Abstract
It is shown how results on Carleson embeddings induced by the Laplace transform can be used to derive new and more general results concerning the weighted (infinite-time) admissibility of control and observation operators for linear semigroup systems with $q$-Riesz bases of eigenvectors. As an example, the heat equation is considered. Next, a new Carleson embedding result is proved, which gives further results on weighted admissibility for analytic semigroups. Finally, controllability by smoother inputs is characterized by means of a new result about weighted interpolation.
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