Nonlinear theories

Recently a rich theory of Sobolev spaces on metric spaces has been developed. It. turas
out that many relevant results from the classical theory have their counterparts in the
mcnic setting ( cf. [P. Hajlasz and P. Koskela. Sobokv met Poincare), Mern. Arner. Math.
Soc. 145 (2000), no. 6888, x+101pp]). In this thesis we prove sharp Sobolev inequalities
in the context of metric spaces. Our approach is ba....,ed on two recent papers, [J. Baster·o
and M. Milman and F. Ruiz, A note on L(oc, q) spaces and Sobolev embeddings, Indiana
Univ. Math. J. 52 (2003), no. 5, 1215- 1230] and [J. Martfn and M. Milman and E.
Pustylnik, Sobolev inequalities: symmetrization and self improvement via truncation, to
appear in J. Funct. Anal.]. These authors establish sharp Sobolev embeddings in the
Euclidean setting using symmetrization. Using suitable maximal operators and covering
lemmas we show that these symmetrization inequalities of Bastero-Milman-Ruiz remain
valid m the metric setting. We also show that the symmetrization by truncation method of
Martfn-Milman-Pustylnik can be implemented in our generalized setting. Furthermore we
also show that our methods can be adapted to deal with non-doubling measures.

Singularly perturbed systems with or without delays commonly appear in mathematical modeling of physical and chemical processes, engineering applications, and increasingly, in mathematical biology. There has been intensive work for singularly
perturbed systems, yet most of the work so far focused on systems without
delays. In this thesis, we provide a new set of tools for the stability analysis for
singularly perturbed control systems with time delays.

This thesis discusses the coupling of a mechanical and electrical oscillator, an arrangement that is often encountered in mechatronics actuators and sensors. The dynamics of this coupled system is mathematically modeled and a low pass equivalent model is presented. Numerical simulations are then performed, for various input signals to characterize the nonlinear relationship between the electrical current and the displacement of the mass. Lastly a framework is proposed to estimate the mass position without the use of a position sensor, enabling the sensorless control of the coupled system and additionally providing the ability for the system to act as an actuator or a sensor. This is of value for health monitoring, diagnostics and prognostics, actuation and power transfer of a number of interconnected machines that have more than one electrical system, driving corresponding mechanical subsystems while being driven by the same voltage source and at the same time being spectrally separated and independent.

System identification methods are frequently used to obtain appropriate models for the purpose of control, fault detection, pattern recognition, prediction, adaptive filtering and other purposes. A number of techniques exist for the identification of linear systems. However, real-world and complex systems are often nonlinear and there exists no generic methodology for the identification of nonlinear systems with unknown structure. A recent approach makes use of highly interconnected networks of simple processing elements, which can be programmed to approximate nonlinear functions to identify nonlinear dynamic systems. This thesis takes a detailed look at identification of nonlinear systems with neural networks. Important questions in the application of neural networks for nonlinear systems are identified; concerning the excitation properties of input signals, selection of an appropriate neural network structure, estimation of the neural network weights, and the validation of the identified model. These questions are subsequently answered. This investigation leads to a systematic procedure for identification using neural networks and this procedure is clearly illustrated by modeling a complex nonlinear system; the components of the space shuttle main engine. Additionally, the neural network weights are determined by using a general purpose optimization technique known as evolutionary programming which is based on the concept of simulated evolution. The evolutionary programming algorithm is modified to include self-adapting step sizes. The effectiveness of the evolutionary programming algorithm as a general purpose optimization algorithm is illustrated on a test suite of problems including function optimization, neural network weight optimization, optimal control system synthesis and reinforcement learning control.

This thesis is concerned with nonlinear dynamical systems subject to random or combined random and deterministic excitations. To this end, a systematic procedure is first developed to obtain the exact stationary probability density for the response of a nonlinear system under both additive and multiplicative excitations of Gaussian white noises. This procedure is applicable to a class of systems called the class of generalized stationary potential. The basic idea is to separate the circulatory probability flow from the noncirculatory flow, thus obtaining two sets of equations for the probability potential. It is shown that previously published exact solutions are special cases of this class. For those nonlinear systems not belonging to the class of generalized stationary potential, an approximate solution technique is developed on the basis of weighted residuals. The original system is replaced by the closest system belonging to the class of generalized stationary potential, in the sense that the statistically weighted residuals are zero for some suitably selected weighting functions. The consistency of the approximation technique is proved in terms of certain statistical moments. The above exact and approximate solution techniques are extended to two types of nonlinear systems: one subjected to non-Gaussian impulsive noise excitations and another subjected to combined harmonic and broad-band random excitations. Approximation procedures are devised to obtain stationary probabilistic solutions for these two types of problems. Monte Carlo simulations are performed to substantiate the accuracy of the approximate solution procedures.

The synchronization of coupled semiconductor lasers with delay is investigated by numerical simulations of the nonlinear dynamic models complemented by a stability analysis of the linearized system. The equations used in the dissertation are based on the well known "Lang-Kobayashi" model modified to include unidirectional and bidirectional coupling. Stability diagrams are calculated and supplemented by numerically integrated time series. Synchronization is determined and quantified by computing the cross-correlation function. It is found that synchronized states are achievable for a wide range of coupling constants and delay times. These findings have implications for experiment and technological applications, notably in cryptography.

Sustained resonance in a linear oscillator is achievable with a drive whose constant frequency matches the resonant frequency of the oscillator. In oscillators with nonlinear restoring forces, i.e., Dung-type oscillators, resonant frequency changes with amplitude, so a constant frequency drive generates a beat oscillation instead of sustained resonance. Dung-type oscillators can be driven into sustained resonance, called autoresonance (AR), when drive frequency is swept in time to match the changing resonant frequency of the oscillator. It is found that near-optimal drive linear sweep rates for autoresonance can be estimated from the beat oscillation resulting from constant frequency excitation. Specically, a least squares estimate of the slope of the Teager-Kaiser instantaneous frequency versus time plot for the rising half-cycle of the beat response to a stationary drive provides a near-optimal estimate of the linear drive sweep rate that sustains resonance in the pendulum, Dung and Dung-Van der Pol oscillators. These predictions are confirmed with model-based numerical simulations. A closed-form approximation to the AM-FM nonlinear resonance beat response of a Dung oscillator driven at its low-amplitude oscillator frequency is obtained from a solution to an associated Mathieu equation. AR time responses are found to evolve along a Mathieu equation primary resonance stability boundary. AR breakdown occurs at sweep rates just past optimal and map to a single stable point just off the Mathieu equation primary resonance stability boundary. Optimal AR sweep rates produce oscillating phase dierences with extrema near 90 degrees, allowing extended time in resonance. AR breakdown occurs when phase difference equals 180 degrees. Nonlinear resonance of the van der Pol type may play a role in the extraordinary sensitivity of the human ear.

Entrepreneurship occupies a curious place in economic theory. On one hand, the importance of entrepreneurship is widely recognized, particularly as it pertains to economic growth. However, the entrepreneur lacks a broadly accepted economic theory, and suffers from a dearth of literature on the subject. We believe that this is due to economics' heavy reliance on linear mathematical theory. In this thesis, we use nonlinear mathematics to construct a model of the entrepreneur that captures the sudden destabilization of a steady state, the unpredictability of a creative action, the possibility of entrepreneurial failure, and sensitivity to small changes in environment.