Thesis (M.S.)--Virginia Polytechnic Institute and State University, 1988. / Vita. Abstract. Includes bibliographical references (leaves 154-155). Also available via the Internet.

Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2009. / Committee Chair: Yi, Yingfei; Committee Member: Chow, Shui-Nee; Committee Member: Dieci, Luca; Committee Member: Verriest, Erik; Committee Member: Weiss, Howie. Part of the SMARTech Electronic Thesis and Dissertation Collection.

In this thesis we present research into linear perturbations in Lemaître-Tolman-Bondi (LTB) and Assisted Coupled Quintessence (ACQ) Cosmologies. First we give a brief overview of the standard model of cosmology. We then introduce Cosmological Perturbation Theory (CPT) at linear order for a at Friedmann-Robertson-Walker (FRW) cosmology. Next we study linear perturbations to a Lemaître-Tolman-Bondi (LTB) background spacetime. Studying the transformation behaviour of the perturbations under gauge transformations, we construct gauge invariant quantities in LTB. We show, using the perturbed energy conservation equation, that there is a conserved quantitiy in LTB which is conserved on all scales. We then briefly extend our discussion to the Lemaître spacetime, and construct gauge-invariant perturbations in this extension of LTB spacetime. We also study the behaviour of linear perturbations in assisted coupled quintessence models in a FRW background. We provide the full set of governing equations for this class of models, and solve the system numerically. The code written for this purpose is then used to evolve growth functions for various models and parameter values, and we compare these both to the standard CDM model and to current and future observational bounds. We also examine the applicability of the "small scale approximation", often used to calculate growth functions in quintessence models, in light of upcoming experiments such as SKA and Euclid. We nd the results of the full equations deviates from the approximation by more than the experimental uncertainty for these future surveys. The construction of the numerical code, Pyessence, written in Python to solve the system of background and perturbed evolution equations for assisted coupled quintessence, is also discussed.

Film blowing is a highly complex industrial process used to manufacture thin plastic films for uses in a wide range of applications; for example, plastic bags. The mathematical modelling of this process involves the analysis of highly nonlinear differential equations describing the complex phenomena arising in the film blowing process, and requires a sophisticated mathematical approach. This dissertation applies an innovative combination of tools, namely analytic, numerical and heuristic mathematical techniques to the analysis of the film blowing process. The research undertaken examines, in particular, a two-point boundary value problem arising from the modelling of the radial profile of the polymer film. For even the simplest modelling of this process, namely the isothermal Newtonian model, the resulting differential equation is a highly nonlinear, second order one, with an extra degree of difficulty due to the presence of a small parameter multiplying the highest derivative. Thus, the problem falls into the category of a nonlinear singular perturbation problem. Analytic techniques are applied to the isothermal Newtonian blown film model to obtain a closed form explicit approximation to the film bubble radius. This is then used as a base approximation for an iterative numerical scheme to obtain an improved numerical solution of the problem. The process is extended to include temperature variations, varying viscosity (Power law model) and viscoelastic effects (Maxwell model). As before, closed form approximations are constructed for these models which are used to launch numerical schemes, whose solutions display good accuracy. The results compare well with results obtained by purely numerical solutions in the literature.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1997. / Includes bibliographical references. Also available on the Internet.

A novel design methodology of a decentralized adaptive sliding mode control scheme for a class of large-scale systems with mismatched disturbances and uncertainties in each subsystem and interconnections is proposed in this thesis. The main idea of this new method is that the design of the switching surface of each subsystem is through the design of a pseudo-feedback controller which can stabilize the dynamics when system is in the sliding mode. The feedback gain of the pseudo controller then becomes a important parameter of switching surface. The proposed controllers of each subsystem contain three parts. The first part is measurable feedback signals, and the second part is an adaptive control mechanism, which is used for overcoming the disturbances and uncertainties of each subsystem and interconnections among subsystems. The information of upper bound of those disturbances and uncertainties are not required. The third part of the decentralized controllers is used for adjusting the convergent rate of state variables of the controlled system. The asymptotical stability is guaranteed for each subsystem even if the mismatched perturbations exist when employing the proposed control scheme. An example is demonstrated for showing the feasibility of the proposed methodology.

Based on the Lyapunov stability theorem, a decentralized adaptive sliding mode control scheme is proposed in this thesis for a class of mismatched perturbed large-scale systems containing dead-zone input to solve regulation problems. The main idea is that some adaptive mechanisms are embedded both in the sliding surface and in the controllers, so that not only the mismatched perturbations are suppressed during the sliding mode, but also the information of upper bound of perturbations is not required. The sliding surface function is firstly designed through the usage of a pseudo controller which is capable of stabilizing the reduced-order systems. The second step is to design the controllers so that the trajectories of the controlled systems are able to reach sliding surface in a finite time. Once the controlled system enters the sliding mode, the asymptotical stability is guaranteed for each subsystem even the mismatched perturbations exist. A numerical example and a practical example are given to demonstrate the feasibility of the proposed design technique.

In this thesis the synchronization of two different chaotic systems with matched and mismatched perturbations are developed by utilizing adaptive backstepping control technique. The adaptive mechanisms embeded in the proposed control scheme is used to adapt the unknown upper bounds of the perturbations. The resultant robust backstepping tracking controller with adaptive mechanisms can indeed drive the trajectories of the slave system to track those of the master system. Two numerical examples and simulations are given to illustrate the correctness of theoretical analyses.

Based on the Lyapunov stability theorem, an adaptive sliding mode control scheme is proposed in this thesis for a class of systems with mismatched state-delayed perturbations to solve regulation problems. The main idea is that some adaptive mechanisms are embedded both in the sliding surfaces and in the controllers, so that not only the mismatched perturbations are suppressed during the sliding mode, but also the information of upper bound of perturbations is not required. The sliding surface functions are firstly designed through the usage of designed pseudo controllers, which is capable of stabilizing the reduced-order systems. The number of the sliding surface functions required by the proposed control scheme depends on the relationship between systems's dimension and number of inputs. The second step is to design the controllers so that the trajectories of the controlled system are able to reach sliding surface in a finite time. Once the controlled system enters the sliding mode, the asymptotical stability is guaranteed. Two numerical examples and one practical experiment are given for demonstrating the feasibility of the proposed control scheme.

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