An algorithm is presented which computes a state feedback for a standard linear system which not only stabilizes, but also dampens the closed-loop system dynamics. In other words, a feedback gain vector is computed such that the eigenvalues of the closed-loop state matrix are within the region of the left half-plane where the magnitude of the real part of each eigenvalue is greater than the imaginary part. This may be accomplished by solving one periodic algebraic Riccati equation and one degenerate Riccati equation. The solution to these equations are computed using numerically robust algorithms. Finally, the periodic Riccati equation is unusual in that it produces one symmetric and one skew symmetric solution, and as a result two different state feedbacks. Both feedbacks dampen the system dynamics, but produce different closed-loop eigenvalues, giving the controller designer greater freedom in choosing a desired feedback.

The Log-Periodic Loop Antenna with Ground Reflector (LPLA-GR) is investigated as a new type of antenna, which provides wide bandwidth, broad beamwidth, and high gain. This antenna has smaller transverse dimensions (by a factor of 2/pi) than a log-periodic dipole antenna with comparable radiation characteristics. Several geometries with different parameters are analyzed numerically using ESP code, which is based on the method of moments. A LPLA-GR with 6 turns and a cone angle of 30* offers the most promising radiation characteristics. This antenna yields 47.6 % gain bandwidth and 12 dB gain according to the numerical analysis. The LPLA-GR also provides linear polarization and unidirectional patterns. Three prototype antennas were constructed and measured in the Virginia Tech Antenna Laboratory. Far-field patterns and input impedance were measured over a wide range of frequencies. The measured results agree well with the calculated results. Because of its wide bandwidth, high gain, and small size, the LPLA is expected to find applications as feeds for reflector antennas, as detectors in EMC scattering range, and as mobile communication antennas. / Master of Science

Frequency domain models for power electronic circuits are either based on iterative techniques such as Newton's method or linearised around an operating point. Iterative frequency domain models provide great accuracy as they are capable of calculating the exact switching instants of the device. On the other hand, the accuracy of a linearised frequency domain model relies on the magnitude of input waveform to be small so that the circuit's operating point does not vary or varies very little. However, an important advantage of a linearised model is its ability to provide insight into waveform distortion interaction, more specifically, the frequency cross-coupling around a power electronic circuit. In general, a linearised model for harmonic analysis would not normally include the description of feedback control. Likewise a linearised model for control analysis would usually disregard frequency interactions above the fundamental (or the most significant component); that is assuming the cross-coupling between harmonic frequencies does not affect the dynamics of control. However, this thesis proposes that a linearised model for control analysis shall also include the complete description of frequency cross-coupling between harmonics to produce the correct dynamic response. This thesis presents a harmonic state-space (HSS) model of an HVdc converter that incorporates the full effect of varying switching instants, both through control and commutation period dynamics, while remaining within the constraints of a linear time-invariant (LTI) system. An example is given using the HSS model to explain how a close to fifth harmonic resonance contributes to the dominant system response through the frequency cross-coupling of the converter and the controller feedback loop. The response of the system is validated against a time domain model built in PSCAD/EMTDC, and more importantly, the correct response cannot be produced without including the harmonic interactions beyond the fundamental frequency component.

Hyperkalemic periodic paralysis, HyperKPP, is an inherited progressive disorder of the muscles caused by mutations in the voltage gated sodium channel (NaV1.4). The objectives of this thesis were to develop a technique for measurement symptoms in vivo using electromyography (EMG) and to determine the mechanism by which Ca2+ alleviates HyperKPP symptoms, since this is unknown. Increasing extracellular [Ca2+] ([Ca2+]e) from 1.3 to 4 mM did not result in any increases in45Ca2+ influx suggesting no increase in intracellular [Ca2+] ([Ca2+]i) acting on an intracellular signaling pathway or on an ion channel such as the Ca2+sensitive K+ channels. HyperKPP muscles have larger TTX-sensitive22Na+ influx than wild type muscles because of the defective NaV1.4 channels. When [Ca2+] was increased from 1.3 to 4 mM, the abnormal 22Na+ influx was completely abolished. Thus, one mechanism by which Ca2+alleviates HyperKPP symptoms is by reducing the abnormal Na+ influx caused by the mutation in the NaV1.4 channel.

The subject of this thesis is scattering of electromagneticwaves from planar and curved periodic structures. The problemspresented are solved in the frequency domain. Scattering from planar structures with two-dimensionalperiodic dependence of constitutive parameters is treated. Theconstitutive parameters are assumed to vary continuously orstepwise in a cross section of a periodically repeating cell.The variation along a longitudinal coordinate z is arbitrary. Ageneral skew lattice is assumed. In the numerical examples, lowloss and high loss dielectric materials are considered. Theproblem is solved by expanding the .elds and constitutiveparameters in quasi-periodic and periodic functionsrespectively, which are inserted into Maxwell’s equations.Through various inner products de.ned with respect to the cell,and elimination of the longitudinal vector components, a linearsystem of ordinary di.erential equations for the transversecomponents of the .elds is obtained. After introducing apropagator, which maps the .elds from one transverse plane toanother, the system is solved by backward integration.Conventional thin metallic FSS screens of patch or aperturetype are included by obtaining generalised transmission andre.ection matrices for these surfaces. The transmission andre.ection matrices are obtained by solving spectral domainintegral equations. Comparisons of the obtained results aremade with experimental results (in one particular case), andwith results obtained using a computer code based on afundamentally di.erent time domain approach. Scattering from thin singly curved structures consisting ofdielectric materials periodic in one dimension is alsoconsidered. Both the thickness and the period are assumed to besmall. The .elds are expanded in an asymptotic power series inthe thickness of the structure, and a scaled wave equation issolved. A propagator mapping the tangential .elds from one sideto the other of the structure is derived. An impedance boundarycondition for the structure coated on a perfect electricconductor is obtained. Keywords:electromagnetic scattering, periodicstructure, frequency selective structure, frequency selectivesurface, grating, coupled wave analysis, electromagneticbandgap, photonic bandgap, asymptotic boundary condition,impedance boundary condition, spectral domain method,homogenisation

We present a periodic technique for measuring the thermal conductivity and diffusivity of thin samples simultaneously. In samples of this type, temperature measurements must be made across the sample faces and are therefore subject to large error due to the interface resistance between the temperature sensor and the sample. The technique uses measurements of the amplitude and phase of the periodic temperature across both a reference sample and the unknown material at several different frequencies. Modeling of the heat flow in the sample allows the simultaneous determination of the thermal parameters of the sample as well as the interface resistance. Data will be presented for standard materials to show the viability of the technique.

Ce travail porte sur des équations aux dérivées partielles issues de la physique mathématique, plus particulièrement sur celles régissant la supraconductivité. Ainsi, la majorité du travail concerne le modèle de Ginzburg-Landau, qui est un modèle macroscopique de supraconducteurs de type-II. Ce travail est divisé en deux parties principales: La première partie se focalise sur l'analyse des vortex du modèle de Ginzburg-Landau en deux dimensions pour les supraconducteurs de type-II, modèle conduisant à une estimation de la variation du nombre de vortex et à l'optimalité du réseau d'Abrikosov parmi les réseaux de Bravais. Nous avons également étudié certains modèles de stuctures des matériaux comme ceux de Lennard-Jones et de Thomas-Fermi. La seconde partie est consacrée à la fonctionnelle de Ginzburg-Landau en dimension $n$. Deux résultats principaux sont obtenus. L'un porte sur l'énergie renormalisée pour les minimiseurs de la fonctionnelle de Ginzburg-Landau. L'autre concerne les limites des solutions de l'équation de Ginzburg-Landau. Ces deux résultats sont fortement reliés aux applications $n$-harmoniques / Our work focus on the elliptic partial differential Equations arising from the mathematical physics, especially from the superconductivity. Therefore most of our work is on the Ginzburg-Landau model, which is a macroscopic model for the type-II superconductors. The work is divided into two big parts : this first part is on the vortices analysis of the Ginzburg-Landau model of Type-II superconductors in 2 dimensions, including the variations of the number of vortices and optimality of Abrikosov lattices among Bravais Lattices. We also have done some work related to the material structure, for example, the Lennard-Jones model and the Thomas-Fermi model. This second part is on the Ginzburg-Landau functional in $n$-dimensional case. Two main results are contained in this part: One is on the renormalized energy for minimizer of $n$-dimensional Ginzburg-Landau functional; The other one is on the limits of solutions to Ginzburg-Landau equations in $n$ dimension. Both of these two results are closely related to the p-harmonic maps

Demonstramos o seguinte teorema: Seja M uma variedade Riemanniana compacta, conexa e sem bordo. Dados um endomorismo f : M ightarrow M, uma função contínua \\phi: M ightarrow R e \\epsilon > 0, então existe um endomorísmo \\tilde f : M ightarrow M tal que d(f; \\tide f) = \\max_{x \\in M} d(f(x); \\tilde f(x)) < \\epsilon, e existe uma medida \\phi-maximizante para \\tilde f que está suportada em uma orbita periodica. Este teorema e uma generalização dos resultados obtidos por S. Addas-Zanatta e F. Tal. / We prove the following theorem: Let M be a bondaryless, compact and connected Riemannian Manifold. Given an endomorphism f on M, a continuous function \\phi : M ightarrow R and \\epsilon > 0, then there exist an endomorphism \\tilde f on M with d(f; \\tilde f) < \\epsilon such that, some \\phi-maximizing measure for \\tilde f is supported on a periodic orbit. This theorem is a generalization of the results obtained by S. Addas-Zanatta and F. Tal.

O propósito deste trabalho é estudar a existência de solução periódica para problemas de oscilação não linear de barras submetidas a forças periódicas. Estudaremos concretamente dois problemas, que serão interpretados como equações diferenciais abstratas de segunda ordem cuja classe foi considerada em Ceron e Lopes [1]. Para garantir a existência de solução periódica dos problemas considerados, mostraremos que a aplicação de Poincaré S é limitada dissipativa e alfa-contração. Isso garante a existência de um atrator invariante compacto e a existência de um ponto fixo de S, o que é equivalente a existência da solução periódica. / Our aim in this work is to study the existence of periodic solution to oscillation in nonlinear problems of beams submitted to periodic forcing. We will study concretely two problems, which can be interpreted as an abstract second order diferential equation studied by Ceron and Lopes [1]. Our intention is to prove the existence of periodic solution to these problems. To this end, we will show that the Poincaré map S is uniform ultimately bounded and alpha-contraction. Thus we have the existence of invariant compact attractor, therefore S have a fixed point, which is equivalent the existence of a periodic solution.

Estudamos um caso especial do problema restrito dos três corpos, chamado problema circular de Sitnikov, quando dois corpos de massas iguais (chamadas de primárias) estão em uma órbita circular (configuração central de dois corpos), enquanto que um terceiro corpo de massa negligenciada (chamada infinitesimal) oscila sobre uma reta perpendicular ao plano das primárias (chamaremos esse movimento de vertical periódico). Aqui analisamos a estabilidade desse movimento periódico, com relação a pequenas perturbações nas direções ortogonais a reta onde ocorre o movimento. Chamaremos a atenção ao fenômeno de alternância entre estabilidade e instabilidade na família do movimento periódico vertical, conforme variamos a amplitude do movimento. / We studied a special case of the restricted three-body problem, named circular problem of Sitnikov, when two body of equal mass (called primaries) moving around each other on circular motion (central configuration of two body), while the third body of negligible mass (called infinitesimal) performs along a straight line orthogonal to the plane of the primaries (so called periodic vertical motions). We analyze the stability of the periodic vertical motions with respect to small perturbations orthogonal to the straight line where the motions occurs. We call attention to the phenomenom of alternation of stability and instability within the family of periodic vertical motions, whenever their amplitude is varied in a continuous manner.