Boundary feedback control of Navier–Stokes equations around arbitrary bluff body with prescribed drag and lift coefficients

Abstract

This paper presents a boundary feedback control method for the two- and three-dimensional Navier–Stokes equations around an arbitrary bluff body, with prescribed drag and lift coefficients. In order to determine the feedback control law, we consider an extended system coupling the equations governing the Navier–Stokes problem with an equation satisfied by the control on the bluff body, which is a part of the domain boundary. We utilize the artificial compressibility method, which approximates the Navier–Stokes equations in the limit, along with a priori estimation techniques, to establish a boundary control. Such a control law ensures that the prescriptions of the drag and lift coefficients are attained. Subsequently, a compactness result allows to let the compressibility parameter tend to zero. The application of the Faedo-Galerkin method is essential in reaching this result.