Solvability of the inverse extremum problems for stationary equations of magnetic hydrodynamics of viscous fluid with mixed boundary conditions

Inverse extremum problems for stationary equations of magnetic hydrodynamics under mixed boundary conditions for velocity and electric and magnetic fields are considered. The solvability of the original boundary problem is proved, inverse extremum problems are formulated and existence of solutions of these problems is proved. Application of Lagrange principle is justified, optimality systems are obtained and analyzed both for arbitrary and for specific cost functionals.