The subject of this thesis is the motion planning algorithm known as the continuation method. To solve motion planning problems, the continuation method proceeds by lifting curves in state space to curves in control space; the lifted curves are the solutions of special initial value problems called path-lifting equations. To validate this procedure, three distinct obstructions must be overcome. The first obstruction is that the endpoint maps of the control system under study must be twice continuously differentiable. By extending a result of A. Margheri, we show that this differentiability property is satisfied by an inclusive class of time-varying fully nonlinear control systems. The second obstruction is the existence of singular controls, which are simply the singular points of a fixed endpoint map. Rather than attempting to completely characterize such controls, we demonstrate how to isolate control systems for which no controls are singular. To this end, we build on the work of S. A. Vakhrameev to obtain a necessary and sufficient condition. In particular, this result accommodates time-varying fully nonlinear control systems. The final obstruction is that the solutions of path-lifting equations may not exist globally. To study this problem, we work under the standing assumption that the control system under study is control-affine. By extending a result of Y. Chitour, we show that the question of global existence can be resolved by examining Lie bracket configurations and momentum functions. Finally, we show that if the control system under study is completely unobstructed with respect to a fixed motion planning problem, then its corresponding endpoint map is a fiber bundle. In this sense, we obtain a necessary condition for unobstructed motion planning by the continuation method. / Thesis (Ph.D, Chemical Engineering) -- Queen's University, 2012-12-18 20:53:43.272

The principal aim of this thesis is to provide a comprehensive account of the meaning of keep V-ing constructions, see (1a) and (1b). (1) a. Mary kept winning (again and again). (1) b. John kept running (for another ten minutes). On the basis of a systematic study of combinations of keep with predicates of different aktionsarts, it is shown that keep can give rise to two different readings which share the overall meaning of ‘continued activity’. It is argued that the two readings of keep V-ing arise from different aspectual properties of the predicate in the complement clause. Under the first reading, labelled the continuative-iterative reading, (1a), the event in the complement clause is telic, and the interpretation is an iterative reading. Under the second reading, labelled the continuative reading, (1b), the event in the complement clause is atelic, and the interpretation is a reading of nonstop continuation. It is argued that keep combines with activity predicates in the relevant construction type, that is, with dynamic, durative and atelic events, and that keep has the ability to induce aspect shift when combining with predicates that are not inherent activities. Thus, in (1a), a punctual and telic winning event is iterated, creating a series which in itself is durative and atelic. In (1b), the running event is already durative and atelic. By comparing keep V-ing with the progressive construction be V-ing, (2), and with two other continuative constructions, continue V-ing, (3), and V on, (4), it is shown that keep readily shifts a telic predicate into an atelic reading by taking scope over the entire event, (1a), but cannot take scope over an internal part of a telic event. Both be V-ing, (2), and continue V-ing, (3), are able to take scope over an internal part of a telic event. (2) John was building the house. (3) John continued building the house. (4) John ran on. In addition, unlike continue V-ing and V on, keep V-ing does not necessarily denote continuation of an event which has already been initiated.

Neste trabalho, estudamos propriedades de continuação única para as soluções da equação tipo Schrödinger com um ponto interação centrado em x=0, \\partial_tu=i(\\Delta_Z+V)u, onde V=V(x,t) é uma função de valor real e -\\Delta_Z é o operador escrito formalmente como \\[-\\Delta_Z=-\\frac\\frac{d^2}{dx^2}+Z\\delta_0,\\] sendo \\delta_0 a delta de Dirac centrada em zero e Z qualquer número real. Logo, usamos estes resultados para ver o possível fenômeno de concentração das soluções, que explodem, da equação de tipo Schrödinger não linear com um ponto de interação em x=0, \\[\\partial_tu=i(\\Delta_Zu+|u|^u),\\] com ho>5. Também, mostramos que para certas condições sobre o potencial dependente do tempo V, a equação linear em cima tem soluções não triviais. / In this work, we study unique continuation properties for solutions of the Schrödinger equations with an point interaction centered at $x=0$, \\begin\\label \\partial_tu=i(\\Delta_Z+V)u, \\end where $V=V(x,t)$ is real value function and $-\\Delta_Z$ is the operator formally written \\[-\\Delta_Z=-\\frac\\frac{d^2}{dx^2}+Z\\delta_0,\\] and $\\delta_0$ is Dirac\'s delta centered at zero and $Z$ is a real number. Next, we use these results in order to study the possible profile of the concentration of blow up solutions for the non linear Schrödinger equation with a point interaction at $x=0$, \\[\\partial_tu=i(\\Delta_Zu+|u|^u),\\] with $ho>5$. Besides, we show that the equation above has non trivial solutions for some conditions on the time dependent potencial $V$.

As equações diferencias parciais tem origem na modelagem do problemas nas ciências e engenharia, tais como a equação do calor, equação da onda, equação de Poisson, entre outras. Para muitas destas equações não é tão simples obter uma técnica analítica para achar sua solução e nestes casos é necessário uso de soluções aproximadas obtidas pelo computador. Existem técnicas tradicionais para solução numérica de uma grande classe de equações diferenciais, mas quando esta equação está na forma implícita, muitas destas técnicas já não podem ser aplicadas. Frequentemente as equações diferenciais parciais de segunda ordem tem maior estudo que as equações de primeira ordem sendo uma das razões que os modelos envolvem derivadas de segunda ordem. No caso das equações diferenciais parciais de primeira ordem implícitas a não linearidade em alguns casos não permite determinar uma solução de forma simples. O trabalho desenvolvido faz uma revisão do método das características para estabelecer as condições necessárias e suficientes, que permitam encontrar uma solução, ao mesmo tempo evidencia a complexidade de determinar uma solução clássica. Dentro das aplicações existentes relacionadas com as Equações Diferenciais Parciais Implícitas de Primeira Ordem, podemos mencionar a Equação cinemática e a Equação de Hamilton-Jacobi que podem-se associar com o movimento de partículas. Para a solução de uma Equação Diferencial Implícita de Primeira Ordem o método das características tem uma estrutura de solução que permite resolver a equação de forma analítica e numérica, desde que se verifique o Teorema de Cauchy. O objetivo deste trabalho de mestrado é obter um método numérico para a solução de equações diferenciais parciais de primeira ordem implícitas. Nós propomos um método numérico do tipo previsor-corretor que resolve uma EDP de primeira ordem implícita, utilizando o sistema característico em conjunto com as condições de banda, para reduzir o erro global nas iterações. / Partial differential equations arise in the modeling of problems in science and engineering, such as the heat equation, wave equation, Poisson equation, among others. For many of these equations it is not so simple to obtain an analytical technique to find a solution in these cases and it is necessary to use a computer to obtain approximate solutions. There are traditional techniques for numerical solution of a large class of differential equations, but when this equation is in implicit form, many of these techniques can no longer be applied. Often partial differential equations of second order are more studied than first order equations the reason being that one of the models involve secondorder derivatives. In the case of implicit partial differential equations of first order the non-linearity in some cases does not allow for a solution in simple from to be determined. The work reviews the method of characteristics to establish the necessary and sufficient conditions that will find a solution at the same time demonstrates the complexity of determining classical solution. Within existing applications related to Partial Differential Equations of First Order Implicit, we can mention the textit kinematic equation and textit equation Hamilton-Jacobi that can be associated with the movement of particles. For the solution of a differential equation First Implicit Order the method of characteristics has a solution framework that enables solve the equation analytically and numerically, provided there is the Cauchy theorem. The objective of this master thesis is to obtain a numerical method for the solution of partial differential equations first order implicit. We propose a numerical method of predictor-corrector type that resolves a EDP first implicate order, using the characteristic system in conjunction with the band conditions, to reduce the overall error in iterations.

Newton-Picard methods are iterative methods that work well for computing roots of nonlinear equations within a continuation framework. This project presents one of these methods and includes the results of a computation involving the Brusselator problem performed by an implementation of the method. This work was done in collaboration with Andrew Salinger at Sandia National Laboratories.

As equações diferencias parciais tem origem na modelagem do problemas nas ciências e engenharia, tais como a equação do calor, equação da onda, equação de Poisson, entre outras. Para muitas destas equações não é tão simples obter uma técnica analítica para achar sua solução e nestes casos é necessário uso de soluções aproximadas obtidas pelo computador. Existem técnicas tradicionais para solução numérica de uma grande classe de equações diferenciais, mas quando esta equação está na forma implícita, muitas destas técnicas já não podem ser aplicadas. Frequentemente as equações diferenciais parciais de segunda ordem tem maior estudo que as equações de primeira ordem sendo uma das razões que os modelos envolvem derivadas de segunda ordem. No caso das equações diferenciais parciais de primeira ordem implícitas a não linearidade em alguns casos não permite determinar uma solução de forma simples. O trabalho desenvolvido faz uma revisão do método das características para estabelecer as condições necessárias e suficientes, que permitam encontrar uma solução, ao mesmo tempo evidencia a complexidade de determinar uma solução clássica. Dentro das aplicações existentes relacionadas com as Equações Diferenciais Parciais Implícitas de Primeira Ordem, podemos mencionar a Equação cinemática e a Equação de Hamilton-Jacobi que podem-se associar com o movimento de partículas. Para a solução de uma Equação Diferencial Implícita de Primeira Ordem o método das características tem uma estrutura de solução que permite resolver a equação de forma analítica e numérica, desde que se verifique o Teorema de Cauchy. O objetivo deste trabalho de mestrado é obter um método numérico para a solução de equações diferenciais parciais de primeira ordem implícitas. Nós propomos um método numérico do tipo previsor-corretor que resolve uma EDP de primeira ordem implícita, utilizando o sistema característico em conjunto com as condições de banda, para reduzir o erro global nas iterações. / Partial differential equations arise in the modeling of problems in science and engineering, such as the heat equation, wave equation, Poisson equation, among others. For many of these equations it is not so simple to obtain an analytical technique to find a solution in these cases and it is necessary to use a computer to obtain approximate solutions. There are traditional techniques for numerical solution of a large class of differential equations, but when this equation is in implicit form, many of these techniques can no longer be applied. Often partial differential equations of second order are more studied than first order equations the reason being that one of the models involve secondorder derivatives. In the case of implicit partial differential equations of first order the non-linearity in some cases does not allow for a solution in simple from to be determined. The work reviews the method of characteristics to establish the necessary and sufficient conditions that will find a solution at the same time demonstrates the complexity of determining classical solution. Within existing applications related to Partial Differential Equations of First Order Implicit, we can mention the textit kinematic equation and textit equation Hamilton-Jacobi that can be associated with the movement of particles. For the solution of a differential equation First Implicit Order the method of characteristics has a solution framework that enables solve the equation analytically and numerically, provided there is the Cauchy theorem. The objective of this master thesis is to obtain a numerical method for the solution of partial differential equations first order implicit. We propose a numerical method of predictor-corrector type that resolves a EDP first implicate order, using the characteristic system in conjunction with the band conditions, to reduce the overall error in iterations.

Orientador: Dílson Amâncio Alves / Banca: Luiz Fernando Bovolato / Banca: Luis Carlos Origa de Oliveira / Banca: Vivaldo Fernando da Costa / Banca: Eduardo Nobuhiro Asada / Resumo: Este trabalho apresenta uma metodologia alternativa para a melhoria da margem de carregamento e redução da perda total de potência ativa com base no método da continuação. Para atingir esta meta, uma equação de parametrização baseada na perda de potência ativa total e as equações da potência reativa nas barras de geração são acrescentadas às equações de fluxo de carga convencional. As tensões nas barras PV são consideradas como variáveis de controle e um novo parâmetro é escolhido para reduzir as perdas de potência ativa nas linhas de transmissão. Os resultados mostram que este procedimento, em geral, conduz a um aumento no ponto de máximo carregamento e por conseguinte, melhoria na margem estática da estabilidade de tensão. Este procedimento também leva a uma redução nos custos operacionais e, simultaneamente, uma melhoria no perfil da tensão. / Abstract: This work presents an alternative methodology for loading margin improvement and total real power losses reduction by using a continuation method. In order to attain this goal, a parameterizing equation based on the total real power losses and the equations of the reactive power at the slack and generation buses are added to the conventional Power Flow equations. The voltages at these buses are considered as control variables and a new parameter is chosen with to reduce the real power losses in the transmission lines. The results show that this procedure leads to maximum loading point increase and consequently, in static voltage stability margin improvement. Besides, this procedure also takes to a reduction in the operational costs and, simultaneously, to voltage profile improvement. / Doutor

This thesis consists in the study of a periodically forced slow-fast system in both its excitable and oscillatory regimes. The slow-fast system under consideration is the FitzHugh-Nagumo model, and the periodic forcing consists of a train of Gaussian-shaped pulses, the width of which is much shorter than the action potential duration. This system is a qualitative model for both an excitable cell and a spontaneously beating cell submitted to periodic electrical stimulation. Such a configuration has often been studied in cardiac electrophysiology, due to the fact that it constitutes a simplified model of the situation of a cardiac cell in the intact heart, and might therefore contribute to the understanding of cardiac arrhythmias. Using continuation methods (AUTO software), we compute periodic-solution branches for the periodically forced system, taking the stimulation period as bifurcation parameter. We then study the evolution of the resulting bifurcation diagram as the stimulation amplitude is raised. In both the excitable and the oscillatory regimes, we find that a critical amplitude of stimulation exists below which the behaviour of the system is trivial: in the excitable case, the bifurcation diagram is restricted to a stable subthreshold period-1 branch, and in the oscillatory case, all the stable periodic solutions belong to isolated loops (i.e., to distinct closed solution branches). Due to the slow-fast nature of the system, the changes that take place in the bifurcation diagram as the critical amplitude is crossed are drastic, while the way the bifurcation diagram re-simplifies above some second amplitude is much more gentle. In the oscillatory case, we show that the critical amplitude is also the amplitude at which the topology of phase-resetting changes type. We explain the origin of this coincidence by considering a one-dimensional discrete map of the circle derived from the phase-resetting curve of the oscillator (the phase-resetting map), map which constitutes a good approximation of the original differential equations under certain conditions. We show that the bifurcation diagram of any such circle map, where the bifurcation parameter appears only in an additive fashion, is always characterized by the period-1 solutions belonging to isolated loops when the topological degree of the map is one, while these period-1 solutions belong to a unique branch when the topological degree of the map is zero.

L'objet de ce travail est d'étendre l'utilisation des méthodes de continuation au cas des problèmes à frontière libre du type (on l'écrit formellement F(λ, y, u) = 0) en présence de bifurcation. Les méthodes de continuation ont été largement étudiées dans le cadre de la résolution de problèmes aux limites non-linéaires, posés sur un domaine fixe, dépendant d'un paramètre. On se réfère aux travaux initialisés par H.B. Keller : (KEL77) (RHE80] (MIT80] Pour adapter ces méthodes à la résolution de problèmes à frontière libre, il a fallu considérer ces derniers comme des problèmes non-linéaires dont l'inconnue est le couple formé de la solution et d'un paramétrage de la frontière libre. Nous nous sommes inspirés des travaux de A. Dervieux consacrés à la perturbation de la solution d'un problème aux limites par rapport à son domaine géométrique [DER81], pour lier entre elles, au moins formellement, les variations de la frontière libre et celles de la solution.

The theory of solving polynomial systems by homotopy continuation method has been proposed by Garcia, Zangwill and Drexler, and the most typical method in this category is total degree homotpy. The numerical implementation of tracking homotopy curves can be taken as two parts: prediction and correction. In this thesis we compare the performance of several prediction methods in the total degree homotopy, including Runge-Kutta method, Adams-Bashforth method and cubic Hermite method. In addition, we design an adaptive stepsize control algorithm in path tracking, which is based on the information obtained during Newton correction process. The numerical experiment shows that the stepsize control algorithm is quite efficient and reliable in path tracking. In the end we employ the algorithm for solving eigenvalue problems by random product homotopy method