Observer-based feedback boundary stabilization of the Navier-Stokes equations

This paper aims at designing an observer-based feedback law which locally stabilizes the solution to the two dimensional Navier-Stokes equations with mixed boundary conditions. We consider a finite number of controls acting on a portion of the boundary through Robin boundary conditions and construct a linear Luenberger observer based on the point observations of the linearized Navier-Stokes equations. The sensor location for the point observations is determined by the response of feedback functional gains. We prove that the nonlinear system coupled with the observer through the feedback law is locally exponentially stable. Numerical experiments based on a Taylor-Hood finite element method are presented to illustrate the design for different Reynolds numbers.