Ce manuscrit est dédié à l'étude de la stabilité et la stabilisation en temps fini des systèmes dynamiques interconnectés et problème de consensus en temps fini. Après une large introduction, la première partie de ce mémoire se focalise sur la stabilité et stabilisation en temps fini des systèmes dynamiques perturbés et des systèmes dynamiques interconnectés. La deuxième partie de cette thèse est consacrée aux problèmes de : consensus en temps fini, consensus moyen en temps fini et stabilisation en temps fini des systèmes multi-agents. Cette notion a été abordé en ciblant les systèmes dynamiques contrôlés non linéaires et complexes de type avec et sans terme de dérive et affine en la commande. Des protocoles sont mis en exergue résolvant les problèmes de consensus/formation en temps fini entre les états de tels systèmes. De nombreuses applications avec des simulations permettent de confirmer les protocoles proposés. / This manuscript is dedicated to the study of finite time stability and stabilization of interconnected dynamical systems and finite time consensus problem. After a general introduction, the first part of this thesis focuses on finite time stability and stabilization of perturbed systems and interconnected systems. The second part of this thesis is devoted to the problems of: finite-time consensus, average consensus and finite time stabilization of multi-agent systems. This concept has been addressed by targeting non-linear controlled dynamical systems: with and without drift term. Some protocols are proposed to solve the finite time consensus problem. Many applications and simulations are illustrated.

In order to understand water quality in the coastal ocean and its effects on human health, the necessity arises to locate the sources of contaminants and track their transport throughout the ocean. Dynamical systems methods are applied to the study of transport of enterococci as an indicator of microbial concentration in the vicinity of Hobie Beach, an urban, subtropical beach in Miami, FL that is used for recreation and bathing on a daily basis. Previous studies on water quality have shown that Hobie Beach has high microbial levels despite having no known point source. To investigate the cause of these high microbial levels, a combination of measured surface drifter trajectories and numerically simulated flows in the vicinity of Hobie Beach is used. The numerically simulated flows are used to identify Lagrangian Coherent Structures (LCSs), which provide a template for transport in the study area. Surface drifter trajectories are shown to be consistent with the simulated flows and the LCS structure. LCSs are then used to explain the persistent water contamination and unusually high concentrations of microbes in the water off of this beach as compared with its neighboring beaches. From the drifter simulations, as well as field experiments, one can see that passive tracers are trapped in the area along the coastline by LCS. The Lagrangian circulation of Hobie Beach, influenced primarily by tide and land geometry causes a high retention rate of water near the shore, and can be used to explain the elevated levels of enterococci in the water.

In this dissertation, we present some analysis of Ricci flow on complete noncompact manifolds. The first half of the dissertation concerns the formation of Type-II singularity in Ricci flow on [mathematical equation]. For each [mathematical equation] , we construct complete solutions to Ricci flow on [mathematical equation] which encounter global singularities at a finite time T such that the singularities are forming arbitrarily slowly with the curvature blowing up arbitrarily fast at the rate [mathematical equation]. Near the origin, blow-ups of such a solution converge uniformly to the Bryant soliton. Near spatial infinity, blow-ups of such a solution converge uniformly to the shrinking cylinder soliton. As an application of this result, we prove that there exist standard solutions of Ricci flow on [mathematical equation] whose blow-ups near the origin converge uniformly to the Bryant soliton. In the second half of the dissertation, we fully analyze the structure of the Lichnerowicz Laplacian of a Bergman metric g[subscript B] on a complex hyperbolic space [mathematical equation] and establish the linear stability of the curvature-normalized Ricci flow at such a geometry in complex dimension [mathematical equation]. We then apply the maximal regularity theory for quasilinear parabolic systems to prove a dynamical stability result of Bergman metric on the complete noncompact CH[superscript m] under the curvature-normalized Ricci flow in complex dimension [mathematical equation]. We also prove a similar dynamical stability result on a smooth closed quotient manifold of [mathematical symbols]. In order to apply the maximal regularity theory, we define suitably weighted little Hölder spaces on a complete noncompact manifold and establish their interpolation properties. / text

Asymptotic stability is a key notion of system stability for controlled dynamical systems as it guarantees that the system trajectories are bounded in a neighborhood of a given isolated equilibrium point and converge to this equilibrium over the infinite horizon. In some applications, however, asymptotic stability is not the appropriate notion of stability. For example, for systems with a continuum of equilibria, every neighborhood of an equilibrium contains another equilibrium and a nonisolated equilibrium cannot be asymptotically stable. Alternatively, in stabilization of spacecraft dynamics via gimballed gyroscopes, it is desirable to find state- and output-feedback control laws that guarantee partial-state stability of the closed-loop system, that is, stability with respect to part of the system state. Furthermore, we may additionally require finite-time stability of the closed-loop system, that is, convergence of the system's trajectories to a Lyapunov stable equilibrium in finite time. The Hamilton-Jacobi-Bellman optimal control framework provides necessary and sufficient conditions for the existence of state-feedback controllers that minimize a given performance measure and guarantee asymptotic stability of the closed-loop system. In this research, we provide extensions of the Hamilton-Jacobi-Bellman optimal control theory to develop state-feedback control laws that minimize nonlinear-nonquadratic performance criteria and guarantee semistability, partial-state stability, finite-time stability, and finite-time partial state stability of the closed-loop system.

The aim of this thesis is to introduce a general framework for what is informally referred to as finite-time dynamics. Within this framework, we study hyperbolicity of reference trajectories, existence of invariant manifolds as well as normal hyperbolicity of invariant manifolds called Lagrangian Coherent Structures. We focus on a simple derivation of analytical results. At the same time, our approach together with the analytical results has strong impact on the numerical implementation by providing calculable expressions for known functions and continuity results that ensure robust computation. The main results of the thesis are robustness of finite-time hyperbolicity in a very general setting, finite-time analogues to classical linearization theorems, an approach to the computation of so-called growth rates and the generalization of the variational approach to Lagrangian Coherent Structures.

The aim of this thesis is to introduce a general framework for what is informally referred to as finite-time dynamics. Within this framework, we study hyperbolicity of reference trajectories, existence of invariant manifolds as well as normal hyperbolicity of invariant manifolds called Lagrangian Coherent Structures. We focus on a simple derivation of analytical results. At the same time, our approach together with the analytical results has strong impact on the numerical implementation by providing calculable expressions for known functions and continuity results that ensure robust computation. The main results of the thesis are robustness of finite-time hyperbolicity in a very general setting, finite-time analogues to classical linearization theorems, an approach to the computation of so-called growth rates and the generalization of the variational approach to Lagrangian Coherent Structures.

The contributions of this thesis lie in the area of global finite-time observer design for a class of nonlinear systems with bounded rational and mixed rational powers imposed on the incremental rate of the nonlinear terms whose solutions exist and are unique for all positive time. In the thesis, two different kinds of nonlinear global finite-time observers are designed by employing of finite-time theory and homogeneity properties with different methods. The global finite-time stability of both proposed observers is derived on the basis of Lyapunov theory. For a class of nonlinear systems with rational and mixed rational powers imposed on the nonlinearities, the first global finite-time observers are designed, where the global finite-time stability of the observation systems is achieved from two parts by combining asymptotic stability and local finitetime stability. The proposed observers can only be designed for the class of nonlinear systems with dimensions greater than 3. The observers have a dynamic high gain and two homogenous terms, one homogeneous of degree greater than 1 and the other of degree less than 1. In order to prove the global finite-time stability of the proposed results, two homogeneous Lyapunov functions are provided, corresponding with the two homogeneous items. One is homogeneous of degree greater than 1, which makes the observation error systems converging into a spherical area around the origin, and the other is of degree less than 1, which ensures local finite-time stability. The second global finite-time observers are also proposed based on the high-gain technique, which does not place any limitation on the dimension of the nonlinear systems. Compared with the first global finite-time observers, the newly designed observers have only one homogeneous term and a new gain update law where two new terms are introduced to dominate some terms in the nonlinearities and ensure global finite-time stability as well. The global finite-time stability is obtained directly based on a sufficient condition of finite-time stability and only one Lyapunov function is employed in the proof. The validity of the two kinds of global finite-time observers that have been designed is illustrated through some simulation results. Both of them can make the observation error systems converge to the origin in finite-time. The parameters, initial conditions as well as the high gain do have some impact on the convergence time, where the high gain plays a stronger role. The bigger the high gain is, the shorter the time it needs to converge. In order to show the performance of the two kinds of observers more clearly, two examples are provided and some comparisons are made between them. Through these, it can be seen that under the same parameters and initial conditions, although the amplitude of the observation error curve is slightly greater, the global finite-time observers with a new gain update law can make the observation error systems converge much more quickly than the global finite-time observers with two homogeneous terms. In the simulation results, one can see that, as a common drawback of high gain observers, they are noise-sensitive. Finding methods to improve their robustness and adaptiveness will be quite interesting, useful and challenging. / Thesis (PhD)--University of Pretoria, 2013. / gm2014 / Electrical, Electronic and Computer Engineering / unrestricted

Les phénomènes thermoélectriques sont un moyen de convertir directement l’énergie thermique en énergie électrique ; ils sont à ce titre au cœur de nombreuses recherches dans le domaine de l’énergétique. Au-delà de l’optimisation des matériaux constituants les générateurs thermoélectriques, il est également nécessaire de mener une réflexion sur la manière dont ces générateurs sont utilisés. La contribution des contacts thermiques entre le générateur et les réservoirs thermiques est un facteur qui va modifier les conditions de fonctionnement optimales du générateur. En utilisant la notion de courant thermique convectif, développée par Thomson il y a plus de 150 ans, nous généralisons les expressions classiques du fonctionnement à puissance maximum pour le générateur pour ce cas de figure. Nous constatons toutefois que ces conditions se réduisent à une adaptation d’impédance, à la fois thermique et électrique Outre son intérêt pratique, le générateur thermoélectrique est également un système modèle de choix pour étudier la théorie du transport couplé et des phénomènes irréversibles. En utilisant la description donnée par Ioffe de ce système, nous montrons que l’efficacité à maximum de puissance, un coefficient de performance au cœur de la thermodynamique à temps fini, s’exprime comme une fonction relativement simple des paramètres du système. La nouveauté de ce travail repose sur une prise en compte appropriée des dissipations internes associées au processus de conversion d’énergie. Les résultats sont généralisés enfin aux cas d’autres machines thermiques telle que la roue à rochet de Feynman. / Thermoelectric phenomena are a way to directly convert thermal energy into electrical energy; they thus are at the heart of several researches in the field of energy conversion. The optimization of the thermoelectric generators includes materials improvement but a reflection on their working conditions is also mandatory. The contribution of the thermal contacts between the generator and the heat reservoirs is a factor that will change the optimum operating conditions of the generator. Using the concept of convective heat flow, developed by Thomson more than 150 years ago, we generalize the classical expression of maximum power conditions. Moreover, we note that these conditions may be reduced to impedance matching conditions, both thermal and electrical. In addition to its practical interest, the thermoelectric generator is also an ideal model system to study the theory of coupled transport and of irreversible phenomena. Using the description of this system given by Ioffe, we show that the maximum power efficiency, a coefficient of performance at the heart of finite time thermodynamics, expressed as a simple function of the system parameters. The novelty of this work is based on a proper consideration of internal dissipation associated with the energy conversion process. The results are then generalized to other thermal engines such as the Feynman ratchet.

Neste trabalho, estudamos a teoria quântica de correção de erros, um dos principais métodos de prevenção de perda de informação num computador quântico. Este método, porém, normalmente é estudado considerando-se condições ideais em que a atuação das portas lógicas que constituem o algoritmo quântico não interfere com o tipo de erro que o sistema sofre. Além disso, as medidas de síndrome empregadas no método tradicional são consideradas instantâneas. Nossos objetivos neste trabalho serão avaliar como a alteração dessas duas suposições modificaria o processo de correção de erros. Com relação ao primeiro objetivo, verificamos que, para erros causados por ambientes externos, a atuação de uma porta lógica simultânea ao ruído pode gerar erros que, a princípio, podem não ser corrigíveis pelo código empregado. Propomos em seguida um método de correção a pequenos passos que pode ser usado para tornar desprezíveis os erros incorrigíveis, além de poder ser usado para reduzir a probabilidade de erros corrigíveis. Para o segundo objetivo, estudamos primeiro como medidas de tempo finito afetam a descoerência de apenas um qubit, concluindo que esse tipo de medida pode na verdade proteger o estado que está sendo medido. Motivados por isso, mostramos que, em certos casos, medidas de síndrome finitas realizadas conjuntamente ao ruído são capazes de proteger o estado dos qubits contra os erros mais eficientemente do que se as medidas fossem realizadas instantaneamente ao fim do processo. / In this work, we study the theory of quantum error correction, one of the main methods of preventing loss of information in a quantum computer. This method, however, is normally studied under ideal conditions in which the operation of the quantum gates that constitute the quantum algorithm do not interefere with the kind of error the system undergoes. Moreover, the syndrome measurements employed in the traditional method are considered instantaneous. Our aims with this work are to evaluate how altering these two suppositions would modify the quantum error correction process. In respect with the first objective, we verify that, for errors caused by external environments, the action of a logical gate simultaneously to the noise can provoke errors that, in principle, may not be correctable by the code employed. We subsequently propose a short-step correction method that can be used to render negligible the uncorrectable errors, besides being capable of reducing the probability of occurrence of correctable errors. For the second objective, we first study how finite-time measurements affect the decoherence of a single qubit, concluding that this kind of measurement can actually protect the state under scrutiny. Motivated by that, we demonstrate, that, in certain cases, finite syndrome measurements performed concurrently with the noise are capable of protecting more efficiently the state of the qubits against errors than if the measurements had been performed instantaneously at the the end of the process.

Considered in this research is a framework for effective formation control of multirobot systems in dynamic environments. The basic formation control involves two important considerations: (1) Real-time trajectory generation algorithms for distributed control based on nominal agent models, and (2) robust tracking of reference trajectories under model uncertainties. Proposed is a two-layer hierarchical architecture for collectivemotion control ofmultirobot nonholonomic systems. It endows robotic systems with the ability to simultaneously deal with multiple tasks and achieve typical complex formation missions, such as collisionfree maneuvers in dynamic environments, tracking certain desired trajectories, forming suitable patterns or geometrical shapes, and/or varying the pattern when necessary. The study also addresses real-time formation tracking of reference trajectories under the presence of model uncertainties and proposes robust control laws such that over each time interval any tracking errors due to system uncertainties are driven down to zero prior to the commencement of the subsequent computation segment. By considering a class of nonlinear systems with favorable finite-time convergence characteristics, sufficient conditions for exponential finite-time stability are established and then applied to distributed formation tracking controls. This manifests in the settling time of the controlled system being finite and no longer than the predefined reference trajectory segment computing time interval, thus making tracking errors go to zero by the end of the time horizon over which a segment of the reference trajectory is generated. This way the next segment of the reference trajectory is properly initialized to go into the trajectory computation algorithm. Consequently this could lead to a guarantee of desired multi-robot motion evolution in spite of system uncertainties. To facilitate practical implementation, communication among multi-agent systems is considered to enable the construction of distributed formation control. Instead of requiring global communication among all robots, a distributed communication algorithm is employed to eliminate redundant data propagation, thus reducing energy consumption and improving network efficiency while maintaining connectivity to ensure the convergence of formation control.