Publikationen am Institut für Mathematik

Anzahl der Treffer: 1929
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Derkach, Volodymyr; Hassi, Seppo; Malamud, Mark;
Generalized boundary triples, II : some applications of generalized boundary triples and form domain invariant Nevanlinna functions. - In: Mathematische Nachrichten, ISSN 1522-2616, Bd. 295 (2022), 6, S. 1113-1162

The paper is a continuation of Part I and contains several further results on generalized boundary triples, the corresponding Weyl functions, and applications of this technique to ordinary and partial differential operators. We establish a connection between Post's theory of boundary pairs of closed nonnegative forms on the one hand and the theory of generalized boundary triples of nonnegative symmetric operators on the other hand. Applications to the Laplacian operator on bounded domains with smooth, Lipschitz, and even rough boundary, as well as to mixed boundary value problem for the Laplacian are given. Other applications concern with the momentum, Schrödinger, and Dirac operators with local point interactions. These operators demonstrate natural occurrence of ES$ES$-generalized boundary triples with domain invariant Weyl functions and essentially selfadjoint reference operators A0.



https://doi.org/10.1002/mana.202000049
Kirchhoff, Jonas;
Linear port-Hamiltonian systems are generically controllable. - In: IEEE transactions on automatic control, ISSN 1558-2523, Bd. 67 (2022), 6, S. 3220-3222

The new concept of relative generic subsets is introduced. It is shown that the set of controllable linear finite-dimensional port-Hamiltonian systems is a relative generic subset of the set of all linear finite-dimensional port-Hamiltonian systems. This implies that a random, continuously distributed port-Hamiltonian system is almost surely controllable.



https://doi.org/10.1109/TAC.2021.3098176
Berger, Thomas; Dennstädt, Dario;
Funnel MPC with feasibility constraints for nonlinear systems with arbitrary relative degree. - In: IEEE control systems letters, ISSN 2475-1456, Bd. 6 (2022), S. 2804-2809

We study tracking control for nonlinear systems with known relative degree and stable internal dynamics by the recently introduced technique of Funnel MPC. The objective is to achieve the evolution of the tracking error within a prescribed performance funnel. We propose a novel stage cost for Funnel MPC, extending earlier designs to the case of arbitrary relative degree, and show that the control objective as well as initial and recursive feasibility are always achieved - without requiring any terminal conditions or a sufficiently long prediction horizon. We only impose an additional feasibility constraint in the optimal control problem.



https://doi.org/10.1109/LCSYS.2022.3178478
Eichfelder, Gabriele; Quintana, Ernest; Rocktäschel, Stefan;
A vectorization scheme for nonconvex set optimization problems. - In: SIAM journal on optimization, ISSN 1095-7189, Bd. 32 (2022), 2, S. 1184-1209

In this paper, we study a solution approach for set optimization problems with respect to the lower set less relation. This approach can serve as a base for numerically solving set optimization problems by using established solvers from multiobjective optimization. Our strategy consists of deriving a parametric family of multiobjective optimization problems whose optimal solution sets approximate, in a specific sense, that of the set-valued problem with arbitrary accuracy. We also examine particular classes of set-valued mappings for which the corresponding set optimization problem is equivalent to a multiobjective optimization problem in the generated family. Surprisingly, this includes set-valued mappings with a convex graph.



https://doi.org/10.1137/21M143683X
Öztürk, Emrah; Rheinberger, Klaus; Faulwasser, Timm; Worthmann, Karl; Preißinger, Markus;
Aggregation of demand-side flexibilities: a comparative study of approximation algorithms. - In: Energies, ISSN 1996-1073, Bd. 15 (2022), 7, 2501, S. 1-14

Traditional power grids are mainly based on centralized power generation and subsequent distribution. The increasing penetration of distributed renewable energy sources and the growing number of electrical loads is creating difficulties in balancing supply and demand and threatens the secure and efficient operation of power grids. At the same time, households hold an increasing amount of flexibility, which can be exploited by demand-side management to decrease customer cost and support grid operation. Compared to the collection of individual flexibilities, aggregation reduces optimization complexity, protects households' privacy, and lowers the communication effort. In mathematical terms, each flexibility is modeled by a set of power profiles, and the aggregated flexibility is modeled by the Minkowski sum of individual flexibilities. As the exact Minkowski sum calculation is generally computationally prohibitive, various approximations can be found in the literature. The main contribution of this paper is a comparative evaluation of several approximation algorithms in terms of novel quality criteria, computational complexity, and communication effort using realistic data. Furthermore, we investigate the dependence of selected comparison criteria on the time horizon length and on the number of households. Our results indicate that none of the algorithms perform satisfactorily in all categories. Hence, we provide guidelines on the application-dependent algorithm choice. Moreover, we demonstrate a major drawback of some inner approximations, namely that they may lead to situations in which not using the flexibility is impossible, which may be suboptimal in certain situations.



https://doi.org/10.3390/en15072501
Hörsch, Florian;
Checking the admissibility of odd-vertex pairings is hard. - In: Discrete applied mathematics, Bd. 317 (2022), S. 42-48

Nash-Williams proved that every graph has a well-balanced orientation. A key ingredient in his proof is admissible odd-vertex pairings. We show that for two slightly different definitions of admissible odd-vertex pairings, deciding whether a given odd-vertex pairing is admissible is co-NP-complete. This resolves a question of Frank. We also show that deciding whether a given graph has an orientation that satisfies arbitrary local arc-connectivity requirements is NP-complete.



https://doi.org/10.1016/j.dam.2022.04.004
De Santis, Marianna; Eichfelder, Gabriele; Patria, Daniele;
On the exactness of the ε-constraint method for biobjective nonlinear integer programming. - In: Operations research letters, ISSN 0167-6377, Bd. 50 (2022), 3, S. 356-361

The ε-constraint method is a well-known scalarization technique used for multiobjective optimization. We explore how to properly define the step size parameter of the method in order to guarantee its exactness when dealing with biobjective nonlinear integer problems. Under specific assumptions, we prove that the number of subproblems that the method needs to address to detect the complete Pareto front is finite. We report numerical results on portfolio optimization instances built on real-world data and show a comparison with an existing criterion space algorithm.



https://doi.org/10.1016/j.orl.2022.04.007
Grundel, Sara; Heyder, Stefan; Hotz, Thomas; Ritschel, Tobias K. S.; Sauerteig, Philipp; Worthmann, Karl;
How much testing and social distancing is required to control COVID-19? : some insight based on an age-differentiated compartmental model. - In: SIAM journal on control and optimization, ISSN 1095-7138, Bd. 60 (2022), 2, S. S145-S169

In this paper, we provide insights on how much testing and social distancing is required to control COVID-19. To this end, we develop a compartmental model that accounts for key aspects of the disease: incubation time, age-dependent symptom severity, and testing and hospitalization delays; the model's parameters are chosen based on medical evidence, and, for concreteness, adapted to the German situation. Then, optimal mass-testing and age-dependent social distancing policies are determined by solving optimal control problems both in open loop and within a model predictive control framework. We aim to minimize testing and/or social distancing until herd immunity sets in under a constraint on the number of available intensive care units. We find that an early and short lockdown is inevitable but can be slowly relaxed over the following months.



https://doi.org/10.1137/20M1377783
Grüne, Lars; Schaller, Manuel; Schiela, Anton;
Efficient model predictive control for parabolic PDEs with goal oriented error estimation. - In: SIAM journal on scientific computing, ISSN 1095-7197, Bd. 44 (2022), 1, S. A471-A500

We show how a posteriori goal oriented error estimation can be used to efficiently solve the subproblems occurring in a model predictive control (MPC) algorithm. In MPC, only an initial part of a computed solution is implemented as a feedback, which motivates grid refinement particularly tailored to this context. To this end, we present a truncated cost functional as an objective for goal oriented adaptivity and prove under stabilizability assumptions that error indicators decay exponentially outside the support of this quantity. This leads to very efficient time and space discretizations for MPC, which we will illustrate by means of various numerical examples.



https://doi.org/10.1137/20M1356324
Schmitz, Philipp; Faulwasser, Timm; Worthmann, Karl;
Willems' fundamental lemma for linear descriptor systems and its use for data-driven output-feedback MPC. - In: IEEE control systems letters, ISSN 2475-1456, Bd. 6 (2022), S. 2443-2448

In this letter we investigate data-driven predictive control of discrete-time linear descriptor systems. Specifically, we give a tailored variant of Willems' fundamental lemma, which shows that for descriptor systems the non-parametric modeling via a Hankel matrix requires less data compared to linear time-invariant systems without algebraic constraints. Moreover, we use this description to propose a data-driven framework for optimal control and predictive control of discrete-time linear descriptor systems. For the latter, we provide a sufficient stability condition for receding-horizon control before we illustrate our findings with an example.



https://doi.org/10.1109/LCSYS.2022.3161054