  Motivated by various applications, this article develops the notion of
boundary control for Maxwell's equations in the frequency domain. Surface curl
is shown to be the appropriate regularization in order for the optimal control
problem to be well-posed. Since, all underlying variables are assumed to be
complex valued, the standard results on differentiability do not directly
apply. Instead, we extend the notion of Wirtinger derivatives to complexified
Hilbert spaces. Optimality conditions are rigorously derived and higher order
boundary regularity of the adjoint variable is established. The state and
adjoint variables are discretized using higher order N\'ed\'elec finite
elements. The finite element space for controls is identified, as a space,
which preserves the structure of the control regularization. Convergence of the
fully discrete scheme is established. The theory is validated by numerical
experiments, in some cases, motivated by realistic applications.
