Limit-case admissibility for positive infinite-dimensional systems

In the context of positive infinite-dimensional linear systems, we systematically study L^p-admissible control and observation operators with respect to the limit-cases p=∞ and p=1, respectively. This requires an in-depth understanding of the order structure on the extrapolation space X₋₁, which we provide. These properties of X₋₁ also enable us to discuss when zero-class admissibility is automatic. While those limit-cases are the weakest form of admissibility on the L^p-scale, it is remarkable that they sometimes directly follow from order theoretic and geometric assumptions. Our assumptions on the geometries of the involved spaces are minimal.