Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control

Numerical methods that preserve geometric invariants of the system, such as energy, momentum or the symplectic form, are called geometric integrators. In this paper we present a method to construct symplectic-momentum integrators for higher-order Lagrangian systems. Given a regular higher-order Lagr...

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Detalles Bibliográficos
Autores: Colombo, Leonardo Jesus, Ferraro, Sebastián José, Martin de Diego, David
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2016
País:Argentina
Institución:Consejo Nacional de Investigaciones Científicas y Técnicas
Repositorio:CONICET Digital (CONICET)
Idioma:inglés
OAI Identifier:oai:ri.conicet.gov.ar:11336/48751
Acceso en línea:http://hdl.handle.net/11336/48751
Access Level:acceso abierto
Palabra clave:DISCRETE VARIATIONAL CALCULUS
HIGHER-ORDER MECHANICS
OPTIMAL CONTROL
VARIATIONAL INTEGRATORS
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
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repository_id_str
spelling Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal ControlColombo, Leonardo JesusFerraro, Sebastián JoséMartin de Diego, DavidDISCRETE VARIATIONAL CALCULUSHIGHER-ORDER MECHANICSOPTIMAL CONTROLVARIATIONAL INTEGRATORShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Numerical methods that preserve geometric invariants of the system, such as energy, momentum or the symplectic form, are called geometric integrators. In this paper we present a method to construct symplectic-momentum integrators for higher-order Lagrangian systems. Given a regular higher-order Lagrangian L: T( k )Q→ R with k≥ 1 , the resulting discrete equations define a generally implicit numerical integrator algorithm on T( k - 1 )Q× T( k - 1 )Q that approximates the flow of the higher-order Euler–Lagrange equations for L. The algorithm equations are called higher-order discrete Euler–Lagrange equations and constitute a variational integrator for higher-order mechanical systems. The general idea for those variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the invariants of the original system, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. We construct an exact discrete Lagrangian Lde using the locally unique solution of the higher-order Euler–Lagrange equations for L with boundary conditions. By taking the discrete Lagrangian as an approximation of Lde, we obtain variational integrators for higher-order mechanical systems. We apply our techniques to optimal control problems since, given a cost function, the optimal control problem is understood as a second-order variational problem.Fil: Colombo, Leonardo Jesus. University of Michigan; Estados UnidosFil: Ferraro, Sebastián José. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; ArgentinaFil: Martin de Diego, David. Instituto de Ciencias Matemáticas; España. Consejo Superior de Investigaciones Científicas; EspañaSpringer2016-12-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/48751Colombo, Leonardo Jesus; Ferraro, Sebastián José; Martin de Diego, David; Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control; Springer; Journal Of Nonlinear Science; 26; 6; 1-12-2016; 1615-16500938-89741432-1467CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1007/s00332-016-9314-9info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007%2Fs00332-016-9314-9info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1410.5766info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2024-05-08T14:03:39Zoai:ri.conicet.gov.ar:11336/48751instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982024-05-08 14:03:39.927CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control
title Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control
spellingShingle Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control
Colombo, Leonardo Jesus
DISCRETE VARIATIONAL CALCULUS
HIGHER-ORDER MECHANICS
OPTIMAL CONTROL
VARIATIONAL INTEGRATORS
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
title_short Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control
title_full Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control
title_fullStr Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control
title_full_unstemmed Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control
title_sort Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control
dc.creator.none.fl_str_mv Colombo, Leonardo Jesus
Ferraro, Sebastián José
Martin de Diego, David
author Colombo, Leonardo Jesus
author_facet Colombo, Leonardo Jesus
Ferraro, Sebastián José
Martin de Diego, David
author_role author
author2 Ferraro, Sebastián José
Martin de Diego, David
author2_role author
author
dc.subject.none.fl_str_mv DISCRETE VARIATIONAL CALCULUS
HIGHER-ORDER MECHANICS
OPTIMAL CONTROL
VARIATIONAL INTEGRATORS
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
topic DISCRETE VARIATIONAL CALCULUS
HIGHER-ORDER MECHANICS
OPTIMAL CONTROL
VARIATIONAL INTEGRATORS
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
description Numerical methods that preserve geometric invariants of the system, such as energy, momentum or the symplectic form, are called geometric integrators. In this paper we present a method to construct symplectic-momentum integrators for higher-order Lagrangian systems. Given a regular higher-order Lagrangian L: T( k )Q→ R with k≥ 1 , the resulting discrete equations define a generally implicit numerical integrator algorithm on T( k - 1 )Q× T( k - 1 )Q that approximates the flow of the higher-order Euler–Lagrange equations for L. The algorithm equations are called higher-order discrete Euler–Lagrange equations and constitute a variational integrator for higher-order mechanical systems. The general idea for those variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the invariants of the original system, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. We construct an exact discrete Lagrangian Lde using the locally unique solution of the higher-order Euler–Lagrange equations for L with boundary conditions. By taking the discrete Lagrangian as an approximation of Lde, we obtain variational integrators for higher-order mechanical systems. We apply our techniques to optimal control problems since, given a cost function, the optimal control problem is understood as a second-order variational problem.
publishDate 2016
dc.date.none.fl_str_mv 2016-12-01
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/48751
Colombo, Leonardo Jesus; Ferraro, Sebastián José; Martin de Diego, David; Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control; Springer; Journal Of Nonlinear Science; 26; 6; 1-12-2016; 1615-1650
0938-8974
1432-1467
CONICET Digital
CONICET
url http://hdl.handle.net/11336/48751
identifier_str_mv Colombo, Leonardo Jesus; Ferraro, Sebastián José; Martin de Diego, David; Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control; Springer; Journal Of Nonlinear Science; 26; 6; 1-12-2016; 1615-1650
0938-8974
1432-1467
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1007/s00332-016-9314-9
info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007%2Fs00332-016-9314-9
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1410.5766
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 15,81155

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