Polyhedral Lyapunov functions structurally ensure global asymptotic stability of dynamical networks iff the Jacobian is non-singular
(2017) In Automatica 86. p.183-191- Abstract
- For a vast class of dynamical networks, including chemical reaction networks (CRNs) with monotonic reaction rates, the existence of a polyhedral Lyapunov function (PLF) implies structural (i.e., parameter-free) local stability. Global structural stability is ensured under the additional assumption that each of the variables (chemical species concentrations in CRNs) is subject to a spontaneous infinitesimal dissipation. This paper solves the open problem of global structural stability in the absence of the infinitesimal dissipation, showing that the existence of a PLF structurally ensures global convergence if and only if the system Jacobian passes a structural non-singularity test. It is also shown that, if the Jacobian is structurally... (More)
- For a vast class of dynamical networks, including chemical reaction networks (CRNs) with monotonic reaction rates, the existence of a polyhedral Lyapunov function (PLF) implies structural (i.e., parameter-free) local stability. Global structural stability is ensured under the additional assumption that each of the variables (chemical species concentrations in CRNs) is subject to a spontaneous infinitesimal dissipation. This paper solves the open problem of global structural stability in the absence of the infinitesimal dissipation, showing that the existence of a PLF structurally ensures global convergence if and only if the system Jacobian passes a structural non-singularity test. It is also shown that, if the Jacobian is structurally non-singular, under positivity assumptions for the system partial derivatives, the existence of an equilibrium is guaranteed. For systems subject to positivity constraints, it is shown that, if the system admits a PLF, under structural non-singularity assumptions, global convergence within the positive orthant is structurally ensured, while the existence of an equilibrium can be proven by means of a linear programming test and the computation of a piecewise-linear-in-rate Lyapunov function.
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Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/3c7aafa6-6723-4b30-b6bf-52e81d779f1b
- author
- Blanchini, Franco and Giordano, Giulia LU
- organization
- publishing date
- 2017-12
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Automatica
- volume
- 86
- pages
- 183 - 191
- publisher
- Pergamon Press Ltd.
- external identifiers
-
- scopus:85029601564
- ISSN
- 0005-1098
- DOI
- 10.1016/j.automatica.2017.08.022
- language
- English
- LU publication?
- yes
- id
- 3c7aafa6-6723-4b30-b6bf-52e81d779f1b
- date added to LUP
- 2017-08-22 11:30:42
- date last changed
- 2024-04-14 16:02:53
@article{3c7aafa6-6723-4b30-b6bf-52e81d779f1b,
abstract = {{For a vast class of dynamical networks, including chemical reaction networks (CRNs) with monotonic reaction rates, the existence of a polyhedral Lyapunov function (PLF) implies structural (i.e., parameter-free) local stability. Global structural stability is ensured under the additional assumption that each of the variables (chemical species concentrations in CRNs) is subject to a spontaneous infinitesimal dissipation. This paper solves the open problem of global structural stability in the absence of the infinitesimal dissipation, showing that the existence of a PLF structurally ensures global convergence if and only if the system Jacobian passes a structural non-singularity test. It is also shown that, if the Jacobian is structurally non-singular, under positivity assumptions for the system partial derivatives, the existence of an equilibrium is guaranteed. For systems subject to positivity constraints, it is shown that, if the system admits a PLF, under structural non-singularity assumptions, global convergence within the positive orthant is structurally ensured, while the existence of an equilibrium can be proven by means of a linear programming test and the computation of a piecewise-linear-in-rate Lyapunov function.<br/><br/>}},
author = {{Blanchini, Franco and Giordano, Giulia}},
issn = {{0005-1098}},
language = {{eng}},
pages = {{183--191}},
publisher = {{Pergamon Press Ltd.}},
series = {{Automatica}},
title = {{Polyhedral Lyapunov functions structurally ensure global asymptotic stability of dynamical networks iff the Jacobian is non-singular}},
url = {{http://dx.doi.org/10.1016/j.automatica.2017.08.022}},
doi = {{10.1016/j.automatica.2017.08.022}},
volume = {{86}},
year = {{2017}},
}