Publications at the Institute of Mathematics

Results: 2079
Created on: Tue, 23 Apr 2024 23:07:23 +0200 in 0.0574 sec


Hoff, Daniel; Mehlitz, Patrick
Notes on the value function approach to multiobjective bilevel optimization. - In: Optimization, ISSN 1029-4945, Bd. 0 (2024), 0, S. 1-37

This paper is concerned with the value function approach to multiobjective bilevel optimization which exploits a lower-level frontier-type mapping in order to replace the hierarchical model of two interdependent multiobjective optimization problems by a single-level multiobjective optimization problem. As a starting point, different value-function-type reformulations are suggested and their relations are discussed. Here, we focus on the situations where the lower-level problem is solved up to efficiency or weak efficiency, and an intermediate solution concept is suggested as well. We study the graph-closedness of the associated efficiency-type and frontier-type mappings. These findings are then used for two purposes. First, we investigate existence results in multiobjective bilevel optimization. Second, for the derivation of necessary optimality conditions via the value function approach, it is inherent to differentiate frontier-type mappings in a generalized way. Here, we are concerned with the computation of upper coderivative estimates for the frontier-type mapping associated with the setting where the lower-level problem is solved up to weak efficiency. We proceed in two ways, relying, on the one hand, on a weak domination property and, on the other hand, on a scalarization approach. Illustrative examples visualize our findings and some flaws in the related literature.



https://doi.org/10.1080/02331934.2024.2323107
Hahn-Klimroth, Maximilian Grischa; Parczyk, Olaf; Person, Yury
Minimum degree conditions for containing an r-regular r-connected spanning subgraph. - In: European journal of combinatorics, Bd. 118 (2024), 103940, S. 1-23

We study optimal minimum degree conditions when an n-vertex graph G contains an r-regular r-connected spanning subgraph. We prove for r fixed and n large the condition to be δ (G) ≥ n+r-2 / 2 when nr ≡ 0 (mod 2). This answers a question of M. Kriesell.



https://doi.org/10.1016/j.ejc.2024.103940
Abreu, Zita; Lieb, Julia; Pinto, Raquel; Rosenthal, Joachim
Criteria for the construction of MDS convolutional codes with good column distances. - In: Advances in mathematics of communications, ISSN 1930-5338, Bd. 18 (2024), 2, S. 595-613

Maximum-distance separable (MDS) convolutional codes are characterized by the property that their free distance reaches the generalized Singleton bound. In this paper, new criteria to construct MDS convolutional codes are presented. These codes also possess optimal first (reverse) column distances. The new criteria allow to relate the construction of MDS convolutional codes to those of reverse superregular Toeplitz matrices. Moreover, using the new criteria as well as the help of computer search, examples for MDS convolutional codes over small finite fields are given.



https://doi.org/10.3934/amc.2023060
Faulwasser, Timm; Flaßkamp, Kathrin; Röbenack, Klaus; Worthmann, Karl
Optimale Steuerung und Regelung - Analyse, Algorithmen und Anwendungen :
Optimal control - analysis, algorithms and applications. - In: Automatisierungstechnik, ISSN 2196-677X, Bd. 72 (2024), 2, S. 77-79

Optimal control has been at the center of many pivotal developments in systems and control in the 20th century. This includes the twin breakthroughs of Richard E. Bellman’s Dynamic Programming and Lew S. Pontryagin’s Maximimum Principle as well as the optimality does not imply stability punchline by Rudolf E. Kalman. Likewise the dissipativity notion for open systems conceived by Jan C. Willems is deeply routed and closely linked to optimal control theory. Moreover, model predictive control can be regarded as an industrially impactful attempt to overcome the difficulties of analytic computation of feedback laws for constrained systems by numerical online computation. First formulations of receding-horizon ideas for optimal control can be traced back to the 1960s. With this pretext one might be tempted to conclude that contemporary research on optimal control is limited to applications. This special issue on optimal control with its particular focus on analysis, algorithms as well as applications falsifies any adhoc conclusion of this kind. Indeed, it combines different contributions which cover a wide array of topics – ranging from hydropower plants and bicycle dynamics to port-Hamiltonian formulations for adaptive structures, distributed predictive control, and moving horizon estimation. Hence, even without drawing upon the currently prevailing trends of data-driven and learning-based control – which also admit optimization-based research avenues – optimal control continues to be a supporting pillar of modern systems and control research with manifold prospects for fundamental analysis, performant algorithms, and challenging applications. Following the established structure of the journal the articles of this special issue are clustered into two categories – methods and applications.



https://doi.org/10.1515/auto-2023-0235
Schaller, Manuel; Zeller, Amelie; Böhm, Michael; Sawodny, Oliver; Tarín, Cristina; Worthmann, Karl
Energie-optimale Steuerung adaptiver Gebäude :
Energy-optimal control of adaptive structures. - In: Automatisierungstechnik, ISSN 2196-677X, Bd. 72 (2024), 2, S. 107-119

Adaptive structures are equipped with sensors and actuators to actively counteract external loads such as wind. This can significantly reduce resource consumption and emissions during the life cycle compared to conventional structures. A common approach for active damping is to derive a port-Hamiltonian model and to employ linear-quadratic control. However, the quadratic control penalization lacks physical interpretation and merely serves as a regularization term. Rather, we propose a controller, which achieves the goal of vibration damping while acting energy-optimal. Leveraging the port-Hamiltonian structure, we show that the optimal control is uniquely determined, even on singular arcs. Further, we prove a stable long-time behavior of optimal trajectories by means of a turnpike property. Last, the proposed controller’s efficiency is evaluated in a numerical study.



https://doi.org/10.1515/auto-2023-0090
Espuny Díaz, Alberto; Morris, Patrick; Perarnau, Guillem; Serra, Oriol
Speeding up random walk mixing by starting from a uniform vertex. - In: Electronic journal of probability, ISSN 1083-6489, Bd. 29 (2024), 26, S. 1-25

The theory of rapid mixing random walks plays a fundamental role in the study of modern randomised algorithms. Usually, the mixing time is measured with respect to the worst initial position. It is well known that the presence of bottlenecks in a graph hampers mixing and, in particular, starting inside a small bottleneck significantly slows down the diffusion of the walk in the first steps of the process. The average mixing time is defined to be the mixing time starting at a uniformly random vertex and hence is not sensitive to the slow diffusion caused by these bottlenecks. In this paper we provide a general framework to show logarithmic average mixing time for random walks on graphs with small bottlenecks. The framework is especially effective on certain families of random graphs with heterogeneous properties. We demonstrate its applicability on two random models for which the mixing time was known to be of order (log n)2, speeding up the mixing to order logn. First, in the context of smoothed analysis on connected graphs, we show logarithmic average mixing time for randomly perturbed graphs of bounded degeneracy. A particular instance is the Newman-Watts small-world model. Second, we show logarithmic average mixing time for supercritically percolated expander graphs. When the host graph is complete, this application gives an alternative proof that the average mixing time of the giant component in the supercritical Erd˝os-Rényi graph is logarithmic.



https://doi.org/10.1214/24-EJP1091
Bartel, Andreas; Clemens, Markus; Günther, Michael; Jacob, Birgit; Reis, Timo
Port-Hamiltonian systems’ modelling in electrical engineering. - In: Scientific computing in electrical engineering, (2024), S. 133-143

The port-Hamiltonian (pH) modelling framework allows for models that preserve essential physical properties such as energy conservation or dissipative inequalities. If all subsystems are modelled as pH systems and the inputs are related to the output in a linear manner, the overall system can be modelled as a pH system, too, which preserves the properties of the underlying subsystems. If the coupling is given by a skew-symmetric matrix, as usual in many applications, the overall system can be easily derived from the subsystems without the need of introducing dummy variables and therefore artificially increasing the complexity of the system. Hence the framework of pH systems is especially suitable for modelling multiphysical systems.



https://doi.org/10.1007/978-3-031-54517-7_15
âCurgus, Branko; Derkach, Volodymyr; Trunk, Carsten
Indefinite Sturm-Liouville operators in polar form. - In: Integral equations and operator theory, ISSN 1420-8989, Bd. 96 (2024), 2, S. 1-58

https://doi.org/10.1007/s00020-023-02746-3
Drücker, Svenja; Lanza, Lukas; Berger, Thomas; Reis, Timo; Seifried, Robert
Experimental validation for the combination of funnel control with a feedforward control strategy. - In: Multibody system dynamics, ISSN 1573-272X, Bd. 0 (2024), 0, S. 1-19

Current engineering design trends, such as lightweight machines and human-machine interaction, often lead to underactuated systems. Output trajectory tracking of such systems is a challenging control problem. Here, we use a two-design-degree of freedom control approach by combining funnel feedback control with feedforward control based on servo-constraints. We present experimental results to verify the approach and demonstrate that the addition of a feedforward controller mitigates drawbacks of the funnel controller. We also present new experimental results for the real-time implementation of a feedforward controller based on servo-constraints on a minimum phase system.



https://doi.org/10.1007/s11044-024-09976-2
Honecker, Maria Christine; Gernandt, Hannes; Wulff, Kai; Trunk, Carsten; Reger, Johann
Feedback rectifiable pairs and stabilization of switched linear systems. - In: Systems & control letters, ISSN 1872-7956, Bd. 186 (2024), 105755, S. 1-10

We address the feedback design problem for switched linear systems. In particular we aim to design a switched state-feedback such that the resulting closed-loop subsystems share the same eigenstructure. To this effect we formulate and analyse the feedback rectification problem for pairs of matrices. We present necessary and sufficient conditions for the feedback rectifiability of pairs for two subsystems and give a constructive procedure to design stabilizing state-feedback for a class of switched systems. In particular the proposed algorithm provides sets of eigenvalues and corresponding eigenvectors for the closed-loop subsystems that guarantee stability for arbitrary switching. Several examples illustrate the characteristics of the problem considered and the application of the proposed design procedure.



https://doi.org/10.1016/j.sysconle.2024.105755