{"id":"https://openalex.org/W3010838774","doi":"https://doi.org/10.1109/cdc40024.2019.9030282","title":"On Almost Lyapunov Functions for Systems with Inputs","display_name":"On Almost Lyapunov Functions for Systems with Inputs","publication_year":2019,"publication_date":"2019-12-01","ids":{"openalex":"https://openalex.org/W3010838774","doi":"https://doi.org/10.1109/cdc40024.2019.9030282","mag":"3010838774"},"language":"en","primary_location":{"id":"doi:10.1109/cdc40024.2019.9030282","is_oa":false,"landing_page_url":"https://doi.org/10.1109/cdc40024.2019.9030282","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2019 IEEE 58th Conference on Decision and Control (CDC)","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/A5017797974","display_name":"Shenyu Liu","orcid":"https://orcid.org/0000-0002-4880-8848"},"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":"Shenyu Liu","raw_affiliation_strings":["Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, U. S. A"],"raw_orcid":null,"affiliations":[{"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/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":["Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, U. S. A"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, U. S. A","institution_ids":["https://openalex.org/I157725225"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.1686,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.53834156,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":null,"issue":null,"first_page":"468","last_page":"473"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12772","display_name":"Control and Stability of Dynamical Systems","score":0.9991000294685364,"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/T12772","display_name":"Control and Stability of Dynamical Systems","score":0.9991000294685364,"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/T10040","display_name":"Adaptive Control of Nonlinear Systems","score":0.9945999979972839,"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/T10621","display_name":"Gene Regulatory Network Analysis","score":0.9922000169754028,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/lyapunov-function","display_name":"Lyapunov function","score":0.7528411149978638},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.7168450355529785},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6359877586364746},{"id":"https://openalex.org/keywords/stability-theory","display_name":"Stability theory","score":0.5826809406280518},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.5680351257324219},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.5660110116004944},{"id":"https://openalex.org/keywords/perturbation","display_name":"Perturbation (astronomy)","score":0.5582653880119324},{"id":"https://openalex.org/keywords/exponential-stability","display_name":"Exponential stability","score":0.553951621055603},{"id":"https://openalex.org/keywords/lyapunov-equation","display_name":"Lyapunov equation","score":0.5174232721328735},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.46674591302871704},{"id":"https://openalex.org/keywords/exponential-growth","display_name":"Exponential growth","score":0.4601254463195801},{"id":"https://openalex.org/keywords/trajectory","display_name":"Trajectory","score":0.45366671681404114},{"id":"https://openalex.org/keywords/work","display_name":"Work (physics)","score":0.43421459197998047},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2770787477493286},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.21957501769065857},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.08909532427787781},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.0825059711933136}],"concepts":[{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.7528411149978638},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.7168450355529785},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6359877586364746},{"id":"https://openalex.org/C41949839","wikidata":"https://www.wikidata.org/wiki/Q1756677","display_name":"Stability theory","level":3,"score":0.5826809406280518},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.5680351257324219},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5660110116004944},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.5582653880119324},{"id":"https://openalex.org/C167964875","wikidata":"https://www.wikidata.org/wiki/Q17011487","display_name":"Exponential stability","level":3,"score":0.553951621055603},{"id":"https://openalex.org/C25854792","wikidata":"https://www.wikidata.org/wiki/Q1028945","display_name":"Lyapunov equation","level":4,"score":0.5174232721328735},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.46674591302871704},{"id":"https://openalex.org/C75235859","wikidata":"https://www.wikidata.org/wiki/Q582659","display_name":"Exponential growth","level":2,"score":0.4601254463195801},{"id":"https://openalex.org/C13662910","wikidata":"https://www.wikidata.org/wiki/Q193139","display_name":"Trajectory","level":2,"score":0.45366671681404114},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.43421459197998047},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2770787477493286},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.21957501769065857},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.08909532427787781},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0825059711933136},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","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/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","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/C1276947","wikidata":"https://www.wikidata.org/wiki/Q333","display_name":"Astronomy","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/cdc40024.2019.9030282","is_oa":false,"landing_page_url":"https://doi.org/10.1109/cdc40024.2019.9030282","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2019 IEEE 58th Conference on Decision and Control (CDC)","raw_type":"proceedings-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":14,"referenced_works":["https://openalex.org/W1963987478","https://openalex.org/W1983032944","https://openalex.org/W1987141063","https://openalex.org/W2002964896","https://openalex.org/W2009668633","https://openalex.org/W2010883041","https://openalex.org/W2017493230","https://openalex.org/W2094426806","https://openalex.org/W2134078068","https://openalex.org/W2155689575","https://openalex.org/W2569353530","https://openalex.org/W2589273562","https://openalex.org/W2964117770","https://openalex.org/W3010838774"],"related_works":["https://openalex.org/W2349796700","https://openalex.org/W2383986032","https://openalex.org/W1504912364","https://openalex.org/W2030955003","https://openalex.org/W2374426356","https://openalex.org/W1671909755","https://openalex.org/W2059628530","https://openalex.org/W2210722882","https://openalex.org/W170012745","https://openalex.org/W2963043651"],"abstract_inverted_index":{"In":[0,70],"this":[1,96],"work":[2,11,171],"an":[3,163,175],"almost":[4,74],"Lyapunov":[5,44,75],"function":[6,45,76,82,97],"theorem":[7],"from":[8,168],"our":[9,71,154,169],"recent":[10],"is":[12,19,60,77,110,133,141,159,189],"generalized":[13],"to":[14,67,112,122,124,192],"systems":[15],"with":[16,65,128,139],"inputs.":[17,69],"It":[18,132],"shown":[20,121],"that":[21,31,136,180],"if":[22,145],"for":[23],"any":[24],"inputs":[25,140],"and":[26,88,105,166,172],"initial":[27],"conditions,":[28],"the":[29,34,43,58,68,73,89,93,99,101,107,125,151,178,193],"time":[30,84,100],"solutions":[32,119],"of":[33,83,92,95],"system":[35,59,138,148],"can":[36,182],"stay":[37],"in":[38,153,177],"a":[39,52,81,137],"\"bad\"":[40,108],"region":[41,109],"where":[42],"does":[46],"not":[47],"decrease":[48],"fast":[49],"enough":[50],"has":[51],"sufficiently":[53],"small":[54],"upper":[55,90],"bound,":[56],"then":[57,160],"globally":[61],"exponentially":[62,142],"stable":[63,144,195],"uniformly":[64],"respect":[66],"analysis,":[72],"directly":[78],"expressed":[79],"as":[80],"along":[85],"arbitrary":[86],"solution":[87,102],"bound":[91],"ratio":[94],"at":[98],"trajectory":[103],"leaves":[104],"enters":[106],"found":[111],"be":[113,184],"less":[114],"than":[115],"1.":[116],"Consequently":[117],"all":[118,150],"are":[120],"converge":[123],"origin":[126],"asymptotically":[127],"some":[129],"careful":[130],"justification.":[131],"also":[134],"concluded":[135],"input-to-state":[143],"its":[146],"auxiliary":[147],"satisfies":[149],"hypotheses":[152],"main":[155],"theorem.":[156],"The":[157],"result":[158],"applied":[161],"on":[162],"example":[164],"adopted":[165],"modified":[167],"previous":[170],"it":[173],"shows":[174],"improvement":[176],"sense":[179],"stability":[181],"still":[183],"verified":[185],"even":[186],"when":[187],"there":[188],"stronger":[190],"perturbation":[191],"example's":[194],"dynamics.":[196]},"counts_by_year":[{"year":2019,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}