Mehdi Badra

Lyapunov Function and Local Feedback Boundary Stabilization of the Navier-Stokes Equations

We study the local exponential stabilization, near a given steady-state flow, of solutions of the Navier-Stokes equations in a bounded domain. The control is performed through a Dirichlet boundary condition. We apply a linear feedback controller, provided by a well-posed infinite-dimensional Riccati equation. We give a characterization of the domain of the closed-loop operator which is obtained from the closed-loop linearized Navier-Stokes system. We give a class of initial data for which a Lyapunov function is obtained. For all

DETECTING AN OBSTACLE IMMERSED IN A FLUID BY SHAPE OPTIMIZATION METHODS

s\in[0,1/2[

Stabilization of Parabolic Nonlinear Systems with Finite Dimensional Feedback or Dynamical Controllers: Application to the Navier-Stokes System

, the stabilization of the two-dimensional Navier-Stokes equations is proved for initial data in

Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control

\mathbf{H}^s(\Omega)\cap V_n^0(\Omega)

Local controllability to trajectories for non-homogeneous incompressible Navier–Stokes equations

, where

On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems

V_n^0(\Omega)

A singular parabolic equation: Existence, stabilization

is the space in which the Stokes operator is defined. We also obtain a three-dimensional stabilization result but only for a very specific set of initial data.

Feedback stabilization of a simplified 1d fluid–particle system

The paper presents a theoretical study of an identification problem by shape optimization methods. The question is to detect an object immersed in a fluid. Here, the problem is modeled by the Stokes equations and treated as a nonlinear least-squares problem. We consider both the Dirichlet and Neumann boundary conditions. Firstly, we prove an identifiability result. Secondly, we prove the existence of the first-order shape derivatives of the state, we characterize them and deduce the gradient of the least-squares functional. Moreover, we study the stability of this setting. We prove the existence of the second-order shape derivatives and we give the expression of the shape Hessian. Finally, the compactness of the Riesz operator corresponding to this shape Hessian is shown and the ill-posedness of the identification problem follows. This explains the need of regularization to numerically solve this problem.