Control of magnetohydrodynamic flow in the formation of a magnetic field with a prescribed configuration

Control problems for the viscous incompressible magnetohydrodynamic equations are considered. The goal is to produce a magnetic field with a prescribed structure at a given time by applying electromotive forces so as to minimize Joule heating or the work done on the conduction currents. Based on estimates obtained for the solution of a subdifferential Cauchy problem for the Navier–Stokes-type equations, the possibility of producing the required magnetic field is proved and the solvability conditions for the control problems are found.