The Bolza-form of the finite-time constrained optimal control problem leads to the Hamilton-Jacobi-Bellman (HJB) equation with terminal boundary conditions and tobe- determined parameters. In general, it is a formidable task to obtain analytical and/or numerical solutions to the HJB equation. This dissertation presents two novel polynomial expansion methodologies for solving optimal feedback control problems for a class of polynomial nonlinear dynamical systems with terminal constraints. The first approach uses the concept of higher-order series expansion methods. Specifically, the Series Solution Method (SSM) utilizes a polynomial series expansion of the cost-to-go function with time-dependent coefficient gains that operate on the state variables and constraint Lagrange multipliers. A significant accomplishment of the dissertation is that the new approach allows for a systematic procedure to generate optimal feedback control laws that exactly satisfy various types of nonlinear terminal constraints. The second approach, based on modified Galerkin techniques for the solution of terminally constrained optimal control problems, is also developed in this dissertation. Depending on the time-interval, nonlinearity of the system, and the terminal constraints, the accuracy and the domain of convergence of the algorithm can be related to the order of truncation of the functional form of the optimal cost function. In order to limit the order of the expansion and still retain improved midcourse performance, a waypoint scheme is developed. The waypoint scheme has the dual advantages of reducing computational efforts and gain-storage requirements. This is especially true for autonomous systems. To illustrate the theoretical developments, several aerospace application-oriented examples are presented, including a minimum-fuel orbit transfer problem. Finally, the series solution method is applied to the solution of a class of partial differential equations that arise in robust control and differential games. Generally, these problems lead to the Hamilton-Jacobi-Isaacs (HJI) equation. A method is presented that allows this partial differential equation to be solved using the structured series solution approach. A detailed investigation, with several numerical examples, is presented on the Nash and Pareto-optimal nonlinear feedback solutions with a general terminal payoff. Other significant applications are also discussed for one-dimensional problems with control inequality constraints and parametric optimization.

In this dissertation we prove the homogenization for two very different classes of nonlinear partial differential equations and nonlinear elliptic integro-differential equations. The first result covers the homogenization of convex and superlinear Hamilton-Jacobi equations with stationary ergodic dependence in time and space simultaneously. This corresponds to equations of the form: [mathematical equation]. The second class of equations is nonlinear integro-differential equations with periodic coefficients in space. These equations take the form, [mathematical equation]. / text

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Thesis (Ph.D.)--York University, 2006. Graduate Programme in Applied Mathematics. / Typescript. Includes bibliographical references (leaves 134-142). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:NR19766

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

Thesis (M.S.)--Brigham Young University. Dept. of Mathematics, 2007. / Includes bibliographical references (p. 48).

Thesis (Ph. D.)--University of Texas at Austin, 2009. / Title from PDF title page (University of Texas Digital Repository, viewed on Sept. 9, 2009). Vita. Includes bibliographical references.

Thesis (Ph.D.)--Brown University, 2005. / Vita. Thesis advisor: Chi-Wang Shu. Includes bibliographical references (leaves 83-87). Also available online.

Th.--Sci. math.--Besançon, 1984. N°: 183.

Made available in DSpace on 2015-12-10T14:23:02Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-03-07. Added 1 bitstream(s) on 2015-12-10T14:27:52Z : No. of bitstreams: 1 000852795.pdf: 576204 bytes, checksum: 28ede436e9367885bc3b672b1903caad (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / A teoria de Hamilton-Jacobi geralmente é apresentada como uma extensão da teoria de Hamilton através das transformações canônicas. No entanto, o matemático Constantin Carathéodory mostrou que essa teoria, sua existência e validade, independem do formalismo hamiltoniano. Neste trabalho, apresentaremos a abordagem de Carathéodory para a teoria de Hamilton-Jacobi. Partindo desse procedimento, construiremos uma teoria de vínculos para que se possa resolver problemas com vínculos involutivos e não-involutivos. Para isso, analisaremos a integrabilidade das equações e introduziremos a operação dos parênteses generalizados que, no lugar do parênteses de Poisson, passará a descrever a dinâmica de sistemas vinculados. Mostraremos uma aplicação dessa teoria de vínculos no modelo BF da teoria de campos. Para finalizar, trataremos da Termodinâmica Axiomática de Carathéodory e também da teoria de Hamilton-Jacobi na Termodinâmica, o que é válido para ilustrar a grande abrangência desse formalismo / The Hamilton-Jacobi theory is usually presented as an extension of the Hamilton's theory through the canonical transformations. However, the mathematician Constantin Carathéodory showed this theory, its existence and validity, is independent of the Hamiltonian formalism. In this work, we present the Caratheodory's approach to the Hamilton-Jacobi theory. From this procedure, we build a theory of constraints which can solve problems with involutive and non-involutive constraints. For this, we analyze the integrability of the equations and introduce the operation of the generalized brackets that, instead of Poisson brackets, will describe the dynamics of constrained systems. We show an application of this theory in BF model of the field theory. Finally, we will discuss the Carathéodory's Axiomatic Thermodynamics and also show the Hamilton-Jacobi theory in Thermodynamics, which is valid to illustrate the wide coverage of this formalism / CNPq: 133488/2011-0