An investigation into structures of bi-integrable and tri-integrable couplings is undertaken. Our study is based on semi-direct sums of matrix Lie algebras. By introducing new classes of matrix loop Lie algebras, we form new Lax pairs and generate several new bi-integrable and tri-integrable couplings of soliton hierarchies through zero curvature equations. Moreover, we discuss properties of the resulting bi-integrable couplings, including infinitely many commuting symmetries and conserved densities. Their Hamiltonian structures are furnished by applying the variational identities associated with the presented matrix loop Lie algebras. The goal of this dissertation is to demonstrate the efficiency of our approach and discover rich structures of bi-integrable and tri-integrable couplings by manipulating matrix Lie algebras.

Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the “intersection” of these two categories is isomorphic to Frölicher spaces, another generalisation of smooth structures. We then give examples of such spaces, as well as examples of diffeological and differential spaces that do not fall into this category. We apply the theory of diffeological spaces to differential forms on a geometric quotient of a compact Lie group. We show that the subcomplex of basic forms is isomorphic to the complex of diffeological forms on the geometric quotient. We apply this to symplectic quotients coming from a regular value of the momentum map, and show that diffeological forms on this quotient are isomorphic as a complex to Sjamaar differential forms. We also compare diffeological forms to those on orbifolds, and show that they are isomorphic complexes as well. We apply the theory of differential spaces to subcartesian spaces equipped with families of vector fields. We use this theory to show that smooth stratified spaces form a full subcategory of subcartesian spaces equipped with families of vector fields. We give families of vector fields that induce the orbit-type stratifications induced by a Lie group action, as well as the orbit-type stratifications induced by a Hamiltonian group action.

This thesis comprises modeling and simulation of an AC-DC converter (Battery charger). An AC-DC converter may cause a high frequency distortion in the electrical power network or augment the existing distortion caused by other devices connected to the network. The goal is to design a controller for suppressing this noise at a reasonable level. We hope the thesis can be considered as a step forward to solve the original problem. One needs an accurate model of the AC-DC converter, to design such a controller. This study tries to clarify the effects of theline inductance on the performance of the converter by modeling and simulating the converter during the commutation time. The idea is to model and simulate the converter for two different conditions; first in the Normal condition by neglecting the effect of line impedance, second in the Commutation condition by considering the effect of the line impedance on commutation of the diodes. One can perform a complete simulation of the converter with combining these two models. The thesis deals with AC-DC converters, Hamiltonian-port modeling, simulation and MATLAB programming using the functionality of the S-function and SIMULINK.

In this thesis we consider mathematical problems related to different aspects of hard sphere systems. In the first part we study planar billiards, which arise in the context of hard sphere systems when only one or two spheres are present. In particular we investigate the possibility of elliptic periodic orbits in the general construction of hyperbolic billiards. We show that if non-absolutely focusing components are present there can be elliptic periodic orbits with arbitrarily long free paths. Furthermore, we show that smooth stadium like billiards have elliptic periodic orbits for a large range of separation distances. In the second part we consider hard sphere systems with a large number of particles, which we model by the Boltzmann equation. We develop a new approach to derive hydrodynamic limits, which is based on classical methods of geometric singular perturbation theory of ordinary differential equations. This method provides new geometric and dynamical interpretations of hydrodynamic limits, in particular, for the of the dissipative Boltzmann equation.

Covariance assignment theory, introduced in the late 1980s, provided the only means to directly control the steady-state error properties of a linear system subject to Gaussian white noise and parameter uncertainty. This theory, however, does not extend to control of the transient uncertainties and to date there exist no practical engineering solutions to the problem of directly and optimally controlling the uncertainty in a linear system from one Gaussian distribution to another. In this thesis I design a dual-mode Receding Horizon Controller (RHC) that takes a controllable, deterministic linear system from an arbitrary initial covariance to near a desired stationary covariance in finite time. The RHC solves a sequence of free-time Optimal Control Problems (OCP) that directly controls the fundamental solution matrices of the linear system; each problem is a right-invariant OCP on the matrix Lie group GLn of invertible matrices. A terminal constraint ensures that each OCP takes the system to the desired covariance. I show that, by reducing the Hamiltonian system of each OCP from T?GLn to gln? x GLn, the transversality condition corresponding to the terminal constraint simplifies the two-point Boundary Value Problem (BVP) to a single unknown in the initial or final value of the costate in gln?. These results are applied in the design of a dual-mode RHC. The first mode repeatedly solves the OCPs until the optimal time for the system to reach the de- sired covariance is less than the RHC update time. This triggers the second mode, which applies covariance assignment theory to stabilize the system near the desired covariance. The dual-mode controller is illustrated on a planar system. The BVPs are solved using an indirect shooting method that numerically integrates the fundamental solutions on R4 using an adaptive Runge-Kutta method. I contend that extension of the results of this thesis to higher-dimensional systems using either in- direct or direct methods will require numerical integrators that account for the Lie group structure. I conclude with some remarks on the possible extension of a classic result called Lie?s method of reduction to receding horizon control.

Thesis (Ph.D.) - University of Glasgow, 2008. / Ph.D. thesis submitted to the Faculty of Information and Mathematical Sciences, Department of Mathematics, 2008. Includes bibliographical references. Print version also available.

Thesis (Ph. D.)--Open University. BLDSC no. DX176663.

Thesis (Ph. D.)--Ohio University, June, 2004. / Includes bibliographical references (p. 167-171).

Thesis (Ph.D.)--Ohio University, June, 2004. / Title from PDF t.p. Includes bibliographical references (p. 167-171)

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.