{"id":"https://openalex.org/W3044089286","doi":"https://doi.org/10.1137/19m1304106","title":"Bilinear Systems---A New Link to $\\mathcal H_2$-norms, Relations to Stochastic Systems, and Further Properties","display_name":"Bilinear Systems---A New Link to $\\mathcal H_2$-norms, Relations to Stochastic Systems, and Further Properties","publication_year":2021,"publication_date":"2021-01-01","ids":{"openalex":"https://openalex.org/W3044089286","doi":"https://doi.org/10.1137/19m1304106","mag":"3044089286"},"language":"en","primary_location":{"id":"doi:10.1137/19m1304106","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m1304106","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":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1910.14427","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5053852920","display_name":"Martin Redmann","orcid":null},"institutions":[{"id":"https://openalex.org/I2801009184","display_name":"Luther University","ror":"https://ror.org/015dhy417","country_code":"KR","type":"education","lineage":["https://openalex.org/I2801009184"]}],"countries":["KR"],"is_corresponding":true,"raw_author_name":"Martin Redmann","raw_affiliation_strings":["Martin Luther University of Halle - Wittenberg"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Martin Luther University of Halle - Wittenberg","institution_ids":["https://openalex.org/I2801009184"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5053852920"],"corresponding_institution_ids":["https://openalex.org/I2801009184"],"apc_list":null,"apc_paid":null,"fwci":0.116,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.37319588,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"59","issue":"4","first_page":"2477","last_page":"2497"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":1.0,"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"}},"topics":[{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":1.0,"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/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9975000023841858,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.9939000010490417,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/reachability","display_name":"Reachability","score":0.785973310470581},{"id":"https://openalex.org/keywords/gramian-matrix","display_name":"Gramian matrix","score":0.7652488350868225},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6518084406852722},{"id":"https://openalex.org/keywords/bilinear-interpolation","display_name":"Bilinear interpolation","score":0.6068555116653442},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.5304906368255615},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5073768496513367},{"id":"https://openalex.org/keywords/reduction","display_name":"Reduction (mathematics)","score":0.4808693528175354},{"id":"https://openalex.org/keywords/lemma","display_name":"Lemma (botany)","score":0.445454865694046},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.4173596203327179},{"id":"https://openalex.org/keywords/probabilistic-logic","display_name":"Probabilistic logic","score":0.4123764634132385},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3926299512386322},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.35063308477401733},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.1461673080921173},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.10902050137519836},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08072617650032043}],"concepts":[{"id":"https://openalex.org/C136643341","wikidata":"https://www.wikidata.org/wiki/Q1361526","display_name":"Reachability","level":2,"score":0.785973310470581},{"id":"https://openalex.org/C77246614","wikidata":"https://www.wikidata.org/wiki/Q1409400","display_name":"Gramian matrix","level":3,"score":0.7652488350868225},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6518084406852722},{"id":"https://openalex.org/C205203396","wikidata":"https://www.wikidata.org/wiki/Q612143","display_name":"Bilinear interpolation","level":2,"score":0.6068555116653442},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.5304906368255615},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5073768496513367},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.4808693528175354},{"id":"https://openalex.org/C2777759810","wikidata":"https://www.wikidata.org/wiki/Q149316","display_name":"Lemma (botany)","level":3,"score":0.445454865694046},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.4173596203327179},{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.4123764634132385},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3926299512386322},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.35063308477401733},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.1461673080921173},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.10902050137519836},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08072617650032043},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"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/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/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","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/C46757340","wikidata":"https://www.wikidata.org/wiki/Q43238","display_name":"Poaceae","level":2,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/19m1304106","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m1304106","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"},{"id":"pmh:oai:arXiv.org:1910.14427","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1910.14427","pdf_url":"https://arxiv.org/pdf/1910.14427","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"mag:3044089286","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1910.14427v5","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.1910.14427","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1910.14427","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1910.14427","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1910.14427","pdf_url":"https://arxiv.org/pdf/1910.14427","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[{"score":0.8500000238418579,"id":"https://metadata.un.org/sdg/7","display_name":"Affordable and clean energy"}],"awards":[{"id":"https://openalex.org/G3876516530","display_name":null,"funder_award_id":"FOR 2402","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320310969","display_name":"Technische Universit\u00e4t Kaiserslautern","ror":"https://ror.org/04zrf7b53"},{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3044089286.pdf","grobid_xml":"https://content.openalex.org/works/W3044089286.grobid-xml"},"referenced_works_count":29,"referenced_works":["https://openalex.org/W37887877","https://openalex.org/W651734400","https://openalex.org/W1511438772","https://openalex.org/W1584666263","https://openalex.org/W1842378573","https://openalex.org/W1966992674","https://openalex.org/W2002858896","https://openalex.org/W2003814439","https://openalex.org/W2019854158","https://openalex.org/W2033314416","https://openalex.org/W2061433628","https://openalex.org/W2069049088","https://openalex.org/W2080209002","https://openalex.org/W2091160474","https://openalex.org/W2102316513","https://openalex.org/W2111221533","https://openalex.org/W2137874355","https://openalex.org/W2146247463","https://openalex.org/W2155711822","https://openalex.org/W2530996620","https://openalex.org/W2614367992","https://openalex.org/W2618826795","https://openalex.org/W2743905828","https://openalex.org/W2808648567","https://openalex.org/W2900536097","https://openalex.org/W2962748632","https://openalex.org/W2963756918","https://openalex.org/W3029200679","https://openalex.org/W3105351150"],"related_works":["https://openalex.org/W2923281596","https://openalex.org/W2752152889","https://openalex.org/W2972470817","https://openalex.org/W3178693814","https://openalex.org/W2257960631","https://openalex.org/W2299320942","https://openalex.org/W3027542739","https://openalex.org/W2610527164","https://openalex.org/W2307333708","https://openalex.org/W1960771713","https://openalex.org/W3037114412","https://openalex.org/W2277381926","https://openalex.org/W2182275140","https://openalex.org/W166706841","https://openalex.org/W3117115292","https://openalex.org/W837044083","https://openalex.org/W2557748596","https://openalex.org/W2796192311","https://openalex.org/W3042824819","https://openalex.org/W2896602708"],"abstract_inverted_index":{"In":[0],"this":[1,59,128],"paper,":[2,60],"we":[3,27],"prove":[4],"several":[5,81],"new":[6,10,64,132],"results":[7,133],"that":[8,144],"give":[9],"insights":[11],"into":[12],"bilinear":[13,21,35,76],"systems.":[14,77],"We":[15],"show":[16],"under":[17],"which":[18],"conditions":[19],"a":[20,29,39,63,137],"system":[22],"is":[23,62,145],"asymptotically":[24],"stable.":[25],"Moreover,":[26,108],"provide":[28],"global":[30],"characterization":[31],"of":[32,58,74,86,104],"reachability":[33],"in":[34,83,102,127],"systems":[36],"based":[37,111,135],"on":[38,112,136],"certain":[40],"Gramian.":[41],"Reachability":[42],"energy":[43],"estimates":[44],"using":[45],"the":[46,67,71,84,105,113],"same":[47],"Gramian":[48],"have":[49],"only":[50],"been":[51],"local":[52],"so":[53],"far.":[54],"The":[55],"main":[56],"result":[57,79],"however,":[61],"link":[65],"between":[66],"output":[68,106,109],"error":[69],"and":[70],"$\\mathcal":[72,93,114],"H_2$-error":[73],"two":[75],"This":[78],"has":[80],"consequences":[82],"field":[85],"model":[87,95,123],"order":[88,96,124],"reduction.":[89],"It":[90],"explains":[91],"why":[92],"H_2$-optimal":[94],"reduction":[97,125],"leads":[98],"to":[99],"good":[100],"approximations":[101],"terms":[103],"error.":[107],"errors":[110],"H_2$-norm":[115],"can":[116],"now":[117],"be":[118],"proved":[119],"for":[120,140],"balancing":[121],"related":[122],"schemes":[126],"paper.":[129],"All":[130],"these":[131],"are":[134],"Gronwall":[138],"lemma":[139],"matrix":[141],"differential":[142],"equations":[143],"established":[146],"here.":[147]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":1}],"updated_date":"2026-07-15T18:14:33.161393","created_date":"2025-10-10T00:00:00"}