Maximum principles for control systems described by measure functional differential equations

“…Note that, by measure functional differential equations, the Volterra type equations of form (8) are usually meant in the existing bibliography on the subject (see, e.g., [8,13,22]), whereas equations with more general types of argument deviation are rather scarce (we can cite, perhaps, only [4, page 217]). Comparing (8) with (1), we find that the latter includes non-Volterra cases as well.…”

Section: Introduction Motivation and Problem Settingmentioning

confidence: 99%

Measure Functional Differential Equations in the Space of Functions of Bounded Variation

Afonso

Rontó

2013

Abstract and Applied Analysis

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“…While the theories of measure differential equations (including the more general abstract measure differential equations) and measure delay differential equations are welldeveloped (see [21], [24], [25], [26], [27], [28], [29], [30], [68], [69], [85], [96], [97], for instance), the literature concerning measure functional differential equations is scarce. See [19], [36] and [37]. These equations encompass various types of equations as we will prove later.…”

Section: Measure Fdesmentioning

confidence: 95%

“…where the Kurzweil-Henstock-Stieltjes integral on the right-hand side is taken with respect to a nondecreasing function g : [t 0 , t 0 + ] ! R. These equations have been studied in [19], [36], among others papers.…”

Section: Impulsive Measure Fdes and Measure Fdesmentioning

confidence: 99%

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Equações diferenciais funcionais em medida e equações dinâmicas funcionais impulsivas em escalas temporais

Mesquita¹

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The aim of this work is to investigate and develop the theory of impulsive functional dynamic equations on time scales. We prove that these equations represent a special case of impulsive measure functional differential equations. Moreover, we present a relation between these equations and measure functional differential equations and, also, a correspondence between them and generalized ordinary differential equations. Also, we clarify the relation between measure functional differential equations and functional dynamic equations on time scales. We obtain results on the existence and uniqueness of solutions, continuous dependence on parameters, non-periodic and periodic averaging principles and stability results for all these types of equations. Moreover, we prove some properties concerning regulated functions and equiregulated sets in a Banach space which were essential to our purposes. The new results presented in this work are contained in 7 papers, two of which have already been published and one accepted. See [16], [32], [34], [36], [37], [38] and [84]. viii Resumo O objetivo deste trabalho é investigar e desenvolver a teoria de equações dinâmicas funcionais impulsivas em escalas temporais. Mostramos que estas equações representam um caso especial de equações diferenciais funcionais em medida impulsivas. Também, apresentamos uma relação entre estas equações e as equações diferenciais funcionais em medida e, ainda, mostramos uma relação entre elas e as equações diferenciais ordinárias generalizadas. Relacionamos, também, as equações diferenciais funcionais em medida e as equações dinâmicas funcionais em escalas temporais. Obtemos resultados sobre existência e unicidade de soluções, dependência contínua, método da média periódico e não-periódico bem como resultados de estabilidade para todos os tipos de equações descritos anteriormente. Também, provamos algumas propriedades relativas às funções regradas e aos conjuntos equiregrados em espaços de Banach, que foram essenciais para os nossos propósitos. Os resultados novos apresentados neste trabalho estão contidos em 7 artigos, dos quais dois já foram publicados e um aceito. Veja [16], [32], [34], [36], [37], [38] e [84]. x With the correspondences between the solutions of a functional dynamic equation on time scales and the solutions of a measure functional differential equation and these last ones and the solutions of a generalized ordinary differential equation, we were able to prove several interesting results for these types of equations, such as local existence and uniqueness of solutions, continuous dependence of solutions on parameters and periodic averaging principles. All these results can be found in [36] and they will be described with more details in Chapters 5, 6 and 7. After this, we decided to improve the class of equations on time scales which we were dealing with. Since impulsive differential equations are powerful tools for applications and, Chapter 7 is devoted to averaging principles. It is divided in three sections. The first one concerns n...

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Maximum principles for control systems described by measure functional differential equations

Cited by 3 publications

References 6 publications

Stability of non-linear systems with time-varying delays

Stability of non-linear systems with time-varying delays

Measure Functional Differential Equations in the Space of Functions of Bounded Variation

Equações diferenciais funcionais em medida e equações dinâmicas funcionais impulsivas em escalas temporais

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