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Stability criteria for positive semigroups on ordered Banach spaces

Title data

Glück, Jochen ; Mironchenko, Andrii:
Stability criteria for positive semigroups on ordered Banach spaces.
In: Journal of Evolution Equations. Vol. 25 (2025) Issue 1 . - 12.
ISSN 1424-3202
DOI: https://doi.org/10.1007/s00028-024-01044-8

Review:

Official URL: Volltext

Project information

Project title:
Project's official title
Project's id
Lyapunov theory meets boundary control systems
MI 1886/2-2

Project financing: Deutsche Forschungsgemeinschaft

Abstract in another language

We consider generators of positive C₀-semigroups and, more generally, resolvent positive operators A on ordered Banach spaces and seek for conditions ensuring the negativity of their spectral bound s(A). Our main result characterizes s(A) < 0 in terms of so-called small-gain conditions that describe the behaviour of Ax for positive vectors x. This is new even in case that the underlying space is an $L^p$-space or a space of continuous functions. We also demonstrate that it becomes considerably easier to characterize the property s(A) < 0 if the cone of the underlying Banach space has non-empty interior or if the essential spectral bound of A is negative. To treat the latter case, we discuss a counterpart of a Krein–Rutman theorem for resolvent positive operators. When A is the generator of a positive C₀-semigroup, our results can be interpreted as stability results for the semigroup, and as such, they complement similar results recently proved for the discrete-time case. In the same vein, we prove a Collatz–Wielandt type formula and a logarithmic formula for the spectral bound of generators of positive semigroups.

Further data

Item Type: Article in a journal
Refereed: Yes
Keywords: positive systems; continuous-time systems; stability; small-gain condition; linear systems; semigroup theory; resolvent positive operator; Krein–Rutman theorem
Subject classification: Mathematics Subject Classification Code: 47B65, 47D06, 47A10, 37L15
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics)
Profile Fields > Advanced Fields > Nonlinear Dynamics
Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Profile Fields
Profile Fields > Advanced Fields
Result of work at the UBT: No
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 06 Mar 2025 08:56
Last Modified: 11 Mar 2025 06:18
URI: https://eref.uni-bayreuth.de/id/eprint/92636