{"id":"https://openalex.org/W2122496615","doi":"https://doi.org/10.1109/acc.2007.4282376","title":"Controllability and Observability of Systems of Linear Delay Differential Equations via the Matrix Lambert W Function","display_name":"Controllability and Observability of Systems of Linear Delay Differential Equations via the Matrix Lambert W Function","publication_year":2007,"publication_date":"2007-07-01","ids":{"openalex":"https://openalex.org/W2122496615","doi":"https://doi.org/10.1109/acc.2007.4282376","mag":"2122496615"},"language":"en","primary_location":{"id":"doi:10.1109/acc.2007.4282376","is_oa":false,"landing_page_url":"https://doi.org/10.1109/acc.2007.4282376","pdf_url":null,"source":{"id":"https://openalex.org/S4210168941","display_name":"Proceedings of the ... American Control Conference/Proceedings of the American Control Conference","issn_l":"0743-1619","issn":["0743-1619","2378-5861"],"is_oa":false,"is_in_doaj":false,"is_core":false,"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":"conference"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2007 American Control Conference","raw_type":"proceedings-article"},"type":"conference-paper","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/A5042798397","display_name":"Yi Sun\u0087","orcid":"https://orcid.org/0000-0003-2342-4917"},"institutions":[{"id":"https://openalex.org/I27837315","display_name":"University of Michigan","ror":"https://ror.org/00jmfr291","country_code":"US","type":"education","lineage":["https://openalex.org/I27837315"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Sun Yi","raw_affiliation_strings":["Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA","Michigan Univ., Ann Arbor"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA","institution_ids":["https://openalex.org/I27837315"]},{"raw_affiliation_string":"Michigan Univ., Ann Arbor","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5103166555","display_name":"Patrick W. Nelson","orcid":"https://orcid.org/0000-0001-8751-2719"},"institutions":[{"id":"https://openalex.org/I27837315","display_name":"University of Michigan","ror":"https://ror.org/00jmfr291","country_code":"US","type":"education","lineage":["https://openalex.org/I27837315"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Patrick W. Nelson","raw_affiliation_strings":["Department of Mathematics, University of Michigan, Ann Arbor, MI, USA","Michigan Univ., Ann Arbor"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Michigan, Ann Arbor, MI, USA","institution_ids":["https://openalex.org/I27837315"]},{"raw_affiliation_string":"Michigan Univ., Ann Arbor","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5067097078","display_name":"A. Galip Ulsoy","orcid":"https://orcid.org/0000-0002-0808-0602"},"institutions":[{"id":"https://openalex.org/I27837315","display_name":"University of Michigan","ror":"https://ror.org/00jmfr291","country_code":"US","type":"education","lineage":["https://openalex.org/I27837315"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"A. Galip Ulsoy","raw_affiliation_strings":["Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA","Michigan Univ., Ann Arbor"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA","institution_ids":["https://openalex.org/I27837315"]},{"raw_affiliation_string":"Michigan Univ., Ann Arbor","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I27837315"],"apc_list":null,"apc_paid":null,"fwci":1.4954,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.80684932,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"3","issue":null,"first_page":"5631","last_page":"5636"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12677","display_name":"Sports Dynamics and Biomechanics","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2204","display_name":"Biomedical 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/T12677","display_name":"Sports Dynamics and Biomechanics","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2204","display_name":"Biomedical 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/T12250","display_name":"Experimental and Theoretical Physics Studies","score":0.9904999732971191,"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"}},{"id":"https://openalex.org/T10157","display_name":"Sports Performance and Training","score":0.911899983882904,"subfield":{"id":"https://openalex.org/subfields/2732","display_name":"Orthopedics and Sports Medicine"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/observability","display_name":"Observability","score":0.9581964015960693},{"id":"https://openalex.org/keywords/controllability","display_name":"Controllability","score":0.9253511428833008},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6637625694274902},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.6263143420219421},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.5089316368103027},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.5070383548736572},{"id":"https://openalex.org/keywords/linear-differential-equation","display_name":"Linear differential equation","score":0.4804089367389679},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.46251073479652405},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.4623699188232422},{"id":"https://openalex.org/keywords/matrix-function","display_name":"Matrix function","score":0.4620233476161957},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4506740868091583},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.446646511554718},{"id":"https://openalex.org/keywords/delay-differential-equation","display_name":"Delay differential equation","score":0.43326523900032043},{"id":"https://openalex.org/keywords/distributed-parameter-system","display_name":"Distributed parameter system","score":0.4307730197906494},{"id":"https://openalex.org/keywords/state-transition-matrix","display_name":"State-transition matrix","score":0.42549240589141846},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.40849775075912476},{"id":"https://openalex.org/keywords/symmetric-matrix","display_name":"Symmetric matrix","score":0.2313539683818817},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.17559275031089783},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.12449806928634644}],"concepts":[{"id":"https://openalex.org/C36299963","wikidata":"https://www.wikidata.org/wiki/Q1369844","display_name":"Observability","level":2,"score":0.9581964015960693},{"id":"https://openalex.org/C48209547","wikidata":"https://www.wikidata.org/wiki/Q1331104","display_name":"Controllability","level":2,"score":0.9253511428833008},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6637625694274902},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.6263143420219421},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.5089316368103027},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.5070383548736572},{"id":"https://openalex.org/C13982400","wikidata":"https://www.wikidata.org/wiki/Q1129902","display_name":"Linear differential equation","level":3,"score":0.4804089367389679},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.46251073479652405},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.4623699188232422},{"id":"https://openalex.org/C4263655","wikidata":"https://www.wikidata.org/wiki/Q2699958","display_name":"Matrix function","level":4,"score":0.4620233476161957},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4506740868091583},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.446646511554718},{"id":"https://openalex.org/C147504518","wikidata":"https://www.wikidata.org/wiki/Q2146125","display_name":"Delay differential equation","level":3,"score":0.43326523900032043},{"id":"https://openalex.org/C112714845","wikidata":"https://www.wikidata.org/wiki/Q1482001","display_name":"Distributed parameter system","level":3,"score":0.4307730197906494},{"id":"https://openalex.org/C22629506","wikidata":"https://www.wikidata.org/wiki/Q7602993","display_name":"State-transition matrix","level":4,"score":0.42549240589141846},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.40849775075912476},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.2313539683818817},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.17559275031089783},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.12449806928634644},{"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/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.0},{"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/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/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/acc.2007.4282376","is_oa":false,"landing_page_url":"https://doi.org/10.1109/acc.2007.4282376","pdf_url":null,"source":{"id":"https://openalex.org/S4210168941","display_name":"Proceedings of the ... American Control Conference/Proceedings of the American Control Conference","issn_l":"0743-1619","issn":["0743-1619","2378-5861"],"is_oa":false,"is_in_doaj":false,"is_core":false,"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":"conference"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2007 American Control Conference","raw_type":"proceedings-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":41,"referenced_works":["https://openalex.org/W83533993","https://openalex.org/W185935341","https://openalex.org/W1502089761","https://openalex.org/W1534096605","https://openalex.org/W1607633954","https://openalex.org/W1971357915","https://openalex.org/W1981359623","https://openalex.org/W1988098342","https://openalex.org/W1991625105","https://openalex.org/W2012279439","https://openalex.org/W2014172170","https://openalex.org/W2024419968","https://openalex.org/W2032507058","https://openalex.org/W2038296108","https://openalex.org/W2064733273","https://openalex.org/W2068728923","https://openalex.org/W2069481217","https://openalex.org/W2070065489","https://openalex.org/W2072113078","https://openalex.org/W2077101292","https://openalex.org/W2078864748","https://openalex.org/W2090153174","https://openalex.org/W2094827327","https://openalex.org/W2099846541","https://openalex.org/W2106689163","https://openalex.org/W2118683725","https://openalex.org/W2147919225","https://openalex.org/W2152904839","https://openalex.org/W2161443002","https://openalex.org/W2162509049","https://openalex.org/W2167244272","https://openalex.org/W2171322829","https://openalex.org/W2172014397","https://openalex.org/W2620790511","https://openalex.org/W2740934843","https://openalex.org/W4244098448","https://openalex.org/W4293087381","https://openalex.org/W6632138476","https://openalex.org/W6683568376","https://openalex.org/W6738439506","https://openalex.org/W6741785340"],"related_works":["https://openalex.org/W2388793872","https://openalex.org/W3105466232","https://openalex.org/W1994993030","https://openalex.org/W2111253939","https://openalex.org/W3195236465","https://openalex.org/W4390607321","https://openalex.org/W4380680181","https://openalex.org/W1968962728","https://openalex.org/W2527119172","https://openalex.org/W2145456946"],"abstract_inverted_index":{"Controllability":[0],"and":[1,11,14,73,75,101],"observability":[2],"of":[3,25,34,43,80,89],"linear":[4,44],"time":[5,100],"delayed":[6],"systems":[7,42,79],"have":[8,16],"been":[9,17,49],"studied,":[10],"various":[12],"definitions":[13],"criteria":[15,70],"presented":[18,96],"since":[19],"the":[20,23,32,35,39,52,62,85,90,98],"1960s.":[21],"However,":[22],"lack":[24],"an":[26],"analytical":[27],"solution":[28,40,86],"approach":[29],"has":[30,48],"limited":[31],"applicability":[33],"existing":[36],"theories.":[37],"Recently,":[38],"to":[41,61],"delay":[45,81],"differential":[46,67,82],"equations":[47,83],"derived":[50],"using":[51,84],"matrix":[53,64,91],"Lambert":[54,92],"W":[55,93],"function,":[56],"in":[57,65,87],"a":[58],"form":[59],"similar":[60],"transition":[63],"ordinary":[66],"equations.":[68],"The":[69],"for":[71,78,97],"controllability":[72],"observability,":[74],"their":[76],"Gramians,":[77],"terms":[88],"function":[94],"are":[95],"first":[99],"illustrated":[102],"with":[103],"examples.":[104]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2018,"cited_by_count":2}],"updated_date":"2026-07-29T14:22:42.915294","created_date":"2025-10-10T00:00:00"}