Application of Bernoulli polynomials for solving variable-order fractional optimal control-affine problems
Autor(a) principal: | |
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Data de Publicação: | 2020 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10773/29430 |
Resumo: | We propose two efficient numerical approaches for solving variable-order fractional optimal control-affine problems. The variable-order fractional derivative is considered in the Caputo sense, which together with the Riemann--Liouville integral operator is used in our new techniques. An accurate operational matrix of variable-order fractional integration for Bernoulli polynomials is introduced. Our methods proceed as follows. First, a specific approximation of the differentiation order of the state function is considered, in terms of Bernoulli polynomials. Such approximation, together with the initial conditions, help us to obtain some approximations for the other existing functions in the dynamical control-affine system. Using these approximations, and the Gauss--Legendre integration formula, the problem is reduced to a system of nonlinear algebraic equations. Some error bounds are then given for the approximate optimal state and control functions, which allow us to obtain an error bound for the approximate value of the performance index. We end by solving some test problems, which demonstrate the high accuracy of our results. |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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7160 |
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Application of Bernoulli polynomials for solving variable-order fractional optimal control-affine problemsVariable-order fractional calculusBernoulli polynomialsOptimal control-affine problemsOperational matrix of fractional integrationWe propose two efficient numerical approaches for solving variable-order fractional optimal control-affine problems. The variable-order fractional derivative is considered in the Caputo sense, which together with the Riemann--Liouville integral operator is used in our new techniques. An accurate operational matrix of variable-order fractional integration for Bernoulli polynomials is introduced. Our methods proceed as follows. First, a specific approximation of the differentiation order of the state function is considered, in terms of Bernoulli polynomials. Such approximation, together with the initial conditions, help us to obtain some approximations for the other existing functions in the dynamical control-affine system. Using these approximations, and the Gauss--Legendre integration formula, the problem is reduced to a system of nonlinear algebraic equations. Some error bounds are then given for the approximate optimal state and control functions, which allow us to obtain an error bound for the approximate value of the performance index. We end by solving some test problems, which demonstrate the high accuracy of our results.MDPI2020-10-13T17:20:55Z2020-01-01T00:00:00Z2020info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10773/29430eng10.3390/axioms9040114Nemati, SomayehTorres, Delfim F. M.info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2024-02-22T11:56:54Zoai:ria.ua.pt:10773/29430Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T03:01:46.077089Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
Application of Bernoulli polynomials for solving variable-order fractional optimal control-affine problems |
title |
Application of Bernoulli polynomials for solving variable-order fractional optimal control-affine problems |
spellingShingle |
Application of Bernoulli polynomials for solving variable-order fractional optimal control-affine problems Nemati, Somayeh Variable-order fractional calculus Bernoulli polynomials Optimal control-affine problems Operational matrix of fractional integration |
title_short |
Application of Bernoulli polynomials for solving variable-order fractional optimal control-affine problems |
title_full |
Application of Bernoulli polynomials for solving variable-order fractional optimal control-affine problems |
title_fullStr |
Application of Bernoulli polynomials for solving variable-order fractional optimal control-affine problems |
title_full_unstemmed |
Application of Bernoulli polynomials for solving variable-order fractional optimal control-affine problems |
title_sort |
Application of Bernoulli polynomials for solving variable-order fractional optimal control-affine problems |
author |
Nemati, Somayeh |
author_facet |
Nemati, Somayeh Torres, Delfim F. M. |
author_role |
author |
author2 |
Torres, Delfim F. M. |
author2_role |
author |
dc.contributor.author.fl_str_mv |
Nemati, Somayeh Torres, Delfim F. M. |
dc.subject.por.fl_str_mv |
Variable-order fractional calculus Bernoulli polynomials Optimal control-affine problems Operational matrix of fractional integration |
topic |
Variable-order fractional calculus Bernoulli polynomials Optimal control-affine problems Operational matrix of fractional integration |
description |
We propose two efficient numerical approaches for solving variable-order fractional optimal control-affine problems. The variable-order fractional derivative is considered in the Caputo sense, which together with the Riemann--Liouville integral operator is used in our new techniques. An accurate operational matrix of variable-order fractional integration for Bernoulli polynomials is introduced. Our methods proceed as follows. First, a specific approximation of the differentiation order of the state function is considered, in terms of Bernoulli polynomials. Such approximation, together with the initial conditions, help us to obtain some approximations for the other existing functions in the dynamical control-affine system. Using these approximations, and the Gauss--Legendre integration formula, the problem is reduced to a system of nonlinear algebraic equations. Some error bounds are then given for the approximate optimal state and control functions, which allow us to obtain an error bound for the approximate value of the performance index. We end by solving some test problems, which demonstrate the high accuracy of our results. |
publishDate |
2020 |
dc.date.none.fl_str_mv |
2020-10-13T17:20:55Z 2020-01-01T00:00:00Z 2020 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10773/29430 |
url |
http://hdl.handle.net/10773/29430 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
10.3390/axioms9040114 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
MDPI |
publisher.none.fl_str_mv |
MDPI |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
repository.mail.fl_str_mv |
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1799137673351266304 |