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.