{"id":"https://openalex.org/W2016627250","doi":"https://doi.org/10.1109/tac.2013.2287112","title":"A Limiting Property of the Matrix Exponential","display_name":"A Limiting Property of the Matrix Exponential","publication_year":2013,"publication_date":"2013-10-24","ids":{"openalex":"https://openalex.org/W2016627250","doi":"https://doi.org/10.1109/tac.2013.2287112","mag":"2016627250"},"language":"en","primary_location":{"id":"doi:10.1109/tac.2013.2287112","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tac.2013.2287112","pdf_url":null,"source":{"id":"https://openalex.org/S184954342","display_name":"IEEE Transactions on Automatic Control","issn_l":"0018-9286","issn":["0018-9286","1558-2523","2334-3303"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Automatic Control","raw_type":"journal-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/A5023990842","display_name":"Sebastian Trimpe","orcid":"https://orcid.org/0000-0002-2785-2487"},"institutions":[{"id":"https://openalex.org/I4210135521","display_name":"Max Planck Institute for Intelligent Systems","ror":"https://ror.org/04fq9j139","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210135521"]},{"id":"https://openalex.org/I4210165766","display_name":"Max Planck Institute for Biology","ror":"https://ror.org/0243gzr89","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210165766"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Sebastian Trimpe","raw_affiliation_strings":["Max-Planck-Institute for Intelligent Systems, T\u00fcbingen, Germany","Max Planck Inst. for Intell. Syst., Tubingen, Germany"],"affiliations":[{"raw_affiliation_string":"Max-Planck-Institute for Intelligent Systems, T\u00fcbingen, Germany","institution_ids":["https://openalex.org/I4210135521"]},{"raw_affiliation_string":"Max Planck Inst. for Intell. Syst., Tubingen, Germany","institution_ids":["https://openalex.org/I4210165766"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5050557198","display_name":"Raffaello D\u2019Andrea","orcid":"https://orcid.org/0000-0001-5287-7849"},"institutions":[{"id":"https://openalex.org/I35440088","display_name":"ETH Zurich","ror":"https://ror.org/05a28rw58","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I35440088"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Raffaello D'Andrea","raw_affiliation_strings":["Institute for Dynamic Systems and Control, Zurich, ETH Zurich, Switzerland","[Institute for Dynamic Systems and Control, ETH Zurich, Zurich, Switzerland]"],"affiliations":[{"raw_affiliation_string":"Institute for Dynamic Systems and Control, Zurich, ETH Zurich, Switzerland","institution_ids":["https://openalex.org/I35440088"]},{"raw_affiliation_string":"[Institute for Dynamic Systems and Control, ETH Zurich, Zurich, Switzerland]","institution_ids":["https://openalex.org/I35440088"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5023990842"],"corresponding_institution_ids":["https://openalex.org/I4210135521","https://openalex.org/I4210165766"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.0878575,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":95},"biblio":{"volume":"59","issue":"4","first_page":"1105","last_page":"1110"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T10046","display_name":"Stability and Control of Uncertain Systems","score":0.9973000288009644,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7010782957077026},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5609543323516846},{"id":"https://openalex.org/keywords/matrix-exponential","display_name":"Matrix exponential","score":0.5360159873962402},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.5298722982406616},{"id":"https://openalex.org/keywords/limiting","display_name":"Limiting","score":0.49610647559165955},{"id":"https://openalex.org/keywords/polynomial-matrix","display_name":"Polynomial matrix","score":0.46021923422813416},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.3634452223777771},{"id":"https://openalex.org/keywords/matrix-polynomial","display_name":"Matrix polynomial","score":0.3332196772098541},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.28569626808166504},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.07695308327674866}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7010782957077026},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5609543323516846},{"id":"https://openalex.org/C195906000","wikidata":"https://www.wikidata.org/wiki/Q1191722","display_name":"Matrix exponential","level":3,"score":0.5360159873962402},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.5298722982406616},{"id":"https://openalex.org/C188198153","wikidata":"https://www.wikidata.org/wiki/Q1613840","display_name":"Limiting","level":2,"score":0.49610647559165955},{"id":"https://openalex.org/C126352355","wikidata":"https://www.wikidata.org/wiki/Q2562273","display_name":"Polynomial matrix","level":4,"score":0.46021923422813416},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.3634452223777771},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.3332196772098541},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.28569626808166504},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.07695308327674866},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tac.2013.2287112","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tac.2013.2287112","pdf_url":null,"source":{"id":"https://openalex.org/S184954342","display_name":"IEEE Transactions on Automatic Control","issn_l":"0018-9286","issn":["0018-9286","1558-2523","2334-3303"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Automatic Control","raw_type":"journal-article"},{"id":"pmh:oai:pure.mpg.de:item_2088577","is_oa":false,"landing_page_url":"http://hdl.handle.net/11858/00-001M-0000-0024-C4DF-E","pdf_url":null,"source":{"id":"https://openalex.org/S4306400654","display_name":"MPG.PuRe (Max Planck Society)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I149899117","host_organization_name":"Max Planck Society","host_organization_lineage":["https://openalex.org/I149899117"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"IEEE Transactions on Automatic Control","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":11,"referenced_works":["https://openalex.org/W590529998","https://openalex.org/W636289703","https://openalex.org/W2003137566","https://openalex.org/W2032607425","https://openalex.org/W2068992774","https://openalex.org/W2085804838","https://openalex.org/W2091359949","https://openalex.org/W2106565812","https://openalex.org/W2171912944","https://openalex.org/W2742470390","https://openalex.org/W4240790602"],"related_works":["https://openalex.org/W2056024401","https://openalex.org/W2011489190","https://openalex.org/W3216258873","https://openalex.org/W2149734642","https://openalex.org/W2185931218","https://openalex.org/W4286858160","https://openalex.org/W3214488505","https://openalex.org/W2466390518","https://openalex.org/W2072116916","https://openalex.org/W4200197172"],"abstract_inverted_index":{"A":[0],"limiting":[1,22,40,79],"property":[2,80],"of":[3,12,27,49,69,85,103],"the":[4,10,24,34,46,50,62,66,70,101,104],"matrix":[5,16,25,29,63],"exponential":[6,26],"is":[7,42,76,81],"proven:":[8],"if":[9],"(1,1)-block":[11,51],"a":[13,21,58],"2-by-2":[14],"block":[15],"becomes":[17],"\u201carbitrarily":[18],"small\u201d":[19],"in":[20,33],"process,":[23],"that":[28,44,72,88],"tends":[30,73],"to":[31,53,74],"zero":[32],"(1,1)-,":[35],"(1,2)-,":[36],"and":[37],"(2,1)-blocks.":[38],"The":[39,78,96],"process":[41],"such":[43],"either":[45],"log":[47],"norm":[48],"goes":[52],"negative":[54],"infinity,":[55],"or,":[56],"for":[57,83],"certain":[59],"polynomial":[60],"dependency,":[61],"associated":[64],"with":[65,91],"largest":[67],"power":[68],"variable":[71],"infinity":[75],"stable.":[77],"useful":[82],"simplification":[84],"dynamic":[86],"systems":[87],"exhibit":[89],"modes":[90],"sufficiently":[92],"different":[93],"time":[94],"scales.":[95],"obtained":[97],"limit":[98],"then":[99],"implies":[100],"decoupling":[102],"corresponding":[105],"dynamics.":[106]},"counts_by_year":[{"year":2025,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}