{"id":"https://openalex.org/W2092114072","doi":"https://doi.org/10.1137/s036301299833464x","title":"Brockett's Problem of Classification of Finite-Dimensional Estimation Algebras for Nonlinear Filtering Systems","display_name":"Brockett's Problem of Classification of Finite-Dimensional Estimation Algebras for Nonlinear Filtering Systems","publication_year":2000,"publication_date":"2000-01-01","ids":{"openalex":"https://openalex.org/W2092114072","doi":"https://doi.org/10.1137/s036301299833464x","mag":"2092114072"},"language":"en","primary_location":{"id":"doi:10.1137/s036301299833464x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s036301299833464x","pdf_url":null,"source":{"id":"https://openalex.org/S897311980","display_name":"SIAM Journal on Control and Optimization","issn_l":"0363-0129","issn":["0363-0129","1095-7138"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Control and Optimization","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/A5012568499","display_name":"Shanjian Tang","orcid":"https://orcid.org/0000-0003-3884-042X"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Shanjian Tang","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5012568499"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.13138495,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"39","issue":"3","first_page":"900","last_page":"916"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11236","display_name":"Control Systems and Identification","score":0.9478999972343445,"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/T11236","display_name":"Control Systems and Identification","score":0.9478999972343445,"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/T10046","display_name":"Stability and Control of Uncertain Systems","score":0.9068999886512756,"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":0.8696253299713135},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.7566778063774109},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.7055102586746216},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.6122105121612549},{"id":"https://openalex.org/keywords/state","display_name":"State (computer science)","score":0.5217791199684143},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5197815299034119},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.49507370591163635},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.47282397747039795},{"id":"https://openalex.org/keywords/rank-condition","display_name":"Rank condition","score":0.47074469923973083},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4560298025608063},{"id":"https://openalex.org/keywords/state-space","display_name":"State space","score":0.4434434175491333},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.43758949637413025},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.43143025040626526},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4085652530193329},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.33048850297927856},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.24628418684005737},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.1560986042022705}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8696253299713135},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.7566778063774109},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.7055102586746216},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.6122105121612549},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.5217791199684143},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5197815299034119},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.49507370591163635},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.47282397747039795},{"id":"https://openalex.org/C2778095854","wikidata":"https://www.wikidata.org/wiki/Q4420961","display_name":"Rank condition","level":3,"score":0.47074469923973083},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4560298025608063},{"id":"https://openalex.org/C72434380","wikidata":"https://www.wikidata.org/wiki/Q230930","display_name":"State space","level":2,"score":0.4434434175491333},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.43758949637413025},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.43143025040626526},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4085652530193329},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.33048850297927856},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.24628418684005737},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.1560986042022705},{"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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","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/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","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/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C48209547","wikidata":"https://www.wikidata.org/wiki/Q1331104","display_name":"Controllability","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/s036301299833464x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s036301299833464x","pdf_url":null,"source":{"id":"https://openalex.org/S897311980","display_name":"SIAM Journal on Control and Optimization","issn_l":"0363-0129","issn":["0363-0129","1095-7138"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Control and Optimization","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":18,"referenced_works":["https://openalex.org/W238977847","https://openalex.org/W1479099019","https://openalex.org/W1492158924","https://openalex.org/W1598534740","https://openalex.org/W1965581829","https://openalex.org/W1967008202","https://openalex.org/W1968152720","https://openalex.org/W1981762376","https://openalex.org/W1996077195","https://openalex.org/W1996374249","https://openalex.org/W1997032888","https://openalex.org/W2009360307","https://openalex.org/W2036922159","https://openalex.org/W2047912374","https://openalex.org/W2084400252","https://openalex.org/W2102291236","https://openalex.org/W2123359798","https://openalex.org/W4243913600"],"related_works":["https://openalex.org/W2087734572","https://openalex.org/W4230292203","https://openalex.org/W2105325102","https://openalex.org/W1603450860","https://openalex.org/W1975300386","https://openalex.org/W2157593928","https://openalex.org/W4210496372","https://openalex.org/W2035756021","https://openalex.org/W186536637","https://openalex.org/W2092114072"],"abstract_inverted_index":{"In":[0,62],"his":[1],"talk":[2],"at":[3],"the":[4,50,53,74,86,99,102,122],"International":[5],"Congress":[6],"of":[7,46,52,69,82,118],"Mathematicians":[8],"in":[9,20,112],"1982,":[10],"Brockett":[11],"proposed":[12],"to":[13,60],"classify":[14],"all":[15,42,114],"finite-dimensional":[16,43,115],"estimation":[17,44,75,116],"algebras":[18,45,117],"arising":[19],"nonlinear":[21],"filters.":[22],"Recently,":[23],"Chen,":[24],"Yau,":[25],"and":[26,77,108],"Leung":[27],"[":[28],"SIAM":[29],"J.":[30],"Control":[31],"Optim.,":[32],"35":[33],"(1997),":[34],"pp.":[35],"1132--1141]":[36],"claimed":[37],"that":[38,101],"they":[39],"had":[40],"classified":[41],"maximal":[47,119],"rank":[48,120],"when":[49],"dimension":[51,96,124],"state-space":[54,123],"is":[55,104],"less":[56],"than":[57],"or":[58],"equal":[59],"four.":[61],"this":[63],"paper,":[64],"we":[65,78,91,110],"introduce":[66],"a":[67,89,105],"series":[68],"completely":[70],"new":[71,83],"computations":[72],"about":[73,85],"algebra,":[76],"find":[79],"two":[80],"sets":[81],"equations":[84],"$\\Omega$-matrix.":[87],"As":[88],"consequence,":[90],"can":[92],"prove,":[93],"without":[94],"any":[95],"assumption":[97],"on":[98],"state-space,":[100],"$\\Omega$-matrix":[103],"constant":[106],"matrix,":[107],"thus":[109],"succeed":[111],"classifying":[113],"with":[121],"being":[125],"arbitrary.":[126]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}