<?xml version="1.0" encoding="utf-8"?>
<TEI xmlns="http://www.tei-c.org/ns/1.0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:hal="http://hal.archives-ouvertes.fr/" xmlns:gml="http://www.opengis.net/gml/3.3/" xmlns:gmlce="http://www.opengis.net/gml/3.3/ce" version="1.1" xsi:schemaLocation="http://www.tei-c.org/ns/1.0 http://api.archives-ouvertes.fr/documents/aofr-sword.xsd">
  <teiHeader>
    <fileDesc>
      <titleStmt>
        <title>HAL TEI export of hal-01633155</title>
      </titleStmt>
      <publicationStmt>
        <distributor>CCSD</distributor>
        <availability status="restricted">
          <licence target="https://creativecommons.org/publicdomain/zero/1.0/">CC0 1.0 - Universal</licence>
        </availability>
        <date when="2026-05-17T18:33:49+02:00"/>
      </publicationStmt>
      <sourceDesc>
        <p part="N">HAL API Platform</p>
      </sourceDesc>
    </fileDesc>
  </teiHeader>
  <text>
    <body>
      <listBibl>
        <biblFull>
          <titleStmt>
            <title xml:lang="en">Finding Non-Polynomial Positive Invariants and Lyapunov Functions for Polynomial Systems through Darboux Polynomials</title>
            <author role="aut">
              <persName>
                <forename type="first">Eric</forename>
                <surname>Goubault</surname>
              </persName>
              <email type="md5">34691a3c485cac901a2837195944f0e2</email>
              <email type="domain">lix.polytechnique.fr</email>
              <idno type="idhal" notation="string">eric-goubault</idno>
              <idno type="idhal" notation="numeric">1172095</idno>
              <idno type="halauthorid" notation="string">208888-1172095</idno>
              <idno type="ORCID">https://orcid.org/0000-0002-3198-1863</idno>
              <affiliation ref="#struct-40217"/>
            </author>
            <author role="aut">
              <persName>
                <forename type="first">Jacques-Henri</forename>
                <surname>Jourdan</surname>
              </persName>
              <email type="md5">b4e45c9f9b0894a511cd9acf51057868</email>
              <email type="domain">normalesup.org</email>
              <idno type="idhal" notation="string">jacques-henri-jourdan</idno>
              <idno type="idhal" notation="numeric">10849</idno>
              <idno type="halauthorid" notation="string">37694-10849</idno>
              <idno type="ORCID">https://orcid.org/0000-0002-9781-7097</idno>
              <idno type="GOOGLE SCHOLAR">https://scholar.google.fr/citations?user=2iT-_QoAAAAJ&amp;hl=fr</idno>
              <idno type="IDREF">https://www.idref.fr/194193454</idno>
              <affiliation ref="#struct-454413"/>
            </author>
            <author role="aut">
              <persName>
                <forename type="first">Sylvie</forename>
                <surname>Putot</surname>
              </persName>
              <email type="md5">623d4bf532ce4189123d269bca70b26b</email>
              <email type="domain">lix.polytechnique.fr</email>
              <idno type="idhal" notation="string">sylvie-putot</idno>
              <idno type="idhal" notation="numeric">184976</idno>
              <idno type="halauthorid" notation="string">119451-184976</idno>
              <idno type="ORCID">https://orcid.org/0000-0001-5624-3755</idno>
              <idno type="IDREF">https://www.idref.fr/181622769</idno>
              <affiliation ref="#struct-40217"/>
            </author>
            <author role="aut">
              <persName>
                <forename type="first">Sriram</forename>
                <surname>Sankaranarayanan</surname>
              </persName>
              <idno type="halauthorid">789073-0</idno>
              <affiliation ref="#struct-302215"/>
            </author>
            <editor role="depositor">
              <persName>
                <forename>Jacques-Henri</forename>
                <surname>Jourdan</surname>
              </persName>
              <email type="md5">a6a8d8fe8ecc3f06f85aae7b62e09f04</email>
              <email type="domain">cnrs.fr</email>
            </editor>
          </titleStmt>
          <editionStmt>
            <edition n="v1" type="current">
              <date type="whenSubmitted">2017-11-11 16:04:54</date>
              <date type="whenModified">2025-07-22 08:34:02</date>
              <date type="whenReleased">2017-11-13 11:13:25</date>
              <date type="whenProduced">2014-06-04</date>
              <date type="whenEndEmbargoed">2017-11-11</date>
              <ref type="file" target="https://hal.science/hal-01633155v1/document">
                <date notBefore="2017-11-11"/>
              </ref>
              <ref type="file" subtype="author" n="1" target="https://hal.science/hal-01633155v1/file/goubault2014finding.pdf" id="file-1633155-1678485">
                <date notBefore="2017-11-11"/>
              </ref>
            </edition>
            <respStmt>
              <resp>contributor</resp>
              <name key="434273">
                <persName>
                  <forename>Jacques-Henri</forename>
                  <surname>Jourdan</surname>
                </persName>
                <email type="md5">a6a8d8fe8ecc3f06f85aae7b62e09f04</email>
                <email type="domain">cnrs.fr</email>
              </name>
            </respStmt>
          </editionStmt>
          <publicationStmt>
            <distributor>CCSD</distributor>
            <idno type="halId">hal-01633155</idno>
            <idno type="halUri">https://hal.science/hal-01633155</idno>
            <idno type="halBibtex">goubault:hal-01633155</idno>
            <idno type="halRefHtml">&lt;i&gt;American Control Conference (ACC)&lt;/i&gt;, Jun 2014, Portland, United States</idno>
            <idno type="halRef">American Control Conference (ACC), Jun 2014, Portland, United States</idno>
            <availability status="restricted">
              <licence target="https://about.hal.science/hal-authorisation-v1/">HAL Authorization<ref corresp="#file-1633155-1678485"/></licence>
            </availability>
          </publicationStmt>
          <seriesStmt>
            <idno type="stamp" n="CEA">CEA - Commissariat à l'énergie atomique</idno>
            <idno type="stamp" n="INRIA">INRIA - Institut National de Recherche en Informatique et en Automatique</idno>
            <idno type="stamp" n="INRIA-ROCQ">INRIA Paris - Rocquencourt</idno>
            <idno type="stamp" n="GALLIUM">Collection HAL Gallium</idno>
            <idno type="stamp" n="LORIA2">Publications du LORIA</idno>
            <idno type="stamp" n="TESTALAIN1">TESTALAIN1</idno>
            <idno type="stamp" n="DRT" corresp="CEA">Direction de la recherche technologique</idno>
            <idno type="stamp" n="INRIA2">INRIA 2</idno>
            <idno type="stamp" n="TDS-MACS">Réseau de recherche en Théorie des Systèmes Distribués, Modélisation, Analyse et Contrôle des Systèmes</idno>
            <idno type="stamp" n="LIST" corresp="CEA">Laboratoire d'Intégration des Systèmes et des Technologies</idno>
            <idno type="stamp" n="GS-SPORT-HUMAN-MOVEMENT">Graduate School Sport, Mouvement, Facteurs Humains</idno>
            <idno type="stamp" n="INRIA-ETATSUNIS">Copublications Inria-Etats-Unis</idno>
          </seriesStmt>
          <notesStmt>
            <note type="audience" n="2">International</note>
            <note type="invited" n="0">No</note>
            <note type="popular" n="0">No</note>
            <note type="peer" n="1">Yes</note>
            <note type="proceedings" n="0">No</note>
          </notesStmt>
          <sourceDesc>
            <biblStruct>
              <analytic>
                <title xml:lang="en">Finding Non-Polynomial Positive Invariants and Lyapunov Functions for Polynomial Systems through Darboux Polynomials</title>
                <author role="aut">
                  <persName>
                    <forename type="first">Eric</forename>
                    <surname>Goubault</surname>
                  </persName>
                  <email type="md5">34691a3c485cac901a2837195944f0e2</email>
                  <email type="domain">lix.polytechnique.fr</email>
                  <idno type="idhal" notation="string">eric-goubault</idno>
                  <idno type="idhal" notation="numeric">1172095</idno>
                  <idno type="halauthorid" notation="string">208888-1172095</idno>
                  <idno type="ORCID">https://orcid.org/0000-0002-3198-1863</idno>
                  <affiliation ref="#struct-40217"/>
                </author>
                <author role="aut">
                  <persName>
                    <forename type="first">Jacques-Henri</forename>
                    <surname>Jourdan</surname>
                  </persName>
                  <email type="md5">b4e45c9f9b0894a511cd9acf51057868</email>
                  <email type="domain">normalesup.org</email>
                  <idno type="idhal" notation="string">jacques-henri-jourdan</idno>
                  <idno type="idhal" notation="numeric">10849</idno>
                  <idno type="halauthorid" notation="string">37694-10849</idno>
                  <idno type="ORCID">https://orcid.org/0000-0002-9781-7097</idno>
                  <idno type="GOOGLE SCHOLAR">https://scholar.google.fr/citations?user=2iT-_QoAAAAJ&amp;hl=fr</idno>
                  <idno type="IDREF">https://www.idref.fr/194193454</idno>
                  <affiliation ref="#struct-454413"/>
                </author>
                <author role="aut">
                  <persName>
                    <forename type="first">Sylvie</forename>
                    <surname>Putot</surname>
                  </persName>
                  <email type="md5">623d4bf532ce4189123d269bca70b26b</email>
                  <email type="domain">lix.polytechnique.fr</email>
                  <idno type="idhal" notation="string">sylvie-putot</idno>
                  <idno type="idhal" notation="numeric">184976</idno>
                  <idno type="halauthorid" notation="string">119451-184976</idno>
                  <idno type="ORCID">https://orcid.org/0000-0001-5624-3755</idno>
                  <idno type="IDREF">https://www.idref.fr/181622769</idno>
                  <affiliation ref="#struct-40217"/>
                </author>
                <author role="aut">
                  <persName>
                    <forename type="first">Sriram</forename>
                    <surname>Sankaranarayanan</surname>
                  </persName>
                  <idno type="halauthorid">789073-0</idno>
                  <affiliation ref="#struct-302215"/>
                </author>
              </analytic>
              <monogr>
                <meeting>
                  <title>American Control Conference (ACC)</title>
                  <date type="start">2014-06-04</date>
                  <date type="end">2014-06-06</date>
                  <settlement>Portland</settlement>
                  <country key="US">United States</country>
                </meeting>
                <imprint>
                  <date type="datePub">2014-06</date>
                </imprint>
              </monogr>
            </biblStruct>
          </sourceDesc>
          <profileDesc>
            <langUsage>
              <language ident="en">English</language>
            </langUsage>
            <textClass>
              <classCode scheme="halDomain" n="info.info-pl">Computer Science [cs]/Programming Languages [cs.PL]</classCode>
              <classCode scheme="halDomain" n="info.info-pf">Computer Science [cs]/Performance [cs.PF]</classCode>
              <classCode scheme="halDomain" n="math.math-ds">Mathematics [math]/Dynamical Systems [math.DS]</classCode>
              <classCode scheme="halDomain" n="math.math-oc">Mathematics [math]/Optimization and Control [math.OC]</classCode>
              <classCode scheme="halTypology" n="COMM">Conference papers</classCode>
              <classCode scheme="halOldTypology" n="COMM">Conference papers</classCode>
              <classCode scheme="halTreeTypology" n="COMM">Conference papers</classCode>
            </textClass>
            <abstract xml:lang="en">
              <p>In this paper, we focus on finding positive invari-ants and Lyapunov functions to establish reachability and stability properties, respectively, of polynomial ordinary differential equations (ODEs). In general, the search for such functions is a hard problem. As a result, numerous techniques have been developed to search for polynomial differential variants that yield positive invariants and polynomial Lyapunov functions that prove stability, for systems defined by polynomial differential equations. However, the systematic search for non-polynomial functions is considered a much harder problem, and has received much less attention. In this paper, we combine ideas from computer algebra with the Sum-Of-Squares (SOS) relaxation for polynomial positive semi-definiteness to find non polynomial differential variants and Lyapunov functions for polynomial ODEs. Using the well-known concept of Darboux polynomials, we show how Darboux polynomials can, in many instances, naturally lead to specific forms of Lyapunov functions that involve rational function, logarithmic and exponential terms. We demonstrate the value of our approach by deriving non-polynomial Lyapunov functions for numerical examples drawn from the literature.</p>
            </abstract>
          </profileDesc>
        </biblFull>
      </listBibl>
    </body>
    <back>
      <listOrg type="structures">
        <org type="laboratory" xml:id="struct-40217" status="VALID">
          <idno type="IdRef">156836882</idno>
          <idno type="ISNI">0000 0004 0405 1788</idno>
          <idno type="RNSR">200118591H</idno>
          <idno type="ROR">https://ror.org/000dbcc61</idno>
          <idno type="Wikidata">Q30299467</idno>
          <orgName>Laboratoire d'Intégration des Systèmes et des Technologies</orgName>
          <orgName type="acronym">LIST (CEA)</orgName>
          <desc>
            <address>
              <addrLine>DRT/LISTNano-INNOVAvenue de la Vauve91120 Palaiseau</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www-list.cea.fr/</ref>
          </desc>
          <listRelation>
            <relation name="DRT/LIST" active="#struct-440043" type="direct"/>
            <relation name="DRT" active="#struct-300016" type="indirect"/>
          </listRelation>
        </org>
        <org type="researchteam" xml:id="struct-454413" status="OLD">
          <idno type="RNSR">200618331D</idno>
          <orgName>Langages de programmation, types, compilation et preuves</orgName>
          <orgName type="acronym">GALLIUM</orgName>
          <date type="start">2016-05-01</date>
          <date type="end">2019-07-31</date>
          <desc>
            <address>
              <addrLine>2 rue Simone Iff -CS 42112 -75589 Paris Cedex 12</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.inria.fr/equipes/gallium</ref>
          </desc>
          <listRelation>
            <relation active="#struct-454310" type="direct"/>
            <relation active="#struct-300009" type="indirect"/>
          </listRelation>
        </org>
        <org type="institution" xml:id="struct-302215" status="VALID">
          <idno type="ROR">https://ror.org/02ttsq026</idno>
          <orgName>University of Colorado [Boulder]</orgName>
          <desc>
            <address>
              <addrLine>Boulder, Colorado 80309</addrLine>
              <country key="US"/>
            </address>
            <ref type="url">http://www.colorado.edu/</ref>
          </desc>
        </org>
        <org type="regrouplaboratory" xml:id="struct-440043" status="VALID">
          <idno type="IdRef">067087930</idno>
          <idno type="ISNI">0000000121157881</idno>
          <idno type="RNSR">199018589D</idno>
          <idno type="ROR">https://ror.org/02ggzyd20</idno>
          <idno type="Wikidata">Q30299418</idno>
          <orgName>Direction de Recherche Technologique (CEA)</orgName>
          <orgName type="acronym">DRT (CEA)</orgName>
          <desc>
            <address>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.drt-cea.com/</ref>
          </desc>
          <listRelation>
            <relation name="DRT" active="#struct-300016" type="direct"/>
          </listRelation>
        </org>
        <org type="institution" xml:id="struct-300016" status="VALID">
          <idno type="IdRef">026372061</idno>
          <idno type="ISNI">0000000122998025</idno>
          <idno type="ROR">https://ror.org/00jjx8s55</idno>
          <idno type="Wikidata">Q868550</idno>
          <orgName>Commissariat à l'énergie atomique et aux énergies alternatives</orgName>
          <orgName type="acronym">CEA</orgName>
          <desc>
            <address>
              <addrLine>Centre de SaclayCentre de GrenobleCentre de Cadaracheetc</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.cea.fr/</ref>
          </desc>
        </org>
        <org type="laboratory" xml:id="struct-454310" status="VALID">
          <idno type="IdRef">241614864</idno>
          <idno type="RNSR">196718247G</idno>
          <idno type="ROR">https://ror.org/05eyd5d35</idno>
          <orgName>Centre Inria de Paris</orgName>
          <date type="start">2016-03-10</date>
          <desc>
            <address>
              <addrLine>48 Rue Barrault, 75013 Paris</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.inria.fr/centre/paris</ref>
          </desc>
          <listRelation>
            <relation active="#struct-300009" type="direct"/>
          </listRelation>
        </org>
        <org type="institution" xml:id="struct-300009" status="VALID">
          <idno type="ROR">https://ror.org/02kvxyf05</idno>
          <orgName>Institut National de Recherche en Informatique et en Automatique</orgName>
          <orgName type="acronym">Inria</orgName>
          <desc>
            <address>
              <addrLine>Domaine de VoluceauRocquencourt - BP 10578153 Le Chesnay Cedex</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.inria.fr/en/</ref>
          </desc>
        </org>
      </listOrg>
    </back>
  </text>
</TEI>