We study the equilibrium solutions of an integrodifferential equation used to model one-dimensional biological swarms. We assume that the motion of the swarm is governed by pairwise interactions, or a convolution in the continuous setting, and derive a continuous model from conservation laws. The steady-state solution found for the model is compactly supported and is shown to be an attractive equilibrium solution via linear perturbation theory. Numerical simulations support that the steady-state solution is attractive for all initial swarm distributions. Some initial results for the model in higher dimensions are also presented.

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Human T-lymphotropic virus type I (HTLV-I) is a persistent human retrovirus characterized by life-long infection and risk of developing HAM/TSP, a progressive neurological and inflammatory disease. Despite extensive studies of HTLV-I, a complete understanding of the viral dynamics has been elusive. Previous mathematical models are unable to fully explain experimental observations. Motivated by a new hypothesis for the mechanism of HTLV-I infection, a three dimensional compartmental model of ordinary differential equations is constructed that focusses on the highly dynamic interactions among populations of healthy, latently infected, and actively infected target cells. Results from mathematical and numerical investigations give rise to relevant biological interpretations. Comparisons of these results with experimental observations allow us to assess the validity of the original hypothesis. Our findings provide valuable insights to the infection and persistence of HTLV-I in vivo and motivate future mathematical and experimental work. / Applied Mathematics

Cette thèse est dédiée au développement d’algorithmes génériques pour l’observation ensembliste de l’état continu et du mode discret des systèmes dynamiques hybrides dans le but de réaliser la détection de défauts. Cette thèse est organisée en deux grandes parties. Dans la première partie, nous avons proposé une méthode rapide et efficace pour le passage ensembliste des gardes. Elle consiste à procéder à la bissection dans la seule direction du temps et ensuite faire collaborer plusieurs contracteurs simultanément pour réduire le domaine des vecteurs d’état localisés sur la garde, durant la tranche de temps étudiée. Ensuite, nous avons proposé une méthode pour la fusion des trajectoires basée sur l'utilisation des zonotopes. Ces méthodes, utilisées conjointement, nous ont permis de caractériser de manière garantie l'ensemble des trajectoires d'état hybride engendrées par un système dynamique hybride incertain sur un horizon de temps fini. La deuxième partie de la thèse aborde les méthodes ensemblistes pour l'estimation de paramètres et pour l'estimation d'état hybride (mode et état continu) dans un contexte à erreurs bornées. Nous avons commencé en premier lieu par décrire les méthodes de détection de défauts dans les systèmes hybrides en utilisant une approche paramétrique et une approche observateur hybride. Ensuite, nous avons décrit deux méthodes permettant d’effectuer les tâches de détection de défauts. Nous avons proposé une méthode basée sur notre méthode d'atteignabilité hybride non linéaire et un algorithme de partitionnement que nous avons nommé SIVIA-H pour calculer de manière garantie l'ensemble des paramètres compatibles avec le modèle hybride, les mesures et avec les bornes d’erreurs. Ensuite, pour l'estimation d'état hybride, nous avons proposé une méthode basée sur un prédicteurcorrecteur construit au dessus de notre méthode d'atteignabilité hybride non linéaire. / This thesis is dedicated to the development of generic algorithms for the set-membership observation of the continuous state and the discrete mode of hybrid dynamical systems in order to achieve fault detection. This thesis is organized into two parts. In the first part, we have proposed a fast and effective method for the set-membership guard crossing. It consists in carrying out bisection in the time direction only and then makes several contractors working simultaneously to reduce the domain of state vectors located on the guard during the study time slot. Then, we proposed a method for merging trajectories based on zonotopic enclosures. These methods, used together, allowed us to characterize in a guaranteed way the set of all hybrid state trajectories generated by an uncertain hybrid dynamical system on a finite time horizon. The second part focuses on set-membership methods for the parameters or the hybrid state (mode and continuous state) of a hybrid dynamical system in a bounded error framework. We started first by describing fault detection methods for hybrid systems using the parametric approach and the hybrid observer approach. Then, we have described two methods for performing fault detection tasks. We have proposed a method for computing in a guaranteed way all the parameters consistent with the hybrid dynamical model, the actual data and the prior error bound, by using our nonlinear hybrid reachability method and an algorithm for partition which we denote SIVIA-H. Then, for hybrid state estimation, we have proposed a method based on a predictor-corrector, which is also built on top of our non-linear method for hybrid reachability.

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Neste trabalho, introduzimos os conceitos de controlabilidade e de observabilidade para equações diferenciais ordinárias generalizadas, apresentamos resultados inéditos sobre condições suficientes e necessárias para controlabilidade e para observabilidade para estas equações e também apresentaremos uma aplicação. Utilizando teoremas de correspondência entre equações diferenciais ordinárias generalizadas e outras equações diferenciais, traduzimos os resultados obtidos para os casos particulares de controlabilidade e observabilidade para equações diferenciais em medida e equações diferencias com impulsos. O fato de trabalharmos no ambiente das equações diferenciais ordinárias generalizadas permitiu que os resultados obtidos pudessem envolver funções com muitas descontinuidades e muito oscilantes, ou seja, de variação ilimitada. Os resultados novos apresentados aqui estão contidos no artigo [21] que se encontra em fase final de redação e será submetido à publicação em breve. / In this work, we introduce concepts of controllability and observability for generalized ordinary differential equations, we present new results on necessary and sufficient conditions for controllability and observability for these equations and we also present an application. Using theorems of correspondence between generalized ordinary differential equations and other differential equations, we translate the results obtained for the particular cases of controllability and observability for measure differential equations and differential equations with impulses. The fact that we work in the framework of generalized ordinary differential equations allows us to obtain results where the functions involved can have many discontinuities and be highly oscillating, that is, of unbounded variation. The new results presented here are contained in the preprint [21] which is under final revision and will soon be submitted for publication.

Estuda-se o espalhamento de doenças contagiosas utilizando modelos suscetível-infectado-recuperado (SIR) representados por equações diferenciais ordinárias (EDOs) e por autômatos celulares probabilistas (ACPs) conectados por redes aleatórias. Cada indivíduo (célula) do reticulado do ACP sofre a influência de outros, sendo que a probabilidade de ocorrer interação com os mais próximos é maior. Efetuam-se simulações para investigar como a propagação da doença é afetada pela topologia de acoplamento da população. Comparam-se os resultados numéricos obtidos com o modelo baseado em ACPs aleatoriamente conectados com os resultados obtidos com o modelo descrito por EDOs. Conclui-se que considerar a estrutura topológica da população pode dificultar a caracterização da doença, a partir da observação da evolução temporal do número de infectados. Conclui-se também que isolar alguns infectados causa o mesmo efeito do que isolar muitos suscetíveis. Além disso, analisa-se uma estratégia de vacinação com base em teoria dos jogos. Nesse jogo, o governo tenta minimizar os gastos para controlar a epidemia. Como resultado, o governo realiza campanhas quase-periódicas de vacinação. / The spreading of contagious diseases is studied by using susceptible-infected-recovered (SIR) models represented by ordinary differential equations (ODE) and by probabilistic cellular automata (PCA) connected by random networks. Each individual (cell) of the PCA lattice experiences the influence of others, where the probability of occurring interaction with the nearest ones is higher. Simulations for investigating how the disease propagation is affected by the coupling topology of the population are performed. The numerical results obtained with the model based on randomly connected PCA are compared to the results obtained with the model described by ODE. It is concluded that considering the topological structure of the population can pose difficulties for characterizing the disease, from the observation of the time evolution of the number of infected individuals. It is also concluded that isolating a few infected subjects can cause the same effect than isolating many susceptible individuals. Furthermore, a vaccination strategy based on game theory is analyzed. In this game, the government tries to minimize the expenses for controlling the epidemic. As consequence, the government implements quasi-periodic vaccination campaigns.

A partial differential equation in space and time represents the physics of rotating beams. Mostly, the numerical solution of such an equation is an available option as analytical solutions are not feasible even for a uniform rotating beam. Although the numerical solutions can be obtained with a number of combinations (in space and time), one tries to seek for a better alternative. In this work, various numerical techniques are applied to the rotating beam problems: finite element method, meshless methods, and B-spline finite element methods. These methods are applied to the governing differential equations of a rotating Euler-Bernoulli beam, rotating Timoshenko beam, rotating Rayleigh beam, and cracked Euler-Bernoulli beam. This work provides some elegant alternatives to the solutions available in the literature, which are more efficient than the existing methods: the p-version of finite element in time for obtaining the time response of periodic ordinary differential equations governing helicopter rotor blade dynamics, the symmetric matrix formulation for a rotating Euler-Bernoulli beam free vibration problem using the Galerkin method, and solution for the Timoshenko beam governing differential equation for free vibration using the meshless methods. Also, the cracked Euler-Bernoulli beam free vibration problem is solved where the importance of higher order polynomial approximation is shown. Finally, the overall response of rotating blades subjected to aerodynamic forcing is obtained in uncoupled trim where the response is independent of the overall helicopter configuration. Stability analysis for the rotor blade in hover and forward flight is also performed using Floquet theory for periodic differential equations.

We discuss two topics in this talk. First we study compact Ricci-flat four dimensional manifolds without boundary and obtain point wise restrictions on curvature( not involving global quantities such as volume and diameter) which force the metric to be flat. We obtain the same conclusion for compact Ricci-flat K¨ahler surfaces with similar but weaker restrictions on holomorphic sectional curvature. Next we study the reaction ODE associated to the evolution of the Riemann curvature operator along the Ricci flow. We analyze the behavior of this ODE near algebraic curvature operators of certain special type that includes the Riemann curvature operators of various(locally) symmetric spaces. We explicitly show the existence of some solution curves to the ODE connecting the curvature operators of certain symmetric spaces. Although the results of these two themes are different, the underlying common feature is the reaction ODE which plays an important role in both.

In this thesis the Ramberg-Osgood nonlinear model for describing the behavior of many different materials is investigated. A brief overview of the model as it is currently used in the literature is undertaken and several misunderstandings and possible pitfalls in its application is pointed out, especially as it pertains to more recent approaches to finding solutions involving the model. There is an investigation of the displacement of a cantilever beam under a combined loading consisting of a distributed load across the entire length of the beam and a point load at its end and new solutions to this problem are provided with a mixture of numerical techniques, which suggest strong mathematical consistency within the model for all theoretical assumptions made. A physical experiment was undertaken and the results prove to be inaccurate when using parameters derived from tensile tests, but when back calculating parameters from the beam test the model has a 14.40% error at its extreme against the experimental data suggesting the necessity for further testing.