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 $p=\infty$ 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.