**Date & Time:** Monday, March 15, 2010, 16:00-17:00.

**Venue:** Ramanujan Hall

**Title:** H-infinity Feedback Boundary Stabilization of 2D Navier-Stokes Equation

**Speaker:** Sheetal Dharmatti, Institut de Mathematique de Toulouse

**Abstract:** 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 $Py$, where $P$ is the so-called Helmholtz projection
operator, 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 equations.