  In this article, we consider the Tikhonov regularization of an optimal
control problem of semilinear partial differential equations with box
constraints on the control. We derive a-priori regularization error estimates
for the control under suitable conditions. These conditions comprise
second-order sufficient optimality conditions as well as regularity conditions
on the control, which consists of a source condition and a condition on the
active sets. In addition, we show that these conditions are necessary for
convergence rates under certain conditions. We also consider sparse optimal
control problems and derive regularization error estimates for them. Numerical
experiments underline the theoretical findings.
