Second order and stability analysis for optimal sparse control of the FitzHugh-Nagumo equation

Optimal sparse control problems are considered for the FitzHugh–Nagumo system including the so-called Schlögl model. The nondifferentiable objective functional of tracking type includes a quadratic Tikhonov regularization term and the L1-norm of the control that accounts for the sparsity. Though the...

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Autores: Casas Rentería, Eduardo|||0000-0002-8364-9416, Ryll, Christopher, Tröltzsch, Fredi
Tipo de recurso: artículo
Fecha de publicación:2015
País:España
Institución:Universidad de Cantabria (UC)
Repositorio:UCrea Repositorio Abierto de la Universidad de Cantabria
Idioma:inglés
OAI Identifier:oai:repositorio.unican.es:10902/7546
Acceso en línea:http://hdl.handle.net/10902/7546
Access Level:acceso abierto
Palabra clave:Optimal control
FitzHugh–Nagumo system
Sparse control
Bang-bang-bang controls
Second order optimality conditions
Weak local minimum
Strong local minimum
Stability
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spelling Second order and stability analysis for optimal sparse control of the FitzHugh-Nagumo equationCasas Rentería, Eduardo|||0000-0002-8364-9416Ryll, ChristopherTröltzsch, FrediOptimal controlFitzHugh–Nagumo systemSparse controlBang-bang-bang controlsSecond order optimality conditionsWeak local minimumStrong local minimumStabilityOptimal sparse control problems are considered for the FitzHugh–Nagumo system including the so-called Schlögl model. The nondifferentiable objective functional of tracking type includes a quadratic Tikhonov regularization term and the L1-norm of the control that accounts for the sparsity. Though the objective functional is not differentiable, a theory of second order sufficient optimality conditions is established for Tikhonov regularization parameter ν > 0 and also for the case ν = 0. In this context, also local minima are discussed that are strong in the sense of the calculus of variations. The second order conditions are used as the main assumption for proving the stability of locally optimal solutions with respect to ν → 0 and with respect to perturbations of the desired state functions. The theory is confirmed by numerical examples that are resolved with high precision to confirm that the optimal solution obeys the system of necessary optimality conditions.This author’s research was partially supported by the Spanish Ministerio de Economía y Competitividad under project MTM2011-22711.Society for Industrial and Applied MathematicsUniversidad de Cantabria20152015-01-01journal articlehttp://purl.org/coar/resource_type/c_6501NAhttp://purl.org/coar/version/c_be7fb7dd8ff6fe43info:eu-repo/semantics/articlehttp://hdl.handle.net/10902/7546SIAM Journal on Control and Optimization, 2015, 53(4), 2168–2202reponame:UCrea Repositorio Abierto de la Universidad de Cantabriainstname:Universidad de Cantabria (UC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:repositorio.unican.es:10902/75462026-06-02T12:39:31Z
dc.title.none.fl_str_mv Second order and stability analysis for optimal sparse control of the FitzHugh-Nagumo equation
title Second order and stability analysis for optimal sparse control of the FitzHugh-Nagumo equation
spellingShingle Second order and stability analysis for optimal sparse control of the FitzHugh-Nagumo equation
Casas Rentería, Eduardo|||0000-0002-8364-9416
Optimal control
FitzHugh–Nagumo system
Sparse control
Bang-bang-bang controls
Second order optimality conditions
Weak local minimum
Strong local minimum
Stability
title_short Second order and stability analysis for optimal sparse control of the FitzHugh-Nagumo equation
title_full Second order and stability analysis for optimal sparse control of the FitzHugh-Nagumo equation
title_fullStr Second order and stability analysis for optimal sparse control of the FitzHugh-Nagumo equation
title_full_unstemmed Second order and stability analysis for optimal sparse control of the FitzHugh-Nagumo equation
title_sort Second order and stability analysis for optimal sparse control of the FitzHugh-Nagumo equation
dc.creator.none.fl_str_mv Casas Rentería, Eduardo|||0000-0002-8364-9416
Ryll, Christopher
Tröltzsch, Fredi
author Casas Rentería, Eduardo|||0000-0002-8364-9416
author_facet Casas Rentería, Eduardo|||0000-0002-8364-9416
Ryll, Christopher
Tröltzsch, Fredi
author_role author
author2 Ryll, Christopher
Tröltzsch, Fredi
author2_role author
author
dc.contributor.none.fl_str_mv Universidad de Cantabria
dc.subject.none.fl_str_mv Optimal control
FitzHugh–Nagumo system
Sparse control
Bang-bang-bang controls
Second order optimality conditions
Weak local minimum
Strong local minimum
Stability
topic Optimal control
FitzHugh–Nagumo system
Sparse control
Bang-bang-bang controls
Second order optimality conditions
Weak local minimum
Strong local minimum
Stability
description Optimal sparse control problems are considered for the FitzHugh–Nagumo system including the so-called Schlögl model. The nondifferentiable objective functional of tracking type includes a quadratic Tikhonov regularization term and the L1-norm of the control that accounts for the sparsity. Though the objective functional is not differentiable, a theory of second order sufficient optimality conditions is established for Tikhonov regularization parameter ν > 0 and also for the case ν = 0. In this context, also local minima are discussed that are strong in the sense of the calculus of variations. The second order conditions are used as the main assumption for proving the stability of locally optimal solutions with respect to ν → 0 and with respect to perturbations of the desired state functions. The theory is confirmed by numerical examples that are resolved with high precision to confirm that the optimal solution obeys the system of necessary optimality conditions.
publishDate 2015
dc.date.none.fl_str_mv 2015
2015-01-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
NA
http://purl.org/coar/version/c_be7fb7dd8ff6fe43
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv http://hdl.handle.net/10902/7546
url http://hdl.handle.net/10902/7546
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.publisher.none.fl_str_mv Society for Industrial and Applied Mathematics
publisher.none.fl_str_mv Society for Industrial and Applied Mathematics
dc.source.none.fl_str_mv SIAM Journal on Control and Optimization, 2015, 53(4), 2168–2202
reponame:UCrea Repositorio Abierto de la Universidad de Cantabria
instname:Universidad de Cantabria (UC)
instname_str Universidad de Cantabria (UC)
reponame_str UCrea Repositorio Abierto de la Universidad de Cantabria
collection UCrea Repositorio Abierto de la Universidad de Cantabria
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.300724

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