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

Applications of Laplace-Carleson embeddings to admissibility and controllability