Numerical solution of the variational PDEs arising in optimal control theory

Detalhes bibliográficos
Autor(a) principal: Costanza,Vicente
Data de Publicação: 2012
Outros Autores: Troparevsky,Maria I., Rivadeneira,Pablo S.
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Computational & Applied Mathematics
Texto Completo: http://old.scielo.br/scielo.php?script=sci_arttext&pid=S1807-03022012000100003
Resumo: An iterative method based on Picard's approach to ODEs' initial-value problems is proposed to solve first-order quasilinear PDEs with matrix-valued unknowns, in particular, the recently discovered variational PDEs for the missing boundary values in Hamilton equations of optimal control. As illustrations the iterative numerical solutions are checked against the analytical solutions to some examples arising from optimal control problems for nonlinear systems and regular Lagrangians in finite dimension, and against the numerical solution obtained through standard mathematical software. An application to the (n + 1)-dimensional variational PDEs associated with the n-dimensional finite-horizon time-variant linear-quadratic problem is discussed, due to the key role the LQR plays in two-degrees-of freedom control strategies for nonlinear systems with generalized costs. Mathematical subject classification: Primary: 35F30; Secondary: 93C10.
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spelling Numerical solution of the variational PDEs arising in optimal control theorynumerical methodsfirst-order PDEsnonlinear systemsoptimal controlHamiltonian equationsboundary-value problemsAn iterative method based on Picard's approach to ODEs' initial-value problems is proposed to solve first-order quasilinear PDEs with matrix-valued unknowns, in particular, the recently discovered variational PDEs for the missing boundary values in Hamilton equations of optimal control. As illustrations the iterative numerical solutions are checked against the analytical solutions to some examples arising from optimal control problems for nonlinear systems and regular Lagrangians in finite dimension, and against the numerical solution obtained through standard mathematical software. An application to the (n + 1)-dimensional variational PDEs associated with the n-dimensional finite-horizon time-variant linear-quadratic problem is discussed, due to the key role the LQR plays in two-degrees-of freedom control strategies for nonlinear systems with generalized costs. Mathematical subject classification: Primary: 35F30; Secondary: 93C10.Sociedade Brasileira de Matemática Aplicada e Computacional2012-01-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersiontext/htmlhttp://old.scielo.br/scielo.php?script=sci_arttext&pid=S1807-03022012000100003Computational & Applied Mathematics v.31 n.1 2012reponame:Computational & Applied Mathematicsinstname:Sociedade Brasileira de Matemática Aplicada e Computacional (SBMAC)instacron:SBMAC10.1590/S1807-03022012000100003info:eu-repo/semantics/openAccessCostanza,VicenteTroparevsky,Maria I.Rivadeneira,Pablo S.eng2012-04-26T00:00:00Zoai:scielo:S1807-03022012000100003Revistahttps://www.scielo.br/j/cam/ONGhttps://old.scielo.br/oai/scielo-oai.php||sbmac@sbmac.org.br1807-03022238-3603opendoar:2012-04-26T00:00Computational & Applied Mathematics - Sociedade Brasileira de Matemática Aplicada e Computacional (SBMAC)false
dc.title.none.fl_str_mv Numerical solution of the variational PDEs arising in optimal control theory
title Numerical solution of the variational PDEs arising in optimal control theory
spellingShingle Numerical solution of the variational PDEs arising in optimal control theory
Costanza,Vicente
numerical methods
first-order PDEs
nonlinear systems
optimal control
Hamiltonian equations
boundary-value problems
title_short Numerical solution of the variational PDEs arising in optimal control theory
title_full Numerical solution of the variational PDEs arising in optimal control theory
title_fullStr Numerical solution of the variational PDEs arising in optimal control theory
title_full_unstemmed Numerical solution of the variational PDEs arising in optimal control theory
title_sort Numerical solution of the variational PDEs arising in optimal control theory
author Costanza,Vicente
author_facet Costanza,Vicente
Troparevsky,Maria I.
Rivadeneira,Pablo S.
author_role author
author2 Troparevsky,Maria I.
Rivadeneira,Pablo S.
author2_role author
author
dc.contributor.author.fl_str_mv Costanza,Vicente
Troparevsky,Maria I.
Rivadeneira,Pablo S.
dc.subject.por.fl_str_mv numerical methods
first-order PDEs
nonlinear systems
optimal control
Hamiltonian equations
boundary-value problems
topic numerical methods
first-order PDEs
nonlinear systems
optimal control
Hamiltonian equations
boundary-value problems
description An iterative method based on Picard's approach to ODEs' initial-value problems is proposed to solve first-order quasilinear PDEs with matrix-valued unknowns, in particular, the recently discovered variational PDEs for the missing boundary values in Hamilton equations of optimal control. As illustrations the iterative numerical solutions are checked against the analytical solutions to some examples arising from optimal control problems for nonlinear systems and regular Lagrangians in finite dimension, and against the numerical solution obtained through standard mathematical software. An application to the (n + 1)-dimensional variational PDEs associated with the n-dimensional finite-horizon time-variant linear-quadratic problem is discussed, due to the key role the LQR plays in two-degrees-of freedom control strategies for nonlinear systems with generalized costs. Mathematical subject classification: Primary: 35F30; Secondary: 93C10.
publishDate 2012
dc.date.none.fl_str_mv 2012-01-01
dc.type.driver.fl_str_mv info:eu-repo/semantics/article
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.uri.fl_str_mv http://old.scielo.br/scielo.php?script=sci_arttext&pid=S1807-03022012000100003
url http://old.scielo.br/scielo.php?script=sci_arttext&pid=S1807-03022012000100003
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv 10.1590/S1807-03022012000100003
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv text/html
dc.publisher.none.fl_str_mv Sociedade Brasileira de Matemática Aplicada e Computacional
publisher.none.fl_str_mv Sociedade Brasileira de Matemática Aplicada e Computacional
dc.source.none.fl_str_mv Computational & Applied Mathematics v.31 n.1 2012
reponame:Computational & Applied Mathematics
instname:Sociedade Brasileira de Matemática Aplicada e Computacional (SBMAC)
instacron:SBMAC
instname_str Sociedade Brasileira de Matemática Aplicada e Computacional (SBMAC)
instacron_str SBMAC
institution SBMAC
reponame_str Computational & Applied Mathematics
collection Computational & Applied Mathematics
repository.name.fl_str_mv Computational & Applied Mathematics - Sociedade Brasileira de Matemática Aplicada e Computacional (SBMAC)
repository.mail.fl_str_mv ||sbmac@sbmac.org.br
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