Optimal control design and implementation for nonlinear systems is a topic of much interest. However, unlike for linear systems, for nonlinear systems explicit analytical solution for optimal feedback control is not available. Numerical techniques, on the other hand, can be used to approximate the solution of the HJB equation to find the optimal control. In this research, a computational approach is developed for finding the optimal control for nonlinear systems with polynomial vector fields based on sum of squares technique. In this research, a numerical technique is developed for optimal control of polynomial nonlinear systems. The approach follows a four-step procedure to obtain both local and approximate global optimality. In the first step, local optimal control is found by using the linearization method and solving the Algebraic Riccati equation with respect to the quadratic part of a given performance index. Next, we utilize the density function method to find a globally stabilizing polynomial nonlinear control for the nonlinear system. In the third step, we find a corresponding Lyapunov function for the designed control in the previous steps based on the Hamilton Jacobi inequality by using semidefinite programming. Finally, to achieve global optimality, we iteratively update the pair of nonlinear control and Lyapunov function based on a state-dependent polynomial matrix inequality. Numerical examples illustrate the effectiveness of the design approach.

A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy to the Faculty of Science, University of the Witwatersrand, Johannesburg, 2019 / We will study the symmetry, invariance properties and conservation laws of partial dif ferential equations (pdes) that arise in a number of situations in mathematical physics. These will be range from Image Processing and noise removal algorithms to Timoshenko beam systems. Furthermore, we will study the invariance properties and approximate conservation laws of some nonlinear Schro¨dinger equation with PT-symmetric potentials with inhomogeneous nonlinearity and some nonlinear Schro¨dinger equation involving a spatially extended system consisting of two coupled elements. / TL (2019)

archives@tulane.edu / The sharing of quantum resources between multiple parties allows for the creation of quantum networks. Traditional four-wave mixing creates twin correlated beams of light. More complex four-wave mixing schemes can create a multitude of correlated beams for use in quantum communication, helping pave the way toward future quantum networks. These correlations can be seen in the intensity-difference squeezing between output modes. In this dissertation, we examine a variety of multi-mode quantum systems. I begin in chapters 2 and 3 by using the noise figure, which compares the signal-to-noise ratios of output modes to input modes, to calculate intensity-difference squeezing and make predictions about phase-sensitivity. In chapter two, I analyze a dual-pump four-wave mixing system yielding three output modes for cases in which a single seed, two asymmetric seeds, and two symmetric seeds are used. In chapter 3, I perform similar calculations for three different cascaded four-wave mixing configurations. Various intensity-difference squeezing combinations are compared for two variations of two cascaded four-wave mixing cells and for three cascaded four-wave mixing cells. Chapter 4 describes a dual pump four-wave mixing scheme with four output modes created experimentally and chapter 5 shows that when only one input mode is seeded this process is phase-insensitive. Interestingly, I find that when only two of the input modes are seeded the system becomes phase-sensitive. Finally, in chapter 6, I describe the simulated and experimental results of using a deep neural network to improve the bit error rates in a classical free-space optical on-off keying scheme, that will eventually be expanded into the quantum regime. / 1 / Sara K Wyllie

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The Ironman triathlon was created in 1978 by combining events with the longest distances for races then contested in Hawaii in swimming, cycling, and running. The Half Ironman triathlon was formed using half the distances of each of the events in the Ironman. The Olympic distance triathlon was created by combining events with the longest distances for races sanctioned by the major federations for swimming, cycling, and running. The relative importance of each event in overall race outcome was not given consideration when determining the distances of each of the races in modern triathlons. Thus, there is a general belief among triathletes that the swimming portion of the standard-distance triathlons is underweighted. We present a nonlinear Bayesian model for triathlon finishing times that models time and standard deviation of time as a function of distance. We use this model to create "fair" triathlons by equating the standard deviations of the times taken to complete the swimming, cycling, and running events. Thus, in these "fair" triathlons, a one standard deviation improvement in any event has an equivalent impact on overall race time.

In many applications involving data collected over time, it is important to get timely estimates and adjustments of the parameters associated with a dynamic model. When the dynamics of the model must be updated, time and computational simplicity are important issues. When the dynamic system is not linear the problem of adaptation and response to feedback are exacerbated. A linear approximation of the process at various levels or “states” may approximate the non-linear system. In this case the approximation is linear within a state and transitions from state to state over time. The transition probabilities are parametrized as a Markov chain, and the within-state dynamics are modeled by an AR time series model. However, in order to make the estimates available almost instantaneously, least squares and weighted least squares estimates are used. This is a modification of the cluster-weighted models proposed by Gershenfeld, Schoner, and Metois (1999). A simulation study compares the models and explores the adequacy of least squares estimators.

Employers rely on judges or raters to accurately rate the potential or performance of candidates through interviews or assessment centre evaluations. As the judgment process places heavy demands on information processing, cognitive ability (of raters) is important to detect and interpret behavioural cues presented by those being rated. A consistent empirical finding is that intelligence is the strongest predictor of rating accuracy, but prior research has largely been based on linear models. However, researchers have yet to investigate whether these variables could be nonlinearly related. By studying nonlinear models in judgment and accuracy, we can not only deepen our understanding of the ‘good judge' in HRM, but we may further enhance methods to select and train raters in applied practice. This secondary research study re-analysed data from a prior published study to evaluate the relationship between rater intelligence and accuracy of interview ratings provided by 146 South African managers. The predictiveness of an ordinary least squares (OLS) linear regression model was compared to two nonlinear models (quadratic and cubic) to determine which statistical approach explained the most variance in rating accuracy scores. Findings provided further support of a linear relationship between intelligence and rating accuracy suggesting no quadratic or cubic interactions. Judges, therefore, produced more accurate ratings at higher levels of intelligence. Possible explanations of the findings include the sample size and task complexity. Study limitations and recommendations for future research are discussed in detail

Laser filamentation is a highly complex and dynamic nonlinear process that is sensitive to many physical parameters. The basic properties that define a filament consist of (i) a narrow, high intensity core that persists for distances much greater than the Rayleigh distance, (ii) a low density plasma channel existing within the filament core, and (iii) a supercontinuum generated over the course of filamentation. However, there remain many questions pertaining to how these basic properties are affected by changes in the conditions in which the filaments are formed; that is the premise of the work presented in this dissertation. To examine the effects of anomalous dispersion and of different multi-photon ionization regimes, filaments were formed in solids with different laser wavelengths. The results provided a better understanding of supercontinuum generation in the anomalous dispersion regime, and of how multi-photon ionization can affect the formation of filaments. Three different experiments were carried out on filamentation in air. The first was an investigation into the effects of geometrical focusing. A simplified theoretical model was derived to determine the transition of filamentation in the linear-focusing and nonlinear- focusing regimes. The second examined the effects of polarization on supercontinuum generation, where a polarization-dependent anomalous spectral broadening phenomenon due to molecular effects was identified. The third involved the characterization of filaments in the ultraviolet. The combination of physical mechanisms responsible for filamentation in the ultraviolet was found to be different from that in the near infrared.

The main goal of this dissertation is to investigate the third-order nonlinearity of organic molecules. This topic contains two aspects: two-photon absorption (2PA) and nonlinear refraction (NLR), which are associated with the imaginary and real part of the third-order nonlinearity (χ (3)) of the material, respectively. With the optical properties tailored through meticulous molecular structure engineering, organic molecules are promising candidates to exhibit large third-order nonlinearities. Both linear (absorption, fluorescence, fluorescence excitation anisotropy) and nonlinear (Z-scan, two-photon fluorescence, pump-probe) techniques are described and utilized to fully characterize the spectroscopic properties of organic molecules in solution or solid-state form. These properties are then analyzed by quantum chemical calculations or other specific quantum mechanical model to understand the origins of the nonlinearities as well as the correlations with their unique molecular structural features. These calculations are performed by collaborators. The 2PA study of organic materials is focused on the structure-2PA property relationships of four groups of dyes with specific molecular design approaches as the following: (1) Acceptor-π-Acceptor dyes for large 2PA cross section, (2) Donor-π-Acceptor dyes for strong solvatochromic effects upon the 2PA spectra, (3) Near-infrared polymethine dyes for a symmetry breaking effect, (4) Sulfur-squaraines vs. oxygen-squaraines to study the role of sulfur atom replacement upon their 2PA spectra. Additionally, the 2PA spectrum of a solid-state single crystal made from a Donor-π-Acceptor dye is measured, and the anisotropic nonlinearity is studied with respect to different incident polarizations. These studies further advance our iv understanding towards an ultimate goal to a predictive capability for the 2PA properties of organic molecules. The NLR study on molecules is focused on the temporal and spectral dispersion of the nonlinear refraction index, n2, of the molecules. Complicated physical mechanisms, originating from either electronic transitions or nuclei movement, are introduced in general. By adopting a prism compressor / stretcher to control the pulsewidth, an evolution of n2 with respect to incident pulsewidth is measured on a simple inorganic molecule –carbon disulfide (CS2) in neat liquid at 700 nm and 1064 nm to demonstrate the pulsewidth dependent nonlinear refraction. The n2 spectra of CS2 and certain organic molecules are measured by femtosecond pulses, which are then analyzed by a 3-level model, a simplified "Sum-over-states" quantum mechanical model. These studies can serve as a precursor for future NLR investigations.