Let f be a C2 local diffeomorphism, of a closed surface M without zero Lyapunov exponents. We have proved that the number of ergodic hyperbolic measures of f with SRB property is less than equal to the number of homoclinic equivalence classes. We use an adaptation of Katok closing lemma for endomorphisms and prove ergodic criterion, introduced in [HHTU], for endomorphisms. We also prove some folklore results on uniqueness of SRB measures, in the presence of topological transitivity / Seja f um endomorfismo C2 non-singular (difeomorfismo local), de uma superfície fechada M e µ uma medida probabilidade Borel f-invariante e ergódica com expoentes de Lyapunov Não nulo. Nós provamos que o número de medidas hiperbólicas com propriedade SRB é para f so menor ou igual ao número de classes equivalentes homoclínicos. Usamos uma adaptaão do closing lema de Katok por endomorfismos e provamos critrio ergódico, introduzido em [HHTU], para endomorfismos. Também provamos alguns resultados folclóricos em unicidade de medidas SRB, na presena de transitividade topológica vii

In this work, our goal is to prove results on prolongation of solutions, uniform boundedness of solutions, uniform stability as well uniform asymptotic stability (in the classical sense of Lyapunov) for measure differential equations and for dynamic equations on time scales. In order to get our results, we employ the theory of generalized ODEs, since these equations encompass measure differential equations and dynamic equations on time scales. Therefore, to get our results, we start by proving the expected result for abstract generalized ODEs. Then, using the correspondence between the solutions of these equations and the solutions of measure differential equations (see [38]), we extend all the results to these the latter. After that, using the correspondence between the solutions of measure differential equations and the solutions of dynamic equations on time scales (see [21]), we extend all the results to these last equations. Finally, we investigate autonomous generalized ODEs and show that these equations do not enlarge the class of classical autonomous ODEs, even when we consider a more general class of functions as right-hand sides. All the new results presented in this work are contained in papers [16, 17, 18, 19]. / Neste trabalho, nosso objetivo e provar resultados sobre prolongamento de soluções, limitação uniforme de soluções, estabilidade uniforme e estabilidade uniforme assintótica (no sentido clássico de Lyapunov) para equações diferenciais em medida e para equações dinâmicas em escalas temporais. A fim de obter os nossos resultados, empregamos a teoria de EDOs generalizadas, uma vez que estas equações abrangem equações diferenciais em medida e equações dinâmicas em escalas temporais. Portanto, para obter nossos resultados, vamos começar por provar, os resultados que queremos para EDOs generalizadas abstratas. Em seguida, usando a correspondência entre as soluções de EDOs generalizadas e soluções de equações diferenciais em medida (ver [38]), estenderemos os resultados para estas ultimas equações. Depois disso, usando a correspondência entre as soluções de equações diferenciais em medida e as soluções de equações dinâmicas em escalas temporais (ver [21]), estenderemos todos os resultados para estas ultimas equações. Finalmente, investigamos EDOs generalizadas autônomas e mostramos que estas equações não aumentam a classe de EDOs autônomas clássicas, mesmo quando consideramos uma classe mais geral de funções nos lados direitos das equações. Os novos resultados encontrados estão contidos em [16, 17, 18, 19].

Ce mémoire de thèse s'articule en deux parties. La première partie s'intéresse à la robustesse du nombre de reproduction de base R0 et du nombre de reproduction type (T), qui sont des seuils pour des systèmes épidémiques. Nous montrons que ces paramètres seuils ne sont pas des jauges fiables pour évaluer la distance qui sépare la Jacobiene (J) du système, calculé au point d'équilibre sans maladie à l'ensemble des matrices stables (S) si J est instable, (respectivement où l'ensemble des matrices instables (U) si J est stable). La deuxième partie se penche sur l'étude d'un modèle déterministe (S V E I R), où S représente les susceptibles, (V) les vaccinés,( E) les latents, (I )les infectieux et( R ) les immuns. Dans le dit modèle, les vaccinés sont considérés comme des « susceptibles dans une moindre mesure » du fait que le vaccin ne garantit pas une immunité totale. Le nombre de reproduction de base Rvac qui assure l'existence et l'unicité de l?équilibre endémique est déterminé. La globale stabilité de l'équilibre endémique est établie en utilisant les techniques de Lyapunov quand Rvac > 1. Ce résultat améliore un résultat de Gumel / This memory is divided in two parts. The first part talk about robustness of basic reproduction number R0 and basic reproduction number type S (T) both of them are thresholds for epidemic systems. We show that those thresholds are not good indicators to evaluate the distance between the jacobian matrix J(DFE) of system at the disease free equilibrium (DFE) and the set of stable ( S_t) or unstable (U ) matrix. The second part talk about an deterministic model ( S V E I R) ; where (S) represent the susceptibles ; (V) the vaccined ; (E) the latents,(I) the infectious, and (R) the removed. In this model, the vaccined are considered like susceptibles, because the vaccine don't confers an perfect immunity. The Basic reproduction number, (R_vac), who ensures the existence and unicity of endemic equilibrium is determined. The global stability of endemic equilibrium point is established using Lyapunov technics when (R_vac) is greater than one (R_vac)> 1

L'objectif de cette thèse est de développer des méthodes de synthèses d'observateurs et des contrôleurs basés sur un observateur pour les systèmes à retard. Différentes classes de systèmes ont été traitées avec différents types de retard. Trois méthodes ont été développées. La première méthode traite des systèmes non linéaires avec des non-linéarités lipschitziennes et consiste à transformer le système d'origine à un système LPV grâce à une reformulation de la propriété classique de Lipschitz. Cette technique est formulée pour les cas continu et discret, respectivement. Nous avons démontré, à travers des exemples numériques, que cette technique offre des conditions de synthèse moins restrictives par rapport aux résultats existants dans la littérature. La seconde méthode est développée pour une classe de systèmes singuliers avec des perturbations. La principale difficulté résidait dans la présence des dérivées des perturbations qui entravent l'analyse de la stabilité et pour laquelle deux approches ont été proposées: une approche $\mathcal{H}_{\infty}$ en utilisant une fonctionnelle de Lyapunov-Krasovskii spéciale dépendante des perturbations et une approche basée sur l'utilisation d'un critère de performance $\mathcal{W}^{1,2}$. La dernière méthode est basée sur l'utilisation des matrices de pondération libres pour résoudre le problème de contrôle des systèmes non-linéaires à retards inconnus. La solution proposée fournit une condition de synthèse LMI garantissant la stabilisation du système en boucle fermée malgré la présence du retard inconnu, au lieu d'une inégalité matricielle linéaire itérative ILMI trouvée habituellement dans la littérature / The objective of this dissertation is to develop observers and observer-based controllers synthesis methods for time-delay systems. Different classes of systems were treated with different types of delay. Three different methods were developed. The first one treats nonlinear systems with Lipschitz nonlinearities and consists in transforming the original system into an LPV system based on a reformulation of the classical Lipschitz property. This technique was formulated for continuous and discrete cases respectively and it was proven to provide less restrictive synthesis conditions when compared to the existing results in the literature. The second method deals with singular systems with disturbances. The main difficulty lay in the presence of the derivatives of the disturbances which hinder the stability analysis and for which two approaches are proposed:~a $\mathcal{H}_{\infty}$ criterion combined with a special Lyapunov-Krasovskii functional depending on disturbances and a $\mathcal{W}^{1,2}$ criterion based on the use of Sobolev norms. The last method is based on the Free Weighting Matrices technique to solve the observation and control problems of a class of nonlinear systems with unknown delays. The proposed solution provides a sufficient LMI synthesis condition ensuring the asymptotic stabilization of the closed loop system, instead of the iterative LMI condition usually found in the literature

Let f be a C2 local diffeomorphism, of a closed surface M without zero Lyapunov exponents. We have proved that the number of ergodic hyperbolic measures of f with SRB property is less than equal to the number of homoclinic equivalence classes. We use an adaptation of Katok closing lemma for endomorphisms and prove ergodic criterion, introduced in [HHTU], for endomorphisms. We also prove some folklore results on uniqueness of SRB measures, in the presence of topological transitivity / Seja f um endomorfismo C2 non-singular (difeomorfismo local), de uma superfície fechada M e µ uma medida probabilidade Borel f-invariante e ergódica com expoentes de Lyapunov Não nulo. Nós provamos que o número de medidas hiperbólicas com propriedade SRB é para f so menor ou igual ao número de classes equivalentes homoclínicos. Usamos uma adaptaão do closing lema de Katok por endomorfismos e provamos critrio ergódico, introduzido em [HHTU], para endomorfismos. Também provamos alguns resultados folclóricos em unicidade de medidas SRB, na presena de transitividade topológica vii

This thesis investigates the average consensus of multi-agent systems with linear dynamics whose interconnections are modelled by balanced digraphs with matrix- weights. The thesis first introduces the notion of balanced digraphs and mirror graphs for matrix weights. Then it proves that the matrix-weight-balanced con- sensus controller is indeed globally asymptotically stable. The Lyapunov stability analysis exploits the properties of the mirror graph of a balanced digraph. Further, the necessary and sufficient condition for the system to achieve average consensus is shown to be positive definiteness of the matrix weights of its balanced digraph. Simulations with robots in SIMULINK verify that positive definite matrix weights on balanced graphs are indeed necessary and sufficient for average consensus. Fi- nally formation control of a multi-robot system is shown to be an application of the matrix-weight-balanced consensus algorithm using real time simulation of Clearpath Ridgeback robots in Gazebo and ROS. / Graduate

O estudo dos pontos singulares em campos vetoriais analíticos é um problema quase completamente resolvido. O único caso que ainda permanece insolúvel é o caso monodrômico, em que as órbitas circundam a singularidade. Em sistemas diferenciais analíticos, se p é singularidade monodrômica, então p ou é um centro, ou é um foco. O problema do centro-foco consiste em determinar condições que diferenciem os casos em que p é um foco, daqueles em que p é um centro. O tema central desta dissertação é a investigação do problema do centro-foco em sistemas diferenciais analíticos com singularidade nilpotente. Este problema é bastante estudado, uma vez que ainda não existe um algoritmo eficiente para este caso, tal como ocorre em sistemas com singularidades não degeneradas. Estudamos duas técnicas bastante distintas. A primeira faz uso da teoria das formas normais e aborda o problema da maneira clássica, dividindo-o na investigação da monodromia e no estudo da estabilidade. O outro método investiga os sistemas diferenciais com singularidades nilpotentes como limite de sistemas com singularidades não degeneradas. A fim de avaliarmos sua eficiência e compreendermos as possíveis obstruções envolvidas, aplicamos os métodos a famílias concretas de sistemas diferenciais / The study of singular points in planar analytic vector fields is a problem almost completely solved. The only case that remains open is the monodromic one, in which the orbits turn around the singularity. In analytic differential systems, if p is a monodromic singular point, then p is either a center or a focus. The center-focus problem consists in determining conditions for distinguishing between a center and a focus. The main purpose of this work is the investigation of the center-focus problem in analytic differential systems with nilpotent singular points. This problem is still widely studied, since there is no algorithm for such case, comparable to the Lyapunov method for the case of non-degenerate singularities. We studied two different methods. The first makes use of the normal form theory and deals with the problem in the classic way, splitting it up in two parts: the investigation of the monodromy and the study of the stability. The latter investigates the differential analytic systems with nilpotent singular points as limit of differential systems with nondegenerate singularities. In order to evaluate the efficiency and understand possible obstructions, we applied the two techniques to concrete families of differential systems

This dissertation investigates vulnerabilities in unmanned vehicles and how to successfully detect and counteract them. As we entrust unmanned vehicles with more responsibilities (e.g. fire-fighting, search and rescue, package delivery), it is crucial to ensure their safe operation. These systems often have not been designed to protect against an intelligent attacker or considering all possible interactions between the physical dynamics and the internal logic. Robust control strategies can verify that the system behaves normally under bounded disturbances, and formal verification methods can check that the system logic operates normally under ideal conditions. However, critical vulnerabilities exist in the intersection of these fields that are addressed in this work. Due to the complex nature of this interaction, only trivial examples have previously been pursued. This work focuses on efficient real-time methods for verification and validation of unmanned vehicles under disturbances and cyberattacks. The efficiency of the verification and validation algorithm is necessary to run it onboard an unmanned vehicle, where it can be used for self diagnosis. We begin with simple linear systems and step to more complex examples with non-linearities. During this progression, new methods are developed to cope with the challenges introduced. We also address how to counter the threat of unmanned aerial systems (UASs) under hostile control by developing and testing an estimation and control scheme for an air-to-air counter UAS system.<br>

Orientador: Suzinei Aparecida Siqueira Marconato / Banca: Maria Aparecida Bená / Banca: Renata Zotin Gomes de Oliveira / Resumo: Neste trabalho apresentamos um estudo sobre estabilidade do equilíbrio nulo de equações diferenciais ordinárias autônomas através do Segundo Método de Liapunov e do Método das Funções Dicotômicas, que é uma extensão do Segundo Método de Liapunov / Abstract: This work presents a study about stability of the null equilibrium of autonomous ordinary differential equations by Liapunov's Second Method and Method of Dichotomic Maps, which is an extension of the Liapunov's Second Method / Mestre

The aim of this effort is to implement a nonlinear flight control architecture, specifically flight path control via command filtered backstepping, for use in AME UAS's Fury® 1500 unmanned flying wing aircraft. Backstepping is a recursive, control-effort minimizing, constructive design procedure that interlaces the choice of a Lyapunov function with the design of feedback control. It allows the use of certain plant states to act as intermediate, virtual controls, for others breaking complex high order systems into a sequence of simpler lower-order design tasks. Work herein is a simplified implementation based on publications by Farrell, Sharma, and Polycarpou. Online approximation is not applied, however command filtering along with two variants of control allocation is. This minimal approach was done to mitigate risk, as adaptation could be added in future work to this baseline. Command filtering assures that control inputs generated meet magnitude, rate, and bandwidth constraints for states and actuators as well as provides command derivatives that reduce computation. Two different forms of control allocation were implemented, the simplest a least-squares pseudo-inverse and the second an optimal quadratic programming method. Two Simulink based simulations successfully flew AME's Fury® 1500 UAS: a nominal case with fully operational actuators and a failure case with an actuator stuck at -10°. Coordinated flight for both cases with outer loop tracking was achieved for a demanding autopilot task of simultaneously varying heading and flight-path angle commands, ±60° and ±10° respectively, for a constant airspeed command of 135 ft/s. Command signals were generated were achievable due to the command filter implementation.