  In this work, an adaptive edge element method is developed for an
H(curl)-elliptic constrained optimal control problem. We use the lowest-order
Nedelec's edge elements of first family and the piecewise (element-wise)
constant functions to approximate the state and the control, respectively, and
propose a new adaptive algorithm with error estimators involving both
residual-type error estimators and lower-order data oscillations. By using a
local regular decomposition for H(curl)-functions and the standard bubble
function techniques, we derive the a posteriori error estimates for the
proposed error estimators. Then we exploit the convergence properties of the
orthogonal $L^2$-projections and the mesh-size functions to demonstrate that
the sequences of the discrete states and controls generated by the adaptive
algorithm converge strongly to the exact solutions of the state and control in
the energy-norm and $L^2$-norm, respectively, by first achieving the strong
convergence towards the solution to a limiting control problem.
Three-dimensional numerical experiments are also presented to confirm our
theoretical results and the quasi-optimality of the adaptive edge element
method.
