Limit-case admissibility for positive infinite-dimensional systems

TL;DR

Addresses limit-case admissibility for positive infinite-dimensional systems by studying order structure on $X_{-1}$ and deriving automatic zero-class admissibility under mild order assumptions. Develops $L^1$- and $L^\infty$-admissibility results for observation operators, and boundary-operator characterizations and positivity criteria for control operators, complemented by Favard-space and ultracontractivity analyses to obtain $L^r$-admissibility ranges. Establishes perturbation results for positive $C_0$-semigroups, showing how zero-class admissibility supports generation under additive and boundary perturbations. The framework relies on minimal geometric assumptions and highlights the interplay between order structures, extrapolation spaces, and perturbation theory in 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.

Theorems & Definitions (79)

• Proposition 2.1
• proof
• Example 2.2: Dual cone of $\mathrm C^k(\overline{\Omega})$
• Theorem 2.3
• proof
• Proposition 2.4
• proof
• Theorem 2.5
• proof : Proof of Theorem \ref{['thm:face-in-extrapolation-space']}
• Theorem 2.6