Control problems for stationary magnetohydrodynamic equations of a viscous heat-conducting fluid under mixed boundary conditions

Control problems for the stationary magnetohydrodynamic model of a viscous heat-conducting fluid are studied under mixed boundary conditions imposed on the velocity, electromagnetic field, and temperature. It is proved that the original boundary value problem and the general control problem are solvable. The application of the Lagrange principle is validated, the regularity of Lagrange multipliers is investigated, and local conditions for the uniqueness of a solution to control problems for specific cost functionals are derived.