A mathematical model for instationary magnetization
processes is considered, where the underlying spatial domain
includes electrically conducting and nonconducting regions. The
model accounts for the magnetic induction law that couples the given
electrical voltage with the induced electrical current in the
induction coil. By a theorem of Showalter on degenerate parabolic
equations, theorems on existence, uniqueness, and regularity of the
solution to the associated Maxwell integrodifferential system are
proved.
Two optimal control problems for instationary magnetization
processes are considered in 3D spatial domains that
include electrically conducting and nonconducting regions. The magnetic
fields are generated by induction coils. In the first model, the induction coil
is considered as part of the conducting region and the electrical current is taken
as control. In the second, the coil is viewed as part of the nonconducting region and the
electrical voltage is the control. Here, an integro-differential equation accounts
for the magnetic induction law that couples the given
electrical voltage with the induced electrical current in the
induction coil.
We derive first-order necessary
optimality condition for the optimal controls of both problems. Based on them,
numerical methods of gradient type are applied. Moreover, we report on the application
of model reduction by POD that lead to tremendous savings. Numerical tests are
presented for academic 3D geometries but also for a real-world application.
An optimal control problem is studied for a quasilinear Maxwell equation of nondegenerate parabolic
type. Well-posedness of the quasilinear state equation, existence of an optimal control, and weak G\^ateaux-differentiability of the control-to-state mapping are proved. Based on these results, first-order necessary optimality conditions and
an associated adjoint calculus are derived.