We first handle some generalizations from the theory of functions of a single complex variable, including results regarding analytic continuation. Several "theorems of continuity" are considered, along with the associated definitions of pseudoconvexity, and these are shown to be equivalent up to a special kind of transformation. By successively applying a form of analytic continuation to a function f , a set of pseudoconvex domains is constructed, and the union of these domains is shown to be the envelope of holomorphy of f . / Science, Faculty of / Mathematics, Department of / Graduate

Feedback is used to control systems whose open-loop behaviour is uncertain. Over the last twenty years a mature theory of robust control has been developed for linear multivariable systems in continuous time. But most practical control systems have constraints such as saturation limits on the actuators, which render the closed-loop nonlinear. Most of the modern controllers are also implemented digitally using computers.The study of this research is divided in two directions: the stability analysis of discrete-time Lur’e systems and the synthesis of static discrete-time anti-windup schemes. With respect to stability analysis, the main contributions of this thesis are the derivations of new LMI-based stability criteria for the discrete-time Lur’e systems with monotonic, slope-restricted nonlinearities via the Lyapunov method. The criteria provide convex stability conditions via LMIs, which can be efficiently computed via convex optimization methods. They are also extended to the general case that includes the non-diagonal MIMO nonlinearities. The importance of extending them to the general case is that it can eventually be applied to the stability analysis of several optimization-based controllers such as an input-constrainedmodel predictive control (MPC), which is inherently discrete. With respect to synthesis, the contribution is the convex formulation of a static discrete-time anti-windup scheme via one of the Jury-Lee criteria (a discrete-time counterpart of Popov criterion), which was previously conjectured to be unachievable. The result is also in the form of LMI, and is extended to several existing static anti-windup schemes with open-loop stable plants.

The purpose of this paper is to investigate some of the properties of convex sets in the plane through synthetic geometry.

This paper defines tropical projective space, TP^n, and the tropical general linear group TPGL(n). After discussing some simple examples of tropical polynomials and their hypersurfaces, a strategy is given for finding all conics in the tropical projective plane. The classification of conics and an analysis of the coefficient space corresponding to such conics is given.

A method that can be used to detect damage in structures is developed. This method is based on the convexity of the geometric interpretation of the eigenvalue problem for undamped positive definite systems. The damage detection scheme establishes various damage scenarios which are used as failure sets. These scenarios are then compared to the structure's actual response by measuring the natural frequencies of the structure and using a Euclideian norm. Mathematical models were developed for application of the method on a cantilever beam. Damage occurring at a single location or in multiple locations was estalished and studied. Experimental verification was performed on serval prismatic beams in which the method provided adequate results. The exact location and extent of damage for several cases was predicted. When the method failed the prediction was very close to the actual condition in the structure. This method is easy to use and does not require a rigorous amount of instrumentation for obtaining the experimental data required in the detection scheme. / Ph. D.

Optimal trajectory planning methods that implement convex optimization techniques are applied to the area of satellite rendezvous and proximity operations. This involves the development of linearized relative orbital motion dynamics and constraints for two satellites, where one maintains a near-circular reference orbit. The result is formulated as a convex optimization problem, where the objective is to minimize the amount of fuel required to transfer from a given initial condition to the desired final conditions. A traditional rendezvous and proximity operations scenario is analyzed, which includes examples of initial approach, inspection, final approach, and docking trajectories. This scenario may include trajectory constraints such as maximum allowable control acceleration levels, approach corridors, and spherical keep-out zones. A second scenario that ensures passive safety, in the event of control failures on the maneuvering satellite. The convex optimization problem is ultimately formulated as a second-order cone program. Algorithm CPU and memory requirements are analyzed. Several examples of resulting optimal trajectories are presented for both scenarios, and these trajectories are implemented in a nonlinear simulation.

No description available.

<p> The closed graph and open mapping theorems are two of the deeper results in the theory of locally convex spaces. They are very rich in their applications in functional analysis. This thesis contains some extensions of these theorems in locally convex spaces. We begin with a study of α-spaces, and γ-spaces, which leads us naturally to a study of δ-spaces. On these spaces, we prove closed graph and open mapping theorems. Similar theorems are also proved for certain classes of Br ('&)-spaces. In particular, a closed graph theorem for B(m)-spaces enables us to characterise certain classes of B( &. y)-spaces. A consideration of countability conditions in locally convex spaces enables us to prove open mapping theorems in Br (J )-spaces. These theorems are then used to relate boundedness of linear mappings and their graphs. </p> / Thesis / Doctor of Philosophy (PhD)

<p> This thesis addresses itself to one of the most general theoretical problems associated with the art of engineering design. Viewed in its entirety the proposed approach integrates the relation between the design and production engineers through the theory of nonlinear optimization. The conventional optimization problem is extended to include the optimal allocation of the upper and lower limits of the random variables of an engineering system. The approach is illustrated by an example using a sequence of increasingly generalized formulations, while the general mathematical theory is also provided. The method appears to offer a practical technique provided a satisfactory cost function can be defined.</p> <p> The thesis presents an analytical approach to full acceptability design conditions as well as less than full acceptability or scrap design conditions. An important distinction between the design and the manufacturing scrap has been introduced and illustrated through examples.</p> <p> The space regionalization technique is utilized to estimate the system design scrap. Optimization strategies are introduced to the mathematically defined upper and lower limits of the regionalization region. This region is then discretized into a number of cells depending upon the probabilistic characteristic of the system random variables.</p> <p> The analytical approach exhibited does not rely explicitly on evaluation of partial derivatives of either the system cost objective or any of its constraints at any point. Moreover, the technique could be applied to engineering systems with either convex or nonconvex feasible regions. It could also be exercised irrespective of the shape of the probabilistic distributions that describe the random variables variation.</p> <p> Industrially oriented design examples are furnished to justify the applicability of the theory in different engineering disciplines.</p> / Thesis / Doctor of Philosophy (PhD)

No description available.