This dissertation is a study of second order, elliptic partial differential equations (PDE) that subject solutions to pointwise gradient constraints. These equations fall into the broad class of scalar non-linear PDE, and therefore, we interpret solutions in the viscosity sense and use methods from the theory of viscosity solutions. These equations are also naturally associated to free boundary problems as the boundary of the region where the gradient constraint is strictly satisfied cannot, in general, be determined before a solution of the PDE has been obtained. Consequently, we also employ techniques from PDE theory developed for free boundary problems.

In addition, we identify connections with control theory. Each solution of the PDE we consider has a probabilistic interpretation as an optimal value of a stochastic control problem. A distinguishing feature of these optimization problems is that the controlled processes have sample paths of bounded variation and thus may be ``singular" with respect to Lebesgue measure on the real line. The theory of stochastic singular control has been used to model spacecraft control, queueing systems, and financial markets in the presence of transaction costs. Our work makes considerable progress at rigorously interpreting the PDE that arise in these applications.

Viscosity Solution Methods in Risk Analysis

The ordinary differential equation for the motion of an inverted pendulum whose support is subjected to a vertical harmonic excitation with high frequency and small amplitude is studied. A mathematical insight is provided as asymptotic methods are employed to derive the limiting equation when the excitation frequency tends to infinity. These show that as the frequency becomes high and certain parametric conditions are satisfied, the inverted pendulum theoretically can be stabilized. A new and rigorous proof using weak convergence methods is given, to confirm the conclusions of the formal asymptotics. Then the analysis is generalized to a single inverted pendulum on an oscillatory base driven at other angles, a single inverted pendulum on a flywheel and a double inverted pendulum on a vertically oscillatory base and, in each case, if the pendulum or pendula can be stabilized, parametric conditions for stability are given.

Then we also use this method to shed light on the `Paul Trap' which is a method of levitating charged particles via an oscillating electric field. Last, PDE analogues are studied in the same manner.

In this thesis, we consider a class of optimal control problems known as coefficient control problems.Such problems are constrained by uniformly elliptic PDE in which the controls appear as some of the coefficients in the differential operator. We begin with a brief review of standard optimal control theory and show that it does not apply to control coefficient problems generally by means of two counterexamples. We then present existence results for the solution to such problems under two different assumptions regarding the controls: Lipschitz continuity with a bounded Lipschitz constant and the case in which the admissible set of controls is closed under H-convergence. We then present a new maximum principle for the latter class of problems which we subsequently use to characterize the nature and behavior of optimal solutions.

In this thesis, we explore some mathematical properties of the electro-chemical model for lithium-ion batteries, which is a nonlinear system of algebraically coupled diffusion equations. We prove a local existence and uniqueness theorem. Furthermore, we prove the asymptotic stability of a modified version, the so called Single Particle Model for several materials.

In this dissertation, I provide a fairly simple and direct proof of the asymptotics of the scaled Kramers-Smoluchowski equation in one dimension. I further generalize this result to the cases where the potential function H (1) has three or more wells, (2) is periodic, and (3) has infinitely many wells.

This paper explores the performance of signal timing plans calibrated with perfect or imperfect information. The arrival information considered include arrival rates and arrival distributions. The study is conducted for different levels of arrival rates and different forms of arrival distributions under a wide range of arrival information inaccuracy, traffic intensity, and intersections with balanced and unbalanced flows.

Our results indicate that the increase in delay, the number of stops, and queue length is in general insignificant when the arrival distribution used to calibrate the optimal timing plan is over- or under-estimated. For the same level of over- or under-estimation, the increase in delay and other measures can be high only when the flow level is very high. The effect, however, is attenuated for overestimation of flow since the cycle length is bounded by a predetermined upper bound. Overall, underestimation of flow appears to be more serious than overestimation of flow in terms of increase in total delay.