A comnutative diagram among discrete and continuous Neumann boundary optimal control problems

Autores
Tarzia, Domingo Alberto
Año de publicación
2014
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We consider a bounded domain nR⊂Ω whose regular boundary 21ΓΓ=Ω∂=Γ∪ consists of the union of two disjoint portions 1Γ and 2Γ with positive measures. The convergence of a family of continuous Neumann boundary mixed elliptic optimal control problems (),αP governed by elliptic variational equalities, when the parameter α of the family (the heat transfer coefficient on the portion of the boundary )1Γ goes to infinity was studied in Gariboldi-Tarzia [15], being the control variable the heat flux on the boundary .2Γ It has been proved that the optimal control, and their corresponding system and adjoint system states are strongly convergent, in adequate functional spaces, to the optimal control, and the system and adjoint states of another Neumann boundary mixed elliptic optimal control problem ()P governed also by an elliptic variational equality with a different boundary condition on the portion of the boundary .1ΓWe consider the discrete approximations ()αhP and ()hP of the optimal control problems ()αP and (),P respectively, for each 0>h and for each ,0>α through the finite element method with Lagrange’s triangles of type 1 with parameter h (the longest side of the triangles). We also discrete the elliptic variational equalities which define the system and their adjoint system states, and the corresponding cost functional of the Neumann boundary optimal control problems ()αP and ().P The goal of this paper is to study the convergence of this family of discrete Neumann boundary mixed elliptic optimal control problems ()αhP when the parameter α goes to infinity. We prove the convergence of the discrete optimal controls, the discrete system and adjoint system states of the family ()αhP to the corresponding discrete Neumann boundary mixed elliptic optimal control problem ()hP when ∞→α for each ,0>h in adequate functional spaces. We also study the convergence when 0→h and we obtain a commutative diagram which relates the continuous and discrete Neumann boundary mixed elliptic optimal control problems (),αhP(),αP()hP and ()P by taking the limits 0→h and ,+∞→α respectively.
Fil: Tarzia, Domingo Alberto. Universidad Austral. Facultad de Cs.empresariales. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Materia
Optimal control problems
numerical analysis
commutative diagram
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/31054

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network_name_str CONICET Digital (CONICET)
spelling A comnutative diagram among discrete and continuous Neumann boundary optimal control problemsTarzia, Domingo AlbertoOptimal control problemsnumerical analysiscommutative diagramhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We consider a bounded domain nR⊂Ω whose regular boundary 21ΓΓ=Ω∂=Γ∪ consists of the union of two disjoint portions 1Γ and 2Γ with positive measures. The convergence of a family of continuous Neumann boundary mixed elliptic optimal control problems (),αP governed by elliptic variational equalities, when the parameter α of the family (the heat transfer coefficient on the portion of the boundary )1Γ goes to infinity was studied in Gariboldi-Tarzia [15], being the control variable the heat flux on the boundary .2Γ It has been proved that the optimal control, and their corresponding system and adjoint system states are strongly convergent, in adequate functional spaces, to the optimal control, and the system and adjoint states of another Neumann boundary mixed elliptic optimal control problem ()P governed also by an elliptic variational equality with a different boundary condition on the portion of the boundary .1ΓWe consider the discrete approximations ()αhP and ()hP of the optimal control problems ()αP and (),P respectively, for each 0>h and for each ,0>α through the finite element method with Lagrange’s triangles of type 1 with parameter h (the longest side of the triangles). We also discrete the elliptic variational equalities which define the system and their adjoint system states, and the corresponding cost functional of the Neumann boundary optimal control problems ()αP and ().P The goal of this paper is to study the convergence of this family of discrete Neumann boundary mixed elliptic optimal control problems ()αhP when the parameter α goes to infinity. We prove the convergence of the discrete optimal controls, the discrete system and adjoint system states of the family ()αhP to the corresponding discrete Neumann boundary mixed elliptic optimal control problem ()hP when ∞→α for each ,0>h in adequate functional spaces. We also study the convergence when 0→h and we obtain a commutative diagram which relates the continuous and discrete Neumann boundary mixed elliptic optimal control problems (),αhP(),αP()hP and ()P by taking the limits 0→h and ,+∞→α respectively.Fil: Tarzia, Domingo Alberto. Universidad Austral. Facultad de Cs.empresariales. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaPushpa Publishing House2014-12info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/31054A comnutative diagram among discrete and continuous Neumann boundary optimal control problems; Pushpa Publishing House; Advances in Differential Equations and Control Processes; 14; 1; 12-2014; 23-540974-3243CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/http://www.pphmj.com/article.php?act=art_references_show&art_id=8721&flag=nextinfo:eu-repo/semantics/altIdentifier/arxiv/https://arxiv.org/abs/1412.6491info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T09:47:18Zoai:ri.conicet.gov.ar:11336/31054instacron: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:34982025-09-29 09:47:18.638CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv A comnutative diagram among discrete and continuous Neumann boundary optimal control problems
title A comnutative diagram among discrete and continuous Neumann boundary optimal control problems
spellingShingle A comnutative diagram among discrete and continuous Neumann boundary optimal control problems
Tarzia, Domingo Alberto
Optimal control problems
numerical analysis
commutative diagram
title_short A comnutative diagram among discrete and continuous Neumann boundary optimal control problems
title_full A comnutative diagram among discrete and continuous Neumann boundary optimal control problems
title_fullStr A comnutative diagram among discrete and continuous Neumann boundary optimal control problems
title_full_unstemmed A comnutative diagram among discrete and continuous Neumann boundary optimal control problems
title_sort A comnutative diagram among discrete and continuous Neumann boundary optimal control problems
dc.creator.none.fl_str_mv Tarzia, Domingo Alberto
author Tarzia, Domingo Alberto
author_facet Tarzia, Domingo Alberto
author_role author
dc.subject.none.fl_str_mv Optimal control problems
numerical analysis
commutative diagram
topic Optimal control problems
numerical analysis
commutative diagram
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv We consider a bounded domain nR⊂Ω whose regular boundary 21ΓΓ=Ω∂=Γ∪ consists of the union of two disjoint portions 1Γ and 2Γ with positive measures. The convergence of a family of continuous Neumann boundary mixed elliptic optimal control problems (),αP governed by elliptic variational equalities, when the parameter α of the family (the heat transfer coefficient on the portion of the boundary )1Γ goes to infinity was studied in Gariboldi-Tarzia [15], being the control variable the heat flux on the boundary .2Γ It has been proved that the optimal control, and their corresponding system and adjoint system states are strongly convergent, in adequate functional spaces, to the optimal control, and the system and adjoint states of another Neumann boundary mixed elliptic optimal control problem ()P governed also by an elliptic variational equality with a different boundary condition on the portion of the boundary .1ΓWe consider the discrete approximations ()αhP and ()hP of the optimal control problems ()αP and (),P respectively, for each 0>h and for each ,0>α through the finite element method with Lagrange’s triangles of type 1 with parameter h (the longest side of the triangles). We also discrete the elliptic variational equalities which define the system and their adjoint system states, and the corresponding cost functional of the Neumann boundary optimal control problems ()αP and ().P The goal of this paper is to study the convergence of this family of discrete Neumann boundary mixed elliptic optimal control problems ()αhP when the parameter α goes to infinity. We prove the convergence of the discrete optimal controls, the discrete system and adjoint system states of the family ()αhP to the corresponding discrete Neumann boundary mixed elliptic optimal control problem ()hP when ∞→α for each ,0>h in adequate functional spaces. We also study the convergence when 0→h and we obtain a commutative diagram which relates the continuous and discrete Neumann boundary mixed elliptic optimal control problems (),αhP(),αP()hP and ()P by taking the limits 0→h and ,+∞→α respectively.
Fil: Tarzia, Domingo Alberto. Universidad Austral. Facultad de Cs.empresariales. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
description We consider a bounded domain nR⊂Ω whose regular boundary 21ΓΓ=Ω∂=Γ∪ consists of the union of two disjoint portions 1Γ and 2Γ with positive measures. The convergence of a family of continuous Neumann boundary mixed elliptic optimal control problems (),αP governed by elliptic variational equalities, when the parameter α of the family (the heat transfer coefficient on the portion of the boundary )1Γ goes to infinity was studied in Gariboldi-Tarzia [15], being the control variable the heat flux on the boundary .2Γ It has been proved that the optimal control, and their corresponding system and adjoint system states are strongly convergent, in adequate functional spaces, to the optimal control, and the system and adjoint states of another Neumann boundary mixed elliptic optimal control problem ()P governed also by an elliptic variational equality with a different boundary condition on the portion of the boundary .1ΓWe consider the discrete approximations ()αhP and ()hP of the optimal control problems ()αP and (),P respectively, for each 0>h and for each ,0>α through the finite element method with Lagrange’s triangles of type 1 with parameter h (the longest side of the triangles). We also discrete the elliptic variational equalities which define the system and their adjoint system states, and the corresponding cost functional of the Neumann boundary optimal control problems ()αP and ().P The goal of this paper is to study the convergence of this family of discrete Neumann boundary mixed elliptic optimal control problems ()αhP when the parameter α goes to infinity. We prove the convergence of the discrete optimal controls, the discrete system and adjoint system states of the family ()αhP to the corresponding discrete Neumann boundary mixed elliptic optimal control problem ()hP when ∞→α for each ,0>h in adequate functional spaces. We also study the convergence when 0→h and we obtain a commutative diagram which relates the continuous and discrete Neumann boundary mixed elliptic optimal control problems (),αhP(),αP()hP and ()P by taking the limits 0→h and ,+∞→α respectively.
publishDate 2014
dc.date.none.fl_str_mv 2014-12
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/11336/31054
A comnutative diagram among discrete and continuous Neumann boundary optimal control problems; Pushpa Publishing House; Advances in Differential Equations and Control Processes; 14; 1; 12-2014; 23-54
0974-3243
CONICET Digital
CONICET
url http://hdl.handle.net/11336/31054
identifier_str_mv A comnutative diagram among discrete and continuous Neumann boundary optimal control problems; Pushpa Publishing House; Advances in Differential Equations and Control Processes; 14; 1; 12-2014; 23-54
0974-3243
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/http://www.pphmj.com/article.php?act=art_references_show&art_id=8721&flag=next
info:eu-repo/semantics/altIdentifier/arxiv/https://arxiv.org/abs/1412.6491
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Pushpa Publishing House
publisher.none.fl_str_mv Pushpa Publishing House
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
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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