Checking conditions for p-admissibility

When checking for admissibility of a given control operator with respect to L^p, there is a strong contrast between the case p = 2 and the case p ≠ 2. This is due to missing Hilbert space structure in the latter case. However, there are natural systems which are not L²-admissible. For example, the Dirichlet controlled heat equation on a compact interval I with state space L²(I) is p-admissible only for p > 4 even in one dimension. Among others, two typical approaches to verifying p-admissibility are  (i) sufficient conditions for analytic semigroups by mapping properties of traces,  (ii) conditions arising from Laplace-Carleson embeddings for diagonal semigroups. We discuss that (i) even leads to a characterization of admissibility using abstract interpolation spaces and we extend (ii) to more general normal and multiplication semigroups. More precisely, L^p-admissibility of b ∈ ℒ(ℂ,X₋₁) is equivalent to boundedness of the Laplace transform as a mapping 𝔏: L^p(0,∞) → L²(ℂ_>0,µ) for a measure µ related to the control operator b. The measure µ is given by µ = ∑_k |b_k|² δ_−λk in the diagonal case, i.e. in the presence of a Riesz basis of eigenvectors of the generator with eigenvalues (λ_k)_k∈ℕ, cf. results by Jacob, Partington and Pott. However, the conditions given in their work are not applicable if the generator does not have compact resolvents. In the case of a normal generator A with spectral measure E, a suitable generalization of the discrete measure is the measure µ = ⟨Eb, b⟩. We also discuss a related result for semigroups generated by multiplication operators on L^q spaces, which generalize the setting of q-Riesz bases. As an application, we investigate p-admissibility for several examples arising from partial differential equations. The systems are based on the heat equation on bounded and unbounded domains. To illustrate the unbounded case, we consider a controlled heat equation on the full Euclidean space or a half-space, leading to the typical case of non-compact resolvents of the generator. This is a collaboration with Felix Schwenninger.