{"id":"https://openalex.org/W4233121385","doi":"https://doi.org/10.1137/s0363012901385198","title":"Asymptotic Control of Pairs of Oscillators Coupled by a Repulsion, with Nonisolated Equilibria I: The Regular Case","display_name":"Asymptotic Control of Pairs of Oscillators Coupled by a Repulsion, with Nonisolated Equilibria I: The Regular Case","publication_year":2002,"publication_date":"2002-01-01","ids":{"openalex":"https://openalex.org/W4233121385","doi":"https://doi.org/10.1137/s0363012901385198"},"language":"en","primary_location":{"id":"doi:10.1137/s0363012901385198","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0363012901385198","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/A5110183023","display_name":"Alexandre Cabot","orcid":null},"institutions":[{"id":"https://openalex.org/I65806277","display_name":"Universit\u00e9 de Limoges","ror":"https://ror.org/02cp04407","country_code":"FR","type":"education","lineage":["https://openalex.org/I65806277"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Alexandre Cabot","raw_affiliation_strings":["LACO - Laboratoire d'Arithm\u00e9tique, de Calcul formel et d'Optimisation (Universit\u00e9 de Limoges Facult\u00e9 des Sciences et Techniques 123 av A. Thomas\r\n87060 LIMOGES cedex\r\n - France)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"LACO - Laboratoire d'Arithm\u00e9tique, de Calcul formel et d'Optimisation (Universit\u00e9 de Limoges Facult\u00e9 des Sciences et Techniques 123 av A. Thomas\r\n87060 LIMOGES cedex\r\n - France)","institution_ids":["https://openalex.org/I65806277"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5071460705","display_name":"Marc-Olivier Czarnecki","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Marc-Olivier Czarnecki","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":5.3793,"has_fulltext":false,"cited_by_count":7,"citation_normalized_percentile":{"value":0.95096109,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":null,"biblio":{"volume":"41","issue":"4","first_page":"1254","last_page":"1280"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11110","display_name":"Stability and Controllability of Differential Equations","score":0.9998999834060669,"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/T11110","display_name":"Stability and Controllability of Differential Equations","score":0.9998999834060669,"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 methods for differential equations","score":0.9878000020980835,"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/T10940","display_name":"Navier-Stokes equation solutions","score":0.9872000217437744,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/nabla-symbol","display_name":"Nabla symbol","score":0.9516156911849976},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6230369210243225},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.5761398077011108},{"id":"https://openalex.org/keywords/zero","display_name":"Zero (linguistics)","score":0.5700410604476929},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.5426180958747864},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4370574951171875},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.234019935131073},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.22788524627685547},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.10033634305000305},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.06982976198196411}],"concepts":[{"id":"https://openalex.org/C54207081","wikidata":"https://www.wikidata.org/wiki/Q937750","display_name":"Nabla symbol","level":3,"score":0.9516156911849976},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6230369210243225},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.5761398077011108},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.5700410604476929},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5426180958747864},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4370574951171875},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.234019935131073},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.22788524627685547},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.10033634305000305},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.06982976198196411},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"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/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","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":2,"locations":[{"id":"doi:10.1137/s0363012901385198","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0363012901385198","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:HAL:hal-00068877v1","is_oa":false,"landing_page_url":"https://hal.science/hal-00068877","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Control and Optimization, 2002, 41, pp.1254-1280","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":13,"referenced_works":["https://openalex.org/W199410564","https://openalex.org/W1580724525","https://openalex.org/W1967598881","https://openalex.org/W1970033879","https://openalex.org/W1970602566","https://openalex.org/W2028739379","https://openalex.org/W2035565712","https://openalex.org/W2053487511","https://openalex.org/W2063357812","https://openalex.org/W2075034671","https://openalex.org/W2076292739","https://openalex.org/W2091760073","https://openalex.org/W2569816327"],"related_works":["https://openalex.org/W2229925025","https://openalex.org/W1539596337","https://openalex.org/W33469805","https://openalex.org/W1855007440","https://openalex.org/W2788247819","https://openalex.org/W2914699178","https://openalex.org/W4301243416","https://openalex.org/W2024985363","https://openalex.org/W2156911440","https://openalex.org/W2357336191"],"abstract_inverted_index":{"Let":[0],"$\\phi":[1],":":[2],"H\\rightarrow":[3],"\\R$":[4],"be":[5,20],"a":[6,10,21,138,189,243],"$\\mathcal{C}^1$":[7],"function":[8,34],"on":[9],"real":[11],"Hilbert":[12],"space":[13],"H":[14,129],"and":[15,31,82,89,112,185,199],"let":[16],"$\\gamma":[17],">":[18],"0$":[19,88,93],"positive":[22],"damping":[23],"parameter.":[24],"For":[25,124],"any":[26,32],"repulsive":[27],"potential":[28,108],"$V:H\\to":[29],"\\R_+$":[30,37],"control":[33],"$\\varepsilon":[35],":\\R_+\\to":[36],"which":[38,254],"tends":[39],"to":[40,161,202,231,241],"zero":[41,162],"as":[42,165],"$t\\to":[43,95,166],"+\\infty$,":[44,96,167],"we":[45],"study":[46],"the":[47,51,54,100,107,116,169,176,180,196,206,212,222,247],"asymptotic":[48],"behavior":[49],"of":[50,53,58,205,211,214,246],"trajectories":[52,177,197],"coupled":[55],"dissipative":[56],"system":[57],"nonlinear":[59],"oscillators":[60],"$$":[61,75,143,153],"\\left":[62],"\\{":[63],"\\begin{array}{l}":[64],"\\ddot{x}+\\gamma":[65],"\\dot{x}+\\nabla\\phi(x)+\\varepsilon(t)\\nabla":[66],"V(x-y)=0,\\\\":[67],"\\ddot{y}+\\gamma":[68],"\\dot{y}+\\nabla\\phi(y)-\\varepsilon(t)\\nabla":[69],"V(x-y)=0.":[70],"\\end{array}":[71],"\\right.":[72],"\\leqno{\\rm":[73],"(HBFC^2)}":[74],"We":[76,250],"first":[77],"provide":[78,251],"general":[79],"existence":[80],"results":[81,258],"show":[83],"that":[84,99,106,113,132,142],"$\\nabla":[85,90],"\\phi":[86,91],"(x(t))\\to":[87],"(y(t))\\to":[92],"when":[94,183,227],"assuming":[97],"either":[98],"trajectory":[101],"(x,y)":[102],"is":[103,110,188,229,237],"bounded,":[104],"or":[105,219],"V":[109],"bounded":[111],"$\\phi$":[114,215,236],"satisfies":[115],"following":[117],"limit":[118],"condition:":[119],"\\vspace*{\\abovedisplayskip}":[120],"\\begin{itemize}":[121],"\\item[\\rm":[122],"(LIM)]":[123],"every":[125],"sequence":[126],"$":[127,130,133],"(z_n)\\subset":[128],"such":[131,141],"\\lim_{n\\to":[134],"+\\infty}|z_n|=+\\infty,$":[135],"there":[136],"exists":[137],"subsequence":[139],"$(z_{\\varphi(n)})$":[140],"\\lim_{n":[144,150],"\\to":[145,151],"+\\infty}{\\phi}(z_{\\varphi(n)})=+\\infty":[146],"\\qquad":[147,149],"\\mbox{or}":[148],"+\\infty}\\nabla{\\phi}(z_{\\varphi(n)})=0.":[152],"\\end{itemize}":[154],"\\noindent":[155],"If":[156],"$\\varepsilon(t)$":[157],"does":[158],"not":[159],"tend":[160],"too":[163],"rapidly":[164],"then":[168,195],"term":[170],"$\\varepsilon(t)":[171],"\\nabla":[172],"V(x-y)$":[173],"asymptotically":[174],"repulses":[175],"one":[178],"from":[179],"other.":[181],"Precisely,":[182],"$H=\\R$,":[184],"if":[186,235],"$\\varepsilon$":[187],"``slow'":[190],"control,":[191],"i.e.,":[192],"$\\int_0^{+\\infty}":[193],"\\varepsilon(t)dt=+\\infty$,":[194],"x":[198],"y":[200],"converge":[201],"extremal":[203],"points":[204],"set":[207,248],"$S=\\{\\lambda\\in":[208],"\\R,":[209],"\\nabla\\phi(\\lambda)=0\\}$":[210],"equilibria":[213],"(when":[216],"$S\\ne":[217],"\\emptyset$),":[218],"they":[220],"have":[221],"same":[223],"limit.":[224],"In":[225],"particular,":[226],"S":[228],"reduced":[230],"an":[232],"interval---for":[233],"example,":[234],"convex---this":[238],"allows":[239],"us":[240],"obtain":[242],"global":[244],"description":[245],"S.":[249],"numerical":[252],"experiments":[253],"make":[255],"our":[256],"convergence":[257],"more":[259],"precise.":[260]},"counts_by_year":[],"updated_date":"2026-08-26T07:47:46.906454","created_date":"2025-10-10T00:00:00"}
