Optimal control of low-frequency electromagnetic fields in multiply connected conductors

A class of optimal control problems for electromagnetic fields is considered. Special emphasis is laid on a non-standard H-based formulation of the equations of low-frequency electromagnetism in multiply connected conductors. By this technique, the low-frequency 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 σ−1 curl operator, an elliptic equation with the div μ∇ operator is set up for a potential ψ in the isolator. Both equations are coupled by appropriate interface conditions. In all problems, the electrical current is controlled in the conducting domain. We discuss two optimal control problems with distributed control. A standard quadratic tracking type objective functional is minimized in the first problem, while a convex nondifferentiable functional with L1-sparsity term is considered in the second. For all problems, the associated sensitivity analysis is performed.