  This paper is concerned with second-order optimality conditions for Tikhonov
regularized optimal control problems governed by the obstacle problem. Using a
simple observation that allows to characterize the structure of optimal
controls on the active set, we derive various conditions that guarantee the
local/global optimality of first-order stationary points and/or the
local/global quadratic growth of the reduced objective function. Our analysis
extends and refines existing results from the literature, and also covers those
situations where the problem at hand involves additional box-constraints on the
control. As a byproduct, our approach shows in particular that Tikhonov
regularized optimal control problems for the obstacle problem can be
reformulated as state-constrained optimal control problems for the Poisson
equation, and that problems involving a subharmonic obstacle and a convex
objective function are uniquely solvable. The paper concludes with three
counterexamples which illustrate that rather peculiar effects can occur in the
analysis of second-order optimality conditions for optimal control problems
governed by the obstacle problem, and that necessary second-order conditions
for such problems may be hard to derive.

Optimization and Control