Sparse initial data identification for parabolic PDE and its finite element approximations

We address the problem of inverse source identication for parabolic equations from the optimal control viewpoint employing measures of minimal norm as initial data. We adopt the point of view of approximate controllability so that the target is not required to be achieved exactly but only in an appr...

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Authors: Casas, E., Vexler, B., Zuazua, E.
Format: article
Status:Versión aceptada para publicación
Publication Date:2015
Country:España
Institution:Basque Center for Applied Mathematics (BCAM)
Repository:BIRD. BCAM's Institutional Repository Data
OAI Identifier:oai:bird.bcamath.org:20.500.11824/244
Online Access:http://hdl.handle.net/20.500.11824/244
Access Level:Open access
Keyword:Approximate controllability
Borel measures
Parabolic equations
Sparse controls
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spelling Sparse initial data identification for parabolic PDE and its finite element approximationsCasas, E.Vexler, B.Zuazua, E.Approximate controllabilityBorel measuresParabolic equationsSparse controlsWe address the problem of inverse source identication for parabolic equations from the optimal control viewpoint employing measures of minimal norm as initial data. We adopt the point of view of approximate controllability so that the target is not required to be achieved exactly but only in an approximate sense. We prove an approximate inversion result and derive a characterization of the optimal initial measures by means of duality and the minimization of a suitable quadratic functional on the solutions of the adjoint system. We prove the sparsity of the optimal initial measures showing that they are supported in sets of null Lebesgue measure. As a consequence, approximate controllability can be achieved efficiently by means of controls that are activated in a nite number of pointwise locations. Moreover, we discuss the nite element numerical approximation of the control problem providing a convergence result of the corresponding optimal measures and states as the discretization parameters tend to zero.201620162015info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersionapplication/pdfhttp://hdl.handle.net/20.500.11824/244reponame:BIRD. BCAM's Institutional Repository Datainstname:Basque Center for Applied Mathematics (BCAM)Ingléshttp://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=11431Reconocimiento-NoComercial-CompartirIgual 3.0 Españahttp://creativecommons.org/licenses/by-nc-sa/3.0/es/info:eu-repo/semantics/openAccessoai:bird.bcamath.org:20.500.11824/2442026-06-19T12:47:47Z
dc.title.none.fl_str_mv Sparse initial data identification for parabolic PDE and its finite element approximations
title Sparse initial data identification for parabolic PDE and its finite element approximations
spellingShingle Sparse initial data identification for parabolic PDE and its finite element approximations
Casas, E.
Approximate controllability
Borel measures
Parabolic equations
Sparse controls
title_short Sparse initial data identification for parabolic PDE and its finite element approximations
title_full Sparse initial data identification for parabolic PDE and its finite element approximations
title_fullStr Sparse initial data identification for parabolic PDE and its finite element approximations
title_full_unstemmed Sparse initial data identification for parabolic PDE and its finite element approximations
title_sort Sparse initial data identification for parabolic PDE and its finite element approximations
dc.creator.none.fl_str_mv Casas, E.
Vexler, B.
Zuazua, E.
author Casas, E.
author_facet Casas, E.
Vexler, B.
Zuazua, E.
author_role author
author2 Vexler, B.
Zuazua, E.
author2_role author
author
dc.subject.none.fl_str_mv Approximate controllability
Borel measures
Parabolic equations
Sparse controls
topic Approximate controllability
Borel measures
Parabolic equations
Sparse controls
description We address the problem of inverse source identication for parabolic equations from the optimal control viewpoint employing measures of minimal norm as initial data. We adopt the point of view of approximate controllability so that the target is not required to be achieved exactly but only in an approximate sense. We prove an approximate inversion result and derive a characterization of the optimal initial measures by means of duality and the minimization of a suitable quadratic functional on the solutions of the adjoint system. We prove the sparsity of the optimal initial measures showing that they are supported in sets of null Lebesgue measure. As a consequence, approximate controllability can be achieved efficiently by means of controls that are activated in a nite number of pointwise locations. Moreover, we discuss the nite element numerical approximation of the control problem providing a convergence result of the corresponding optimal measures and states as the discretization parameters tend to zero.
publishDate 2015
dc.date.none.fl_str_mv 2015
2016
2016
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/acceptedVersion
format article
status_str acceptedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.11824/244
url http://hdl.handle.net/20.500.11824/244
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=11431
dc.rights.none.fl_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:BIRD. BCAM's Institutional Repository Data
instname:Basque Center for Applied Mathematics (BCAM)
instname_str Basque Center for Applied Mathematics (BCAM)
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