Contributions to the study of a class of optimal control problems on the orthogonal groups SO(3) and SO(4)

In this thesis we investigate a class of invariant optimal control problems, and their associated quadratic Hamilton-Poisson systems, on the orthogonal groups SO(3) and SO(4). Specifically, we are concerned with the class of left-invariant control affine systems. We begin by classifying all cost-extended systems on SO(3) under cost equivalence. (Cost-extended systems are closely related to optimal control problems.) A classification of all quadratic Hamilton-Poisson systems on the (minus) Lie-Poisson space so(3)*, under affine equivalence, is also obtained. For the normal forms obtained in our classification (of Hamilton-Poisson systems) we investigate the (Lyapunov) stability nature of the equilibria using spectral and energy-Casimir methods. For a subclass of these systems, we obtain analytic expressions for the integral curves of the associated Hamiltonian vector fields in terms of (basic) Jacobi elliptic functions. The explicit relationship between the classification of cost-extended systems on SO(3) and the classification of quadratic Hamilton- Poisson systems on so(3)* is provided. On SO(4), a classification of all left-invariant control affine systems under L-equivalence is obtained. We then determine which of these representatives are controllable, thus obtaining a classification under detached feedback equivalence. We also obtain a partial classification of quadratic Hamilton-Poisson systems on the Lie-Poisson space so(4)*. An investigation of the stability nature of the equilibria for a subclass of these systems is also done. Several illustrative examples of optimal control problems on the orthogonal group SO(3) are provided. More specifically, we consider an optimal control problem corresponding to a representative of our classification (of cost-extended system) for each possible number of control inputs. For each of these problems, we obtain explicit expressions for the extremal trajectories on the homogeneous space S2 by projecting the extremal trajectories on the group SO(3). The examples provided show how our classifications of cost-extended systems and Hamilton-Poisson systems can be used to obtain the optimal controls and the extremal trajectories corresponding to a large class of optimal control problems on SO(3). An example of a four-input optimal control problem on SO(4) is also provided. This example is provided to show how the solutions of certain problems on SO(4) can be related to the solutions of certain optimal control problems on SO(3)

Whose Vote Counts? Minority Vote Dilution and Election Rights

What is the value of one man\u27s vote? While the obvious answer is one man, one vote, some votes are worth less. Over the last one hundred years, minorities have battled to receive equal rights and opportunities

Leaving College: Why Students Withdrew from a University

The purpose of this study was to determine the reasons why students withdrew during a semester from a mid-sized, comprehensive university located in the Midwest. Six hundred forty-five students were asked to complete the ACT Withdrawing/Non-returning Student Survey during the 1992-93 academic year and summer semester. Three hundred sixty-five completed surveys were returned for a 57% response rate. Respondents indicated many different reasons for leaving which varied by year in school and whether or not the respondent was a graduate or undergraduate student. There was no typical withdrawing student and there were many reasons students withdrew over which the university has little or no control. The report concludes with a discussion of Vincent Tinto\u27s (1993) ideas concerning institutional departure. The retention and persistence of students in higher education has been the focus of serious intellectual inquiry for many years. Various concepts of institutional departure, persistence and models for programmatic interventions to reduce departure have been developed. (For example, see Pascarella & Terenzini, 1991; Stage & Rushin, 1993; Steele, Kennedy, & Gordon, 1993; Tinto, 1993; Wolfe, 1993.) The purpose of this study was to focus on one aspect of student attrition, and. to investigate the reasons and general trends as . to why students withdrew during a semester from a midsized comprehensive university located in the Midwest. This information could then be used to guide institutional action

Isomorphic Substitution in Vanadinite [Pb5(VO4)3C1] - Raman Spectroscopic Study

The Raman and infrared spectroscopy of three vanadinite [Pb5(VO4)Cl] specimens from three localities has been studied. Single crystal Raman spectra were obtained and the spectra were found to be both orientation and temperature dependent. Measurement of the Raman spectra at liquid nitrogen temperature enabled better band separation and increased intensities of weak bands through band narrowing enabling, the assignment of the bands in vanadinites to be made. Two types of isomorphous substitution are identified (a) substitution in the bulk of the crystal where electron beam microanalysis identifies the presence of calcium and copper. (b) surface substitution where infrared spectroscopy shows the isomorphous substitution of vanadate by phosphate and of chloride by hydroxyl groups

Paraoxonase gene polymorphisms and haplotype analysis in a stroke population

Peer reviewedPublisher PD