The solution of optimal control problems through the Hamilton-Jacobi-Bellman (HJB) equation offers guaranteed satisfaction of both the necessary and sufficient conditions for optimality. However, finding an exact solution to the HJB equation is a near impossible task for many optimal control problems. This thesis presents an approximation method for solving finite-horizon optimal control problems involving nonlinear dynamical systems. The method uses finite-order approximations of the partial derivatives of the cost-to-go function, and successive higher-order differentiations of the HJB equation. Natural byproducts of the proposed method provide sensitivities of the controls to changes in the initial states, which can be used to approximate the solution to neighboring optimal control problems. For highly nonlinear problems, the method is modified to calculate control sensitivities about a nominal trajectory. In this framework, the method is shown to provide accurate control sensitivities at much lower orders of approximation. Several numerical examples are presented to illustrate both applications of the approximation method.

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.

This text is concerned with the Fredholm theory and stable approximation of bounded linear operators generated by a class of infinite matrices $(a_{ij})$ that are either banded or have certain decay properties as one goes away from the main diagonal. The operators are studied on $\ell^p$ spaces of functions $\Z^N\to X$, where $p\in[1,\infty]$, $N\in\N$ and $X$ is a complex Banach space. The latter means that our matrix entries $a_{ij}$ are indexed by multiindices $i,j\in\Z^N$ and that every $a_{ij}$ is itself a bounded linear operator on $X$. Our main focus lies on the case $p=\infty$, where new results are derived, and it is demonstrated in both general theory and concrete operator equations from mathematical physics how advantage can be taken of these new $p=\infty$ results in the general case $p\in[1,\infty]$.

Techn. Univ., Habil.-Schr.--Chemnitz, 2006.

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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The goal of this thesis research is to examine the Patty Ann Farms site using noninvasive techniques, such as a magnetic gradiometer. The Patty Ann Farms site, 12H1169, located in northeastern Hamilton County Indiana, is a multicomponent archaeological site spanning all periods of prehistory. Diagnostic artifacts from the Paleo-Indian, Archaic, and Woodland periods have been surface collected by the land owner. The land owner’s collection was documented, and the site was recorded at the Division of Historic Preservation and Archaeology--Department of Natural Resources, in 2004. Since then, a controlled surface survey has been conducted identifying three areas of high artifact density and preliminary soil phosphate tests have been conducted. / Department of Anthropology

In 1975, Sheehan conjectured that every simple 4-regular hamiltonian graph has a second Hamilton cycle. If Sheehan's Conjecture holds, then the result can be extended to all simple d-regular hamiltonian graphs with d at least 3. First, we survey some previous results which verify the existence of a second Hamilton cycle if d is large enough. We will then demonstrate some techniques for finding a second Hamilton cycle that will be used throughout this paper. Finally, we use these techniques and show that for certain 4-regular Hamiltonian graphs whose automorphism group is large enough, a second Hamilton cycle exists.

Charles Darwin proclaimed, “It is not the strongest of the species that survives, nor the most intelligent that survives. It is the one that is the most adaptable to change”. At the time, he probably did not fathom the relevance of his statement to the economy of cities. As the manufacturing sector dissipates, industrial cities strive to adapt by diversifying their local economy. This research provides a narrative of Hamilton’s industrial development and its transformation in search of a new identity. It examines the city’s economical, social and physical decay and its current urban regeneration that is based on the re-appropriation of its cultural landscape. This thesis argues that when cities focus only on the economic dimension of development, it may have an adverse influence on their inherent cultural identity which serves to undermine their ability to adapt and diversify. For Hamilton, a case in point is urban transformation of James Street North in a city that was recently subject to decades of neglect. James Street North has become the centre of a bourgeoning arts scene that is beginning to revitalize its neighbourhood. The thesis proposes the adaptive re-use of a deteriorated yet historically significant urban block within the area. The design intervention advocates an urban intensification intended to materialize a social and aesthetic identity derived from the urban agendas of Jane Jacobs, Charles Landry, and Sharon Zukin. The design synthesis proposes to establish a ‘creative milieu’ that becomes a catalyst for social cohesion, sustainable regeneration and an incubator for creativity. The design strategy consists of a hybrid building typology that is able to intensify diversity, exhibit creativity and engage dialogue among its occupants.