Boundary stabilization ofthe Navier-Stokes equations with feedback controller via a Galerkin method

In this work we study the exponential stabilization of the two and three-dimensionalNavier-Stokes equations in a bounded domain Ω, around a given steady-state flow,by means of a boundary control. In order to determine a feedback law, we consider anextended system coupling the Navier-Stokes equations with an equation satisfied by thecontrol on the domain boundary. While most traditional approaches apply a feedbackcontroller via an algebraic Riccati equation, the Stokes-Oseen operator orextension operators, a Galerkin method is proposed instead in this study.The Galerkin method permits to construct a stabilizing boundary controland by using energy a priori estimation technics, the exponential decay is obtained.A compactness result then allows us to pass to the limit in the system satisfied bythe approximated solutions. The resulting feedback control is proven to be globallyexponentially stabilizing the steady states of the two and three-dimensionalNavier-Stokes equations.