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

Abstract

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 of a nonmagnetic control region. The second part of the paper deals with the finite element analysis based on the edge elements of Nédélec's first family for the control and state discretization, and the continuous piecewise linear ansatz for the Lagrange multiplier discretization. The discrete Helmholtz decomposition and the discrete compactness property of the Nédélec edge elements are the main tools for the finite element analysis. Our final results include the convergence and a priori error estimates for the finite element approximation. Numerical results illustrating the theoretical findings are presented.