Abstract
We study an optimal control problem for the stationary Stokes equations with variable density and viscosity in a 2D bounded domain under mixed boundary conditions. On in-flow and out-flow parts of the boundary, nonhomogeneous Dirichlet boundary conditions are used, while on the solid walls of the flow domain, the impermeability condition and the Navier slip condition are provided. We control the system by the external forces (distributed control) as well as the velocity boundary control acting on a fixed part of the boundary. We prove the existence of weak solutions of the state equations, by firstly expressing the fluid density in terms of the stream function (Frolov formulation). Then, we analyze the control problem and prove the existence of global optimal solutions. Using a Lagrange multipliers theorem in Banach spaces, we derive an optimality system. We also establish a second-order sufficient optimality condition and show that the marginal function of this control system is lower semi-continuous.
Highlights
In this work, we study an optimal control problem for the stationary Stokes equations with variable density and viscosity in a domain Ω ⊂ R2, which is assumed to be a bounded and connected set with boundary Γ d=ef ∂Ω of class C1
We have studied an optimal control problem for the 2D stationary Stokes equations with variable density and viscosity, using nonhomogeneous Dirichlet boundary conditions on one part of the boundary of the flow domain and the Navier slip boundary conditions on the other part
Expressing the fluid density in terms of the stream function, we proved the existence and uniqueness of weak solutions of the dynamical equations with the same regularity (H1-regularity) as weak solutions of the classical Stokes system with constant density and viscosity under Dirichlet boundary conditions acting on the whole boundary
Summary
We study an optimal control problem for the stationary Stokes equations with variable density and viscosity in a domain Ω ⊂ R2, which is assumed to be a bounded and connected set with boundary Γ d=ef ∂Ω of class C1. The systems that model the behavior of viscous and incompressible fluids with constant density can have two types of character: either elliptical, for the steady state, or parabolic, for the non-stationary state In both cases, when an approximation argument is used, such as the Galerkin method, for instance, higher-order estimates are usually obtained for the approximate solutions, which facilitates the passage to the limit in the nonlinear terms and, it is more simple to achieve the required results. The beginning of the study of such problems dates back to the paper of Illarionov [17] He investigated optimal boundary control for a model of 2D steady-state flows of a nonhomogeneous incompressible fluid under the assumption that the viscosity μ is constant. For the reader’s convenience, in Appendix A (see Table A1), we collect the main symbols used in this paper and explain their meaning
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