Quadratic Lyapunov functions for stability of the generalized proportional fractional differential equations with applications to neural networks
Autor(a) principal: | |
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Data de Publicação: | 2021 |
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/10773/32687 |
Resumo: | A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordinary derivatives and Caputo-type fractional derivatives). We define the exponential stability and the Mittag–Leffler stability of the equilibrium. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality for the generalized proportional Caputo fractional derivative of the quadratic Lyapunov function. Several sufficient conditions are presented to guarantee these types of stability. Finally, two numerical examples are presented to illustrate the effectiveness of our theoretical results. |
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Quadratic Lyapunov functions for stability of the generalized proportional fractional differential equations with applications to neural networksGeneralized Caputo proportional fractional derivativeStabilityExponential stabilityMittag–Leffler stabilityQuadratic Lyapunov functionsHopfield neural networksA fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordinary derivatives and Caputo-type fractional derivatives). We define the exponential stability and the Mittag–Leffler stability of the equilibrium. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality for the generalized proportional Caputo fractional derivative of the quadratic Lyapunov function. Several sufficient conditions are presented to guarantee these types of stability. Finally, two numerical examples are presented to illustrate the effectiveness of our theoretical results.MDPI2021-12-02T11:02:12Z2021-01-01T00:00:00Z2021info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10773/32687eng10.3390/axioms10040322Almeida, RicardoAgarwal, Ravi P.Hristova, SnezhanaO’Regan, Donalinfo: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:RCAAP2024-02-22T12:02:53Zoai:ria.ua.pt:10773/32687Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T03:04:15.135271Repositório 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 |
Quadratic Lyapunov functions for stability of the generalized proportional fractional differential equations with applications to neural networks |
title |
Quadratic Lyapunov functions for stability of the generalized proportional fractional differential equations with applications to neural networks |
spellingShingle |
Quadratic Lyapunov functions for stability of the generalized proportional fractional differential equations with applications to neural networks Almeida, Ricardo Generalized Caputo proportional fractional derivative Stability Exponential stability Mittag–Leffler stability Quadratic Lyapunov functions Hopfield neural networks |
title_short |
Quadratic Lyapunov functions for stability of the generalized proportional fractional differential equations with applications to neural networks |
title_full |
Quadratic Lyapunov functions for stability of the generalized proportional fractional differential equations with applications to neural networks |
title_fullStr |
Quadratic Lyapunov functions for stability of the generalized proportional fractional differential equations with applications to neural networks |
title_full_unstemmed |
Quadratic Lyapunov functions for stability of the generalized proportional fractional differential equations with applications to neural networks |
title_sort |
Quadratic Lyapunov functions for stability of the generalized proportional fractional differential equations with applications to neural networks |
author |
Almeida, Ricardo |
author_facet |
Almeida, Ricardo Agarwal, Ravi P. Hristova, Snezhana O’Regan, Donal |
author_role |
author |
author2 |
Agarwal, Ravi P. Hristova, Snezhana O’Regan, Donal |
author2_role |
author author author |
dc.contributor.author.fl_str_mv |
Almeida, Ricardo Agarwal, Ravi P. Hristova, Snezhana O’Regan, Donal |
dc.subject.por.fl_str_mv |
Generalized Caputo proportional fractional derivative Stability Exponential stability Mittag–Leffler stability Quadratic Lyapunov functions Hopfield neural networks |
topic |
Generalized Caputo proportional fractional derivative Stability Exponential stability Mittag–Leffler stability Quadratic Lyapunov functions Hopfield neural networks |
description |
A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordinary derivatives and Caputo-type fractional derivatives). We define the exponential stability and the Mittag–Leffler stability of the equilibrium. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality for the generalized proportional Caputo fractional derivative of the quadratic Lyapunov function. Several sufficient conditions are presented to guarantee these types of stability. Finally, two numerical examples are presented to illustrate the effectiveness of our theoretical results. |
publishDate |
2021 |
dc.date.none.fl_str_mv |
2021-12-02T11:02:12Z 2021-01-01T00:00:00Z 2021 |
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/10773/32687 |
url |
http://hdl.handle.net/10773/32687 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
10.3390/axioms10040322 |
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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1799137698065154049 |