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This thesis presents an approximate solution of second order relative motion equations. The equations of motion for a Keplerian orbit in spherical coordinates are expanded in Taylor series form using reference conditions consistent with that of a circular orbit. Only terms that are linear or quadratic in state variables are kept in the expansion. A perturbation method is employed to obtain an approximate solution of the resulting nonlinear differential equations. This new solution is compared with the previously known solution of the linear case to show improvement, and with numerical integration of the quadratic differential equation to understand the error incurred by the approximation. In all cases, the comparison is made by computing the difference of the approximate state (analytical or numerical) from numerical integration of the full nonlinear Keplerian equations of motion. / Master of Science

Three robust tracking control design strategies are proposed in this dissertation for different classes of nonlinear MIMO dynamic systems with mismatched perturbations. The first controller design method is proposed for a class of nonlinear MIMO dynamic systems in canonical form. The second design procedure of controller is for the nonlinear MIMO dynamic systems without canonical form. A decentralized controller is presented in the last for perturbed large-scale systems with time-varying delay interconnections, where the knowledge of the exact function of time-delay is not required. These robust tracking controllers with a perturbation estimating scheme and an adaptive control mechanism embedded are designed by means of the variable structure control technique and Lyapunov stability theorem. The adaptive control mechanism is used to adapt the unknown upper-bound of perturbation estimation error, so that the knowledge of upper-bounds of perturbation as well as perturbation estimation error is not required. The chattering phenomenon is effectively alleviated, for the gain of the proposed controllers, which needs only to overcome the perturbation estimation error, is in general smaller than those of the traditional sliding mode controllers. Furthermore, the stability of the overall controlled systems is proved, and the desired tracking accuracy can be achieved by adjusting the design parameters of the proposed controller schemes. A numerical example for each controller's design is provided for demonstrating the feasibility of the proposed control schemes.

Thèse de doctorat : Mathématiques : Paris, Faculté des sciences : 1840. / Titre provenant de l'écran-titre.

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Yuen Pui-Ho. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 158-159). / Text in English; abstracts in English and Chinese. / Yuen Pui-Ho. / ACKNOWLEDGEMENTS --- p.1 / ABSTRACT --- p.2 / CONTENTS --- p.4 / LIST OF FIGURES --- p.7 / Chapter CHAPTER 1 - --- INTRODUCTION --- p.13 / Chapter 1-1 --- Interaction with EM-field --- p.15 / Chapter 1-2 --- Objective of this study --- p.17 / Chapter 1-3 --- Thesis arrangement --- p.18 / Chapter CHAPTER 2 - --- LITERATURE REVIEW --- p.20 / Chapter 2-1 --- Numerical Method --- p.20 / Chapter 2-1-1 --- Classical calculation --- p.20 / Chapter 2-1-2 --- Quantum mechanical calculation --- p.21 / Chapter 2-1-2-1 --- Finite Difference Method --- p.21 / Chapter 2-1-2-2 --- Non-finite difference method --- p.23 / Chapter 2-1-3 --- Calculation of dissociation probability --- p.24 / Chapter 2-2 --- Morse oscillator under constant driving frequency --- p.26 / Chapter 2-2-1 --- Classical and quantum dynamics comparison --- p.26 / Chapter 2-2-2 --- Resonance condition and factors affecting dissociation probability --- p.27 / Chapter 2-3 --- MORSE OSCILLATOR UNDER CHIRPED DRIVING FREQUENCY --- p.33 / Chapter 2-3-1 --- Bucket Dynamics --- p.35 / Chapter 2-3-2 --- Floquet Analysis --- p.38 / Chapter 2-4 --- Summaries --- p.43 / Chapter CHAPTER 3 - --- METHODOLOGY --- p.46 / Chapter 3-1 --- The model --- p.46 / Chapter 3-2 --- An approximation to sinusoidal variation --- p.47 / Chapter 3-3 --- Approximation to chirping frequency --- p.51 / Chapter 3-4 --- Calculating time evolution operator for piecewise constant --- p.56 / Chapter 3-5 --- Performance of the piecewise constant approach --- p.58 / Chapter 3-5-1 --- Error vs. time --- p.59 / Chapter 3-5-2 --- Accuracy dependence on number of term included --- p.62 / Chapter 3-5-3 --- Accuracy dependence on applied E-field --- p.64 / Chapter 3-5-4 --- "Drawbacks of the "" piecewise constant approach ´ح" --- p.65 / Chapter CHAPTER 4 - --- BEHAVIOR OF MORSE OSCILLATOR UNDER SEMI-CLASSICAL EM- FIELD --- p.67 / Chapter 4-1 --- Morse oscillator in monochromatic laser light --- p.67 / Chapter 4-1-1 --- Theoretical analysis --- p.67 / Chapter 4-1-2 --- Numerical simulation --- p.70 / Chapter 4-1-2-1 --- Dependence on resonant levels --- p.70 / Chapter 4-1-2-2 --- Dependence on Intensity --- p.76 / Chapter 4-1-2-3 --- Frequency deviation from resonant frequency --- p.79 / Chapter 4-2 --- Morse oscillator driven by chirped laser pulse --- p.89 / Chapter 4-2-1 --- Chirping scheme --- p.89 / Chapter 4-2-2 --- Level climbing and residual --- p.92 / Chapter 4-2-3 --- Factors affecting dissociation probability --- p.98 / Chapter 4-2-3-1 --- Dependence on driving amplitude --- p.98 / Chapter 4-2-3-2 --- Dependence on rate of frequency change and initial driving frequency --- p.99 / Chapter 4-2-3-3 --- Dependency on starting state --- p.104 / Chapter 4-3 --- Chapter Summary --- p.106 / Chapter CHAPTER 5 - --- "BUCKET DYNAMICS AND QUANTUM MECHANICAL ""PHASE SPACE""" --- p.108 / Chapter 5-1 --- Bucket dynamics in classical Morse oscillator --- p.109 / Chapter 5-2 --- "Quantum mechanical ""phase space""" --- p.114 / Chapter 5-2-1 --- Wigner function and its properties --- p.115 / Chapter 5-2-2 --- Q-function and its properties --- p.121 / Chapter 5-2-3 --- Q-functions of the energy eigenstates of Morse oscillator --- p.123 / Chapter 5-3 --- Evolution of Wigner function and Q-function --- p.125 / Chapter 5-3-1 --- Evolution of Wigner function and Q-function under periodic driving field --- p.125 / Chapter 5-3-2 --- Evolution of Wigner function and Q-function under driving field with chirped frequency --- p.132 / Chapter 5-4 --- "Bucket dynamics and, quantum mechanical ""phase space""" --- p.138 / Chapter 5-5 --- Chapter Summary --- p.141 / Chapter CHAPTER 6 - --- CONCLUSIONS AND FURTHER STUDIES --- p.143 / Chapter 6-1 --- Morse oscillator driven by monochromic laser light --- p.143 / Chapter 6-1-1 --- Factors affecting the behavior of Morse oscillator in monochromic laser light --- p.144 / Chapter 6-1-1-1 --- Effect of laser intensity --- p.144 / Chapter 6-1-1-2 --- Effect of driving frequency --- p.144 / Chapter 6-2 --- Morse oscillator driven by chirped laser pulse --- p.145 / Chapter 6-3 --- "Bucket dynamics and quantum mechanical ""phase space""" --- p.146 / Chapter 6-4 --- Wigner function and Q-function of driven Morse oscillator --- p.147 / Chapter 6-5 --- Conclusions and Discussions --- p.149 / APPENDIX A - EQUIVALENCE OF QUANTUM SYSTEMS WITH DIFFERENT GAUGES --- p.150 / APPENDIX B - CALCULATION OF TRANSITION MATRIX ELEMENT TMN --- p.152 / "APPENDIX C - CALCULATION OF PERTURBED MATRIX ELEMENTS, H´ةKN" --- p.154 / BIBLIOGRAPHY --- p.158

Cosmological perturbation theory (CPT) is an important tool with which inhomogeneities that seed the observed structure of our universe can be studied. This thesis introduces the subject of CPT and discusses applications of this at both linear and second order. At linear order the evolution of the curvature perturbation around horizon crossing is examined. We study single field inflation models numerically, and compare the curvature perturbation at horizon crossing to that at the end of inflation. In addition, linear-order CPT is extended to the case of a multi-fluid system and an approximation for the velocities of the baryons and photons in the early universe as well as the strength of the electric field is found. We use second order CPT to study magnetogenesis. By using fully relativistic, non-linear CPT we show how magnetic fields are generated. This is done by presenting the first fully analytical calculation of the magnetic field at second order. Our results suggest that magnetic fields with strengths of the order of 10²⁷G and with scale dependence M ∝ k⁴ may be generated - findings which are largely in agreement with previous numerical results. We end by outlining possible extensions to this work, in particular related to the study of primordial magnetogenesis.