  In this paper we consider the convergence analysis of adaptive finite element
method for elliptic optimal control problems with pointwise control
constraints. We use variational discretization concept to discretize the
control variable and piecewise linear and continuous finite elements to
approximate the state variable. Based on the well-established convergence
theory of AFEM for elliptic boundary value problems, we rigorously prove the
convergence and quasi-optimality of AFEM for optimal control problems with
respect to the state and adjoint state variables, by using the so-called
perturbation argument. Numerical experiments confirm our theoretical analysis.
