Digital Commons Network^™

Variational Analysis And Optimal Control Of The Sweeping Process, Hoang Dinh Nguyen

Wayne State University Dissertations

We formulate and study an optimal control problem for the sweeping(Moreau) process, where control functions enter the moving sweeping

set. To the best of our knowledge, this is the first study in the literature devoted to optimal control of the sweeping process. We first establish an existence theorem of optimal solutions and then derive necessary optimality conditions for this optimal control problem of a new type, where the dynamics is governed by discontinuous differential inclusions with variable right-hand sides. Our approach to necessary optimality conditions is based on the method of discrete approximations and advanced tools of variational analysis and …

Moduli Spaces And Cw Structures Arising From Morse Theory, Lizhen Qin

Wayne State University Dissertations

In this dissertation, we study the moduli spaces and CW Structures arising from Morse theory.

Suppose M is a smooth manifold and f is a Morse function on it. We consider the negative gradient flow of f. Suppose the flow satisfies transversality. This naturally defines the moduli spaces of flow lines and gives a stratication of M by its unstable manifolds. The gluing of broken flow lines can also be constructed.

We prove that, under certain assumptions, these moduli spaces can be compactified and the compactified spaces are smooth manifolds with corners. Moreover, these compactified manifolds satisfy certain orientation formulas. …

Spectral Collocation Method For Compact Integral Operators, Can Huang

Wayne State University Dissertations

We propose and analyze a spectral collocation method for integral

equations with compact kernels, e.g. piecewise smooth kernels and

weakly singular kernels of the form $\frac{1}{|t-s|^\mu}, \;

0<\mu<1. $ We prove that 1) for integral equations, the convergence

rate depends on the smoothness of true solutions $y(t)$. If $y(t)$

satisfies condition (R): $\|y^{(k)}\|_{L^\infty[0,T]}\leq

ck!R^{-k}$}, we obtain a geometric rate of convergence; if $y(t)$

satisfies condition (M): $\|y^{(k)}\|_{L^{\infty}[0,T]}\leq cM^k $,

we obtain supergeometric rate of convergence for both Volterra

equations and Fredholm equations and related integro differential

equations; 2) for eigenvalue problems, the convergence rate depends

on the smoothness of eigenfunctions. The same convergence rate for

the largest modulus eigenvalue approximation …

Genus 0, 1, 2 Actions Of Some Almost Simple Groups Of Lie Rank 2, Xianfen Kong

Wayne State University Dissertations

Please see the paper.

Thanks.

Numerical Methods For Problems Arising In Risk Management And Insurance, Zhuo Jin

Wayne State University Dissertations

In this dissertation we investigate numerical methods for problems annuity purchasing and dividend optimization arising in risk management and insurance. We consider the models with Markov regime-switching process. The regime-switching model contains both continuous and discrete components in their evolution and is referred to as a hybrid system. The discrete events are used to model the random factors that cannot formulated by differential equations. The switching process between regimes is modulated as a finite state Markov chain.

As is widely recognized, this regime-switching model appears to be more versatile and more realistic. However, because of the regime switching and the …

Asymptotic Expansions And Stability Of Hybrid Systems With Two-Time Scales, Dung Tien Nguyen

Wayne State University Dissertations

In this dissertation, we consider solutions of hybrid systems in which both continuous dynamics and discrete events coexists. One

of the main ingredients of our models is the two-time-scale formulation. Under broad conditions, asymptotic expansions are developed for the solutions of the systems of backward equations for switching diffusion in two classes of models, namely, fast switching systems and fast diffusion systems. To prove the validity of the asymptotic expansions, uniform error bounds are obtained.

In the second part of the dissertation, a singular linear system is considered. Again a two-time-scale formulation is used. Under suitable conditions, the system has …

New Variational Principles With Applications To Optimization Theory And Algorithms, Hung Minh Phan

Wayne State University Dissertations

In this dissertation we investigate some applications of variational analysis in optimization theory and algorithms. In the first part we develop new extremal principles in variational analysis that deal with finite and infinite systems of convex and nonconvex sets. The results obtained, under the name of tangential extremal principles and rated extremal principles, combine primal and dual approaches to the study of variational systems being in fact first extremal principles applied to infinite systems of sets. These developments are in the core geometric theory of variational analysis. Our study includes the basic theory and applications to problems of semi-infinite programming …