Real Processes and Realizability of a Stabilization Method for the Navier—Stokes Equations by Boundary Feedback Control

2002 ◽  
pp. 137-177 ◽  
Author(s):  
Andrei V. Fursikov
Keyword(s):  
Feedback Control ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Stabilization Method ◽  
Navier Stokes Equations ◽  
Boundary Feedback Control ◽  
Boundary Feedback

scholarly journals Stabilization of the non-homogeneous Navier–Stokes equations in a 2d channel

2019 ◽  
Vol 25 ◽  
pp. 66
Author(s):  
Sourav Mitra
Keyword(s):  
Feedback Control ◽  
Poiseuille Flow ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Boundary Stabilization ◽  
Navier Stokes Equations ◽  
Local Boundary ◽  
Hyperbolic Dynamics ◽  
Finite Dimensional ◽  
Dimensional Range

In this article, we study the local boundary stabilization of the non-homogeneous Navier–Stokes equations in a 2d channel around Poiseuille flow which is a stationary solution for the system under consideration. The feedback control operator we construct has finite dimensional range. The homogeneous Navier–Stokes equations are of parabolic nature and the stabilization result for such system is well studied in the literature. In the present article we prove a stabilization result for non-homogeneous Navier–Stokes equations which involves coupled parabolic and hyperbolic dynamics by using only one boundary control for the parabolic part.

Stabilization of Solution to Navier-Stokes Equations by Feedback Control Defined on the Boundary

2002 ◽  
Author(s):  
Andrei V. Fursikov
Keyword(s):  
Fluid Flow ◽  
Feedback Control ◽  
Stokes Equations ◽  
Time Moment ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Velocity Vector Field ◽  
Navier Stokes Equations ◽  
Real Process ◽  
Mathematical Construction

Let vˆ be a velocity vector field of steady-state fluid flow in a bounded container. We do not suppose that vˆ is stable. For each fluid flow which is close to vˆ at time moment t = 0 we propose a mathematical construction of feedback control from the boundary of the container which stabilize to vˆ this flow, i.e. which forces this flow to tend to vˆ with prescribed exponential rate. We introduce a notion of “real process” which is an abstract analog of fluid flow or (in other version) of numerical solution of Navier-Stokes equations. Real process differs from exact solution of three-dimensional Navier-Stokes equaitons on some small fluctuatons. Alhtough construction of feedback control is based on precise solving of Navier-Stokes equations, feedback control obtained by this method can react on unpredictable fluctuations mentioned above damping them. Such construction can be useful for numerical calculation because there fluctuations appear always.

scholarly journals Surfing the edge: using feedback control to find nonlinear solutions

2017 ◽  
Vol 831 ◽  
pp. 579-591 ◽  
Author(s):  
A. P. Willis ◽  
Y. Duguet ◽  
O. Omel’chenko ◽  
M. Wolfrum
Keyword(s):  
Feedback Control ◽  
State Space ◽  
Stokes Equations ◽  
Travelling Waves ◽  
Edge State ◽  
Navier Stokes ◽  
Edge States ◽  
Navier Stokes Equations ◽  
Unstable Solutions ◽  
Feedback Control Strategy

Many transitional wall-bounded shear flows are characterised by the coexistence in state space of laminar and turbulent regimes. Probing the edge boundary between the two attractors has led in the last decade to the numerical discovery of new (unstable) solutions to the incompressible Navier–Stokes equations. However, the iterative bisection method used to compute edge states can become prohibitively costly for large systems. Here we suggest a simple feedback control strategy to stabilise edge states, hence accelerating their numerical identification by several orders of magnitude. The method is illustrated for several configurations of cylindrical pipe flow. Travelling waves solutions are identified as edge states, and can be isolated rapidly in only one short numerical run. A new branch of solutions is also identified. When the edge state is a periodic orbit or chaotic state, the feedback control does not converge precisely to solutions of the uncontrolled system, but nevertheless brings the dynamics very close to the original edge manifold in a single run. We discuss the opportunities offered by the speed and simplicity of this new method to probe the structure of both state space and parameter space.

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