Control Problem Related to 2D Stokes Equations with Variable Density and Viscosity

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.

Theoretical Analysis of Boundary Value Problems for Generalized Boussinesq Model of Mass Transfer with Variable Coefficients

“…Among the papers devoted to the study of various models generalizing the Boussinesq approximation, we note [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. In [13,14] the global solvability of the stationary boundary value problem for nonlinear heat transfer equations is proved in the case, when the viscosity coefficient depends on temperature.…”

“…It should be noted that the cycle of articles by E.S. Baranovskii with co-authors [18][19][20][21] are devoted to the study of boundary and extremum problems for stationary models of the dynamics of viscous incompressible fluid. In detail, the model of non-isothermal creeping flows of an incompressible fluid is considered in [18].…”

“…In [20], the control problem for 2D Stokes equations with variable density and viscosity is studied. In [21], the existence of an optimal solution for the problem of boundary control of non-isothermal stationary flows of low-concentration aqueous polymer solutions in a limited three-dimensional domain is proved.…”

Boundary Feedback Stabilization of Two-Dimensional Shallow Water Equations with Viscosity Term

“…It also discusses the associated challenges, such as start-control, impulse-control and distributed-control laws, which have been studied for the Oseen and Navier-Stokes equations. Recently, global solution as well as an optimality system and a second-order sufficient optimality condition were obtained for the stationary two-dimensional Stokes equations [15,16], while an optimal controllability of a stationary two-dimensional non-Newtonian fluid in a pipeline network is studied in [17].…”