Extended boundary approach for optimal control of incompressible steady MHD equations

The interest in Magnetohydrodynamics (MHD) flow control arises from a wide range of applications, such as aluminum casting, liquid metal cooled fission reactors and nuclear fusion technologies. In fact one can control the velocity of a conductive fluid through the adjustment of an external magnetic field and obtain a desired flow profile. In this paper we present a new approach for solving optimal flow control problems involving MHD equations where the external field control is achieved through its boundary conditions. To this purpose we introduce a new approach that transforms the optimal boundary control problem into an extended distributed one by using lifting functions which extend the controlled magnetic field from the boundary into the fluid domain. The adoption of lifting functions brings several theoretical and numerical advantages. With this approach boundary controls can be sought in their natural functional spaces, integral constraints on the boundary magnetic field can be implicitly taken into account and robust numerical algorithms developed for distributed control techniques can be adopted. We study some theoretical aspects of this optimal control approach and analyze a simple numerical example in order to show its effectiveness.