Limit-case admissibility for positive infinite-dimensional systems

Abstract

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 follow from order theoretic and geometric assumptions. Our assumptions on the geometries of the involved spaces are minimal.

Weakend the assumption "the closed unit ball is upwards directed" to "the open unit ball be upwards directed". Added new references. Added some details to the proof of Theorem~4.5(a). Wording in certain places has been modified