Aachen, Techn. Hochsch., Diss., 2004

Techn. University, Diss., 2004--Darmstadt.

Techn. Universiẗat, Diss., 2005.

In this thesis the stability and the region of attraction of nonlinear dynamical systems' equilibrium points are considered. Methods from linear programming are combined with theorems from the Lyapunov theory of dynamical systems to develop numerical algorithms. These algorithms deliver non-trivial information about the stability-behaviour of an equilibrium of a continuous, autonomous, nonlinear system. Two linear programs, LP1 and LP2, are developed. LP1 depends on a simply connected open neighborhood N of the equilibrium at the origin and two constants, a and m. The construction of LP1 implies that if it does not possess a feasible solution, then the corresponding system is not a,m-exponentially stable on N. LP2 has the property that every feasible solution of the linear program defines a piecewise-affine (piecewise-linear) Lyapunov function or a Lyapunov-like function V for the system.

We consider immersed hypersurfaces in euclidean $R^{n+1}$ which are stable with respect to an elliptic parametric functional with integrand $F=F(N)$ depending on normal directions only. We prove an integral curvature estimate provided that $F$ is sufficiently close to the area integrand, extending the classical curvature estimate of Schoen, Simon and Yau for stable minimal hypersurfaces in $R^{n+1}$. As a crucial point of our analysis we derive a generalized Simons inequality for the laplacian of the length of a weighted second fundamental form with respect to an abstract metric associated with $F$. Using Moser's iteration technique we finally prove a pointwise curvature estimate for $n leq 5$. As an application we obtain a new Bernstein result for complete stable hypersurfaces of dimension $n leq 5$.

Let $I^2:=[-1,1] imes[-1,1]$ be the unit square and let $U$ be a subspace of $C(I^2)$. If $f$ is a continuous function, then $u^{ast}in U$ is said to be a {it best one--sided $L^1$--approximation to f in $U$ from above} if $u^{ast}geq f$ and $|f-u^{ast}|_1leq |f-u|$ for every $u in U$ with $ugeq f$. In this paper we consider the problem of characterization of such best approximants for the case where $U$ consists of all (quasi--)blending--functions of order $(m,1)$.

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This thesis deals with the structure of the no-wait job-shop problem as well as the derivation of fast approximation algorithms. These algorithms are based on a decomposition approach into a sequencing and a timetabling problem that was initially introduced by Macchiaroli et al. (1999). In the thesis the problems are derived from a mixed integer formulation of the original problem and proved to be NP-hard in the strong sense. After presenting a fast heuristic approach for the timetabling problem, the focus lies on the sequencing problem for which several local search algorithms are presented. The algorithms are tested on a wide variety of benchmark problems for the classical job-shop problem. Among the algorithms, the tabu search approach outperforms all other algorithms that can be found in the literature in objective value as well as computation time.

Stochastic programming offers a way to treat uncertainty in decision problems. In particular, it allows the minimization of risk. We consider mean-risk models involving deviation measures, as for instance the standard deviation and the semideviation, and discuss these risk measures in the framework of stochastic dominance as well as in the framework of coherent risk measures. We derive statements concerning the structure and the stability of the resulting optimization problems whereby we emphasize on models including integrality requirements on some decision variables. Then we propose decomposition algorithms for the mean-risk models under consideration and present numerical results for two stochastic programming applications.

This thesis deals with regualrity questions of elliptic and parabolic systems of partial differential equations of second order. With the help of a Harnack inequality it is shown that bounded weak solutions of certain parabolic systems are Hölder continous. In the second chapter we prove regularity theorems for weak harmonic mappings in the interior and at the boundary, an important tool for these theorems are again two Harnack inequalities. The last two chapters deal with degenerate elliptic systems, for certain degenerate elliptic coefficients (e.g. in the Muckenhouptclass A2) we prove two Harnack inequalities and show with these inequalities some regularity theorems for degenerate elliptic systems.