Compactness of Fixed Point Maps and the Ball-Marsden-Slemrod Conjecture

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 $p=1$.