Neue Modelle und Steuerung vernetzter Probleme für ereignisbasierte und kontinuierliche Dynamiken

• Antragstellerinnen / Antragsteller: Professorin Dr. Simone Göttlich; Professor Dr. Michael Herty
• Fachliche Zuordnung: Mathematik
• Förderung: Förderung von 2016 bis 2020
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 298682575

• Erstellungsjahr: 2020

Zusammenfassung der Projektergebnisse

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.

Projektbezogene Publikationen (Auswahl)

• Space mapping techniques for the optimal inflow control of transmission lines, Optimization Methods and Software, Vol. 33(1), pp. 120–139, 2018
  Göttlich, S. & Teuber, C.
  
• Modeling of a diffusion with aggregation: Rigorous derivation and numerical simulation, ESAIM, Vol. 52(2), pp. 567–593, 2018
  Chen, Li; Göttlich, Simone & Knapp, Stephan
  
• Discretized Feedback Control for Systems of Linearized Hyperbolic Balance Laws, Mathematical Control and Related Fields, Vol. 9, No. 3, pp. 517–539, 2019
  Gerster, Stephan & Herty, Michael
  
• Hyperbolic Stochastic Galerkin Formulation for the p-System, Journal of Computational Physics, Vol. 395, pp. 186–204, 2019
  Gerster, Stephan; Herty, Michael & Sikstel, Aleksey
  
• On the limits of stabilizability for networks of strings, Systems & Control Letters, Vol. 131, 2019
  Gugat, Martin & Gerster, Stephan
  
• Optimal control of electricity input given an uncertain demand, Mathematical Methods of Operations Research, 2019
  Göttlich, Simone; Korn, Ralf & Lux, Kerstin
  
• The Euler scheme for stochastic differential equations with discontinuous drift coefficient: A numerical study of the convergence rate, Advances in Difference Equations, 429, 2019
  Göttlich, S.; Lux, K. & Neuenkirch, A.
  
• Convex quadratic mixed-integer problems with quadratic constraints, Operations Research Proceedings, 2019
  Göttlich, Simone; Hameister, Kathinka & Herty, Michael
  
• Entropies and Symmetrization of Hyperbolic Stochastic Galerkin Formulations, Communications in Computational Physics, 2019
  Stephan Gerster, Stephan Gerster & Michael Herty, Michael Herty