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URN to cite this document: urn:nbn:de:bvb:703-epub-8284-9
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 der Verlagsversion: https://doi.org/10.1007/s00028-024-01044-8
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Project information
| Project title: |
Project's official title Project's id Lyapunov theory meets boundary control systems MI 1886/2-2 |
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| Project financing: |
Deutsche Forschungsgemeinschaft |
Abstract
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 |
|---|---|
| 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 |
| DDC Subjects: | 500 Science > 510 Mathematics |
| 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 |
| Language: | English |
| Originates at UBT: | Yes |
| URN: | urn:nbn:de:bvb:703-epub-8284-9 |
| Date Deposited: | 11 Mar 2025 06:17 |
| Last Modified: | 11 Mar 2025 06:17 |
| URI: | https://epub.uni-bayreuth.de/id/eprint/8284 |
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