Physical Sciences and Mathematics Commons^™

On A Multi-Dimensional Singular Stochastic Control Problem: The Parabolic Case, Nhat Do Minh Nguyen

Wayne State University Dissertations

This dissertation considers a stochastic dynamic system which is governed by a multidimensional diffusion process with time dependent coefficients. The control acts additively on the state of the system. The objective is to minimize the expected cumulative cost associated with the position of the system and the amount of control exerted. It is proved that Hamilton-Jacobi-Bellman’s equation of the problem has a solution, which corresponds to the optimal cost of the problem. We also investigate the smoothness of the free boundary arising from the problem.

In the second part of the dissertation, we study the backward parabolic problem for a …

New Characterizations Of Sobolev Spaces On Heisenberg And Carnot Groups And High Order Sobolev Spaces On Eucliean Spaces, Xiaoyue Cui

Wayne State University Dissertations

This dissertation focuses on new characterizations of Sobolev spaces .

It encompasses an in-depth study of Sobolev spaces on Heisenberg groups, as well as Carnot groups, second order and high order Sobolev spaces on Euclidean spaces.

Recovery Techniques For Finite Element Methods And Their Applications, Hailong Guo

Wayne State University Dissertations

Recovery techniques are important post-processing methods to obtain improved approximate solutions from primary data with reasonable cost. The practical us- age of recovery techniques is not only to improve the quality of approximation, but also to provide an asymptotically exact posteriori error estimators for adaptive meth- ods. This dissertation presents recovery techniques for nonconforming finite element methods and high order derivative as well as applications of gradient recovery.

Our first target is to develop a systematic gradient recovery technique for Crouzeix- Raviart element. The proposed method uses finite element solution to build a better approximation of the exact gradient based …

Well-Posedness Properties In Variational Analysis And Its Applications, Wei Ouyang

Wayne State University Dissertations

This dissertation focuses on the study and applications of some significant properties in well-posedness and sensitivity analysis, among which the notions of uniform metric regularity , higher-order metric subregularity and its strong subregularity counterpart play an essential role in modern variational analysis. We derived verifiable sufficient conditions and necessary conditions for those notions in terms of appropriate generalized differential as well as geometric constructions of variational analysis. Concrete examples are provided to illustrate the behavior and compare the results. Optimality conditions of parametric variational systems (PVS) under equilibrium constraints are also investigated via the terms of coderivatives. We derived necessary …

Finite-Difference Approximations And Optimal Control Of Differential Inclusions, Yuan Tian

Wayne State University Dissertations

This dissertation concerns the study of the generalized Bolza type problem for dynamic systems governed by constrained differential inclusions. We develop finite-discrete approximations of differential inclusions by using the implicit Euler scheme and the Runge-Kutta scheme for approximating time derivatives, while an appropriate well-posedness of such approximations is justified. Our principal result establishes the uniform approximation of strong local minimizers for the continuous-time Bolza problem by optimal solutions to the corresponding discretized finite-difference systems by the strengthen $W^{1,2}$-norm approximation of this type in the case ``intermediate" (between strong and weak minimizers) local minimizers under additional assumptions. Especially the implicitly discrete …