Advanced

Separable Lyapunov functions for monotone systems: Constructions and limitations.

Dirr, Gunther; Ito, Hiroshi; Rantzer, Anders LU and Rüffer, Björn (2015) In Discrete and Continuous Dynamical Systems. Series B 20(8). p.2497-2526
Abstract
For monotone systems evolving on the positive orthant, two types of Lyapunov functions are considered: Sum- and max-separable Lyapunov functions. One can be written as a sum, the other as a maximum of functions of scalar arguments. Several constructive existence results for both types are given. Notably, one construction provides a max-separable Lyapunov function that is defined at least on an arbitrarily large compact set, based on little more than the knowledge about one trajectory. Another construction for a class of planar systems yields a global sum-separable Lyapunov function, provided the right hand side satisfies a small-gain type condition. A number of examples demonstrate these methods and shed light on the relation between the... (More)
For monotone systems evolving on the positive orthant, two types of Lyapunov functions are considered: Sum- and max-separable Lyapunov functions. One can be written as a sum, the other as a maximum of functions of scalar arguments. Several constructive existence results for both types are given. Notably, one construction provides a max-separable Lyapunov function that is defined at least on an arbitrarily large compact set, based on little more than the knowledge about one trajectory. Another construction for a class of planar systems yields a global sum-separable Lyapunov function, provided the right hand side satisfies a small-gain type condition. A number of examples demonstrate these methods and shed light on the relation between the shape of sublevel sets and the right hand side of the system equation. Negative examples show that there are indeed globally asymptotically stable systems that do not admit either type of Lyapunov function. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Discrete and Continuous Dynamical Systems. Series B
volume
20
issue
8
pages
2497 - 2526
publisher
Amer Inst Mathematical Sciences
external identifiers
  • wos:000362747200009
  • scopus:84939825936
ISSN
1553-524X
DOI
10.3934/dcdsb.2015.20.2497
project
LCCC
language
English
LU publication?
yes
id
35dbbf10-7332-4c7b-af47-833575cc6fde (old id 7753319)
date added to LUP
2015-08-05 11:03:48
date last changed
2017-08-20 03:02:17
@article{35dbbf10-7332-4c7b-af47-833575cc6fde,
  abstract     = {For monotone systems evolving on the positive orthant, two types of Lyapunov functions are considered: Sum- and max-separable Lyapunov functions. One can be written as a sum, the other as a maximum of functions of scalar arguments. Several constructive existence results for both types are given. Notably, one construction provides a max-separable Lyapunov function that is defined at least on an arbitrarily large compact set, based on little more than the knowledge about one trajectory. Another construction for a class of planar systems yields a global sum-separable Lyapunov function, provided the right hand side satisfies a small-gain type condition. A number of examples demonstrate these methods and shed light on the relation between the shape of sublevel sets and the right hand side of the system equation. Negative examples show that there are indeed globally asymptotically stable systems that do not admit either type of Lyapunov function.},
  author       = {Dirr, Gunther and Ito, Hiroshi and Rantzer, Anders and Rüffer, Björn},
  issn         = {1553-524X},
  language     = {eng},
  number       = {8},
  pages        = {2497--2526},
  publisher    = {Amer Inst Mathematical Sciences},
  series       = {Discrete and Continuous Dynamical Systems. Series B},
  title        = {Separable Lyapunov functions for monotone systems: Constructions and limitations.},
  url          = {http://dx.doi.org/10.3934/dcdsb.2015.20.2497},
  volume       = {20},
  year         = {2015},
}