Boundary Velocity Suboptimal Control of Incompressible Flow in Cylindrically Perforated Domains

Abstract

Optimal control problems (OCPs) for the Navier–Stokes equations have been the subject of extensive study in recent years. A systematic mathematical and numerical analysis of OCPs of different types (e.g., having Dirichlet, Neumann, and distributed controls) for the steady-state Navier–Stokes system was given by Abergel and Temam (1990), Fursikov, Gunzburger, and Hou (2000, 1998), Hou and Ravindran (1998), Gunzburger, Hou, and Svobodny (1992), Ivanenko and Mel’nik (1988), and Zgurovsky and Mel’nik (2004). Dirichlet controls (i.e., boundary velocity controls or boundary mass flux control) are common in applications (Temam 1984). However, as is shown in (Hou and Svobodny 1993), even though the admissible controls are smooth, the optimality systems for optimal Dirichlet control problems involve a boundary Laplacian or a boundary biharmonic equation. This circumstance makes the numerical resolution of the optimality systems, and hence the numerical calculation of an optimal control for such systems, very complicated. So, much effort has been made for the development of penalty, approximation, and relaxation methods for solving optimal Dirichlet control problems (see Abergel and Temam (1990), Babuška (1973), Hou and Ravindran (1998)).