  This paper is concerned with an optimal control problem subject to the
$H^1$-critical defocusing semilinear wave equation on a smooth and bounded
domain in three spatial dimensions. Due to the criticality of the nonlinearity
in the wave equation, unique solutions to the PDE obeying energy bounds are
only obtained in special function spaces related to Strichartz estimates and
the nonlinearity. The optimal control problem is complemented by
pointwise-in-time constraints of Trust-Region type $\|u(t)\|_{L^2(\Omega)} \leq
\omega(t)$. We prove existence of globally optimal solutions to the optimal
control problem and give optimality conditions of both first- and second order
necessary as well as second order sufficient type. A nonsmooth regularization
term for the natural control space $L^1(0,T;L^2(\Omega))$, which also promotes
sparsity in time of an optimal control, is used in the objective functional.
