Asymptotic analysis of an optimal control problem for a viscous incompressible fluid with Navier slip boundary conditions

Article type: Research Article

Authors: Gariboldi, Claudia^a | Takahashi, Takéo^{b; *}

Affiliations: [a] Departamento de Matemática, FCEFQyN, Universidad Nacional de Río Cuarto, Ruta 36 Km 601, 5800 Río Cuarto, Argentina | [b] Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France

Correspondence: [*] Corresponding author. E-mail: takeo.takahashi@inria.fr.

Abstract: We consider an optimal control problem for the Navier–Stokes system with Navier slip boundary conditions. We denote by α the friction coefficient and we analyze the asymptotic behavior of such a problem as α→∞. More precisely, we prove that if we take an optimal control for each α, then there exists a sequence of optimal controls converging to an optimal control of the same optimal control problem for the Navier–Stokes system with the Dirichlet boundary condition. We also show the convergence of the corresponding direct and adjoint states.

Keywords: Optimal control, Navier–Stokes system, Navier slip boundary condition

DOI: 10.3233/ASY-211685

Journal: Asymptotic Analysis, vol. 126, no. 3-4, pp. 379-399, 2022