Operator-theoretic characterization of eventually monotone systems

Aivar Sootla, Alexandre Mauroy

Research output: Contribution to journalArticle

Abstract

Monotone systems are dynamical systems whose solutions preserve a partial order in the initial condition for all positive times. It stands to reason that some systems may preserve a partial order only after some initial transient. These systems are usually called eventually monotone. While monotone systems have a characterization in terms of their vector fields (i.e. Kamke-M¨ uller condition), eventually monotone systems have not been characterized in such an explicit manner. In order to provide a characterization, we drew inspiration from the results for linear systems, where eventually monotone (positive) systems are studied using the spectral properties of the system (i.e. Perron-Frobenius property). In the case of nonlinear systems, this spectral characterization is not straightforward, a fact that explains why the class of eventually monotone systems has received little attention to date. In this paper, we show that a spectral characterization of nonlinear eventually monotone systems can be obtained through the Koopman operator framework. We consider a number of biologically inspired examples to illustrate the potential applicability of eventual monotonicity.
LanguageEnglish
Number of pages13
JournalControl Systems Letters (L-CSS)
Volume2
Issue number3
DOIs
StatePublished - Jul 2018

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Monotone Systems
Operator
Partial Order
Perron-Frobenius
Positive Systems
Spectral Properties
Monotonicity
Vector Field
Monotone
Initial conditions
Nonlinear Systems
Dynamical system
Linear Systems

Cite this

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title = "Operator-theoretic characterization of eventually monotone systems",
abstract = "Monotone systems are dynamical systems whose solutions preserve a partial order in the initial condition for all positive times. It stands to reason that some systems may preserve a partial order only after some initial transient. These systems are usually called eventually monotone. While monotone systems have a characterization in terms of their vector fields (i.e. Kamke-M¨ uller condition), eventually monotone systems have not been characterized in such an explicit manner. In order to provide a characterization, we drew inspiration from the results for linear systems, where eventually monotone (positive) systems are studied using the spectral properties of the system (i.e. Perron-Frobenius property). In the case of nonlinear systems, this spectral characterization is not straightforward, a fact that explains why the class of eventually monotone systems has received little attention to date. In this paper, we show that a spectral characterization of nonlinear eventually monotone systems can be obtained through the Koopman operator framework. We consider a number of biologically inspired examples to illustrate the potential applicability of eventual monotonicity.",
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Operator-theoretic characterization of eventually monotone systems. / Sootla, Aivar; Mauroy, Alexandre.

In: Control Systems Letters (L-CSS), Vol. 2, No. 3, 07.2018.

Research output: Contribution to journalArticle

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