Stabilization and Best Actuator Location for the Navier--Stokes Equations

Abstract

We study the numerical approximation of the boundary stabilization of the Navier--Stokes equations with mixed Dirichlet/Neumann boundary conditions, around an unstable stationary solution in a two dimensional domain. We first derive a semidiscrete controlled system, coming from a finite element approximation of the Navier--Stokes equations, which is new in the literature. We propose a new strategy for finding a boundary feedback control law able to stabilize the nonlinear semidiscrete controlled system in the presence of boundary disturbances. We determine the best control location. Next, we study the degree of stabilizability of the different real generalized eigenspaces of the controlled system. Based on that analysis, we determine an invariant subspace $\mathbb{Z}_u$ and the projection of the controlled system onto $\mathbb{Z}_u$. The projected system is used to determine feedback control laws. Our numerical results show that this control strategy is quite efficient when applied to the Navier--Stokes system for a Reynolds number $R_e=150$ with boundary perturbations.