Conservation Laws with Uncertainty on Networks: Traffic Flow and Supply Systems

This thesis investigates conservation laws under uncertainty, focusing on two key applications: vehicular traffic accident models and supply systems with demand uncertainty. While deterministic conservation laws are well-established, real-world systems are often affected by unpredictable events such as traffic accidents or fluctuating consumer demand, which introduces randomness into these models. In the first part, the thesis addresses how random traffic accidents influence vehicular traffic flow. Two modeling approaches are considered: microscopic models, which describe individual vehicles governed by a system of ordinary differential equations, and macroscopic models, which describe the evolution of traffic density over time and space using hyperbolic partial differential equations. Accidents reduce the road capacity and contribute to congestion. Their randomness is captured using stochastic processes and probability measures quantifying the likelihood of accidents. The study also investigates the relationship between microscopic and macroscopic models, particularly as the number of vehicles in the microscopic model approaches infinity. The analysis is extended from single-lane roads to entire traffic networks. The second application examines supply systems under uncertainty, where conservation laws model the transport of goods, like energy or water. Uncertainty arises from unpredictable consumer behavior at the network sinks. The thesis aims to determine the optimal inflow into supply networks and distribution parameters at junctions to minimize the deviation between supply and demand. Various assumptions regarding information availability of demand, and extensions to bilevel optimization problems are proposed. For both applications numerical simulations using suitable approximation schemes are used to assess the performance of the networks under uncertainty.