  In this paper, we study optimal control problems of semilinear elliptic and
parabolic equations. A tracking cost functional, quadratic in the control and
state variables, is considered. No control constraints are imposed. We prove
that the corresponding state equations are well-posed for controls in $L^2$.
However, it is well-known that in the $L^2$ framework the mappings involved in
the control problem are not Frechet differentiable in general, which makes any
analysis of the optimality conditions challenging. Nevertheless, we prove that
every $L^2$ optimal control belongs to $L^\infty$, and consequently standard
optimality conditions are available.
