math.OC — Optimization and Control

Continuity of solutions to abstract linear control systems

Contributed by Frédéric Marbach

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 $1 \le p < \infty$).