{"id":"https://openalex.org/W1482650705","doi":"https://doi.org/10.1007/0-387-33882-9_8","title":"Riemannian Metric of the Averaged Controlled Kepler Equation","display_name":"Riemannian Metric of the Averaged Controlled Kepler Equation","publication_year":2006,"publication_date":"2006-10-29","ids":{"openalex":"https://openalex.org/W1482650705","doi":"https://doi.org/10.1007/0-387-33882-9_8","mag":"1482650705"},"language":"en","primary_location":{"id":"doi:10.1007/0-387-33882-9_8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/0-387-33882-9_8","pdf_url":"https://link.springer.com/content/pdf/10.1007%2F0-387-33882-9_8.pdf","source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IFIP International Federation for Information Processing","raw_type":"book-chapter"},"type":"conference-paper","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://link.springer.com/content/pdf/10.1007%2F0-387-33882-9_8.pdf","any_repository_has_fulltext":null},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5107838661","display_name":"Bernard Bonnard","orcid":null},"institutions":[{"id":"https://openalex.org/I177064439","display_name":"Universit\u00e9 de Bourgogne","ror":"https://ror.org/03k1bsr36","country_code":"FR","type":"education","lineage":["https://openalex.org/I177064439"]},{"id":"https://openalex.org/I4210112333","display_name":"Institut de Math\u00e9matiques de Bourgogne","ror":"https://ror.org/021f0sa24","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I4210112333","https://openalex.org/I4405253280"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"B. Bonnard","raw_affiliation_strings":["Institut de math\u00e9matiques de Bourgogne, UMR CNRS 5584, Dijon","IMB - Institut de Math\u00e9matiques de Bourgogne [Dijon] (Universit\u00e9 de Bourgogne - 9, avenue Alain Savary - B.P. 47 870 - 21078 Dijon Cedex - France)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut de math\u00e9matiques de Bourgogne, UMR CNRS 5584, Dijon","institution_ids":["https://openalex.org/I4210112333"]},{"raw_affiliation_string":"IMB - Institut de Math\u00e9matiques de Bourgogne [Dijon] (Universit\u00e9 de Bourgogne - 9, avenue Alain Savary - B.P. 47 870 - 21078 Dijon Cedex - France)","institution_ids":["https://openalex.org/I177064439","https://openalex.org/I4210112333"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5028391099","display_name":"Jean\u2010Baptiste Caillau","orcid":"https://orcid.org/0000-0002-1719-2016"},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I134560555","display_name":"Universit\u00e9 Toulouse III - Paul Sabatier","ror":"https://ror.org/02v6kpv12","country_code":"FR","type":"education","lineage":["https://openalex.org/I134560555"]},{"id":"https://openalex.org/I177064439","display_name":"Universit\u00e9 de Bourgogne","ror":"https://ror.org/03k1bsr36","country_code":"FR","type":"education","lineage":["https://openalex.org/I177064439"]},{"id":"https://openalex.org/I205747304","display_name":"Institut National Polytechnique de Toulouse","ror":"https://ror.org/033p9g875","country_code":"FR","type":"education","lineage":["https://openalex.org/I205747304"]},{"id":"https://openalex.org/I3131550300","display_name":"Universit\u00e9 Toulouse-I-Capitole","ror":"https://ror.org/0443n9e75","country_code":"FR","type":"education","lineage":["https://openalex.org/I3131550300"]},{"id":"https://openalex.org/I4210112333","display_name":"Institut de Math\u00e9matiques de Bourgogne","ror":"https://ror.org/021f0sa24","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I4210112333","https://openalex.org/I4405253280"]},{"id":"https://openalex.org/I4210119061","display_name":"Institut de Recherche en Informatique de Toulouse","ror":"https://ror.org/01rx4qw44","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I205747304","https://openalex.org/I4210119061","https://openalex.org/I4210152422","https://openalex.org/I4405258862","https://openalex.org/I4405259414"]},{"id":"https://openalex.org/I4210152422","display_name":"Universit\u00e9 Toulouse - Jean Jaur\u00e8s","ror":"https://ror.org/04ezk3x31","country_code":"FR","type":"education","lineage":["https://openalex.org/I4210152422"]},{"id":"https://openalex.org/I4210160189","display_name":"Institut Polytechnique de Bordeaux","ror":"https://ror.org/054qv7y42","country_code":"FR","type":"education","lineage":["https://openalex.org/I4210160189"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"J. -B. Caillau","raw_affiliation_strings":["ENSEEIHT-IRIT, UMR CNRS 5505, Toulouse","IMB - Institut de Math\u00e9matiques de Bourgogne [Dijon] (Universit\u00e9 de Bourgogne - 9, avenue Alain Savary - B.P. 47 870 - 21078 Dijon Cedex - France)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"ENSEEIHT-IRIT, UMR CNRS 5505, Toulouse","institution_ids":["https://openalex.org/I1294671590","https://openalex.org/I134560555","https://openalex.org/I205747304","https://openalex.org/I3131550300","https://openalex.org/I4210119061","https://openalex.org/I4210152422","https://openalex.org/I4210160189"]},{"raw_affiliation_string":"IMB - Institut de Math\u00e9matiques de Bourgogne [Dijon] (Universit\u00e9 de Bourgogne - 9, avenue Alain Savary - B.P. 47 870 - 21078 Dijon Cedex - France)","institution_ids":["https://openalex.org/I177064439","https://openalex.org/I4210112333"]}]},{"author_position":"last","author":{"id":null,"display_name":"R. Dujol","orcid":null},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I134560555","display_name":"Universit\u00e9 Toulouse III - Paul Sabatier","ror":"https://ror.org/02v6kpv12","country_code":"FR","type":"education","lineage":["https://openalex.org/I134560555"]},{"id":"https://openalex.org/I205747304","display_name":"Institut National Polytechnique de Toulouse","ror":"https://ror.org/033p9g875","country_code":"FR","type":"education","lineage":["https://openalex.org/I205747304"]},{"id":"https://openalex.org/I3131550300","display_name":"Universit\u00e9 Toulouse-I-Capitole","ror":"https://ror.org/0443n9e75","country_code":"FR","type":"education","lineage":["https://openalex.org/I3131550300"]},{"id":"https://openalex.org/I4210119061","display_name":"Institut de Recherche en Informatique de Toulouse","ror":"https://ror.org/01rx4qw44","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I205747304","https://openalex.org/I4210119061","https://openalex.org/I4210152422","https://openalex.org/I4405258862","https://openalex.org/I4405259414"]},{"id":"https://openalex.org/I4210152422","display_name":"Universit\u00e9 Toulouse - Jean Jaur\u00e8s","ror":"https://ror.org/04ezk3x31","country_code":"FR","type":"education","lineage":["https://openalex.org/I4210152422"]},{"id":"https://openalex.org/I4210160189","display_name":"Institut Polytechnique de Bordeaux","ror":"https://ror.org/054qv7y42","country_code":"FR","type":"education","lineage":["https://openalex.org/I4210160189"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"R. Dujol","raw_affiliation_strings":["ENSEEIHT-IRIT, UMR CNRS 5505, Toulouse"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"ENSEEIHT-IRIT, UMR CNRS 5505, Toulouse","institution_ids":["https://openalex.org/I1294671590","https://openalex.org/I134560555","https://openalex.org/I205747304","https://openalex.org/I3131550300","https://openalex.org/I4210119061","https://openalex.org/I4210152422","https://openalex.org/I4210160189"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":9,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.02302632,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":"79","last_page":"89"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9879000186920166,"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/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9879000186920166,"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.9837999939918518,"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"}},{"id":"https://openalex.org/T11565","display_name":"Thermoelastic and Magnetoelastic Phenomena","score":0.9728999733924866,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"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":0.7671018838882446},{"id":"https://openalex.org/keywords/modulo","display_name":"Modulo","score":0.5951309204101562},{"id":"https://openalex.org/keywords/hamiltonian","display_name":"Hamiltonian (control theory)","score":0.5855682492256165},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.5550190806388855},{"id":"https://openalex.org/keywords/riemannian-geometry","display_name":"Riemannian geometry","score":0.5398141741752625},{"id":"https://openalex.org/keywords/kepler-problem","display_name":"Kepler problem","score":0.5214673280715942},{"id":"https://openalex.org/keywords/fundamental-theorem-of-riemannian-geometry","display_name":"Fundamental theorem of Riemannian geometry","score":0.4898223578929901},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4850675165653229},{"id":"https://openalex.org/keywords/curvature","display_name":"Curvature","score":0.4836781919002533},{"id":"https://openalex.org/keywords/degenerate-energy-levels","display_name":"Degenerate energy levels","score":0.479672908782959},{"id":"https://openalex.org/keywords/prescribed-scalar-curvature-problem","display_name":"Prescribed scalar curvature problem","score":0.4793480336666107},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.46131420135498047},{"id":"https://openalex.org/keywords/camassa\u2013holm-equation","display_name":"Camassa\u2013Holm equation","score":0.4114390015602112},{"id":"https://openalex.org/keywords/sectional-curvature","display_name":"Sectional curvature","score":0.3801506757736206},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.35626012086868286},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.253250390291214},{"id":"https://openalex.org/keywords/ricci-curvature","display_name":"Ricci curvature","score":0.2096080482006073},{"id":"https://openalex.org/keywords/scalar-curvature","display_name":"Scalar curvature","score":0.18067869544029236},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1752295196056366},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.1509452760219574},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.13676458597183228},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.10657945275306702},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.06962946057319641}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7671018838882446},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.5951309204101562},{"id":"https://openalex.org/C130787639","wikidata":"https://www.wikidata.org/wiki/Q5645293","display_name":"Hamiltonian (control theory)","level":2,"score":0.5855682492256165},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.5550190806388855},{"id":"https://openalex.org/C181104567","wikidata":"https://www.wikidata.org/wiki/Q761383","display_name":"Riemannian geometry","level":2,"score":0.5398141741752625},{"id":"https://openalex.org/C20422478","wikidata":"https://www.wikidata.org/wiki/Q2564682","display_name":"Kepler problem","level":2,"score":0.5214673280715942},{"id":"https://openalex.org/C4306028","wikidata":"https://www.wikidata.org/wiki/Q2185349","display_name":"Fundamental theorem of Riemannian geometry","level":4,"score":0.4898223578929901},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4850675165653229},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.4836781919002533},{"id":"https://openalex.org/C72319582","wikidata":"https://www.wikidata.org/wiki/Q584304","display_name":"Degenerate energy levels","level":2,"score":0.479672908782959},{"id":"https://openalex.org/C14196021","wikidata":"https://www.wikidata.org/wiki/Q7240881","display_name":"Prescribed scalar curvature problem","level":5,"score":0.4793480336666107},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.46131420135498047},{"id":"https://openalex.org/C24454023","wikidata":"https://www.wikidata.org/wiki/Q5025006","display_name":"Camassa\u2013Holm equation","level":3,"score":0.4114390015602112},{"id":"https://openalex.org/C42448751","wikidata":"https://www.wikidata.org/wiki/Q855622","display_name":"Sectional curvature","level":4,"score":0.3801506757736206},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.35626012086868286},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.253250390291214},{"id":"https://openalex.org/C12089564","wikidata":"https://www.wikidata.org/wiki/Q1195879","display_name":"Ricci curvature","level":3,"score":0.2096080482006073},{"id":"https://openalex.org/C12520029","wikidata":"https://www.wikidata.org/wiki/Q1147161","display_name":"Scalar curvature","level":3,"score":0.18067869544029236},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1752295196056366},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.1509452760219574},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.13676458597183228},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.10657945275306702},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.06962946057319641},{"id":"https://openalex.org/C21547014","wikidata":"https://www.wikidata.org/wiki/Q1423657","display_name":"Operations management","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/C200741047","wikidata":"https://www.wikidata.org/wiki/Q1957758","display_name":"Integrable system","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/0-387-33882-9_8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/0-387-33882-9_8","pdf_url":"https://link.springer.com/content/pdf/10.1007%2F0-387-33882-9_8.pdf","source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IFIP International Federation for Information Processing","raw_type":"book-chapter"}],"best_oa_location":{"id":"doi:10.1007/0-387-33882-9_8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/0-387-33882-9_8","pdf_url":"https://link.springer.com/content/pdf/10.1007%2F0-387-33882-9_8.pdf","source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IFIP International Federation for Information Processing","raw_type":"book-chapter"},"sustainable_development_goals":[{"display_name":"Affordable and clean energy","score":0.6200000047683716,"id":"https://metadata.un.org/sdg/7"}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1482650705.pdf","grobid_xml":"https://content.openalex.org/works/W1482650705.grobid-xml"},"referenced_works_count":13,"referenced_works":["https://openalex.org/W588617861","https://openalex.org/W1981853621","https://openalex.org/W1993262371","https://openalex.org/W2006331190","https://openalex.org/W2009481231","https://openalex.org/W2058463483","https://openalex.org/W2094984457","https://openalex.org/W2269971458","https://openalex.org/W3013347551","https://openalex.org/W3088867366","https://openalex.org/W4205727720","https://openalex.org/W4292360406","https://openalex.org/W7005701668"],"related_works":["https://openalex.org/W1976330683","https://openalex.org/W3103229299","https://openalex.org/W2952578374","https://openalex.org/W3183653752","https://openalex.org/W1999110602","https://openalex.org/W3118825917","https://openalex.org/W1535135171","https://openalex.org/W4244130316","https://openalex.org/W2075515459","https://openalex.org/W2063715014"],"abstract_inverted_index":{"A":[0],"non-autonomous":[1],"sub-Riemannian":[2],"problem":[3,46,58],"is":[4,14,42,69,93,102],"considered:":[5],"Since":[6],"periodicity":[7],"with":[8],"respect":[9],"to":[10,55,71],"the":[11,19,23,26,29,31,37,48,56,60,64,72,79,83,87,96],"independent":[12],"variable":[13],"assumed,":[15],"one":[16],"can":[17,51],"define":[18],"averaged":[20,32,57,80],"problem.":[21],"In":[22],"case":[24],"of":[25,28,63,82,98],"minimization":[27],"energy,":[30],"Hamiltonian":[33,81],"remains":[34],"quadratic":[35,65],"in":[36,86],"adjoint":[38],"variable.":[39],"When":[40],"it":[41],"non-degenerate,":[43],"a":[44,99],"Riemannian":[45,91],"and":[47,95],"corresponding":[49],"metric":[50,92],"be":[52],"uniquely":[53],"associated":[54],"modulo":[59],"orthogonal":[61],"group":[62],"form.":[66],"The":[67,90],"analysis":[68],"applied":[70],"controlled":[73],"Kepler":[74,84],"equation.":[75],"Explicit":[76],"computations":[77],"provide":[78],"motion":[85],"three-dimensional":[88],"case.":[89],"given,":[94],"curvature":[97],"special":[100],"subsytem":[101],"evaluated.":[103]},"counts_by_year":[],"updated_date":"2026-07-29T14:22:42.915294","created_date":"2025-10-10T00:00:00"}