Analysis of Control Problems for Stationary Magnetohydrodynamics Equations under the Mixed Boundary Conditions for a Magnetic Field

Abstract

The optimal control problems for stationary magnetohydrodynamic equations under the inhomogeneous mixed boundary conditions for a magnetic field and the Dirichlet condition for velocity are considered. The role of controls in the control problems under study is played by normal and tangential components of the magnetic field given on different parts of the boundary and by the exterior current density. Quadratic tracking-type functionals for velocity, magnetic field or pressure are taken as cost functionals. The global solvability of the control problems under consideration is proved, an optimality system is derived and, based on its analysis, a mathematical apparatus for studying the local uniqueness and stability of the optimal solutions is developed. On the basis of the developed apparatus, the local uniqueness of solutions of control problems for specific cost functionals is proved, and stability estimates of optimal solutions are established.

Stability Estimates of Optimal Solutions for the Steady Magnetohydrodynamics-Boussinesq Equations

“…Arguing as in Alekseev ([14], Section 4), or as in Alekseev [58], we show that the nonlinear hydrothermodynamic summands ((u m • ∇)u m , v) and (u m • ∇T m , S) tend, respectively, to ((u * • ∇)u * , v) and (u * • ∇T * , S) as m → ∞, whereas the nonlinear magnetohydrodynamic summand (rot…”

“…The present work continues to develop the direction started in the previous papers by Alekseev [34,42,58]. Its goal is to develop the mathematical apparatus for the study of control problems for the MHD-Boussinesq model of a viscous heat-conducting fluid and to apply the developed apparatus to prove the solvability of the control problems under consideration and to derive unique stability estimates of optimal solutions for certain control problems.…”

“…The use of this approach allowed us to obtain results on the solvability of boundary value problems under weakened conditions on the smoothness of boundary data for the magnetic field. The same idea proved useful when studying control problems for the MHD model (see Alekseev [58]). Its application allowed us to develop in Alekseev [58] an effective mathematical apparatus for the study of multiparameter control problems for the MHD model with minimal requirements for the smoothness of the boundary controls.…”

“…The same idea proved useful when studying control problems for the MHD model (see Alekseev [58]). Its application allowed us to develop in Alekseev [58] an effective mathematical apparatus for the study of multiparameter control problems for the MHD model with minimal requirements for the smoothness of the boundary controls. Their role is played by the normal and tangential components of the magnetic field determined on different parts of the boundary of the flow region.…”

Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer

“…Close questions on the study of the correctness of boundary value or control problems for stationary equations of magnetic hydrodynamics of viscous incompressible or heatconducting liquid in the Boussinesq approximation were investigated in [38][39][40][41][42][43]. In [44], the solvability of the initial-boundary problem for the non-stationary MHD-Boussinesq system is considered under mixed boundary conditions for velocity, magnetic field, and temperature, in the case when the viscosity coefficient, magnetic permeability, electrical conductivity, thermal conductivity, and specific heat of the fluid depend on the temperature.…”