Optimal Control for Incompressible Steady MHD Flows via Constrained Extended Boundary Approach

Optimal boundary control problems associated with the Magnetohydrodynamic (MHD) equations have a wide and important range of applications. If one desires to increase or decrease the velocity inside a channel filled with a conductive fluid or to impose a desired velocity profile near the channel outflow, the control of the magnetic field may represent the best way to reach the objective. The magnetic field inside the domain obeys to the MHD equations and the control by an external field can be achieved only through the control of the boundary conditions. In this work, we study a class of stationary boundary MHD optimal velocity and optimal flow control problems. Standard boundary control approach is not very straightforward to implement numerically in comparison to distributed control and often leads to unnecessary smooth controls. In this paper we present a very different approach from the standard one where the optimal boundary control problem is transformed into an extended distributed problem. This can be achieved by considering boundary controls in the form of lifting functions which extend from the boundary into the inner domain. The optimal solution is then searched by exploring all possible extended functions. This approach gives robustness to the boundary control algorithm which can be solved by standard distributed control techniques over the interior part of the domain. Boundary controls obtained by extended functions have several advantages. The extended function can easily take into account several possible boundary conditions and both Dirichlet and Neumann controls. We can seek these boundary controls in their natural functional spaces differently from the standard approach where the control must be, for feasibility reasons, in smoother spaces. Also in this approach integral constraints on the boundary magnetic field may be implicitly taken into account. Some theoretical aspects of this optimal control approach are investigated and numerical examples of boundary controls in channels are presented in order to show the performance of this approach on conductive flows.