Optimal Control for MHD flows

Catalin Trenchea
University of Pittsburgh

We first consider a viscous, incompressible, electrically conducting
fluid in a bounded two-dimensional domain. We assume that the flow is
governed by the Navier-Stokes equations, and the magnetic field by
Maxwell's equations.  The initial conditions are known and it is
desired to force the system to a given state called the "target". For
this purpose the right hand sides of both equations act as
controls. The optimal control problem seeks to minimize the
discrepancies between flow and target over time.

Existence of optimal solutions is proved and first-order necessary
conditions for optimality are used to derive an optimality system of
partial differential equations whose solutions provide optimal states
and controls.  Finite element approximations are defined and a priori
estimates are used to show their convergence to the exact optimal
solution. Some results of the computational experiments are presented.

We also study a modified Navier-Stokes equation coupled with Maxwell's
equation in three dimensions. The existence of optimal solutions is
shown, the Gateux differentiability for the MHD system with respect to
controls is proved, and the optimality system is derived.