Admissibility of Control Operators for Solution Families to Volterra Integral Equations

We reduce the admissibility of finite-dimensional control operators for an evolution system (satisfying certain mild regularity conditions) to that for a semigroup control system, for which there are plenty of results available in the literature. One such necessary and sufficient condition is the Carleson measure criterion. We prove this condition explicitly for evolution systems without the regularity assumption mentioned above. We also look at some examples which show that our respective hypotheses are necessary.