Loading…

Convergence Assessment of the Trajectories of a Bioreaction System by Using Asymmetric Truncated Vertex Functions

In several open and closed-loop systems, the trajectories converge to a region instead of an equilibrium point. Identifying the convergence region and proving the asymptotic convergence upon arbitrarily large initial values of the state variables are regarded as important issues. In this work, the c...

Full description

Saved in:
Bibliographic Details
Published in:Symmetry (Basel) 2020-04, Vol.12 (4), p.513
Main Authors: Rincón, Alejandro, Florez, Gloria Yaneth, Olivar, Gerard
Format: Article
Language:English
Subjects:
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
cited_by cdi_FETCH-LOGICAL-c336t-4d4eab6af226d7c151f7e724b4531118e7b1a89a09587f8221d1a34d5e2f62aa3
cites cdi_FETCH-LOGICAL-c336t-4d4eab6af226d7c151f7e724b4531118e7b1a89a09587f8221d1a34d5e2f62aa3
container_end_page
container_issue 4
container_start_page 513
container_title Symmetry (Basel)
container_volume 12
creator Rincón, Alejandro
Florez, Gloria Yaneth
Olivar, Gerard
description In several open and closed-loop systems, the trajectories converge to a region instead of an equilibrium point. Identifying the convergence region and proving the asymptotic convergence upon arbitrarily large initial values of the state variables are regarded as important issues. In this work, the convergence of the trajectories of a biological process is determined and proved via truncated functions and Barbalat’s Lemma, while a simple and systematic procedure is provided. The state variables of the process asymptotically converge to a compact set instead of an equilibrium point, with asymmetrical bounds of the compact sets. This convergence is rigorously proved by using asymmetric forms with vertex truncation for each state variable and the Barbalat’s lemma. This includes the definition of the truncated V i functions and the arrangement of its time derivative in terms of truncated functions. The proposed truncated function is different from the common one as it accounts for the model nonlinearities and the asymmetry of the vanishment region. The convergence analysis is valid for arbitrarily large initial values of the state variables, and arbitrarily large size of the convergence regions. The positive invariant nature of the convergence regions is proved. Simulations confirm the findings.
doi_str_mv 10.3390/sym12040513
format article
fullrecord <record><control><sourceid>doaj_cross</sourceid><recordid>TN_cdi_doaj_primary_oai_doaj_org_article_86f3d9c8b4604aef903df02365cb0700</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><doaj_id>oai_doaj_org_article_86f3d9c8b4604aef903df02365cb0700</doaj_id><sourcerecordid>oai_doaj_org_article_86f3d9c8b4604aef903df02365cb0700</sourcerecordid><originalsourceid>FETCH-LOGICAL-c336t-4d4eab6af226d7c151f7e724b4531118e7b1a89a09587f8221d1a34d5e2f62aa3</originalsourceid><addsrcrecordid>eNpNkE1LAzEQhoMoWGpP_oHcpZrP_TjWYrVQ8GDrdZnNTmpKd6NJFPffu7UincvMvPA8h5eQa85upSzZXexbLphimsszMhIsl9OiLNX5yX1JJjHu2DCaaZWxEfmY--4LwxY7g3QWI8bYYpeotzS9IV0H2KFJPjiMhwzovfMBwSTnO_rSx4QtrXu6ia7bDnzftpiCMwP42RlI2NBXDAm_6WL4D1C8IhcW9hEnf3tMNouH9fxpunp-XM5nq6mRMktT1SiEOgMrRNbkhmtuc8yFqpWWnPMC85pDUQIrdZHbQgjecJCq0ShsJgDkmCyP3sbDrnoProXQVx5c9Rv4sK0gJGf2WBWZlU1pinqoRAHaksnGMiEzbWqWMza4bo4uE3yMAe2_j7PqUH51Ur78AbCAeIM</addsrcrecordid><sourcetype>Open Website</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>Convergence Assessment of the Trajectories of a Bioreaction System by Using Asymmetric Truncated Vertex Functions</title><source>Publicly Available Content Database (Proquest) (PQ_SDU_P3)</source><creator>Rincón, Alejandro ; Florez, Gloria Yaneth ; Olivar, Gerard</creator><creatorcontrib>Rincón, Alejandro ; Florez, Gloria Yaneth ; Olivar, Gerard</creatorcontrib><description>In several open and closed-loop systems, the trajectories converge to a region instead of an equilibrium point. Identifying the convergence region and proving the asymptotic convergence upon arbitrarily large initial values of the state variables are regarded as important issues. In this work, the convergence of the trajectories of a biological process is determined and proved via truncated functions and Barbalat’s Lemma, while a simple and systematic procedure is provided. The state variables of the process asymptotically converge to a compact set instead of an equilibrium point, with asymmetrical bounds of the compact sets. This convergence is rigorously proved by using asymmetric forms with vertex truncation for each state variable and the Barbalat’s lemma. This includes the definition of the truncated V i functions and the arrangement of its time derivative in terms of truncated functions. The proposed truncated function is different from the common one as it accounts for the model nonlinearities and the asymmetry of the vanishment region. The convergence analysis is valid for arbitrarily large initial values of the state variables, and arbitrarily large size of the convergence regions. The positive invariant nature of the convergence regions is proved. Simulations confirm the findings.</description><identifier>ISSN: 2073-8994</identifier><identifier>EISSN: 2073-8994</identifier><identifier>DOI: 10.3390/sym12040513</identifier><language>eng</language><publisher>MDPI AG</publisher><subject>asymptotic convergence ; global stability ; invariant set ; Lyapunov-like function ; vertex truncation</subject><ispartof>Symmetry (Basel), 2020-04, Vol.12 (4), p.513</ispartof><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c336t-4d4eab6af226d7c151f7e724b4531118e7b1a89a09587f8221d1a34d5e2f62aa3</citedby><cites>FETCH-LOGICAL-c336t-4d4eab6af226d7c151f7e724b4531118e7b1a89a09587f8221d1a34d5e2f62aa3</cites><orcidid>0000-0002-7381-0560 ; 0000-0003-1862-4842</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27923,27924</link.rule.ids></links><search><creatorcontrib>Rincón, Alejandro</creatorcontrib><creatorcontrib>Florez, Gloria Yaneth</creatorcontrib><creatorcontrib>Olivar, Gerard</creatorcontrib><title>Convergence Assessment of the Trajectories of a Bioreaction System by Using Asymmetric Truncated Vertex Functions</title><title>Symmetry (Basel)</title><description>In several open and closed-loop systems, the trajectories converge to a region instead of an equilibrium point. Identifying the convergence region and proving the asymptotic convergence upon arbitrarily large initial values of the state variables are regarded as important issues. In this work, the convergence of the trajectories of a biological process is determined and proved via truncated functions and Barbalat’s Lemma, while a simple and systematic procedure is provided. The state variables of the process asymptotically converge to a compact set instead of an equilibrium point, with asymmetrical bounds of the compact sets. This convergence is rigorously proved by using asymmetric forms with vertex truncation for each state variable and the Barbalat’s lemma. This includes the definition of the truncated V i functions and the arrangement of its time derivative in terms of truncated functions. The proposed truncated function is different from the common one as it accounts for the model nonlinearities and the asymmetry of the vanishment region. The convergence analysis is valid for arbitrarily large initial values of the state variables, and arbitrarily large size of the convergence regions. The positive invariant nature of the convergence regions is proved. Simulations confirm the findings.</description><subject>asymptotic convergence</subject><subject>global stability</subject><subject>invariant set</subject><subject>Lyapunov-like function</subject><subject>vertex truncation</subject><issn>2073-8994</issn><issn>2073-8994</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><sourceid>DOA</sourceid><recordid>eNpNkE1LAzEQhoMoWGpP_oHcpZrP_TjWYrVQ8GDrdZnNTmpKd6NJFPffu7UincvMvPA8h5eQa85upSzZXexbLphimsszMhIsl9OiLNX5yX1JJjHu2DCaaZWxEfmY--4LwxY7g3QWI8bYYpeotzS9IV0H2KFJPjiMhwzovfMBwSTnO_rSx4QtrXu6ia7bDnzftpiCMwP42RlI2NBXDAm_6WL4D1C8IhcW9hEnf3tMNouH9fxpunp-XM5nq6mRMktT1SiEOgMrRNbkhmtuc8yFqpWWnPMC85pDUQIrdZHbQgjecJCq0ShsJgDkmCyP3sbDrnoProXQVx5c9Rv4sK0gJGf2WBWZlU1pinqoRAHaksnGMiEzbWqWMza4bo4uE3yMAe2_j7PqUH51Ur78AbCAeIM</recordid><startdate>20200401</startdate><enddate>20200401</enddate><creator>Rincón, Alejandro</creator><creator>Florez, Gloria Yaneth</creator><creator>Olivar, Gerard</creator><general>MDPI AG</general><scope>AAYXX</scope><scope>CITATION</scope><scope>DOA</scope><orcidid>https://orcid.org/0000-0002-7381-0560</orcidid><orcidid>https://orcid.org/0000-0003-1862-4842</orcidid></search><sort><creationdate>20200401</creationdate><title>Convergence Assessment of the Trajectories of a Bioreaction System by Using Asymmetric Truncated Vertex Functions</title><author>Rincón, Alejandro ; Florez, Gloria Yaneth ; Olivar, Gerard</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c336t-4d4eab6af226d7c151f7e724b4531118e7b1a89a09587f8221d1a34d5e2f62aa3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>asymptotic convergence</topic><topic>global stability</topic><topic>invariant set</topic><topic>Lyapunov-like function</topic><topic>vertex truncation</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Rincón, Alejandro</creatorcontrib><creatorcontrib>Florez, Gloria Yaneth</creatorcontrib><creatorcontrib>Olivar, Gerard</creatorcontrib><collection>CrossRef</collection><collection>DAOJ: Directory of Open Access Journals</collection><jtitle>Symmetry (Basel)</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Rincón, Alejandro</au><au>Florez, Gloria Yaneth</au><au>Olivar, Gerard</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Convergence Assessment of the Trajectories of a Bioreaction System by Using Asymmetric Truncated Vertex Functions</atitle><jtitle>Symmetry (Basel)</jtitle><date>2020-04-01</date><risdate>2020</risdate><volume>12</volume><issue>4</issue><spage>513</spage><pages>513-</pages><issn>2073-8994</issn><eissn>2073-8994</eissn><abstract>In several open and closed-loop systems, the trajectories converge to a region instead of an equilibrium point. Identifying the convergence region and proving the asymptotic convergence upon arbitrarily large initial values of the state variables are regarded as important issues. In this work, the convergence of the trajectories of a biological process is determined and proved via truncated functions and Barbalat’s Lemma, while a simple and systematic procedure is provided. The state variables of the process asymptotically converge to a compact set instead of an equilibrium point, with asymmetrical bounds of the compact sets. This convergence is rigorously proved by using asymmetric forms with vertex truncation for each state variable and the Barbalat’s lemma. This includes the definition of the truncated V i functions and the arrangement of its time derivative in terms of truncated functions. The proposed truncated function is different from the common one as it accounts for the model nonlinearities and the asymmetry of the vanishment region. The convergence analysis is valid for arbitrarily large initial values of the state variables, and arbitrarily large size of the convergence regions. The positive invariant nature of the convergence regions is proved. Simulations confirm the findings.</abstract><pub>MDPI AG</pub><doi>10.3390/sym12040513</doi><orcidid>https://orcid.org/0000-0002-7381-0560</orcidid><orcidid>https://orcid.org/0000-0003-1862-4842</orcidid><oa>free_for_read</oa></addata></record>
fulltext fulltext
identifier ISSN: 2073-8994
ispartof Symmetry (Basel), 2020-04, Vol.12 (4), p.513
issn 2073-8994
2073-8994
language eng
recordid cdi_doaj_primary_oai_doaj_org_article_86f3d9c8b4604aef903df02365cb0700
source Publicly Available Content Database (Proquest) (PQ_SDU_P3)
subjects asymptotic convergence
global stability
invariant set
Lyapunov-like function
vertex truncation
title Convergence Assessment of the Trajectories of a Bioreaction System by Using Asymmetric Truncated Vertex Functions
url http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-08T13%3A06%3A09IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-doaj_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Convergence%20Assessment%20of%20the%20Trajectories%20of%20a%20Bioreaction%20System%20by%20Using%20Asymmetric%20Truncated%20Vertex%20Functions&rft.jtitle=Symmetry%20(Basel)&rft.au=Rinc%C3%B3n,%20Alejandro&rft.date=2020-04-01&rft.volume=12&rft.issue=4&rft.spage=513&rft.pages=513-&rft.issn=2073-8994&rft.eissn=2073-8994&rft_id=info:doi/10.3390/sym12040513&rft_dat=%3Cdoaj_cross%3Eoai_doaj_org_article_86f3d9c8b4604aef903df02365cb0700%3C/doaj_cross%3E%3Cgrp_id%3Ecdi_FETCH-LOGICAL-c336t-4d4eab6af226d7c151f7e724b4531118e7b1a89a09587f8221d1a34d5e2f62aa3%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_id=info:pmid/&rfr_iscdi=true