  Given a parameter dependent fixed point equation $x = F(x,u)$, we derive an
abstract compactness principle for the fixed point map $u \mapsto x^*(u)$ under
the assumptions that (i) the fixed point equation can be solved by the
contraction principle and (ii) the map $u \mapsto F(x,u)$ is compact for fixed
$x$. This result is applied to infinite-dimensional, semi-linear control
systems and their reachable sets. More precisely, we extend a
non-controllability result of Ball, Marsden, and Slemrod [1] to semi-linear
systems. First we consider $L^p$-controls, $p>1$. Subsequently we analyze the
case $p=1$.
