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Feedback Boundary Stabilization of Two Dimensional Navier-Stokes Equations

We study the robust or $H^{\infty}$ exponential stabilization of the linearized Navier-Stokes equations around an unstable stationary solution in a two-dimensional domain $\Omega$. The disturbance is an unknown perturbation in the boundary condition of the fluid flow. We determine a feedback boundary control law, robust with respect to boundary perturbations, by solving a Max-Min linear quadratic control problem. Next we show that this feedback law locally stabilizes the Navier-Stokes system. We do not assume that the normal component of the control is equal to zero. In that case the state equation, satisfied by the velocity field  $y$ is decoupled into an evolution equation satisfied by $Py$ and  a quasi-stationary elliptic equation satisfied by $(I-P)y$. Using this decomposition we show that the feedback law can be expressed only as a function of $Py$. In the two-dimensional case, we show that the linear feedback law provides a local exponential stabilization of the Navier-Stokes equation.