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Master's Dissertation
DOI
10.11606/D.55.2008.tde-01042008-110225
Document
Author
Full name
Lucas Felipe Rodrigues dos Santos Garcia
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2008
Supervisor
Committee
Federson, Márcia Cristina Anderson Braz (President)
Frasson, Miguel Vinicius Santini
Gadotti, Marta Cilene
Title in Portuguese
Estabilidade para equações diferenciais em medida
Keywords in Portuguese
Equações diferenciais em medida
Equações diferenciais ordinárias generalizadas
Estabilidade variacional
Funcional de Lyapunov
Abstract in Portuguese
Neste trabalho, nós investigamos a estabilidade da solução trivial da seguinte Equação Diferencial em Medida (EDM) Dx = f(x, t) + g(x, t)Du, (1) onde 'B BARRA IND. c' = {'x PERTENCE A' 'R POT. n'; //x// ' < OU=' c}, f : 'B BARRA IND.c' × [a, b] 'SETA' 'R POT.n' e g : 'B BARRA IND. c' × [a, b] 'SETA' ' R POT n', u : [a, b] ' ETA' ! R é uma função de variação limitada em [a, b] e contínua à esquerda em (a, b], f(x, ·) é Lebesgue integrável em [a, b], g(x, ·) é du-integrável em [a, b], f(0, t) = 0 = g(0, t) para todo t e Dx e Du denotam as derivadas distribucionais de x e u no sentido de L. Schwartz. Nós consideramos as funções f e g num contexto bem geral. Assim, para obtermos nossos resultados, nós provamos a correspondência biunívoca entre as soluções da classe de EDMs (1) em tal contexto e as soluções de certa classe de equação diferencial ordinária generalizada (EDOG). Desta forma, foi possível aplicarmos as técnicas e resultados da teoria das equações diferenciais ordinárias generalizadas, como teoremas do tipo Lyapunov e do tipo Lyapunov inverso, para obtermos os resultados correspondentes para a EDM (1). Os resultados apresentados neste trabalho sobre estabilidade da solução trivial da EDM (1) são inéditos. Parte deles foram apresentados no 660 Seminário Brasileiro de Análise. Veja [7]
Title in English
Stability for measure differential equations
Keywords in English
Generalized ordinary differential equations
Lyapunov functional
Measure differential equations
Variational stability
Abstract in English
In this work, we investigate the stability of the trivial solution of the following Measure Differential Equation (MDE) Dx = f(x, t) + g(x, t)Du, (2) where 'B BARRA IND.c' = {x 'PERTENCE A' 'R POT.n'; //x// ' < OU=' c}, f : 'B BARRA IND.c' × [a, b] 'SETA' 'R POT.n' and g : 'B BARRA IND.c' × [a, b] 'SETA' 'R POT. n' , u is function of bounded variation in [a, b] which is also left continuous on (a, b], f(x, ·) is Lebesgue integrable in [a, b] and g(x, ·) is du-integrable in [a, b], f(0, t) = 0 = g(0, t) for all t and Dx, Du denote the derivatives of x and u in the sense of distributions of L. Schwartz. We consider the functions f and g in a general setting. Thus, in order to obtain our results, we prove there is a one-to-one correspondence between the solutions of the MDE 2) in this setting and the solutions of a certain class of generalized ordinary differential equation (GODE). In this manner, it was possible to apply the techniques and results from the teory of GODE's, such as Lyapunov-type and converse Lyapunov-type theorems, to obtain the corresponding results for our MDE (2). The results presented in this work concerning the stability of the trivial solution of the MDE (2) are new. Some of them were presented at the 66th Seminário Brasileiro de Análise. See [7]
 
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Publishing Date
2008-04-01
 
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