Novel models and control for networked problems: from discrete event to continuous dynamics
Final Report Abstract
Within this DFG-project we built upon and extended various models that describe both microscopic as well as macroscopic phenomena. A major part of the extensions includes the consideration of uncertainty in initial and boundary conditions, the dynamics themselves as well as external sources of randomness. We discussed wellposedness of the arising kinetic and hyperbolic problems and we derived appropriate coupling conditions for models on networks. One-dimensional coupling conditions have been compared to two-dimensional counterparts shedding light on the gap between one- and multiple spatial dimensional hyperbolic balance laws. Having set up well-posed models, we addressed questions of stability and control and developed suitable numerical schemes to approximate solutions. Coupled optimization problems with constraints given by fluid-dynamic equations have been extensively analysed. A particular focus has been set on determining an optimal inflow control that matches an uncertain power demand stream. Moreover, we provided insight in phenomena of various fields of applications, where kinetic and hyperbolic equations naturally appear. The major part of the project has been realized as planned. However, the existence and uniqueness of solutions to hyperbolic stochastic Galerkin formulations was shown by using a relative-entropy framework instead of wave-front tracking. The research results obtained motivate the analysis of further related research questions. They form a profound basis for investigations on the stabilization of material equations in a joined project with Prof. Bambach (Brandenburgische Technische Universität). The consideration of further constraints for the optimal inflow into hyperbolic supply systems to enhance supply reliability is a very interesting research task and of high practical relevance, especially in the context of the energy transition.
Publications
- Modeling of a diffusion with aggregation: Rigorous derivation and numerical simulation, ESAIM, Vol. 52(2), pp. 567–593, 2018
L. Chen, S. Göttlich, S. Knapp
(See online at https://doi.org/10.1051/m2an/2018028) - Space mapping techniques for the optimal inflow control of transmission lines, Optimization Methods and Software, Vol. 33(1), pp. 120–139, 2018
S. Göttlich, C. Teuber
(See online at https://doi.org/10.1080/10556788.2016.1278542) - Convex quadratic mixed-integer problems with quadratic constraints, Operations Research Proceedings, 2019
S. Göttlich, K. Hameister, M. Herty
(See online at https://doi.org/10.1007/978-3-030-48439-2_15) - Discretized Feedback Control for Systems of Linearized Hyperbolic Balance Laws, Mathematical Control and Related Fields, Vol. 9, No. 3, pp. 517–539, 2019
S. Gerster, M. Herty
(See online at https://doi.org/10.3934/mcrf.2019024) - Entropies and Symmetrization of Hyperbolic Stochastic Galerkin Formulations, Communications in Computational Physics, 2019
S. Gerster, M. Herty
(See online at https://doi.org/10.4208/cicp.OA-2019-0047) - Hyperbolic Stochastic Galerkin Formulation for the p-System, Journal of Computational Physics, Vol. 395, pp. 186–204, 2019
S. Gerster, M. Herty, A. Sikstel
(See online at https://doi.org/10.1016/j.jcp.2019.05.049) - On the limits of stabilizability for networks of strings, Systems & Control Letters, Vol. 131, 2019
M. Gugat, S. Gerster
(See online at https://doi.org/10.1016/j.sysconle.2019.104494) - Optimal control of electricity input given an uncertain demand, Mathematical Methods of Operations Research, 2019
S. Göttlich, R. Korn, K. Lux
(See online at https://doi.org/10.1007/s00186-019-00678-6) - The Euler scheme for stochastic differential equations with discontinuous drift coefficient: A numerical study of the convergence rate, Advances in Difference Equations, 429, 2019
S. Göttlich, K. Lux, A. Neuenkirch
(See online at https://doi.org/10.1186/s13662-019-2361-4)
