A counterexample in the theory of linear singularly perturbed systems
In this note we compare the bounded solutions of the linear singularly perturbed system ε X' = A (t) x + f (t), with the solutions of the algebraic system A (t) x + f (t) = 0. Here A and f are bounded C1 functions with bounded derivatives. We assume that the eigenvalues of A (t) satisfy | R...
- Autores:
-
Naulin, Raúl
- Tipo de recurso:
- Article of journal
- Fecha de publicación:
- 1991
- Institución:
- Universidad Nacional de Colombia
- Repositorio:
- Universidad Nacional de Colombia
- Idioma:
- spa
- OAI Identifier:
- oai:repositorio.unal.edu.co:unal/43433
- Acceso en línea:
- https://repositorio.unal.edu.co/handle/unal/43433
http://bdigital.unal.edu.co/33531/
- Palabra clave:
- 51 Matemáticas / Mathematics
Bounded solutions
system linear algebraic system
bounded functions
Lipschitz function
Soluciones acotadas
sistema lineal
sistema algebráico
funciones acotadas
función de Lipschitz
- Rights
- openAccess
- License
- Atribución-NoComercial 4.0 Internacional
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Atribución-NoComercial 4.0 InternacionalDerechos reservados - Universidad Nacional de Colombiahttp://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Naulin, Raúl25df13a8-27e4-46d5-b20f-0cea2b9db1743002019-06-28T11:58:22Z2019-06-28T11:58:22Z1991-01-01ISSN: 2357-4100https://repositorio.unal.edu.co/handle/unal/43433http://bdigital.unal.edu.co/33531/In this note we compare the bounded solutions of the linear singularly perturbed system ε X' = A (t) x + f (t), with the solutions of the algebraic system A (t) x + f (t) = 0. Here A and f are bounded C1 functions with bounded derivatives. We assume that the eigenvalues of A (t) satisfy | R e λ (t )| ≥ y and gt; 0. It is known that for small ε, the following estimate is valid hasta ||kε (f) + A-1 f || ≤ εL || f ||1, where kε(f) denotes the bounded solution of ε X' = A(t)x +f (t), || f || = sup R| f(t)|, || f ||1 : = || f || +|| f´ || and L is a constant. We prove that this estimate cannot be replaced by ||kε (f) + A-1 f || ≤ εL || f ||. Futhermore, if, instead of the condition that A be C1, we require that the function be bounded and Lipschitz continuous, we show that the same estimate, ||kε (f) + A-1 || ≤ εL || f ||1, can be obtained.En esta nota se comparan las soluciones acotadas del sistema lineal singularmente perturbado ε X' = A (t) x + f (t), con las soluciones del sistema algebráico A (t) x + f (t) = 0. Aquí A y f son funciones acotadas de clase C1, con derivadas acotadas. Suponemos además que los valores propios de A (t) satisfacen la condición | R e λ (t)| ≥ y and gt; 0. Es sabido que para ϵ C1 y ε suficientemente pequeños vale la siguiente estimación: ||kε (f) + A-1 || ≤ εL || f ||1, donde kε (f) denota la única solución acotada de ε X' = A(t)x +f (t), || f || = sup R| f(t)|, || f ||1 : = || f || +|| f´ || y L es una constante que no depende de f ni de ε. Probaremos que esta estimación no puede ser extendida hasta ||kε (f) + A-1 f || ≤ εL || f ||. Además, si en lugar de exigir que A sea de clase C1 pedimos que A sea una función de Lipschitz acotada, entonces sigue siendo válida la estimación ||kε (f) + A-1 || ≤ εL || f ||1.application/pdfspaUniversidad Nacional de Colombiahttp://revistas.unal.edu.co/index.php/recolma/article/view/33395Universidad Nacional de Colombia Revistas electrónicas UN Revista Colombiana de MatemáticasRevista Colombiana de MatemáticasRevista Colombiana de Matemáticas; Vol. 25, núm. 1-4 (1991); 95-102 0034-7426Naulin, Raúl (1991) A counterexample in the theory of linear singularly perturbed systems. Revista Colombiana de Matemáticas; Vol. 25, núm. 1-4 (1991); 95-102 0034-7426, 25 (1-4). pp. 95-102. ISSN 2357-410051 Matemáticas / MathematicsBounded solutionssystem linear algebraic systembounded functionsLipschitz functionSoluciones acotadassistema linealsistema algebráicofunciones acotadasfunción de LipschitzA counterexample in the theory of linear singularly perturbed systemsArtículo de revistainfo:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1http://purl.org/coar/version/c_970fb48d4fbd8a85Texthttp://purl.org/redcol/resource_type/ARTORIGINAL33395-123875-1-PB.pdfapplication/pdf2112598https://repositorio.unal.edu.co/bitstream/unal/43433/1/33395-123875-1-PB.pdfac05c72c7d3a2de2296ba08dedc22a0aMD51THUMBNAIL33395-123875-1-PB.pdf.jpg33395-123875-1-PB.pdf.jpgGenerated Thumbnailimage/jpeg7359https://repositorio.unal.edu.co/bitstream/unal/43433/2/33395-123875-1-PB.pdf.jpg52f46a5b1e5baf65c776a7e9d45f5cddMD52unal/43433oai:repositorio.unal.edu.co:unal/434332024-02-10 23:06:13.617Repositorio Institucional Universidad Nacional de Colombiarepositorio_nal@unal.edu.co |
dc.title.spa.fl_str_mv |
A counterexample in the theory of linear singularly perturbed systems |
title |
A counterexample in the theory of linear singularly perturbed systems |
spellingShingle |
A counterexample in the theory of linear singularly perturbed systems 51 Matemáticas / Mathematics Bounded solutions system linear algebraic system bounded functions Lipschitz function Soluciones acotadas sistema lineal sistema algebráico funciones acotadas función de Lipschitz |
title_short |
A counterexample in the theory of linear singularly perturbed systems |
title_full |
A counterexample in the theory of linear singularly perturbed systems |
title_fullStr |
A counterexample in the theory of linear singularly perturbed systems |
title_full_unstemmed |
A counterexample in the theory of linear singularly perturbed systems |
title_sort |
A counterexample in the theory of linear singularly perturbed systems |
dc.creator.fl_str_mv |
Naulin, Raúl |
dc.contributor.author.spa.fl_str_mv |
Naulin, Raúl |
dc.subject.ddc.spa.fl_str_mv |
51 Matemáticas / Mathematics |
topic |
51 Matemáticas / Mathematics Bounded solutions system linear algebraic system bounded functions Lipschitz function Soluciones acotadas sistema lineal sistema algebráico funciones acotadas función de Lipschitz |
dc.subject.proposal.spa.fl_str_mv |
Bounded solutions system linear algebraic system bounded functions Lipschitz function Soluciones acotadas sistema lineal sistema algebráico funciones acotadas función de Lipschitz |
description |
In this note we compare the bounded solutions of the linear singularly perturbed system ε X' = A (t) x + f (t), with the solutions of the algebraic system A (t) x + f (t) = 0. Here A and f are bounded C1 functions with bounded derivatives. We assume that the eigenvalues of A (t) satisfy | R e λ (t )| ≥ y and gt; 0. It is known that for small ε, the following estimate is valid hasta ||kε (f) + A-1 f || ≤ εL || f ||1, where kε(f) denotes the bounded solution of ε X' = A(t)x +f (t), || f || = sup R| f(t)|, || f ||1 : = || f || +|| f´ || and L is a constant. We prove that this estimate cannot be replaced by ||kε (f) + A-1 f || ≤ εL || f ||. Futhermore, if, instead of the condition that A be C1, we require that the function be bounded and Lipschitz continuous, we show that the same estimate, ||kε (f) + A-1 || ≤ εL || f ||1, can be obtained. |
publishDate |
1991 |
dc.date.issued.spa.fl_str_mv |
1991-01-01 |
dc.date.accessioned.spa.fl_str_mv |
2019-06-28T11:58:22Z |
dc.date.available.spa.fl_str_mv |
2019-06-28T11:58:22Z |
dc.type.spa.fl_str_mv |
Artículo de revista |
dc.type.coar.fl_str_mv |
http://purl.org/coar/resource_type/c_2df8fbb1 |
dc.type.driver.spa.fl_str_mv |
info:eu-repo/semantics/article |
dc.type.version.spa.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.coar.spa.fl_str_mv |
http://purl.org/coar/resource_type/c_6501 |
dc.type.coarversion.spa.fl_str_mv |
http://purl.org/coar/version/c_970fb48d4fbd8a85 |
dc.type.content.spa.fl_str_mv |
Text |
dc.type.redcol.spa.fl_str_mv |
http://purl.org/redcol/resource_type/ART |
format |
http://purl.org/coar/resource_type/c_6501 |
status_str |
publishedVersion |
dc.identifier.issn.spa.fl_str_mv |
ISSN: 2357-4100 |
dc.identifier.uri.none.fl_str_mv |
https://repositorio.unal.edu.co/handle/unal/43433 |
dc.identifier.eprints.spa.fl_str_mv |
http://bdigital.unal.edu.co/33531/ |
identifier_str_mv |
ISSN: 2357-4100 |
url |
https://repositorio.unal.edu.co/handle/unal/43433 http://bdigital.unal.edu.co/33531/ |
dc.language.iso.spa.fl_str_mv |
spa |
language |
spa |
dc.relation.spa.fl_str_mv |
http://revistas.unal.edu.co/index.php/recolma/article/view/33395 |
dc.relation.ispartof.spa.fl_str_mv |
Universidad Nacional de Colombia Revistas electrónicas UN Revista Colombiana de Matemáticas Revista Colombiana de Matemáticas |
dc.relation.ispartofseries.none.fl_str_mv |
Revista Colombiana de Matemáticas; Vol. 25, núm. 1-4 (1991); 95-102 0034-7426 |
dc.relation.references.spa.fl_str_mv |
Naulin, Raúl (1991) A counterexample in the theory of linear singularly perturbed systems. Revista Colombiana de Matemáticas; Vol. 25, núm. 1-4 (1991); 95-102 0034-7426, 25 (1-4). pp. 95-102. ISSN 2357-4100 |
dc.rights.spa.fl_str_mv |
Derechos reservados - Universidad Nacional de Colombia |
dc.rights.coar.fl_str_mv |
http://purl.org/coar/access_right/c_abf2 |
dc.rights.license.spa.fl_str_mv |
Atribución-NoComercial 4.0 Internacional |
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http://creativecommons.org/licenses/by-nc/4.0/ |
dc.rights.accessrights.spa.fl_str_mv |
info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
Atribución-NoComercial 4.0 Internacional Derechos reservados - Universidad Nacional de Colombia http://creativecommons.org/licenses/by-nc/4.0/ http://purl.org/coar/access_right/c_abf2 |
eu_rights_str_mv |
openAccess |
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application/pdf |
dc.publisher.spa.fl_str_mv |
Universidad Nacional de Colombia |
institution |
Universidad Nacional de Colombia |
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Repositorio Institucional Universidad Nacional de Colombia |
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