Admissibility of control operators for positive semigroups and robustness of input-to-state stability

TL;DR

This work addresses admissibility of input operators and robustness of input-to-state stability (ISS) for linear positive semigroups on Banach lattices. It derives a resolvent-based lower bound criterion that yields $L^1$-admissibility for positive input operators, and it analyzes ISS under Desch-Schappacher perturbations, providing necessary and sufficient conditions via the spectral radius condition $r(R(0,A_{-1})P)<1$ together with $s(A)<0$. An application to a boundary-controlled renewal/transport model with non-local boundary conditions demonstrates the theory and yields a concrete bound $r(R(0,A_{-1})P) \le \frac{1}{\|q\|_\infty}\|\beta\|_\infty$, giving a practical ISS condition $\|\beta\|_\infty<\|q\|_\infty$. Overall, the paper offers verifiable, sharp criteria for admissibility and ISS robustness in positive distributed parameter systems, with potential extensions to broader perturbed boundary-control problems.

Abstract

In this paper, we establish the well-posedness and stability of distributed parameter systems, focusing on linear positive control systems in a Banach lattice setting. We characterize well-posedness and derive a sufficient condition for admissibility based on a lower norm estimate of the resolvent operator on the positive cone. Furthermore, we analyze input-to-state stability (ISS) under boundary perturbations within the domain of the semigroup generator. Notably, we provide necessary and sufficient conditions for the robustness of ISS under Desch-Schappacher perturbations. Our theoretical results are demonstrated through a boundary value-controlled transport equation with non-local boundary conditions.

Theorems & Definitions (37)

• Definition 1.1
• Proposition 2.1
• proof
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Remark 2.1
• Remark 2.2
• Remark 2.3