Analysis of Nonlinear Tridiagonal Cooperative Systems using Totally Positive Linear Differential Systems

Michael Margaliot, Eduardo D. Sontag

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Cooperative tridiagonal dynamical systems appear often in biological and engineering applications. The most important theorem for such systems was arguably one proved by Smillie in 1984, and subsequently refined by other authors. Smillie showed that-under mild technical assumptions-precompact trajectories always converge to equilibria. The key to his proof was the construction of an integer-valued Lyapunov function that certifies that the number of sign variations in the vector of derivatives of states eventually stabilizes. This paper shows how to re-derive Smillie's theorem by appealing to results from Binyamin Schwarz, who analyzed the sign variations in solutions of linear systems whose flows are totally nonnegative or totally positive (meaning that all minors are nonnegative or positive, respectively). The connection is through the variational equation associated to the original system. In addition to connecting two seemingly disparate areas of research, the connection allows one to both simplify proofs and extend the validity of Smillie's Theorem.

Original languageEnglish
Title of host publication2018 IEEE Conference on Decision and Control, CDC 2018
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages3104-3109
Number of pages6
ISBN (Electronic)9781538613955
DOIs
StatePublished - 2 Jul 2018
Event57th IEEE Conference on Decision and Control, CDC 2018 - Miami, United States
Duration: 17 Dec 201819 Dec 2018

Publication series

NameProceedings of the IEEE Conference on Decision and Control
Volume2018-December
ISSN (Print)0743-1546
ISSN (Electronic)2576-2370

Conference

Conference57th IEEE Conference on Decision and Control, CDC 2018
Country/TerritoryUnited States
CityMiami
Period17/12/1819/12/18

Funding

FundersFunder number
National Science Foundation1817936
Air Force Office of Scientific ResearchFA9550-14-1-0060
Defense Advanced Research Projects AgencyFA8650-18-1-7800
United States-Israel Binational Science Foundation
Israel Science Foundation

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