Publikationen am Institut für Mathematik

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Öztürk, Emrah; Faulwasser, Timm; Worthmann, Karl; Preißinger, Markus; Rheinberger, Klaus
Alleviating the curse of dimensionality in Minkowski sum approximations of storage flexibility. - In: IEEE transactions on smart grid, Bd. 0 (2024), 0, S. 1-11

Many real-world applications require the joint optimization of a large number of flexible devices over time. The flexibility of, e.g., multiple batteries, thermostatically controlled loads, or electric vehicles can be used to support grid operation and to reduce operation costs. Using piecewise constant power values, the flexibility of each device over d time periods can be described as a polytopic subset in power space. The aggregated flexibility is given by the Minkowski sum of these polytopes. As the computation of Minkowski sums is in general demanding, several approximations have been proposed in the literature. Yet, their application potential is often objective-dependent and limited by the curse of dimensionality. We show that up to 2d vertices of each polytope can be computed efficiently and that the convex hull of their sums provides a computationally efficient inner approximation of the Minkowski sum. Via an extensive simulation study, we illustrate that our approach outperforms ten state-of-the-art inner approximations in terms of computational complexity and accuracy for different objectives. Moreover, we propose an efficient disaggregation method applicable to any vertex-based approximation. The proposed methods provide an efficient means to aggregate and to disaggregate energy storages in quarter-hourly periods over an entire day with reasonable accuracy for aggregated cost and for peak power optimization.



https://doi.org/10.1109/TSG.2024.3420156
Hotz, Thomas; Le, Huiling; Wood, Andrew T. A.
Central limit theorem for intrinsic Fréchet means in smooth compact Riemannian manifolds. - In: Probability theory and related fields, ISSN 1432-2064, Bd. 189 (2024), 3/4, S. 1219-1246

We prove a central limit theorem (CLT) for the Fréchet mean of independent and identically distributed observations in a compact Riemannian manifold assuming that the population Fréchet mean is unique. Previous general CLT results in this setting have assumed that the cut locus of the Fréchet mean lies outside the support of the population distribution. In this paper we present a CLT under some mild technical conditions on the manifold plus the following assumption on the population distribution: in a neighbourhood of the cut locus of the population Fréchet mean, the population distribution is absolutely continuous with respect to the volume measure on the manifold and in this neighhbourhood the Radon-Nikodym derivative has a version that is continuous. So far as we are aware, the CLT given here is the first which allows the cut locus to have co-dimension one or two when it is included in the support of the distribution. A key part of the proof is establishing an asymptotic approximation for the parallel transport of a certain vector field. Whether or not a non-standard term arises in the CLT depends on whether the co-dimension of the cut locus is one or greater than one: in the former case a non-standard term appears but not in the latter case. This is the first paper to give a general and explicit expression for the non-standard term which arises when the co-dimension of the cut locus is one.



https://doi.org/10.1007/s00440-024-01291-3
Rakhmanov, Saparboy; Trunk, Carsten; Matrasulov, Davronbek
Quantum particle under dynamical confinement: from quantum fermi acceleration to high harmonic generation. - In: Physica scripta, ISSN 1402-4896, Bd. 99 (2024), 7, 075308, S. 1-13

Quantum dynamics of a particle confined in a box with time-dependent wall is revisited by considering some unexplored aspects of the problem. In particular, the case of dynamical confinement in a time-dependent box in the presence of purely time-varying external potential is treated by obtaining exact solution. Also, some external potentials approving separation of space and time variables in the Schrödinger equation with time-dependent boundary conditions are classified. Time-dependence of the average kinetic energy and average quantum force are analyzed. A model for optical high harmonic generation in the presence of dynamical confinement and external monochromatic time-dependent homogeneous electric field is proposed.



https://doi.org/10.1088/1402-4896/ad52c8
Behrndt, Jussi; Schmitz, Philipp; Teschl, Gerald; Trunk, Carsten
Perturbation and spectral theory for singular indefinite Sturm-Liouville operators. - In: Journal of differential equations, ISSN 1090-2732, Bd. 405 (2024), S. 151-178

https://doi.org/10.1016/j.jde.2024.05.043
Ilchmann, Achim; Kirchhoff, Jonas; Schaller, Manuel
Port-Hamiltonian descriptor systems are relative generically controllable and stabilizable. - In: Mathematics of control, signals, and systems, ISSN 1435-568X, Bd. 0 (2024), 0, insges. 37 S.

The present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359-377, 2021) on generic controllability and of Ilchmann and Kirchhoff (Math Control Signals Syst 35:45-76, 2022) on relative generic controllability of linear differential-algebraic equations. We extend the result from general, unstructured differential-algebraic equations to differential-algebraic equations of port-Hamiltonian type. We derive results on relative genericity. These findings are the basis for characterizing relative generic controllability of port-Hamiltonian systems in terms of dimensions. A similar result is proved for relative generic stabilizability.



https://doi.org/10.1007/s00498-024-00392-7
Berger, Thomas; Dennstädt, Dario
Funnel MPC for nonlinear systems with arbitrary relative degree. - In: Automatica, ISSN 0005-1098, Bd. 167 (2024), 111759, S. 1-10

The model predictive control (MPC) scheme funnel MPC enables output tracking of smooth reference signals with prescribed error bounds for nonlinear multi-input multi-output systems with stable internal dynamics. Earlier works achieved the control objective for system with relative degree restricted to one or incorporated additional feasibility constraints in the optimal control problem. Here we resolve these limitations by introducing a modified stage cost function relying on a weighted sum of the tracking error derivatives. The weights need to be sufficiently large and we state explicit lower bounds. Under these assumptions we are able to prove initial and recursive feasibility of the novel funnel MPC scheme for systems with arbitrary relative degree - without requiring any terminal conditions, a sufficiently long prediction horizon or additional output constraints.



https://doi.org/10.1016/j.automatica.2024.111759
Hörsch, Florian; Kaiser, Tomáš; Kriesell, Matthias
Rainbow bases in matroids. - In: SIAM journal on discrete mathematics, ISSN 1095-7146, Bd. 38 (2024), 2, S. 1472-1491

Recently, it was proved by Bérczi and Schwarcz that the problem of factorizing a matroid into rainbow bases with respect to a given partition of its ground set is algorithmically intractable. On the other hand, many special cases were left open. We first show that the problem remains hard if the matroid is graphic, answering a question of Bérczi and Schwarcz. As another special case, we consider the problem of deciding whether a given digraph can be factorized into subgraphs which are spanning trees in the underlying sense and respect upper bounds on the indegree of every vertex. We prove that this problem is also hard. This answers a question of Frank. In the second part of the article, we deal with the relaxed problem of covering the ground set of a matroid by rainbow bases. Among other results, we show that there is a linear function f such that every matroid that can be factorized into k bases for some k ≥ 3 can be covered by f(k) rainbow bases if every partition class contains at most 2 elements.



https://doi.org/10.1137/22M1516750
Kirchhoff, Jonas;
Generic observability for port-Hamiltonian descriptor systems. - In: Mathematics of control, signals, and systems, ISSN 1435-568X, Bd. 0 (2024), 0, insges. 43 S.

The present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359-377, 2021. https://doi.org/10.1007/s00498-021-00287-x), Ilchmann and Kirchhoff (Math Control Signals Syst 35:45-76, 2023. https://doi.org/10.1007/s00498-021-00287-x) on (relative) generic controllability of unstructured linear differential-algebraic systems and of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) on (relative) generic controllability of port-Hamiltonian descriptor systems. We extend their results to (relative) genericity of observability. For unstructured differential-algebraic systems, criteria for (relative) generic observability are derived from Ilchmann and Kirchhoff (Math Control Signals Syst 35:45-76, 2023. https://doi.org/10.1007/s00498-021-00287-x) using duality. This is not possible for port-Hamiltonian systems. Hence, we tweak the results of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) and derive similar criteria as for the unstructured case. Additionally, we consider certain rank constraints on the system matrices.



https://doi.org/10.1007/s00498-024-00388-3
Rakhmanov, Saparboy; Trunk, Carsten; Znojil, Miloslav; Matrasulov, Davronbek
PT-symmetric dynamical confinement: Fermi acceleration, quantum force, and Berry phase. - In: Physical review, ISSN 2469-9934, Bd. 109 (2024), 5, 053519

We consider a quantum particle under the dynamical confinement caused by PT-symmetric box with a moving wall. The latter is described in terms of the time-dependent Schrödinger equation obeying the time-dependent PT-symmetric boundary conditions. The class of the functions, describing time dependence of the wall's position and keeping the system as PT-symmetric is found. Physically observable characteristics, such as average kinetic energy and the average quantum force are calculated as a function of time. It is found that the behavior of the average kinetic energy as a function of time is completely different than that for of Hermitian counterpart of the model, while the average quantum force behaves similarly to that for Hermitian system. Also, geometric phase is calculated for the harmonically oscillating wall regime. The experimental realization of the proposed model is discussed.



https://doi.org/10.1103/PhysRevA.109.053519
Espuny Díaz, Alberto; Lichev, Lyuben; Mitsche, Dieter; Wesolek, Alexandra
Sharp threshold for embedding balanced spanning trees in random geometric graphs. - In: Journal of graph theory, ISSN 1097-0118, Bd. 0 (2024), 0, S. 1-19

A rooted tree is balanced if the degree of a vertex depends only on its distance to the root. In this paper we determine the sharp threshold for the appearance of a large family of balanced spanning trees in the random geometric graph G(n,r,d). In particular, we find the sharp threshold for balanced binary trees. More generally, we show that all sequences of balanced trees with uniformly bounded degrees and height tending to infinity appear above a sharp threshold, and none of these appears below the same value. Our results hold more generally for geometric graphs satisfying a mild condition on the distribution of their vertex set, and we provide a polynomial time algorithm to find such trees.



https://doi.org/10.1002/jgt.23106