A theorem on error estimates for smooth nonlinear programming
problems in Banach spaces is proved that can be used to derive
optimal error estimates for optimal control problems. This theorem is applied
to a class of optimal control problems for quasilinear elliptic equations.
The state equation is approximated by a finite element scheme, while different
discretization methods are used for the control functions. The distance of
locally optimal controls to their discrete approximations is estimated.
Several classes of optimal control of electromagnetic fields are considered. Special emphasis is
laid on a non-standard $H$-based formulation of the equations of electromagnetism in multiply connected conductors. By this technique, the Maxwell equations can be solved with reduced computational complexity. While the magnetic field $H$ in the conductor is obtained from an elliptic equation
with the $\curl \sigma^{-1} \curl$ operator, an elliptic equation with the $\div \mu \nabla$ operator is set up for a potential $\psi$ in the isolator.
Both equations are coupled by appropriate interface conditions. In all problems, the
electrical current is controlled in the conducting domain. Several types of control functions are discussed. In particular, the problem of sparse optimal control is investigated in a package of electrical wires. For all problems, the associated sensitivity
analysis is performed.
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.
A mathematical model is set up that can be useful for controlled voltage excitation in time-dependent electromagnetism.
The well-posedness of the model is proved and an associated optimal control problem is investigated. Here, the control
function is a transient voltage and the aim of the control is the best approximation of desired electric and magnetic fields in
suitable $L^2$-norms.
Special emphasis is laid on an adjoint calculus for first-order necessary optimality conditions.
Moreover, a {peculiar attention is devoted to propose a formulation for which the computational complexity of the finite element solution method is substantially reduced}.
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.