Linear stability for differential equations with infinite delay via semigroup theory

Detalhes bibliográficos
Autor(a) principal: Caetano, Diogo Loureiro
Data de Publicação: 2018
Tipo de documento: Dissertação
Idioma: eng
Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
Texto Completo: http://hdl.handle.net/10451/34646
Resumo: Tese de mestrado, Matemática, Universidade de Lisboa, Faculdade de Ciências, 2018
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spelling Linear stability for differential equations with infinite delay via semigroup theoryEquações diferenciaisAtraso infinitoEstabilidade linearTeoria de semigruposTeoria sun-starTeses de mestrado - 2018Departamento de MatemáticaTese de mestrado, Matemática, Universidade de Lisboa, Faculdade de Ciências, 2018In this dissertation, we provide a proof of a Principle of Linearized Stability for a class of autonomous differential equations with infinite delay. This is done via techniques from functional analysis, namely duality theory for semigroups of bounded linear operators, following the approach of O. Diekmann and M. Gyllenberg in [12]. First, we make a detailed study of some aspects of the theory of strongly continuous semigroups (also called C0 semigroups) of linear operators in Banach spaces. In particular, we prove the classical theorem of Hille-Yosida, characterizing infinitesimal generators of C0 semigroups, and define the adjoint of a strongly continuous semigroup. Since the adjoint semigroup is not necessarily strongly continuous, we study whether it can be restricted to some subdomain where strong continuity holds. This is the starting point for the sun-star calculus, of which we make use throughout the remaining chapters. We introduce some elements of the sun-star theory for linear operators and give meaning to an abstract integral equation, for which we prove existence, uniqueness, continuation and regularity of solutions. We then consider, in a suitable (weighted) space of continuous functions on (-∞;0] that vanish at -∞, an initial value problem for a differential equation with infinite delay and prove an equivalence result between solutions of such equation and the solution semigroup of an abstract integral equation. After that, we study the characteristic equation of the linearized problem, and prove that the roots of this equation are precisely the eigenvalues of the infinitesimal generator of the solution semigroup of the linear equation. Moreover, we show that, on a fixed half-space, there are only finitely many such roots. Consequently, the spectral projection of the resolvent operator induces a decomposition of the phase space as the direct sum of two invariant subspaces - one with finite dimension, and the other where the semigroup is exponentially stable -, to which we can apply a theorem by Desch and Schappacher. As a result, we obtain a proof of the Principle of Linearized Stability, generalizing for this case the well-known result for ordinary and finite-delay differential equations.Faria, Teresa, 1958-Repositório da Universidade de LisboaCaetano, Diogo Loureiro2018-08-29T13:05:56Z201820182018-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesisapplication/pdfhttp://hdl.handle.net/10451/34646TID:201988933enginfo: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:RCAAP2023-11-08T16:30:08Zoai:repositorio.ul.pt:10451/34646Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T21:49:21.960781Repositório 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 Linear stability for differential equations with infinite delay via semigroup theory
title Linear stability for differential equations with infinite delay via semigroup theory
spellingShingle Linear stability for differential equations with infinite delay via semigroup theory
Caetano, Diogo Loureiro
Equações diferenciais
Atraso infinito
Estabilidade linear
Teoria de semigrupos
Teoria sun-star
Teses de mestrado - 2018
Departamento de Matemática
title_short Linear stability for differential equations with infinite delay via semigroup theory
title_full Linear stability for differential equations with infinite delay via semigroup theory
title_fullStr Linear stability for differential equations with infinite delay via semigroup theory
title_full_unstemmed Linear stability for differential equations with infinite delay via semigroup theory
title_sort Linear stability for differential equations with infinite delay via semigroup theory
author Caetano, Diogo Loureiro
author_facet Caetano, Diogo Loureiro
author_role author
dc.contributor.none.fl_str_mv Faria, Teresa, 1958-
Repositório da Universidade de Lisboa
dc.contributor.author.fl_str_mv Caetano, Diogo Loureiro
dc.subject.por.fl_str_mv Equações diferenciais
Atraso infinito
Estabilidade linear
Teoria de semigrupos
Teoria sun-star
Teses de mestrado - 2018
Departamento de Matemática
topic Equações diferenciais
Atraso infinito
Estabilidade linear
Teoria de semigrupos
Teoria sun-star
Teses de mestrado - 2018
Departamento de Matemática
description Tese de mestrado, Matemática, Universidade de Lisboa, Faculdade de Ciências, 2018
publishDate 2018
dc.date.none.fl_str_mv 2018-08-29T13:05:56Z
2018
2018
2018-01-01T00:00:00Z
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/masterThesis
format masterThesis
status_str publishedVersion
dc.identifier.uri.fl_str_mv http://hdl.handle.net/10451/34646
TID:201988933
url http://hdl.handle.net/10451/34646
identifier_str_mv TID:201988933
dc.language.iso.fl_str_mv eng
language eng
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.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
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