{"id":"https://openalex.org/W2010883041","doi":"https://doi.org/10.1109/cdc.2014.7039864","title":"On almost Lyapunov functions","display_name":"On almost Lyapunov functions","publication_year":2014,"publication_date":"2014-12-01","ids":{"openalex":"https://openalex.org/W2010883041","doi":"https://doi.org/10.1109/cdc.2014.7039864","mag":"2010883041"},"language":"en","primary_location":{"id":"doi:10.1109/cdc.2014.7039864","is_oa":false,"landing_page_url":"https://doi.org/10.1109/cdc.2014.7039864","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"53rd IEEE Conference on Decision and Control","raw_type":"proceedings-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5086321981","display_name":"Daniel Liberzon","orcid":"https://orcid.org/0000-0003-2383-0114"},"institutions":[{"id":"https://openalex.org/I157725225","display_name":"University of Illinois Urbana-Champaign","ror":"https://ror.org/047426m28","country_code":"US","type":"education","lineage":["https://openalex.org/I157725225"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Daniel Liberzon","raw_affiliation_strings":["Department of Mathematics, University of Illinois at Urbana-Champaign, U.S.A","Coordinated Science Laboratory, University of Illinois at Urbana, Champaign, U.S.A"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Illinois at Urbana-Champaign, U.S.A","institution_ids":["https://openalex.org/I157725225"]},{"raw_affiliation_string":"Coordinated Science Laboratory, University of Illinois at Urbana, Champaign, U.S.A","institution_ids":["https://openalex.org/I157725225"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5008654137","display_name":"Charles Ying","orcid":null},"institutions":[{"id":"https://openalex.org/I157725225","display_name":"University of Illinois Urbana-Champaign","ror":"https://ror.org/047426m28","country_code":"US","type":"education","lineage":["https://openalex.org/I157725225"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Charles Ying","raw_affiliation_strings":["Department of Mathematics, University of Illinois at Urbana-Champaign, U.S.A","Coordinated Science Laboratory, University of Illinois at Urbana, Champaign, U.S.A"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Illinois at Urbana-Champaign, U.S.A","institution_ids":["https://openalex.org/I157725225"]},{"raw_affiliation_string":"Coordinated Science Laboratory, University of Illinois at Urbana, Champaign, U.S.A","institution_ids":["https://openalex.org/I157725225"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5005202787","display_name":"Vadim Zharnitsky","orcid":"https://orcid.org/0000-0003-3417-8993"},"institutions":[{"id":"https://openalex.org/I157725225","display_name":"University of Illinois Urbana-Champaign","ror":"https://ror.org/047426m28","country_code":"US","type":"education","lineage":["https://openalex.org/I157725225"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Vadim Zharnitsky","raw_affiliation_strings":["University of Illinois at Urbana-Champaign, Urbana, IL, US","Department of Mathematics at the University of Illinois at Urbana-Champaign, U.S.A"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Illinois at Urbana-Champaign, Urbana, IL, US","institution_ids":["https://openalex.org/I157725225"]},{"raw_affiliation_string":"Department of Mathematics at the University of Illinois at Urbana-Champaign, U.S.A","institution_ids":["https://openalex.org/I157725225"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.8208,"has_fulltext":false,"cited_by_count":8,"citation_normalized_percentile":{"value":0.77245022,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":null,"issue":null,"first_page":"3083","last_page":"3088"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11110","display_name":"Stability and Controllability of Differential Equations","score":0.9962000250816345,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11110","display_name":"Stability and Controllability of Differential Equations","score":0.9962000250816345,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9958000183105469,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9940000176429749,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/lyapunov-function","display_name":"Lyapunov function","score":0.7640034556388855},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6632875800132751},{"id":"https://openalex.org/keywords/trajectory","display_name":"Trajectory","score":0.5729701519012451},{"id":"https://openalex.org/keywords/perturbation","display_name":"Perturbation (astronomy)","score":0.5723210573196411},{"id":"https://openalex.org/keywords/lyapunov-exponent","display_name":"Lyapunov exponent","score":0.5388009548187256},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5144784450531006},{"id":"https://openalex.org/keywords/exponential-stability","display_name":"Exponential stability","score":0.5026109218597412},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.46082839369773865},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4598219692707062},{"id":"https://openalex.org/keywords/lyapunov-equation","display_name":"Lyapunov equation","score":0.45742735266685486},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4264596700668335},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.4115101397037506},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.41134709119796753},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1953478455543518},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.13026058673858643}],"concepts":[{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.7640034556388855},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6632875800132751},{"id":"https://openalex.org/C13662910","wikidata":"https://www.wikidata.org/wiki/Q193139","display_name":"Trajectory","level":2,"score":0.5729701519012451},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.5723210573196411},{"id":"https://openalex.org/C191544260","wikidata":"https://www.wikidata.org/wiki/Q1238630","display_name":"Lyapunov exponent","level":3,"score":0.5388009548187256},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5144784450531006},{"id":"https://openalex.org/C167964875","wikidata":"https://www.wikidata.org/wiki/Q17011487","display_name":"Exponential stability","level":3,"score":0.5026109218597412},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.46082839369773865},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4598219692707062},{"id":"https://openalex.org/C25854792","wikidata":"https://www.wikidata.org/wiki/Q1028945","display_name":"Lyapunov equation","level":4,"score":0.45742735266685486},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4264596700668335},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.4115101397037506},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.41134709119796753},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1953478455543518},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.13026058673858643},{"id":"https://openalex.org/C1276947","wikidata":"https://www.wikidata.org/wiki/Q333","display_name":"Astronomy","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C2775924081","wikidata":"https://www.wikidata.org/wiki/Q55608371","display_name":"Control (management)","level":2,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/cdc.2014.7039864","is_oa":false,"landing_page_url":"https://doi.org/10.1109/cdc.2014.7039864","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"53rd IEEE Conference on Decision and Control","raw_type":"proceedings-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":9,"referenced_works":["https://openalex.org/W653291882","https://openalex.org/W1511018260","https://openalex.org/W1528301553","https://openalex.org/W1591868040","https://openalex.org/W1999475301","https://openalex.org/W2479575276","https://openalex.org/W3212877690","https://openalex.org/W4234844164","https://openalex.org/W6635287784"],"related_works":["https://openalex.org/W2349796700","https://openalex.org/W1983507234","https://openalex.org/W2284157877","https://openalex.org/W2383986032","https://openalex.org/W2030955003","https://openalex.org/W1789972284","https://openalex.org/W2374426356","https://openalex.org/W2059628530","https://openalex.org/W2164487476","https://openalex.org/W1979419366"],"abstract_inverted_index":{"We":[0,53],"study":[1],"asymptotic":[2],"stability":[3],"properties":[4],"of":[5,11,30,33,41,74,127,134],"nonlinear":[6],"systems":[7],"in":[8,19,111],"the":[9,28,39,59,66,72,75,78,112,115,131,135,140],"presence":[10],"\u201calmost":[12],"Lyapunov\u201d":[13],"functions":[14],"which":[15,99],"decrease":[16],"along":[17],"solutions":[18,56],"a":[20,31,62,96,101],"given":[21,102,141],"region":[22,60],"not":[23,82],"everywhere":[24],"but":[25],"rather":[26],"on":[27,50,71,87],"complement":[29],"set":[32,43,64,76,113],"small":[34,63],"volume.":[35,52],"Nothing":[36],"specific":[37],"about":[38],"structure":[40],"this":[42],"is":[44,93,118],"assumed":[45],"besides":[46],"an":[47],"upper":[48],"bound":[49],"its":[51],"show":[54],"that":[55,108],"starting":[57],"inside":[58],"approach":[61],"around":[65],"origin":[67],"whose":[68],"volume":[69,73],"depends":[70],"where":[77,114],"Lyapunov":[79,116],"function":[80,117],"does":[81],"decrease,":[83,121],"as":[84,86],"well":[85],"other":[88],"system":[89,103],"parameters.":[90],"The":[91],"result":[92],"established":[94],"by":[95],"perturbation":[97],"argument":[98],"compares":[100],"trajectory":[104],"with":[105],"nearby":[106],"trajectories":[107,129],"lie":[109],"entirely":[110],"known":[119],"to":[120,137],"and":[122],"trades":[123],"off":[124],"convergence":[125],"speed":[126],"these":[128],"against":[130],"expansion":[132],"rate":[133],"distance":[136],"them":[138],"from":[139],"trajectory.":[142]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}