Optimal Control of Quasilinear $\boldsymbol{H}(\mathbf{curl})$-Elliptic Partial Differential Equations in Magnetostatic Field Problems

This paper examines the mathematical and numerical analysis for optimal control problems governed by quasilinear $\boldsymbol{H}(\mathbf{curl})$-elliptic partial differential equations. We consider a mathematical model involving isotropic materials with magnetic permeability depending strongly on the magnetic field. Due to the physical and mathematical nature of the problem, it is necessary to include divergence-free constraints on the state and the control. The divergence-free control constraint is treated as an explicit variational equality constraint, whereas a Lagrange multiplier is included in the state equation to deal with the divergence-free state constraint. We investigate the sensitivity analysis of the control-to-state operator and establish the associated optimality conditions. Here, the key tool for proving the KKT theory is the Helmholtz decomposition. An important consequence of the optimality system is a higher regularity result for the optimal control, which we prove under the assumption ...