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Creating chaotic advection is the most efficient strategy to achieve mixing in a microscale or in a very viscous fluid, and it has many important applications in microfluidic devices, material processing and so on. In this paper, we present a quantitative long-term theory of resonant mixing in 3-D near-integrable flows. We use the flow in the annulus between two coaxial elliptic counter-rotating cylinders as a demonstrative model. We illustrate that such resonance phenomena as resonance and separatrix crossings accelerate mixing by causing the jumps of adiabatic invariants. We calculate the width of the mixing domain and estimate a characteristic time of mixing. We show that the resulting mixing can be described in terms of a single diffusion-type equation with a diffusion coefficient depending on the averaged effect of multiple passages through resonances. We discuss what must be done to accommodate the effects of the boundaries of the chaotic domain. / Mechanical Engineering

This dissertation is primarily concerned with studying the invariant subspaces of left-invertible, weighted shifts, with generalizations to left-invertible operators where applicable. The two main problems that are researched can be stated together as When does a weighted shift have the one-dimensional wandering subspace property for all of its closed, invariant subspaces? This can fail either by having a subspace that is not generated by its wandering subspace, or by having a subspace with an index greater than one. For the former we show that every left-invertible, weighted shift is similar to another weighted shift with a residual space, with respect to being generated by the wandering subspace, of dimension $n$, where $n$ is any finite number. For the latter we derive necessary and sufficient conditions for a pure, left-invertible operator with an index of one to have a closed, invariant subspace with an index greater than one. We use these conditions to show that if a closed, invariant subspace for an operator in a class of weighted shifts has a vector in $l^1$, then it must have an index equal to one, and to produce closed, invariant subspaces with an index of two for operators in another class of weighted shifts. / Ph. D.

This dissertation presents geometric approaches of understanding chaotic transport in phase space that is fundamental across many disciplines in physical sciences and engineering. This approach is based on analyzing phase space transport using boundaries and regions inside these boundaries in presence of perturbation. We present a geometric view of defining such boundaries and study the transport that occurs by crossing such phase space structures. The structure in two dimensional non-autonomous system is the codimension 1 stable and unstable manifolds associated with the hyperbolic fixed points. The manifolds separate regions with varied dynamical fates and their time evolution encodes how the initial conditions in a given region of phase space get transported to other regions. In the context of four dimensional autonomous systems, the corresponding structure is the stable and unstable manifolds of unstable periodic orbits which reside in the bottlenecks of energy surface. The total energy and the cylindrical (or tube) manifolds form the necessary and sufficient condition for global transport between regions of phase space. Furthermore, we adopt the geometric view to define escaping zones for avoiding transition/escape from a potential well using partial control. In this approach, the objective is two fold: finding the minimum control that is required for avoiding escape and obtaining discrete representation called disturbance of continuous noise that is present in physical sciences and engineering. In the former scenario, along with avoiding escape, the control is constrained to be smaller than the disturbance so that it can not exactly cancel out the disturbances. / Ph. D. / The prediction and control of critical events in engineering systems has been a major objective of scientific research in recent years. The multifaceted problems facing the modern society includes critical events such as spread of pathogens and pollutants in atmosphere and ocean, capsize of boats and cruise ships, space exploration and asteroid collision, to name but a few. Although, at first glance they seem to be disconnected problems in different areas of engineering and science, however, they have certain features that are inherently common. This can be studied using the abstraction of phase space which can be thought of as the universe where all possible solutions of the governing equations, derived using principles of physics, live and evolve in time. The <i>phase space</i> can be just 2D, 3D or even infinite dimensional but the critical events manifest themselves as volumes of phase space, which represent solutions at a given instant of time, get transported from one region to another due to the underlying dynamics. This mathematical abstraction is called phase space transport and studied under the umbrella of dynamical systems theory. The geometric view of the solutions that live in the phase space provides insight into the mechanisms of how the critical events occur, and the understanding of these mechanisms is useful in deciding about control strategies. A slightly different view for understanding critical events is to consider a thought experiment where a ball is rolling on a multi-well surface or potential well. As the time evolves, the ball will escape from its initial well and roll into another well, and eventually start exploring all the wells in a seemingly unpredictable way. However, these unpredictable escape/transition can be studied systematically using methods of chaos and dynamical systems. The escape/transition in a potential well implies a dramatic change in the behavior of the system, and hence the significance in prediction and control of <i>escaping dynamics</i>. The control aspect becomes more challenging due to inherent disturbance in the system that is difficult to model and we may not have the equal or more control authority to cancel those disturbances. However, we can usually estimate the maximum values of the disturbance, and try to avoid escaping from the potential well while using a smaller control. This idea is called <i>partial control of escaping dynamics</i> and can guarantee avoidance of escape for <i>ad infinitum</i>. In this doctoral research, we focus on the two mechanisms, phase space transport and escaping dynamics, by considering problems from fluid dynamics and capsize of a ship. The applications are used for numerical demonstration and evidence of the general approach in studying a large class of problems in classical physics.

This dissertation expands across three topics the geometric theory of phase space transit in the circular restricted three-body problem (CR3BP) and its generalizations. The first topic generalizes the low energy transport theory that relies on linearizing the Lagrange points in the CR3BP to time-periodic perturbations of the CR3BP, such as the bicircular problem (BCP) and the elliptic restricted three-body problem (ER3BP). The Lagrange points are no longer invariant under perturbation and are replaced by periodic orbits, which we call Lagrange periodic orbits. Calculating the monodromy matrix of the Lagrange periodic orbit and transforming into eigenbasis coordinates reveals that the transport geometry is a discrete analogue of the continuous transport geometry in the unperturbed problem. The second topic extends the theory of low energy phase space transit in periodically perturbed models using a nonlinear analysis of the geometry. This nonlinear analysis relies on calculating the monodromy tensors, which generalize monodromy matrices in order to encode higher order behavior, about the Lagrange periodic orbit. A nonlinear approximate map can be obtained which can be used to iterate initial conditions within the linear eigenbasis, providing a computationally efficient means of distinguishing transit and nontransit orbits that improves upon the predictions of the linear framework. The third topic demonstrates that the recently-discovered "arches of chaos" that stretch through the solar system, causing substantial phase space divergence for high energy particles, may be identified with the stable and unstable manifolds to the singularities of the CR3BP. We also study the arches in terms of particle orbital elements and demonstrate that the arches correspond to gravity assists in the two-body limit. / Doctor of Philosophy / Suppose that we have a spacecraft and we want to model its motion under gravity. Depending upon what trade-offs we are willing to make between accuracy and complexity, we have several options at our disposal. For example, the restricted three-body problem (R3BP) and its generalizations prove useful in many real-world situations and are rich in theoretical power despite seeming mathematically simple. The simplest restricted three-body problem is the circular restricted three-body problem (CR3BP). In the CR3BP, two masses (like a star and a planet or a planet and a moon) orbit their common center of gravity in circular orbits, while a much smaller body (like a spacecraft) moves freely, influenced by the gravitational fields that the two masses create. If we add in an extra force that acts on the spacecraft in a periodic, cycling way, the regular CR3BP becomes a periodically-perturbed CR3BP. Examples of periodically-perturbed CR3BP's include the bicircular problem (BCP), which adds in a third mass that appears to orbit the center of the system from a distance, and the elliptic restricted three-body problem (ER3BP), which allows the two masses to orbit more realistically as ellipses rather than circles. The purpose of this dissertation is to determine how to select trajectories that move spacecraft between places of interest in restricted three-body models. We generalize existing theories of CR3BP spacecraft motion to periodically-perturbed CR3BP's in the first two topics, and then we investigate some new areas of research in the unperturbed CR3BP in the third topic. We utilize numerical computations and mathematical methods to perform these analyses.

En imagerie couleur, un système d’acquisition capture une scène avec une haute résolution spatiale mais une résolution spectrale limitée. L’imagerie hyperspectrale permet d’acquérir la scène avec une grande résolution spectrale. Un système d’acquisition hyperspectrale est un ensemble complexe et il est difficile de l’utiliser pour acquérir des données dans une situation où les conditions d’imageries ne sont pas contrôlées. De plus, ces systèmes sont chers et souvent encombrants ou difficiles à manipuler. À cause de ces problèmes, l’utilisation de l’imagerie hyperspectrale n’a pas encore été beaucoup utilisée en vision assistée par ordinateur, et la plupart des systèmes de vision utilise l’imagerie couleur.L’imagerie multispectrale propose une solution intermédiaire, elle permet de capturer une information moins résolue selon la dimension spectrale, comparée à l’hyperspectrale, tout en préservant la résolution spatiale. Ces systèmes sont moins encombrants et moins difficiles à maitriser grâce aux récentes avancées technologiques, et arrivent sur le marché en tant que produits commerciaux. On peut citer les matrices de filtres spectraux (spectral filter arrays) qui permettent l’acquisition en temps réel d’images multispectrales grâce à l’utilisation d’unecaméra de complexité similaire à une caméra couleur. Jusqu’ici, les informations capturées par ces systèmes étaient considérées de la même manière que les imageurs hyperspectraux en champ proche, c’est à dire que pour utiliser l’information au mieux, les conditions d’acquisitions devaient être connues et le système calibré, en particulier pour l’éclairage de la scène et la dynamique de la scène.Afin d’élargir l’utilisation de l’imagerie multispectrale pour la vision par ordinateur dans des conditions générales, je propose dans cette thèse de développer les méthodes calculatoires en imagerie couleur (computational color imaging) et de les adapter aux systèmes d’imagerie multispectraux. Une caractéristique très puissante de l’imagerie couleur est de proposer un rendu constant des couleurs de la surface d’un objet à travers différentes conditions d’acquisition via l’utilisation d’algorithmes et divers traitements de l’information.Dans cette thèse, j’étends la notion de constance des couleurs et de balance des blancs de l’imagerie couleur à l’imagerie multispectrale. J’introduis le terme de constance de l’information spectrale (multispectral constancy).Je propose la construction d’un ensemble d’outils permettant la représentation constante de l’information spectrale à travers le changement d’éclairage. La validité de ces outils est évaluée à travers la reconstruction de la réflectance spectrale des objets lorsque l’éclairage change. Nous avons également acquis de nouvelles images hyperspectrales et multispectrales mises à disposition de la communauté.Ces outils et données permettront de favoriser la généralisation de l’utilisation de l’imagerie multispectrale en champ proche dans les applications classiques utilisant traditionnellement l’imagerie couleur et de sortir ce mode d’imagerie des laboratoires. L’avantage en vision par ordinateur est une meilleure analyse de la réflectance de la surface des objets et donc un avantage certain dans les tâches de classification et d’identification de matériaux. / A conventional color imaging system provides high resolution spatial information and low resolution spectral data. In contrast, a multispectral imaging system is able to provide both the spectral and spatial information of a scene in high resolution. A multispectral imaging system is complex and it is not easy to use it as a hand held device for acquisition of data in uncontrolled conditions. The use of multispectral imaging for computer vision applications has started recently but is not very efficient due to these limitations. Therefore, most of the computer vision systems still rely on traditional color imaging and the potential of multispectral imaging for these applications has yet to be explored.With the advancement in sensor technology, hand held multispectral imaging systems are coming in market. One such example is the snapshot multispectral filter array camera. So far, data acquisition from multispectral imaging systems require specific imaging conditions and their use is limited to a few applications including remote sensing and indoor systems. Knowledge of scene illumination during multispectral image acquisition is one of the important conditions. In color imaging, computational color constancy deals with this condition while the lack of such a framework for multispectral imaging is one of the major limitation in enabling the use of multispectral cameras in uncontrolled imaging environments.In this work, we extend some methods of computational color imaging and apply them to the multispectral imaging systems. A major advantage of color imaging is the ability of providing consistent color of objects and surfaces across varying imaging conditions. In this work, we extend the concept of color constancy and white balancing from color to multispectral images, and introduce the term multispectral constancy.The validity of proposed framework for consistent representation of multispectral images is demonstrated through spectral reconstruction of material surfaces from the acquired images. We have also presented a new hyperspectral reflectance images dataset in this work. The framework of multispectral constancy will make it one step closer for the use of multispectral imaging in computer vision applications, where the spectral information, as well as the spatial information of a surface will be able to provide distinctive useful features for material identification and classification tasks.

Le calcul des ensembles atteignables est une approche de facto utilisée dans de nombreuses méthodes de vérification formelles pour les systèmes hybrides. Mais le calcul exact de l'ensemble atteignable est un problème insurmontable pour de nombreux types de systèmes hybrides, soit en raison de l'indécidabilité ou de la complexité de calcul élevée. Alternativement, beaucoup de recherches ont été axées sur l'utilisation de représentations d'ensembles qui peuvent être manipulées efficacement pour calculer une surestimation suffisamment précise de l'ensemble atteignable. Les zonotopes sont une représentation utile de l'ensemble dans l'analyse de l'accessibilité en raison de leur fermeture et de leur faible complexité pour le calcul de la transformation linéaire et des opérations sommaires de Minkowski. Mais pour approximer les ensembles de temps non bornés atteignables par des invariants positifs, les zonotopes ont l'inconvénient suivant. L'efficacité d'une représentation d'ensemble pour calculer un invariant positif dépend de l'encodage efficace des directions de convergence des états vers un équilibre. Dans un système hybride affine, certaines des directions de convergence peuvent être codées par les vecteurs propres à valeur complexe des matrices de transformation. Mais la représentation zonotopique ne peut pas exploiter la structure propre complexe des matrices de transformation car elle n'a que des générateurs à valeur réelle.Par conséquent, nous étendons les zonotopes réels au domaine de valeur complexe d'une manière qui peut capturer la contraction le long de vecteurs évalués complexes. Cela donne une nouvelle représentation d'ensemble appelée zonotope complexe. Géométriquement, les zonotopes complexes représentent une classe plus large d'ensembles qui comprennent des ensembles non polytopiques ainsi que des zonotopes polytopiques. Ils conservent le mérite des zonotopes réels que nous pouvons effectuer efficacement la transformation linéaire et les opérations sommaires de Minkowski et calculer la fonction de support. De plus, nous montrons qu'ils peuvent capturer la contraction le long de vecteurs propres complexes. De plus, nous développons des approximations traitables par calcul pour la vérification d'inclusion et l'intersection avec des demi-espaces. En utilisant ces opérations sur des zonotopes complexes, nous développons des programmes convexes pour vérifier les propriétés d'invariance linéaire des systèmes hybrides affines à temps discret et la stabilité exponentielle des systèmes impulsifs linéaires. Nos expériences sur certains exemples de benchmarks démontrent l'efficacité des techniques de vérification basées sur des zonotopes complexes. / Computing reachable sets is a de facto approach used in many formal verification methods for hybrid systems. But exact computation of the reachable set is an in- tractable problem for many kinds of hybrid systems, either due to undecidability or high computational complexity. Alternatively, quite a lot of research has been focused on using set representations that can be efficiently manipulated to com- pute sufficiently accurate over-approximation of the reachable set. Zonotopes are a useful set representation in reachability analysis because of their closure and low complexity for computing linear transformation and Minkowski sum operations. But for approximating the unbounded time reachable sets by positive invariants, zonotopes have the following drawback. The effectiveness of a set representation for computing a positive invariant depends on efficiently encoding the directions for convergence of the states to an equilibrium. In an affine hybrid system, some of the directions for convergence can be encoded by the complex valued eigen- vectors of the transformation matrices. But the zonotope representation can not exploit the complex eigenstructure of the transformation matrices because it only has real valued generators.Therefore, we extend real zonotopes to the complex valued domain in a way that can capture contraction along complex valued vectors. This yields a new set representation called complex zonotope. Geometrically, complex zonotopes repre- sent a wider class of sets that include some non-polytopic sets as well as polytopic zonotopes. They retain the merit of real zonotopes that we can efficiently perform linear transformation and Minkowski sum operations and compute the support function. Additionally, we show that they can capture contraction along complex valued eigenvectors. Furthermore, we develop computationally tractable approx- imations for inclusion-checking and intersection with half-spaces. Using these set operations on complex zonotopes, we develop convex programs to verify lin- ear invariance properties of discrete time affine hybrid systems and exponential stability of linear impulsive systems. Our experiments on some benchmark exam- ples demonstrate the efficiency of the verification techniques based on complex zonotopes.