Thesis (M. Phil.)--University of Hong Kong, 1998. / Includes bibliographical references (leaves 89-91).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1968. / Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 221-257).

In the classical calculus of variations, the Hamilton - Jacobi theory leads, under general hypotheses, to sufficient conditions for a local minimum. The optimal control problem as well has its own Hamilton -Jacobi approach to sufficient conditions for optimality. In this thesis we extend this approach to the differential inclusion problem; a general, nonconvex, nondifferentiable control problem. In particular, the familiar Hamilton - Jacobi equation is generalized and a corresponding necessary condition (chapter 2) is obtained. The sufficiency condition (chapter 3) is derived and an example is presented where it is shown how this result may lead to considerable simplification. Finally, we show (chapter 4) how the classical theory of canonical transformations may be brought to bear on certain Hamiltonian inclusions associated with the differential inclusion problem. Our main tool will be the generalized gradient, a set valued derivative for Lipschitz functions which reduces to the subdifferential of convex analysis in the convex case and the familiar derivative in the C¹ case. / Science, Faculty of / Mathematics, Department of / Graduate

by Yung Siu-Pang. / Includes bibliographical references / Thesis (M.Ph.)--Chinese University of Hong Kong, 1985

Tai Chi-hung. / Thesis (M.Phil.)--Chinese University of Hong Kong. / Bibliography: leaves 40-41.

In this thesis, we consider the following two important subjects in the modern variational analysis for the corresponding nonconvex/nonmonotone and nonsmooth cases: geometric results and the variational inequality problem. By using the variational technique, we first present several nonsmooth (nonconvex) geometric results (including an approximate projection result, an extended extremal principle, nonconvex separation theorems, a nonconvex generalization of the Bishop-Phelps theorem and a separable point result) which extend some fundamental theorems in linear functional analysis, convex analysis and optimization theory. Then, by transforming the variational inequality problem into equivalent optimization problems, we establish some error bound result for the nonsmooth and nonmonotone variational inequality problem. / Variational arguments are classical techniques whose use can be traced back to the early development of the calculus of variations. Rooted in the physical principle of least action they have evolved greatly in connection with applications in optimization theory and optimal control. Recently, the discovery of modern variational principles and nonsmooth analysis further expand the range of applications of these techniques and give a new way for extending some geometric results in linear functional analysis and convex analysis. / Li, Guoyin. / "August 2007." / Adviser: Kung-Fu Ng. / Source: Dissertation Abstracts International, Volume: 69-02, Section: B, page: 1043. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 80-86). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract in English and Chinese. / School code: 1307.

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Previous work on modal analysis of mooring lines has been performed from different theoretical formulations. Most studies have focused on mooring lines of a single homogeneous material, and the effect of added mass and damping produced by the water has not been examined deeply. The variational formulation approach, employed in this research to perform a modal analysis, has been useful to study the behavior of several realistic mooring lines. The cases presented are composed from segments of materials with different mechanical characteristics, more similar to those in current offshore projects. In the newly proposed formulation, damping produced by transverse motion of the mooring line through the surrounding water has been added to the modal analysis. The modal analysis formulation applied in this work has been verified with calculations from commercial software and the results are sufficiently accurate to understand the global behavior of the dynamics of mooring lines with the damping produced by the sea water. Inclusion of linearized drag damping in the modal analysis showed that the modal periods of the mooring systems studied depend on the amplitude of the transverse motion of the mooring line. When more amplitude in the motion is expected more damping is obtained. Two realistic designs of mooring lines were compared: one made up with a main insert of steel rope, called “Steel System”, and one composed by a main insert of polyester, named “Polyester System”. Comparing the natural periods of both systems, the Steel System appears to be safer because its fundamental natural period is more distant from the wave excitation periods produced by storms. The same happens considering the wave excitation periods produced by prevailing seas. In this case the natural periods of the Polyester System are nearer to the wave excitation periods causing fatigue loads. The transverse mode shapes for lateral motions of the mooring lines are observed to be continuous and smooth across material transitions, such as transitions between chain and wire rope and transitions between chain and polyester rope. This behavior is not always observed in the tangential mode shapes for the Polyester System where significant differences in dynamic tension seem to be present in the specific cases studied.