{"id":"https://openalex.org/W1567799384","doi":"https://doi.org/10.1109/cdc.2002.1184809","title":"Nonsmooth functions and uniform limits of smooth control Lyapunov functions","display_name":"Nonsmooth functions and uniform limits of smooth control Lyapunov functions","publication_year":2003,"publication_date":"2003-08-27","ids":{"openalex":"https://openalex.org/W1567799384","doi":"https://doi.org/10.1109/cdc.2002.1184809","mag":"1567799384"},"language":"en","primary_location":{"id":"doi:10.1109/cdc.2002.1184809","is_oa":false,"landing_page_url":"https://doi.org/10.1109/cdc.2002.1184809","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the 41st IEEE Conference on Decision and Control, 2002.","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/A5051861988","display_name":"Ludovic Faubourg","orcid":null},"institutions":[{"id":"https://openalex.org/I177064439","display_name":"Universit\u00e9 de Bourgogne","ror":"https://ror.org/03k1bsr36","country_code":"FR","type":"education","lineage":["https://openalex.org/I177064439"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"L. Faubourg","raw_affiliation_strings":["LAAO, Universit\u00e9 de Bourgogne, France"],"affiliations":[{"raw_affiliation_string":"LAAO, Universit\u00e9 de Bourgogne, France","institution_ids":["https://openalex.org/I177064439"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5105403343","display_name":"Jean\u2010Baptiste Pomet","orcid":null},"institutions":[{"id":"https://openalex.org/I1326498283","display_name":"Institut national de recherche en informatique et en automatique","ror":"https://ror.org/02kvxyf05","country_code":"FR","type":"funder","lineage":["https://openalex.org/I1326498283"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"J.-B. Pomet","raw_affiliation_strings":["INRIA, Sophia-Antipolis, France"],"affiliations":[{"raw_affiliation_string":"INRIA, Sophia-Antipolis, France","institution_ids":["https://openalex.org/I1326498283"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5051861988"],"corresponding_institution_ids":["https://openalex.org/I177064439"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.05474408,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"2","issue":null,"first_page":"1932","last_page":"1933"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10040","display_name":"Adaptive Control of Nonlinear Systems","score":0.9829000234603882,"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/T10040","display_name":"Adaptive Control of Nonlinear Systems","score":0.9829000234603882,"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/T12772","display_name":"Control and Stability of Dynamical Systems","score":0.972100019454956,"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/T10301","display_name":"Mitochondrial Function and Pathology","score":0.9402999877929688,"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":0.8012349009513855},{"id":"https://openalex.org/keywords/control-lyapunov-function","display_name":"Control-Lyapunov function","score":0.745927095413208},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.69215327501297},{"id":"https://openalex.org/keywords/lyapunov-redesign","display_name":"Lyapunov redesign","score":0.6699888706207275},{"id":"https://openalex.org/keywords/lyapunov-equation","display_name":"Lyapunov equation","score":0.652973473072052},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.6110021471977234},{"id":"https://openalex.org/keywords/lyapunov-exponent","display_name":"Lyapunov exponent","score":0.6077181696891785},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.5334734320640564},{"id":"https://openalex.org/keywords/almost-everywhere","display_name":"Almost everywhere","score":0.5104314684867859},{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.5058555603027344},{"id":"https://openalex.org/keywords/limit-of-a-sequence","display_name":"Limit of a sequence","score":0.49073827266693115},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4636327922344208},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4568539261817932},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.4422692656517029},{"id":"https://openalex.org/keywords/lyapunov-optimization","display_name":"Lyapunov optimization","score":0.4299573302268982},{"id":"https://openalex.org/keywords/limit-of-a-function","display_name":"Limit of a function","score":0.4299180507659912},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.41711798310279846},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.38737955689430237},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.29054152965545654},{"id":"https://openalex.org/keywords/control","display_name":"Control (management)","score":0.18047478795051575},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.15373358130455017},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.10043692588806152},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08507019281387329}],"concepts":[{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.8012349009513855},{"id":"https://openalex.org/C201030206","wikidata":"https://www.wikidata.org/wiki/Q5165805","display_name":"Control-Lyapunov function","level":5,"score":0.745927095413208},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.69215327501297},{"id":"https://openalex.org/C37935115","wikidata":"https://www.wikidata.org/wiki/Q6707085","display_name":"Lyapunov redesign","level":4,"score":0.6699888706207275},{"id":"https://openalex.org/C25854792","wikidata":"https://www.wikidata.org/wiki/Q1028945","display_name":"Lyapunov equation","level":4,"score":0.652973473072052},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.6110021471977234},{"id":"https://openalex.org/C191544260","wikidata":"https://www.wikidata.org/wiki/Q1238630","display_name":"Lyapunov exponent","level":3,"score":0.6077181696891785},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.5334734320640564},{"id":"https://openalex.org/C27142425","wikidata":"https://www.wikidata.org/wiki/Q1139334","display_name":"Almost everywhere","level":2,"score":0.5104314684867859},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.5058555603027344},{"id":"https://openalex.org/C71287196","wikidata":"https://www.wikidata.org/wiki/Q847204","display_name":"Limit of a sequence","level":3,"score":0.49073827266693115},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4636327922344208},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4568539261817932},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.4422692656517029},{"id":"https://openalex.org/C101403955","wikidata":"https://www.wikidata.org/wiki/Q6707083","display_name":"Lyapunov optimization","level":5,"score":0.4299573302268982},{"id":"https://openalex.org/C19448537","wikidata":"https://www.wikidata.org/wiki/Q33540","display_name":"Limit of a function","level":3,"score":0.4299180507659912},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.41711798310279846},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.38737955689430237},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.29054152965545654},{"id":"https://openalex.org/C2775924081","wikidata":"https://www.wikidata.org/wiki/Q55608371","display_name":"Control (management)","level":2,"score":0.18047478795051575},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.15373358130455017},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.10043692588806152},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08507019281387329},{"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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","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/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","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":2,"locations":[{"id":"doi:10.1109/cdc.2002.1184809","is_oa":false,"landing_page_url":"https://doi.org/10.1109/cdc.2002.1184809","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the 41st IEEE Conference on Decision and Control, 2002.","raw_type":"proceedings-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.122.2538","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.122.2538","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.rutgers.edu/~sontag/FTP_DIR/02cdc-papers-refs-eds/355_WeM11-6.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":6,"referenced_works":["https://openalex.org/W593474027","https://openalex.org/W1960239420","https://openalex.org/W1975161889","https://openalex.org/W2041076275","https://openalex.org/W2117431422","https://openalex.org/W6617828207"],"related_works":["https://openalex.org/W3217133416","https://openalex.org/W2383986032","https://openalex.org/W1970635650","https://openalex.org/W2349796700","https://openalex.org/W2546013078","https://openalex.org/W2588600327","https://openalex.org/W3008905280","https://openalex.org/W4380686336","https://openalex.org/W2401318053","https://openalex.org/W2351869311"],"abstract_inverted_index":{"When":[0],"is":[1,18,34,51],"a":[2,8,21,39,56,62,66,73,82,93],"(non-smooth)":[3],"function":[4,25,67,85,96],"the":[5,27,30,36,49,70,98,101],"limit":[6,37,71],"of":[7,10,29,38,41,72,75,100],"sequence":[9,40,74],"smooth":[11,42,76],"(continuously":[12],"differentiable)":[13],"control":[14,23,43,77,94],"Lyapunov":[15,24,44,78,95],"functions?":[16],"It":[17],"known":[19],"that":[20,48,86],"\"non-smooth":[22],"in":[26,97],"sense":[28,99],"Clarke":[31,102],"(convex)":[32],"gradient\"":[33],"indeed":[35],"functions;":[45],"we":[46],"show":[47],"converse":[50],"not":[52],"true":[53],"by":[54],"exhibiting":[55],"counter":[57],"example.":[58],"We":[59],"also":[60],"give":[61],"condition":[63,89],"under":[64],"which":[65],"cannot":[68,90],"be":[69,92],"functions.":[79],"Of":[80],"course,":[81],"(non":[83],"smooth)":[84],"satisfies":[87],"this":[88],"either":[91],"gradient.":[103]},"counts_by_year":[],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}