{"id":"https://openalex.org/W7203852913","doi":null,"title":"Efficient Solutions of Generalized k-Way Lyapunov Equations and their Application to Control of Nonlinear Systems","display_name":"Efficient Solutions of Generalized k-Way Lyapunov Equations and their Application to Control of Nonlinear Systems","publication_year":2026,"publication_date":"2026-08-17","ids":{"openalex":"https://openalex.org/W7203852913"},"language":"en","primary_location":{"id":"pmh:oai:vtechworks.lib.vt.edu:10919/143735","doi":null,"is_oa":true,"landing_page_url":"https://hdl.handle.net/10919/143735","pdf_url":"https://vtechworks.lib.vt.edu/bitstreams/94dfbe04-a1fe-4efc-9934-f79a594e8790/download","source":{"id":"https://openalex.org/S4306400248","display_name":"VTechWorks (Virginia Tech)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I859038795","host_organization_name":"Virginia Tech","host_organization_lineage":["https://openalex.org/I859038795"],"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":"Dissertation"},"volume":null,"type":"dissertation","is_traditional":true,"indexed_in":[],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://vtechworks.lib.vt.edu/bitstreams/94dfbe04-a1fe-4efc-9934-f79a594e8790/download","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5069202698","display_name":"Hamza Adjerid","orcid":"https://orcid.org/0009-0000-9025-0764","observed_orcids":["https://orcid.org/0009-0000-9025-0764"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Adjerid, Hamza","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5069202698"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9550999999046326,"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":0.9550999999046326,"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/T11081","display_name":"Advanced Control Systems Design","score":0.010300000198185444,"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/T11416","display_name":"Numerical Solvers for Ordinary Differential Equations","score":0.008999999612569809,"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/hamilton-jacobi-bellman-equation","display_name":"Hamilton-Jacobi-Bellman equation","score":0.9580000042915344},{"id":"https://openalex.org/keywords/polynomial-approximation","display_name":"polynomial approximation","score":0.8659999966621399},{"id":"https://openalex.org/keywords/differential-algebraic-equations","display_name":"differential-algebraic equations","score":0.6970000267028809},{"id":"https://openalex.org/keywords/preconditioning","display_name":"preconditioning","score":0.6349999904632568},{"id":"https://openalex.org/keywords/kronecker-product","display_name":"Kronecker product","score":0.8450000286102295},{"id":"https://openalex.org/keywords/low-rank-approximation","display_name":"low-rank approximation","score":0.8610000014305115},{"id":"https://openalex.org/keywords/model-reduction","display_name":"model reduction","score":0.7490000128746033},{"id":"https://openalex.org/keywords/balanced-truncation","display_name":"balanced truncation","score":0.9259999990463257},{"id":"https://openalex.org/keywords/optimal-control","display_name":"optimal control","score":0.7139999866485596},{"id":"https://openalex.org/keywords/nonlinear-systems","display_name":"nonlinear systems","score":0.6629999876022339},{"id":"https://openalex.org/keywords/value-function","display_name":"value function","score":0.628000020980835},{"id":"https://openalex.org/keywords/energy-function","display_name":"energy function","score":0.6759999990463257},{"id":"https://openalex.org/keywords/nonlinear-constraints","display_name":"nonlinear constraints","score":0.7089999914169312}],"concepts":[{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.7160999774932861},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6503000259399414},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.612500011920929},{"id":"https://openalex.org/C25854792","wikidata":"https://www.wikidata.org/wiki/Q1028945","display_name":"Lyapunov equation","level":4,"score":0.5769000053405762},{"id":"https://openalex.org/C37935115","wikidata":"https://www.wikidata.org/wiki/Q6707085","display_name":"Lyapunov redesign","level":4,"score":0.5281999707221985},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5084999799728394},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5067999958992004},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.4830000102519989},{"id":"https://openalex.org/C46030957","wikidata":"https://www.wikidata.org/wiki/Q1238125","display_name":"Kronecker product","level":3,"score":0.4092000126838684},{"id":"https://openalex.org/C23917780","wikidata":"https://www.wikidata.org/wiki/Q50698","display_name":"Algebraic equation","level":3,"score":0.3671000003814697},{"id":"https://openalex.org/C91581856","wikidata":"https://www.wikidata.org/wiki/Q2707001","display_name":"Nonlinear control","level":3,"score":0.3644999861717224},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.3379000127315521},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.3379000127315521},{"id":"https://openalex.org/C101403955","wikidata":"https://www.wikidata.org/wiki/Q6707083","display_name":"Lyapunov optimization","level":5,"score":0.3368000090122223},{"id":"https://openalex.org/C201030206","wikidata":"https://www.wikidata.org/wiki/Q5165805","display_name":"Control-Lyapunov function","level":5,"score":0.33500000834465027},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.33489999175071716},{"id":"https://openalex.org/C186219872","wikidata":"https://www.wikidata.org/wiki/Q955889","display_name":"Differential algebraic equation","level":4,"score":0.32580000162124634},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.3188000023365021},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.3140999972820282},{"id":"https://openalex.org/C191544260","wikidata":"https://www.wikidata.org/wiki/Q1238630","display_name":"Lyapunov exponent","level":3,"score":0.3095000088214874},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.3093000054359436},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3050999939441681},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.28130000829696655},{"id":"https://openalex.org/C39482219","wikidata":"https://www.wikidata.org/wiki/Q192826","display_name":"Kronecker delta","level":2,"score":0.27869999408721924},{"id":"https://openalex.org/C94523830","wikidata":"https://www.wikidata.org/wiki/Q11203","display_name":"System of linear equations","level":2,"score":0.27720001339912415},{"id":"https://openalex.org/C147504518","wikidata":"https://www.wikidata.org/wiki/Q2146125","display_name":"Delay differential equation","level":3,"score":0.26989999413490295},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.25200000405311584}],"mesh":[],"locations_count":1,"locations":[{"id":"pmh:oai:vtechworks.lib.vt.edu:10919/143735","doi":null,"is_oa":true,"landing_page_url":"https://hdl.handle.net/10919/143735","pdf_url":"https://vtechworks.lib.vt.edu/bitstreams/94dfbe04-a1fe-4efc-9934-f79a594e8790/download","source":{"id":"https://openalex.org/S4306400248","display_name":"VTechWorks (Virginia Tech)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I859038795","host_organization_name":"Virginia Tech","host_organization_lineage":["https://openalex.org/I859038795"],"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":"Dissertation"}],"best_oa_location":{"id":"pmh:oai:vtechworks.lib.vt.edu:10919/143735","doi":null,"is_oa":true,"landing_page_url":"https://hdl.handle.net/10919/143735","pdf_url":"https://vtechworks.lib.vt.edu/bitstreams/94dfbe04-a1fe-4efc-9934-f79a594e8790/download","source":{"id":"https://openalex.org/S4306400248","display_name":"VTechWorks (Virginia Tech)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I859038795","host_organization_name":"Virginia Tech","host_organization_lineage":["https://openalex.org/I859038795"],"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":"Dissertation"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G1377434643","display_name":null,"funder_award_id":"CMMI-2130695","funder_id":"https://openalex.org/F4320337391","funder_display_name":"Division of Civil, Mechanical and Manufacturing Innovation"},{"id":"https://openalex.org/G8591983444","display_name":"Collaborative Research: Nonlinear Balancing: Reduced Models and Control","funder_award_id":"2130695","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320337391","display_name":"Division of Civil, Mechanical and Manufacturing Innovation","ror":"https://ror.org/028yd4c30"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7203852913.pdf","grobid_xml":"https://content.openalex.org/works/W7203852913.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W2582365613","https://openalex.org/W4395443227","https://openalex.org/W4417264423","https://openalex.org/W7161557279"],"related_works":["https://openalex.org/W4417264423","https://openalex.org/W1522977283","https://openalex.org/W2905953459","https://openalex.org/W7096907900","https://openalex.org/W2066009815","https://openalex.org/W2914732036","https://openalex.org/W4396216078","https://openalex.org/W2188188881","https://openalex.org/W3012130149","https://openalex.org/W2144096549"],"abstract_inverted_index":{"Nonlinear":[0],"systems":[1,23,49,132],"of":[2,10,21,34,47,60,86,147,154,167,234,237,243],"differential":[3,108],"equations":[4,99,110,121,200],"often":[5],"arise":[6],"from":[7],"the":[8,54,57,84,87,101,128,131,145,179,183,215,227,235,244],"discretization":[9],"PDEs":[11],"or":[12,18,122,214],"modeling":[13],"complex":[14],"processes.":[15],"Model":[16],"reduction":[17,46],"optimal":[19,58],"control":[20,59],"these":[22,35,189],"are":[24,104,133,218,247],"essential":[25],"for":[26,97,151,197],"numerous":[27],"applications":[28],"that":[29,143,160,201],"require":[30,115],"simulation":[31],"and/or":[32],"stabilization":[33],"systems.":[36,170,251],"Energy":[37],"functions":[38,78],"play":[39],"a":[40,69,93,193],"central":[41],"role":[42],"when":[43,208],"performing":[44],"model":[45],"nonlinear":[48,63,102],"using":[50,164],"balanced":[51],"truncation.":[52],"On":[53],"other":[55],"hand,":[56],"problems":[61],"with":[62],"constraints":[64],"can":[65,79,161],"be":[66,81,162,186],"expressed":[67,163],"through":[68],"so-called":[70],"value":[71,74],"function.":[72],"The":[73,112,156,171,221,241],"function":[75],"and":[76,229],"energy":[77],"each":[80],"characterized":[82],"as":[83],"solution":[85,236],"Hamilton-Jacobi-Bellman":[88],"(HJB)":[89],"equations.":[90,126,240],"We":[91],"propose":[92,140],"polynomial":[94,113],"approximation":[95],"framework":[96],"HJB":[98],"where":[100,130],"dynamics":[103,184],"described":[105,134],"by":[106,135],"Stokes-type":[107],"algebraic":[109],"(DAEs).":[111],"coefficients":[114],"solutions":[116],"to":[117,185],"either":[118],"k-way":[119,124,198,238],"Lyapunov":[120,125,199,239],"generalized":[123],"For":[127,188],"case":[129],"sparse":[136],"large":[137],"matrices,":[138],"we":[139,191],"preconditioning":[141],"techniques":[142],"enable":[144],"application":[146],"standard":[148],"iterative":[149],"solvers":[150],"two":[152],"classes":[153],"problems.":[155],"first":[157],"involves":[158],"matrices":[159,213],"Kronecker":[165],"products":[166],"symmetric":[168],"tri-diagonal":[169],"second":[172],"is":[173,202],"more":[174,203],"generally":[175],"applicable,":[176],"but":[177],"requires":[178],"linear":[180],"term":[181],"in":[182],"stable.":[187],"systems,":[190],"developed":[192],"recursive":[194],"ADI":[195],"method":[196,223],"attractive":[204],"than":[205],"existing":[206],"algorithms":[207,246],"some":[209],"specific":[210],"original":[211],"system":[212],"cost":[216],"matrix":[217],"low":[219],"rank.":[220],"proposed":[222],"significantly":[224],"speeds":[225],"up":[226],"simulations":[228],"provides":[230],"accurate":[231],"low-rank":[232],"approximations":[233],"effectiveness":[242],"derived":[245],"demonstrated":[248],"on":[249],"benchmark":[250]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-08-21T00:00:00"}