On checking $\mathrm{L}^p$-admissibility for parabolic control systems

In this note we discuss the difficulty of verifying $\mathrm{L}^p$-admissibility for $p\neq 2$ -- that even manifests in the presence of a self-adjoint semigroup generator on a Hilbert space -- and survey tests for $\mathrm{L}^p$-admissibility of given control operators. These tests are obtained by virtue of either mapping properties of boundary trace operators, yielding a characterization of admissibility via abstract interpolation spaces; or through Laplace--Carleson embeddings, slightly extending results from Jacob, Partington and Pott to a class of systems which are not necessarily diagonal with respect to sequence spaces. Special focus is laid on illustrating the theory by means of examples based on the heat equation on various domains.