  In this paper we discuss the numerical solution of elliptic distributed
optimal control problems with state or control constraints when the control is
considered in the energy norm. As in the unconstrained case we can relate the
regularization parameter and the finite element mesh size in order to ensure an
optimal order of convergence which only depends on the regularity of the given
target, also including discontinuous target functions. While in most cases,
state or control constraints are discussed for the more common $L^2$
regularization, much less is known in the case of energy regularizations. But
in this case, and for both control and state constraints, we can formulate
first kind variational inequalities to determine the unknown state, from wich
we can compute the control in a post processing step. Related variational
inequalities also appear in obstacle problems, and are well established both
from a mathematical and a numerical analysis point of view. Numerical results
confirm the applicability and accuracy of the proposed approach.
