in a small subdomain. The velocity field is subject to Lions boundary conditions. Simulations are presented showing the stabilizing performance of the proposed feedback. The results also apply to a class of observer design problems.
Paper Structure (23 sections, 13 theorems, 117 equations, 9 figures)

This paper contains 23 sections, 13 theorems, 117 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Stabilization to trajectories.
  3. The main stabilizability result
  4. On applications to observer design
  5. Further literature
  6. Contents and general notation
  7. Functional setting and families of actuators
  8. Vector and scalar function spaces
  9. The actuators
  10. The vorticity equation
  11. Feedback operator in terms of the vorticity
  12. Main result in vorticity formulation
  13. Proof of main result in vector field formulation
  14. Proof of the main result in vorticity formulation
  15. Dynamics of the different to the targeted trajectory
  16. ...and 8 more sections

Key Result

Lemma 2.1

The mapping $\mathop{\rm curl}\nolimits\colon{\mathbf V} \to H$ is an isometry and a bijection.

Figures (9)

  • Figure 1: Supports $\overline\omega_j^M$ of the actuators as in \ref{['suppAct']}. $M_\sigma=M^2$.
  • Figure 2: Supports of actuators for a triangular (sub)domain. $M_\sigma=4^{M-1}$.
  • Figure 3: Triangulations ${\mathcal{T}}_0$, $M\in\{1,2\}$, and supports of the actuators.
  • Figure 4: Increasing $M$. Data \ref{['data:unk']}.
  • Figure 5: Stream function of the target. Data \ref{['data:unk']}.
  • ...and 4 more figures

Theorems & Definitions (31)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 21 more