This work investigates modeling, approximation, estimation, and control for classes of nonlinear systems whose state evolves in space $mathbb{R}^n times H$, where $mathbb{R}^n$ is a n-dimensional Euclidean space and $H$ is a infinite dimensional Hilbert space. Specifically, two classes of nonlinear systems are studied in this dissertation. The first topic develops a novel framework for adaptive estimation of nonlinear systems using reproducing kernel Hilbert spaces. A nonlinear adaptive estimation problem is cast as a time-varying estimation problem in $mathbb{R}^d times H$. In contrast to most conventional strategies for ODEs, the approach here embeds the estimate of the unknown nonlinear function appearing in the plant in a reproducing kernel Hilbert space (RKHS), $H$. Furthermore, the well-posedness of the framework in the new formulation is established. We derive the sufficient conditions for existence, uniqueness, and stability of an infinite dimensional adaptive estimation problem. A condition for persistence of excitation in a RKHS in terms of an evaluation functional is introduced to establish the convergence of finite dimensional approximations of the unknown function in RKHS. Lastly, a numerical validation of this framework is presented, which could have potential applications in terrain mapping algorithms. The second topic delves into estimation and control of history dependent differential equations. This study is motivated by the increasing interest in estimation and control techniques for robotic systems whose governing equations include history dependent nonlinearities. The governing dynamics are modeled using a specific form of functional differential equations. The class of history dependent differential equations in this work is constructed using integral operators that depend on distributed parameters. Consequently, the resulting estimation and control equations define a distributed parameter system whose state, and distributed parameters evolve in finite and infinite dimensional spaces, respectively. The well-posedness of the governing equations is established by deriving sufficient conditions for existence, uniqueness and stability for the class of functional differential equations. The error estimates for multiwavelet approximation of such history dependent operators are derived. These estimates help determine the rate of convergence of finite dimensional approximations of the online estimation equations to the infinite dimensional solution of distributed parameter system. At last, we present the adaptive sliding mode control strategy developed for the history dependent functional differential equations and numerically validate the results on a simplified pitch-plunge wing model. / Ph. D.

Orientador: Suzete Maria Silva Afonso / Banca: Marta Cilene Gadotti / Banca: Miguel Vinicius Santini Frasson / Resumo: Neste trabalho, serão apresentados, inicialmente, resultados de estabilidade de soluções de equações diferenciais funcionais com retardamento (EDFRs) utilizando a teoria de Lyapunov. Em seguida, resultados similares serão abordados via teoria de pontos fixos. Os teoremas de pontos fixos a serem utilizados serão: Teorema do Ponto Fixo da Contração, Teorema do Ponto Fixo de Schauder e o Teorema do Ponto Fixo de Krasnoselskii. Vários exemplos serão apresentados para ilustrar os resultados exibidos / Abstract: In this work, stability results of solutions of retarded functional differential equations (RFDEs) using the Lyapunov theory will be presented initially. Subsequently, similar results will be approached via fixed-point theory. The fixed-point theorems to be used are: Contraction Fixed-Point Theorem, Schauder's Fixed-Point Theorem and Krasnoselskii Fixed-Point Theorem. Several examples will be presented to illustrate the results displayed / Mestre

The subject of this thesis is lens design using a system of functional differential equations arising from Fermat's Principle in geometrical optics. The emphasis is primarily on existence, uniqueness, and analyticity, properties of solutions to these equations, but some asymptotic methods are developed for special cases. Three specific lens problems are considered in detail: the first is an axial lens having two pairs of foci on the optical axis, the second is an axial lens which focuses light at two different frequencies to two distinct points, the third is a lens symmetric about an axis having foci not on said axis.

In this paper, we obtain convergence of solutions of stochastic differential systems with memory gap to those with full finite memory. More specifically, solutions of stochastic differential systems with memory gap are processes in which the intrinsic dependence of the state on its history goes only up to a specific time in the past. As a consequence of this convergence, we obtain a new existence proof and approximation scheme for stochastic functional differential equations (SFDEs) whose coefficients have linear growth. In mathematical finance, an option pricing formula with full finite memory is obtained through convergence of stock dynamics with memory gap to stock dynamics with full finite memory.

The solutions for a class of Neutral Functional Di erential Equations (NFDE) with weakly singular kernels are studied. Using singular expansion techniques, a representation of the solution of the NFDE is obtained by studing an associated Volterra Integral Equation. We study the Collocation Method as a projection method for the approximation of solutions for Volterra Integral Equations. Particulary, the possibility of achieving higher order ap- proximations is discussed. Special attention is given to the choice of the projection space and its relation to the smoothness of the approximated solution. Finally, we study the identification problem for a parameter appearing in the weakly singular operator of the NFDE. / Ph. D.

Control systems governed by nonlinear neutral functional differential equations are formulated as abstract evolution equations in product spaces. At this point existence and uniqueness of solutions are studied. This formulation is used to develop a general approximation scheme for those systems. Convergence of this scheme is analyzed. It is also shown how spline based approximating methods fall within this general framework. An illustrative example is presented. / Ph. D.

In this thesis, we consider a class of stochastic dynamics running backwards, so called backward stochastic differential equations (BSDEs) in the literature. We demonstrate BSDEs can be reformulated as functional differential equations defined on path spaces, and therefore solving BSDEs is equivalent to solving the associated functional differential equations. With such observation we can solve BSDEs on general filtered probability space satisfying the usual conditions, and in particular without the requirement of the martingale representation. We further solve the above functional differential equations numerically, and propose a numerical scheme based on the time discretization and the Picard iteration. This in turn also helps us solve the associated BSDEs numerically. In the second part of the thesis, we consider a class of BSDEs with quadratic growth (QBSDEs). By using the functional differential equation approach introduced in this thesis and the idea of the Cole-Hopf transformation, we first solve the scalar case of such QBSDEs on general filtered probability space satisfying the usual conditions. For a special class of QBSDE systems (not necessarily scalar) in Brownian setting, we do not use such Cole-Hopf transformation at all, and instead introduce the weak solution method, which is to use the strong solutions of forward backward stochastic differential equations (FBSDEs) to construct the weak solutions of such QBSDE systems. Finally we apply the weak solution method to a specific financial problem in the credit risk setting, where we modify the Merton's structural model for credit risk by using the idea of indifference pricing. The valuation and the hedging strategy are characterized by a class of QBSDEs, which we solve by the weak solution method.

Orientador: Andréa Cristina Prokopczyk Arita / Coorientador: Luís Antônio Fernandes Oliveira / Banca: Michelle Fernanda Pierri Hernandez / Banca: Waldemar Donizete Bastos / Resumo: Esta dissertação é um estudo sobre a controlabilidade de sistemas de controle descritos por equações diferenciais abstratas. Primeiramente, são apresentados alguns resultados de controlabilidade para sistemas lineares e sem retardo. Em seguida, é estabelecido um critério para a controlabilidade aproximada de sistemas lineares com retardo, através da comparação entre o conjunto atingível destes sistemas com o conjunto atingível dos sistemas sem retardo. Por fim, é apresentada uma generalização do resultado anterior para sistemas do tipo neutro com retardo / Abstract: This dissertation is a study on the controllability of control systems described by abstract differential equations. First, some results of controllability for linear systems without delay are presented. Then, a criterion for the approximate controllability of linear systems with delay is established by comparing the reachable set of these systems with the reachable set of the systems without delay. Finally, a generalization of the previous result for systems of neutral type with delay is presented / Mestre

Neste trabalho estudamos a controlabilidade e a estabilização de certos tipos de sistemas com retardo. Obtemos um resultado de estabilização para sistema com retardo periódicos e um resultado que nos permite concluir a controlabilidade do sistema com retardo baseado na controlabilidade do mesmo sistema porém, sem retardo. Apresentamos a equação do calor como exemplo / In this work we study controllability and stabilizability of a type of delayed systems. We get a stabilization result for delayed periodic systems and we get a result that allow us to conclude the controllability of the delayed system based on the controllability of the same system without delay. We present the heat equation like example

O objetivo deste trabalho é investigar propriedades qualitativas de certas equações diferenciais funcionais retardadas de segunda ordem quando lhes são impostos controles de impulsos adequados. Os principais resultados dizem respeito a estabilidade e oscilação por impulsos. Mais especificamente, consideramos algumas equações e provamos que suas soluções triviais podem ser estabilizadas por impulsos. Em seguida, consideramos uma destas equações e provamos que suas soluções podem se tornar oscilatórias com a imposição apropriada de controles de impulsos. Apresentamos alguns exemplos que ilustram nossos resultados. Além do objetivo acima, procuramos produzir um texto que compreendesse a teoria fundamental das equações diferenciais funcionais retardadas impulsivas, teoria esta que, até então, não podia ser encontrada num único texto como este. Desenvolvemos e discutimos existência, unicidade, continuação de soluções, intervalo maximal de existência e dependência contínua de soluções dos valores iniciais para equações diferenciais retardadas impulsivas. / The purpose of this work is to investigate qualitative properties of certain second order delay differential equations when some proper impulse controls are added to them. The main results concern the stability and scillation by impulses. More specifically, we consider some equations and prove that their trivial solutions can be stabilized by impulses. We also consider one of these equations and prove that all solutions oscillate when proper impulse controls are imposed. We give some examples to illustrate our results. Because dealing with systems with both delays and impulses is a recent interest of some mathematicians we also considered producing a text that would encompass the fundamental theory of retarded functional differential equations with impulses. Up to now such theory could not be found in a single text as this one. Therefore we discuss and develop basic aspects of the theory as existence, uniqueness, continuability of solutions, maximal interval of existence and continuous dependence of solutions on initial values for impulsive retarded differential equations.