Modelling and control of balance laws with applications to networks

In this thesis essentially two types of partial differential equations and their applications to networks are considered. The first system is of hyperbolic type with an additional source term. We consider the control of boundary value problems and investigate stability properties. The tool to analyse stability and exponential decay is Lyapunov functions. The theoretical results are transferred to specific applications, such as electric transmission lines and road networks. The second partial differential equation is of parabolic type and describes the movement of an organism under the influence of certain substances. In literature, this process is known as chemotaxis. The chemotaxis model is extended to networks by introducing coupling conditions which ensure the conservation of mass. We prove existence and uniqueness from a numerical point of view. To ensure stability and positivity for the numerical scheme, a network CFL condition is derived by a detailed computation. Finally, several numerical examples are presented

Modelling and control of balance laws with applications to networks

In this thesis essentially two types of partial differential equations and their applications to networks are considered. The first system is of hyperbolic type with an additional source term. We consider the control of boundary value problems and investigate stability properties. The tool to analyse stability and exponential decay is Lyapunov functions. The theoretical results are transferred to specific applications, such as electric transmission lines and road networks. The second partial differential equation is of parabolic type and describes the movement of an organism under the influence of certain substances. In literature, this process is known as chemotaxis. The chemotaxis model is extended to networks by introducing coupling conditions which ensure the conservation of mass. We prove existence and uniqueness from a numerical point of view. To ensure stability and positivity for the numerical scheme, a network CFL condition is derived by a detailed computation. Finally, several numerical examples are presented

Numerical feedback stabilization with applications to networks

The focus is on the numerical consideration of feedback boundary control problems for linear systems of conservation laws including source terms. We explain under which conditions the numerical discretization can be used to design feedback boundary values for network applications such as electric transmission lines or traffic flow systems. Several numerical examples illustrate the properties of the results for different types of networks

A pedestrian flow model with stochastic velocities: Microscopic and macroscopic approaches

We investigate a stochastic model hierarchy for pedestrian flow. Starting from a microscopic social force model, where the pedestrians switch randomly between the two states stop-or-go, we derive an associated macroscopic model of conservation law type. Therefore we use a kinetic mean-field equation and introduce a new problem-oriented closure function. Numerical experiments are presented to compare the above models and to show their similarities.Comment: 23 page