  This paper aims to study the convergence of adaptive finite element method
for control constrained elliptic optimal control problems under $L^2$-norm. We
prove the contraction property and quasi-optimal complexity for the $L^2$-norm
errors of both the control, the state and adjoint state variables with
$L^2$-norm based AFEM, this is in contrast to and improve our previous work
[13] where convergence of AFEM based on energy norm had been studied and
suboptimal convergence for the control variable was obtained and observed
numerically. For the discretization we use variational discretization for the
control and piecewise linear and continuous finite elements for the state and
adjoint state. Under mild assumptions on the initial mesh and the mesh
refinement algorithm to keep the adaptive meshes sufficiently mildly graded we
prove the optimal convergence of AFEM for the control problems, numerical
results are provided to support our theoretical findings.
