We analyze the optimal control of an electromagnetic process in type-II superconductivity. The PDE-constrained optimization problem is to find an optimal applied current density, which steers the electromagnetic fields to the desired ones in the presence of a type-II superconductor. The governing PDE system for the electromagnetic fields consists of hyperbolic evolution Maxwell equations with a nonlinear and nonsmooth constitutive law for the electric field and the current density based on the Bean critical-state model. Through the use of the Maxwell theory, the semigroup theory, Helmholtz decomposition, and results on maximal monotone operators, we develop a mathematical theory including an existence analysis and first-order necessary optimality conditions for the nonsmooth PDE-constrained optimization problem.