Control Problems for Conservation Laws with Traffic Applications

Conservation and balance laws on networks have been the subject of much research interest given their wide range of applications to real-world processes, particularly traffic flow. This open access monograph is the first to investigate different types of control problems for conservation laws that arise in the modeling of vehicular traffic. Four types of control problems are discussed - boundary, decentralized, distributed, and Lagrangian control - corresponding to, respectively, entrance points and tolls, traffic signals at junctions, variable speed limits, and the use of autonomy and communication. Because conservation laws are strictly connected to Hamilton-Jacobi equations, control of the latter is also considered. An appendix reviewing the general theory of initial-boundary value problems for balance laws is included, as well as an appendix illustrating the main concepts in the theory of conservation laws on networks

Dynamic Optimization of Gas Transmission Networks for Storage of Renewable Energy

To ensure security of supply in the presence of highly volatile generation of renewable electric energy, extensive storage is required. In this thesis the application of mathematical optimization methods to gas transmission networks with electricity driven compressor stations, operated as electricity storage, is discussed. Therefore, a transient network model that incorporates the gas dynamics described by the isothermal Euler equations as well as technical network elements is introduced and reviewed as coupled systems of hyperbolic balance laws. For optimization problems on networks these PDEs are commonly discretized by finite differences using an implicit box-scheme. The comparison with finite volume simulations, obtained using a high order ADER method, shows that the finite difference approximations represent sufficiently well the gas dynamics for typical flow situations on transmission networks while requiring much less computational effort. The optimization model is then applied to realistic test problems abstracting parts of the German gas transmission network. The results for different price scenarios, which are used as indicator for the availability of renewable energy, show the potential of using pipelines as short term storage for electric energy but also the limitations. The thesis is concluded by a discussion of the recently proposed distributed optimization algorithm ALADIN and its application to the structured gas network optimization problems. Therefore, an implementation of this algorithm as well as the experiences applying it to the introduced model are presented. Compared to a general purpose interior-point method, this approach of exploiting the problem structure shows promising performance for simple examples but fails for more complicated model instances.Um die Versorgungssicherheit auch bei stark fluktuierender Stromerzeugung aus regenerativen Quellen sicherzustellen, werden umfangreiche Speicher benötigt. In dieser Arbeit wird die Anwendung mathematischer Optimierungsmethoden auf Gasnetze mit elektrisch betriebenen Verdichtern diskutiert, die als Stromspeicher eingesetzt werden sollen. Dafür wird ein transientes Netzmodell, das sowohl die Gasdynamik in Form der isothermen Euler-Gleichungen als auch weitere technische Netzelemente umfasst, eingeführt und anschließend im Kontext von hyperbolischer Erhaltungsgleichungen auf Netzen besprochen. Für Optimierungsprobleme auf Netzen werden diese partiellen Differentialgleichungen üblicherweise mit Finiten Differenzen eines impliziten Boxschemas diskretisiert. Der Vergleich mit den Ergebnissen von Finite-Volumen-Simulationen, die mit einem ADER-Verfahren höherer Ordnung bestimmt werden, zeigt, dass diese Finite-Differenzen-Approximationen die typische Dynamik in Gastransportnetzen genügend genau darstellen und dafür erheblich weniger Rechenaufwand benötigen. Dieses Optimierungsmodell wird dann auf realistische Testprobleme angewendet, die Teile des deutschen Gastransportnetzes abstrahieren. Die Ergebnisse für verschiedene Preissituationen, die als Indikator für das Angebot an erneuerbarer Energie genutzt werden, zeigen das Potential der Nutzung von Gasleitungen als kurzfristiger Stromspeicher aber auch die Einschränkungen. Zum Abschluss dieser Arbeit werden der kürzlich vorgestellte, verteilte Optimierungsalgorithmus ALADIN und seine Anwendung auf die strukturierten Gasnetzoptimierungsprobleme diskutiert. Dazu werden eine konkrete Implementation des Algorithmus und die Erfahrungen aus deren Einsatz für die vorgestellten Netzmodelle präsentiert. Im Vergleich zu einem Standardverfahren zeigt dieser Vorschlag einer Struktur ausnutzenden Lösungsmethode für einfache Beispiele ein vielversprechendes Verhalten, scheitert jedoch für kompliziertere Modellinstanzen

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Topology Reconstruction of Dynamical Networks via Constrained Lyapunov Equations

The network structure (or topology) of a dynamical network is often unavailable or uncertain. Hence, we consider the problem of network reconstruction. Network reconstruction aims at inferring the topology of a dynamical network using measurements obtained from the network. In this technical note we define the notion of solvability of the network reconstruction problem. Subsequently, we provide necessary and sufficient conditions under which the network reconstruction problem is solvable. Finally, using constrained Lyapunov equations, we establish novel network reconstruction algorithms, applicable to general dynamical networks. We also provide specialized algorithms for specific network dynamics, such as the well-known consensus and adjacency dynamics.Comment: 8 page

Non-Linear Lattice

The development of mathematical techniques, combined with new possibilities of computational simulation, have greatly broadened the study of non-linear lattices, a theme among the most refined and interdisciplinary-oriented in the field of mathematical physics. This Special Issue mainly focuses on state-of-the-art advancements concerning the many facets of non-linear lattices, from the theoretical ones to more applied ones. The non-linear and discrete systems play a key role in all ranges of physical experience, from macrophenomena to condensed matter, up to some models of space discrete space-time