Optimal control model for the critical state in superconductors

Abstract

Grounded on a variational principle, we present a generalization of the standard critical-state approach in type-II superconductors. The free energy is minimized with the constraint $|\mathbf{J}(\mathbf{r})|<~{J}_{c}$ for the macroscopic current density, posing the problem in the framework of the optimal control theory. The application of this mathematical tool allows us to determine the critical state in which the system organizes itself. This permits to confirm the critical-state hypothesis for an idealized one-dimensional geometry and to deal with multicomponent field situations, for which additional constitutive laws are provided. A geometrical picture of the field penetration process has been developed and we obtain both analytical and numerical solutions for two-dimensional problems under an applied parallel field and superimposed transport current.