## Mathematik

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- Existence Results for Plasma Physics Models Containing a Fully Coupled Magnetic Field (2009)
- The present thesis concern is the initial value problem for three nonlinear systems of partial differential equations: the Vlasov-Darwin system, the Vlasov-Poisswell system and a version of the latter which is called the modified Vlasov-Poisswell system. These equations belong to kinetic theory, which has proved useful when describing large particle systems in different areas of physics such as kinetic theory of gases, the formation of stellar structures or plasma physics. In the present thesis equations originating in plasma physics are considered which describe the evolution of the time dependent density function f(t,x,v) (t - time, x – position, v - particle velocity) of a large ensemble of charged particles in the (x,v)-phase space influenced by the electromagnetic field created by the particles and when neglecting collisions. The focus of the investigation is on existence and uniqueness questions for solutions of the initial value problem, i.e., it is asked whether there exists a solution f of the system under consideration such that f(t=0)=f0 where f0 is a prescribed initial datum. In order to answer this question further properties of solutions such as energy and charge conservation or decay rates must be taken into account. An important issue is, whether - if necessary under additional hypotheses or by weakening the concept of solution - global solutions, i.e., solutions existing for all t>=0, may be obtained. The most important results are a theorem about local existence and uniqueness of classical solutions of the Vlasov-Poisswell system, a global existence result for weak solutions of the modified Vlasov-Poisswell system, and a global existence theorem for classical solutions of the Vlasov-Darwin system under the assumption of smallness of the initial.

- Primitive central idempotents of finite group rings of symmetric and alternating groups in characteristic 2 (2009)
- The paper contains computational results, the primitive central idempotents of group rings of symmetric and alternating groups of degree smaller or equal 54 in characteristic 2

- Primitive central idempotents of finite group rings of symmetric and alternating groups in characteristic 3 (2009)
- The paper contains computational results, the primitive central idempotents of group rings of symmetric and alternating groups of degree smaller or equal 31 in characteristic 3

- Receding Horizon Control: A Suboptimality-based Approach (2009)
- Within the proposed work we consider analytical, conceptional and implementational issues of so called receding horizon controllers in a sampled-data setting. The principle of such a controller is simple: Given the current state of a system we compute an open-loop control which is optimal for a given costfunctional over a fixed prediction horizon. Then, the control is implemented on the first sampling interval and the basic open-loop optimal control problem is shifted forward in time which allows for a repeated evaluation. The contribution of this thesis is threefold: First, we prove estimates for the performance of a receding horizon control, a concept which we call suboptimality degree. These estimate are online computable and can be applied for stabilizing as well as practically stabilizing receding horizon control laws. Moreover, they not only allow for guaranteeing stability of the closed-loop but also for quantifying the loss of performance of the receding horizon control law compared to the infinite horizon control law. Based on these estimates, we introduce adaptation strategies to modify the underlying receding horizon controller in order to guarantee a certain lower bound on the suboptimality degree while reducing the computing cost/time necessary to solve this problem. Within this analysis, the length of the optimization horizon is the parameter we wish to adapt. To this end, we develop and proof several shortening and prolongation strategies which also allow for an effective implementation. Moreover, extensions of our suboptimality estimates to receding horizon controllers with varying optimization horizon are shown. Last, we present details on our implementation of a receding horizon controller PCC2 (http://www.nonlinearmpc.com) which is on the one hand computationally efficient but also allows for easily incorporating our theoretical results. Since a full analysis of such a controller would exceed the scope of this work, we focus on the main aspects of this algorithm using different examples. In particular, we concentrate on the impact of certain choices of parameters on the computing time. We also consider interactions between these parameters to give a guideline to effectively implement and solve further examples. Moreover, we show applicability and effectiveness of our theoretical results using simulations of standard problems for receding horizon controllers.