  It is shown how results on Carleson embeddings induced by the Laplace
transform can be use to derive new and more general results concerning the
weighted admissibility of control and observation operators for linear
semigroup systems with q-Riesz bases of eigenvectors. Next, a new Carleson
embedding result is proved, which gives further results on weighted
admissibility for analytic semigroups. Finally, controllability by smoother
inputs is characterised by means of a new result about weighted interpolation.
