Continuity of solutions to abstract linear control systems

Abstract — v1

It has long been known that the solutions to abstract linear control systems are continuous in time for controls in $L^p$ with $1 \le p < \infty$. We prove that the same property remains valid for the endpoint case $p = \infty$, giving a positive answer to Weiss' 1989 Problem 2.4. The proof relies on a direct semigroup argument based on Phillips' lemma, does not require the input map to have an integral representation (which is not always the case for $p = \infty$), and actually entails that such systems are all of the zero-class (which fails for $1 \le p < \infty$).