Distributed parameter systems (DPS) are systems defined on infinite dimensional spaces. This includes problems governed by partial differential equations (PDEs) and delay differential equations. In order to numerically implement a controller for a physical system we often first approximate the PDE and the PDE controller using some finite dimensional scheme. However, control design at this level will typically give rise to controllers that are inherently large-scale. This presents a challenge since we are interested in the design of robust, real-time controllers for physical systems. Therefore, a reduction in the size of the model and/or controller must take place at some point. Traditional methods to obtain lower order controllers involve reducing the model from that for the PDE, and then applying a standard control design technique. One such model reduction technique is balanced truncation. However, it has been argued that this type of method may have an inherent weakness since there is a loss of physical information from the high order, PDE approximating model prior to control design. In an attempt to capture characteristics of the PDE controller before the reduction step, alternative techniques have been introduced that can be thought of as controller reduction methods as opposed to model reduction methods. One such technique is LQG balanced truncation. Only recently has theory for LQG balanced truncation been developed in the infinite dimensional setting. In this work, we numerically investigate the viability of LQG balanced truncation as a suitable means for designing low order, robust controllers for distributed parameter systems. We accomplish this by applying both balanced reduction techniques, coupled with LQG, MinMax and central control designs for the low order controllers, to the cable mass, Klein-Gordon, and Euler-Bernoulli beam PDE systems. All numerical results include a comparison of controller performance and robustness properties of the closed loop systems. / Ph. D.

When designing a control for a physical system described by a PDE, it is often necessary to reduce the size of the controller for the PDE system. This is done so that real time control can be achieved. One approach often taken by engineers is to reduce the approximating finite-dimensional system using a balanced reduction method known as balanced truncation and then design a control for the lower order system. The unsettling idea about this method is that it involves discarding information and then designing a control. What if valuable physical information were lost that would have allowed a more effective control to be designed? This paper will explore an alternate balanced reduction method called LQG balancing. This approach allows for the designing of a control on the full order approximating system and then reducing the control. Along the way, the basic ideas of feedback control design will be discussed, including system balancing and model reduction. Following, there will be mention of the linear Klein-Gordon equation and the development of the one-dimensional finite element approximation of the PDE. Finally, simulations and numerical experiments are used to discuss the differences between the two balanced reduction methods. / Master of Science

Many nonlinear Klein-Gordon equations have been studied numerically, and in a few cases, analytical solutions have been found. We used the variational method to study three different equations in this family. The first one to be studied here was the linear equation, Utt - Uzz + U = 0, where U is a real Klein-Gordon field. Attempts to find non-stationary radiative-type solutions of this equation were not successful. Next we studied the nonlinear equation Utt - U:= ± IUl 2U = O, with U complex, which represents a nonlinear massless scalar field. Here we searched for possible stationary solutions using the variational approximation, however to no avail. Next, we added a linear term to this second equation, which then became Utt - Uzll: ± IUl2U + µU = 01 whereµ can always be scaled to ±1. Here we found that we can find approximate variational solutions of the form A(t)e^i{k(x-z0(t))+a)e / 2w2(z) . This third equation is a generalization of the tf,4 equation, which has many physical applications. However, the variational solution found required different signs on the coefficients of this equation than are found in the O4 equation. Properties and features of this variational solution will be discussed.

Nesta dissertação apresentamos um método de quantização matemática e conceitualmente rigoroso para o campo escalar livre de interações. Trazemos de início alguns aspéctos importantes da Teoria de Distribuições e colocamos alguns pontos de geometria Lorentziana. O restante do trabalho é dividido em duas partes: na primeira, estudamos equações de onda em variedades Lorentzianas globalmente hiperbólicas e apresentamos o conceito de soluções fundamentais no contexto de equações locais. Em seguida, progressivamente construímos soluções fundamentais para o operador de onda a partir da distribuição de Riesz. Uma vez estabelecida uma solução para a equação de onda em uma vizinhança de um ponto da variedade, tratamos de construir uma solução global a partir da extensão do problema de Cauchy a toda a variedade, donde as soluções fundamentais dão lugar aos operadores de Green a partir da introdução de uma condição de contorno. Na última parte do trabalho, apresentamos um mínimo da Teoria de Categorias e Funtores para utilizar esse formalismo na contrução de um funtor de segunda quantização entre a categoria de variedades Lorentzianas globalmente hiperbólicas e a categoria de redes de álgebras C* satisfazendo os axiomas de Haag-Kastler. Ao fim, retomamos o caso particular do campo escalar quântico livre. / In this thesis we present a both mathematical and conceptually rigorous quantization method for the neutral scalar field free of interactions. Initially, we introduce some aspects of the Theory of Distributions and we establish some points of Lorentzian geometry. The rest of the work is divided in two parts: in the first one, we study wave equations on globally hyperbolic Lorentzian manifolds, hence presenting the concept of fundamental solutions within the context of locally defined wave equations. Next, we progressively construct fundamental solutions for the wave operator from the Riesz distribution. Once established a solution to the wave equation in a neighbourhood of a point of the manifold, we move forward to produce a global solution from the extension of the Cauchy problem to the whole manifold. At this stage, fundamental solutions are replaced by Green\'s operators by the imposition of appropriate boundary conditions. In the last part, we present a minimum on the Theory of Categories and Functors. This is followed by the use of this formalism in the development of a second-quantization functor between the category of Lorentzian globally hyperbolic manifolds and the category of nets of C*-algebras obeying Haag-Kastler axioms. Finally, we turn our attention to the particular case of the quantum free scalar field.

In this work we study properties of spacetimes with a high degree of symme- try. Particularly, we focus on geometries related to higher-dimensional rotating black-hole spacetimes described by the Kerr-NUT-(A)dS metric. In the first part, we examine spacetimes admitting a separable Klein-Gordon equation. Motivated by Carter's work in four dimensions, we introduce a separable met- ric ansatz in higher dimensions. Analyzing Einstein's equations, we obtain the Kerr-NUT-(A)dS and specific Einstein-K¨ahler metrics. Then we consider a metric ansatz in the form of warped geometries of two Klein-Gordon separable metrics and classify the corresponding solutions. In the second part, we in- vestigate a class of limits of the Kerr-NUT-(A)dS spacetimes where particular roots of metric functions degenerate. Our limiting procedure results in various NUT-like and near-horizon geometries such as the higher-dimensional Taub- NUT-(A)dS spacetime. We demonstrate that the symmetries of the resulting geometries are enhanced, which is manifested by decomposition of Killing ten- sors into Killing vectors. The third part of this work deals with generalized symmetry axes of the Kerr-NUT-(A)dS spacetimes that are formed by fixed points of isometries. We show that some parts of the symmetry axes are sin- gular for nonzero NUT charges....

Le but de cette thèse est d’obtenir des informations sur le spectre discret d’opérateurs non auto-adjoints définis par des perturbations relativement compactes d’opérateurs auto-adjoints. Ces opérateurs auto-adjoints sont choisis parmi les opérateurs classiques de mécanique quantique. Il s’agit des opérateurs de Dirac, de Klein-Gordon et le laplacien fractionnaire qui généralise l’opérateur de Schrödinger habituellement considéré pour de tels problèmes. La principale méthode utilisée ici relève d’un théorème d’analyse complexe donnant une condition de type Blaschke sur les zéros d’une fonction holomorphe du disque unité. Cette condition traduit lecomportement des valeurs propres de l’opérateur perturbé sous forme d’inégalités de type Lieb-Thirring. Une autre méthode venant d’analyse fonctionnelle a été employée pour obtenir de telles inégalités et les deux méthodes sont comparées entre elles. / The topic of this thesis concerns the discrete spectrum of non-selfadjoint operators defined by relatively compact perturbation of selfadjoint operators. These selfadjoint operators are choosen among classical operators of quantum mechanics. These areDirac operator, Klein-Gordon operator, and the fractional Laplacian who generalize the Schrödinger operator. The main method is based on a theorem of complex analysis which gives Blaschke-type condition on the zeros of a holomorphic function on the unit disc. This Blaschke condition gives the information on the behaviour of eigenvalues of the perturbed operator by mean of Lieb-Thirring-type inequalities. Another method using functional analysis is also used to obtain these kind of inequalities and both methods are compared to each other.

Μελετάµε χωρικά εντοπισµένες και χρονικά περιοδικές λύσεις σε διακριτά συστήµατα που εκτείνονται σε µία χωρική διάσταση. Αυτού του είδους οι λύσεις είναι γνωστές µε τον όρο discrete breathers (DB) ή intrinsic localized modes (ILM). Στην ελληνική ϐιϐλιογραϕία, έχουν ονοµαστεί ∆ιακριτές Πνοές. Απαραίτητα χαρακτηριστικά για την εµϕάνιση τέτοιων λύσεων είναι η ύπαρξη ενός άνω φράγµατος του γραµµικού φάσµατος καθώς και η µη γραµµικότητα των εξισώσεων κίνησης, χαρακτηριστικά που συναντάµε σε πολλά φυσικά συστήµατα. Συγκεκριμένα, ασχολούµαστε µε πλέγµατα τύπου Klein Gordon και παρουσιάσουµε μια αποδείξη ύπαρξης τέτοιων λύσεων καθώς και αριθµητικά αποτελέσµατα µελετώντας παράλληλα την ευστάθεια των περιοδικών αυτών λύσεων µέσω της ϑεωρίας Floquet. Πέραν του κλασικού µοντέλου, όπου έχουµε αλληλεπιδράσεις πλησιέστερων γειτόνων, εισάγουµε επίσης ένα νέο µοντέλο µε αλληλεπιδράσεις µακράς εµβέλειας η οποία ελέγχεται µέσω µιας παράµετρου α και µελετάµε τις επιπτώσεις που έχει η μεταβολή του εύρους αλληλεπίδρασης στον χωρικό εντοπισµό και την ευστάθεια ενός DB. / We study time-periodic and spatially localized solutions in discrete dynamical systems describing Hamiltonian lattices in one spatial dimension. These solutions are called discrete breathers (DBs) or intrinsic localized modes (ILM). Necessary conditions for their occurrence are the boundedness of the spectrum of linear oscillations of the system as well as the nonlinearity of the equations of motion. More specifically, we focus on a Klein Gordon lattice and present an existence proof for such solutions, as well as numerical results revealing the stability (or instability) of DBs using Floquet theory. Besides reporting on the classical Klein Gordon model with nearest neighbor interactions, we also introduce long range interactions in our model, which are controlled by a parameter α and study the effect of varying the range of interactions on the spatial localization and the stability of a DB.

Les travaux de cette thèse sont regroupés en trois grandes parties traitant respectivement-des ondes stationnaires des systèmes de Schr"odinger-Maxwell-Proca et de Klein-Gordon-Maxwell-Proca sur une variété riemannienne fermée (compacte sans bord dans toute la thèse),-de systèmes elliptiques de Kirchhoff sur une variété riemannienne fermée,-de phénomènes d'explosion propres aux petites dimensions. / This thesis, divided into three main parts, deals with-standing waves for Schrödinger-Maxwell-Proca and Klein-Gordon-Maxwell-Proca systems on a closed Riemannian manifold (compact without boundary during all the thesis),-elliptic Kirchhoff systems on a closed manifold,-low-dimensional blow-up phenomena.

Made available in DSpace on 2016-06-02T20:27:38Z (GMT). No. of bitstreams: 1 3466.pdf: 565162 bytes, checksum: 770041f07c68eda588bd0c501dabe93d (MD5) Previous issue date: 2011-02-10 / Financiadora de Estudos e Projetos / In this work we analyze the existence of radially symmetric solutions, positive solutions as well as the existence of ground state solutions for a class of Klein-Gordon-Maxwell equations when the nonlinearity exhibits critical behavior. For the positive and ground state solutions we prove existence results when a potential V is introduced. In order to obtain such results, we use variational methods / Neste trabalho analisamos a existência de soluções radialmente simétricas, soluções positivas, bem como a existência de soluções ground state para uma classe de equações do tipo Klein-Gordon-Maxwell quando a não-linearidade exibe comportamento crítico. Para as soluções positivas e do tipo ground state provamos resultados de existência quando um potencial V é introduzido. A fim de obtermos tais resultados, usamos métodos variacionais.

Made available in DSpace on 2015-05-14T12:14:11Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 2344020 bytes, checksum: 1d77972a45b8beef7c3fe6631dfddaa2 (MD5) Previous issue date: 2014-02-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this dissertation we consider the influence of gravitational fields due to the black holes of Kerr-Newman and Kerr-Newman-de Sitter, on a massive scalar field, with and without charge. We obtain the exact solutions of the radial Klein-Gordon equation in the spacetime of Kerr-Newman which are given in terms of the confluent Heun functions. In the particular case of a extreme Kerr-Newman black hole, the solution is given in terms of double confluent Heun functions. We also investigate the solutions close to the exterior event horizon and very far from the black hole. For a charged scalar field, we obtain exact solutions corresponding to the angular Klein-Gordon equation in the Kerr-Newman-de Sitter spacetime which are given in terms of the Heun functions. Using a method due to Damour and Ruffini, we study the Hawking radiation of massive scalar particles. In the Kerr-Newman black hole, we obtain the exact solutions for both the angular and radial Klein-Gordon equations, which are given in terms of the confluent Heun functions. From the radial solution, we obtain the exact wave solutions near to the exterior horizon of the black hole, and discuss the Hawking radiation of charged massive scalar particles. / Nesta dissertação tratamos da influência do campo gravitacional produzido pelos buracos negros de Kerr-Newman e Kerr-Newman-de Sitter sobre um campo escalar massivo com e sem carga. Obtemos as soluções exatas da parte radial da equação de Klein-Gordon em um espaço-tempo de Kerr-Newman, que são dadas em termos das funções confluentes de Heun. No caso particular correspondente ao buraco negro de Kerr-Newman extremo, a solução é dada em termos das funções duplamente confluentes de Heun. Investigamos, também, as soluções nas proximidades do horizonte de evento exterior e longe do buraco negro. Para um campo escalar massivo carregado, obtemos as soluções exatas para a parte angular da equação de Klein-Gordon em um espaço-tempo de Kerr-Newman-de Sitter, que são dadas em temos das funções de Heun. Utilizando o método de Damour & Ruffini, estudamos a radiação Hawking para partículas escalares massivas. No buraco negro de Kerr-Newman, obtemos as soluções exatas de ambas as partes radial e angular da equação de Klein-Gordon, que são dadas em termos das funções confluentes de Heun. A partir da solução radial, obtemos as soluções de ondas exatas próximas ao horizonte exterior do buraco negro e discutimos a radiação Hawking para partículas escalares massivas carregadas.