## Fakultät für Mathematik und Informatik

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Simple closed formulas for endpoint geodesics on Graßmann manifolds are presented. In addition to realizing the shortest distance between two points, geodesics are also essential tools to generate more sophisticated curves that solve higher order interpolation problems on manifolds. This will be illustrated with the geometric de Casteljau construction offering an excellent alternative to the variational approach which gives rise to Riemannian polynomials and splines.

The signal modelling framework JimenaE simulates dynamically Boolean networks. In contrast to SQUAD, there is systematic and not just heuristic calculation of all system states. These specific features are not present in CellNetAnalyzer and BoolNet. JimenaE is an expert extension of Jimena, with new optimized code, network conversion into different formats, rapid convergence both for system state calculation as well as for all three network centralities. It allows higher accuracy in determining network states and allows to dissect networks and identification of network control type and amount for each protein with high accuracy. Biological examples demonstrate this: (i) High plasticity of mesenchymal stromal cells for differentiation into chondrocytes, osteoblasts and adipocytes and differentiation-specific network control focusses on wnt-, TGF-beta and PPAR-gamma signaling. JimenaE allows to study individual proteins, removal or adding interactions (or autocrine loops) and accurately quantifies effects as well as number of system states. (ii) Dynamical modelling of cell–cell interactions of plant Arapidopsis thaliana against Pseudomonas syringae DC3000: We analyze for the first time the pathogen perspective and its interaction with the host. We next provide a detailed analysis on how plant hormonal regulation stimulates specific proteins and who and which protein has which type and amount of network control including a detailed heatmap of the A.thaliana response distinguishing between two states of the immune response. (iii) In an immune response network of dendritic cells confronted with Aspergillus fumigatus, JimenaE calculates now accurately the specific values for centralities and protein-specific network control including chemokine and pattern recognition receptors.

We consider a multi-species gas mixture described by a kinetic model. More precisely, we are interested in models with BGK interaction operators. Several extensions to the standard BGK model are studied.
Firstly, we allow the collision frequency to vary not only in time and space but also with the microscopic velocity. In the standard BGK model, the dependence on the microscopic velocity is neglected for reasons of simplicity. We allow for a more physical description by reintroducing this dependence. But even though the structure of the equations remains the same, the so-called target functions in the relaxation term become more sophisticated being defined by a variational procedure.
Secondly, we include quantum effects (for constant collision frequencies). This approach influences again the resulting target functions in the relaxation term depending on the respective type of quantum particles.
In this thesis, we present a numerical method for simulating such models. We use implicit-explicit time discretizations in order to take care of the stiff relaxation part due to possibly large collision frequencies. The key new ingredient is an implicit solver which minimizes a certain potential function. This procedure mimics the theoretical derivation in the models. We prove that theoretical properties of the model are preserved at the discrete level such as conservation of mass, total momentum and total energy, positivity of distribution functions and a proper entropy behavior. We provide an array of numerical tests illustrating the numerical scheme as well as its usefulness and effectiveness.

Several aspects of the control of large-scale systems communicating over digital channels are considered.
In particular, the issue of delay, quantization, and packet loss is addressed with the help of dynamic quantization.
New small-gain results suitable for networked control systems are introduced and it is shown that many of the known small-gain conditions are equivalent.
The issue of bandwidth limitations is addressed with the help of event-triggered control.
A novel approach termed parsimonious triggering is introduced, which helps to rule out the occurrence of an infinite number of triggering events within finite time.
Moreover, the feasibility of the presented approaches is demonstrated by numerical examples.

The subject of this thesis is the controllability of interconnected linear systems, where the interconnection parameter are the control variables. The study of accessibility and controllability of bilinear systems is closely related to their system Lie algebra. In 1976, Brockett classified all possible system Lie algebras of linear single-input, single-output (SISO) systems under time-varying output feedback. Here, Brockett's results are generalized to networks of linear systems, where time-varying output feedback is applied according to the interconnection structure of the network. First, networks of linear SISO systems are studied and it is assumed that all interconnections are independently controllable. By calculating the system Lie algebra it is shown that accessibility of the controlled network is equivalent to the strong connectedness of the underlying interconnection graph in case the network has at least three subsystems. Networks with two subsystems are not captured by these proofs. Thus, we give results for this particular case under additional assumption either on the graph structure or on the dynamics of the node systems, which are both not necessary. Additionally, the system Lie algebra is studied in case the interconnection graph is not strongly connected. Then, we show how to adapt the ideas of proof to networks of multi-input, multi-output (MIMO) systems. We generalize results for the system Lie algebra on networks of MIMO systems both under output feedback and under restricted output feedback. Moreover, the case with generalized interconnections is studied, i.e. parallel edges and linear dependencies in the interconnection controls are allowed. The new setting demands to distinguish between homogeneous and heterogeneous networks. With this new setting only sufficient conditions can be found to guarantee accessibility of the controlled network. As an example, networks with Toeplitz interconnection structure are studied.