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Tue, 27 Oct 2020 17:23:30 +0100Tue, 27 Oct 2020 17:23:30 +0100Shape turnpike for linear parabolic PDE models
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/357
We introduce and study the turnpike property for time-varying shapes, within the viewpoint of optimal control. We focus here on
second-order linear parabolic equations where the shape acts as a source term and we seek the optimal time-varying shape that
minimizes a quadratic criterion. We first establish existence of optimal solutions under some appropriate sufficient conditions. We
then provide necessary conditions for optimality in terms of adjoint equations and, using the concept of strict dissipativity, we prove
that state and adjoint satisfy the measure-turnpike property, meaning that the extremal time-varying solution remains essentially
close to the optimal solution of an associated static problem. We show that the optimal shape enjoys the exponential turnpike
property in term of Hausdorff distance for a Mayer quadratic cost. We illustrate the turnpike phenomenon in optimal shape design
with several numerical simulations.Lance Gontran; Emmanuel Trélat; Enrique Zuazuaarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/357Tue, 27 Oct 2020 17:23:30 +0100Stochastic optimization methods for the simultaneous control of parameter-dependent systems
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/354
We address the application of stochastic optimization methods for the simultaneous control of parameter-dependent systems. In particular, we focus on the classical Stochastic Gradient Descent (SGD) approach of Robbins and Monro, and on the recently developed Continuous Stochastic Gradient (CSG) algorithm. We consider the problem of computing simultaneous controls through the minimization of a cost functional defined as the superposition of individual costs for each realization of the system. We compare the performances of these stochastic approaches, in terms of their computational complexity, with those of the more classical Gradient Descent (GD) and Conjugate Gradient (CG) algorithms, and we discuss the advantages and disadvantages of each methodology. In agreement with well-established results in the machine learning context, we show how the SGD and CSG algorithms can significantly reduce the computational burden when treating control problems depending on a large amount of parameters. This is corroborated by numerical experiments.Umberto Biccari; Ana Navarro-Quiles; Enrique Zuazuaarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/354Tue, 27 Oct 2020 13:12:01 +0100The Turnpike property and the long-time behavior of the Hamilton-Jacobi equation
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/355
In this work, we analyze the consequences that the so-called turnpike property has on the long-time behavior of the value function corresponding to an optimal control problem. As a by-product, we obtain the long-time behavior of the solution to the associated Hamilton-Jacobi-Bellman equation.
In order to carry out our study, we use the setting of a finite-dimensional linear-quadratic optimal control problem, for which the turnpike property is well understood. We prove that, when the time horizon T tends to infinity, the value function converges to a travelling-front like solution of the form W(x) + c T + λ. In addition, we provide a control interpretation of each of these three terms in the spirit of the turnpike theory. Finally, we compare this asymptotic decomposition with the existing results on long-time behavior for Hamilton-Jacobi equations. We stress that in our case, the Hamiltonian is not coercive in the momentum variable, a case rarely considered in the classical literature about Hamilton-Jacobi equations.Carlos Esteve; H Kouhkouh; Dario Pighin; Enrique Zuazuaarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/355Tue, 27 Oct 2020 13:12:01 +0100Control under constraints for multi-dimensional reaction-diffusion monostable and bistable equations
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/351
Dynamic phenomena in social and biological sciences can often be modeled employing reaction diffusion equations. Frequently in applications, their control plays an important role when avoiding population extinction or propagation of infectious diseases, enhancing multicultural features, etc. When addressing these issues from a mathematical viewpoint one of the main challenges is that, because of the intrinsic nature of the models under consideration, the solution, typically a proportion or a density function, needs to preserve given lower and upper bounds (taking values in [0; 1])).
Controlling the system to the desired final configuration then becomes complex, and sometimes even impossible. In the present work, we analyze the controllability to constant steady states of spatially homogeneous semilinear heat equations, with constraints in the state, and using boundary controls, which is indeed a natural way of acting on the system in the present context. The nonlinearities considered are among the most frequent: monostable and bistable ones. We prove that controlling the system to a constant steadystate may become impossible when the diffusivity is too small (or when the domain is large), due to the existence of barrier functions. When such an obstruction does not arise, we build sophisticated control strategies combining the dissipativity of the system, the existence of traveling waves, some connectivity of the set of steady states. This connectivity allows building paths that the controlled trajectories can follow, in a long time, with small oscillations, preserving the natural constraints of the system.
This kind of strategy was successfully implemented in one space dimension, where phase plane analysis techniques allowed to decode the nature of the set of steady states. These techniques fail in the present multidimensional setting. We employ a fictitious domain technique, extending the system to a larger ball, and building paths of radially symmetric solution that can then be restricted to the original domain. The results are illustrated by numerical simulations of these models that find several applications, such as the extinction of minority languages or the survival of rare species in sufficiently large reserved areas.Domenec Ruiz-Balet; Enrique Zuazuaarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/351Tue, 27 Oct 2020 13:12:00 +0100Model predictive control with random batch methods for a guiding problem
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/352
We model, simulate and control the guiding problem for a herd of evaders under the action of repulsive drivers. The problem is formulated in an optimal control framework, where the drivers (controls) aim to guide the evaders (states) to a desired region of the Euclidean space.
The numerical simulation of such models quickly becomes unfeasible for a large number of interacting agents. To reduce the computational cost, we use the Random Batch Method (RBM), which provides a computationally feasible approximation of the dynamics. At each time step, the RBM randomly divides the set of particles into small subsets (batches), considering only the interactions inside each batch. Due to the averaging effect, the RBM approximation converges to the exact dynamics as the time discretization gets finer. We propose an algorithm that leads to the optimal control of a fixed RBM approximated trajectory using a classical gradient descent. The resulting control is not optimal for the original complete system, but rather for the reduced RBM model. We then adopt a Model Predictive Control (MPC) strategy to handle the error in the dynamics. While the system evolves in time, the MPC strategy consists in periodically updating the state and computing the optimal control over a long-time horizon, which is implemented recursively in a shorter time-horizon. This leads to a semi-feedback control strategy. Through numerical experiments we show that the combination of RBM and MPC leads to a significant reduction of the computational cost, preserving the capacity of controlling the overall dynamics.Dongnam Ko; Enrique Zuazuaarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/352Tue, 27 Oct 2020 13:12:00 +0100Asymptotic behavior of scalar convection-diffusion equations
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/350
In these lecture notes, we address the problem of large-time asymptotic behaviour of the solutions to scalar convection-diffusion equations set in [katex]\mathbb{R}^N[/katex]. The large-time asymptotic behaviour of the solutions to many convection-diffusion equations is strongly linked with the behavior of the initial data at infinity. In fact, when the initial datum is integrable and of mass [katex]M[/katex], the solutions to the equations under consideration oftentimes behave like the associated self-similar profile of mass [katex]M[/katex], thus emphasising the role of scaling variables in these scenarios. However, these equations can also manifest other asymptotic behaviors, including weakly non-linear, linear or strongly non-linear behavior depending on the form of the convective term. We give an exhaustive presentation of several results and techniques, where we clearly distinguish the role of the spatial dimension and the form of the nonlinear convective term.Enrique Zuazuaarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/350Tue, 27 Oct 2020 13:11:59 +0100A Stochastic Approach to the Synchronization of Coupled Oscillators
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/349
This paper deals with an optimal control problem associated with the Kuramoto model describing the dynamical behavior of a network of coupled oscillators. Our aim is to design a suitable control function allowing us to steer the system to a synchronized configuration in which all the oscillators are aligned on the same phase. This control is computed via the minimization of a given cost functional associated with the dynamics considered. For this minimization, we propose a novel approach based on the combination of a standard Gradient Descent (GD) methodology with the recently-developed Random Batch Method (RBM) for the efficient numerical approximation of collective dynamics. Our simulations show that the employment of RBM improves the performances of the GD algorithm, reducing the computational complexity of the minimization process and allowing for a more efficient control calculation.Umberto Biccari; Enrique Zuazuaarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/349Tue, 27 Oct 2020 13:11:58 +0100Classical system theory revisited for Turnpike in standard state space systems and impulse controllable descriptor systems
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/348
The concept of turnpike connects the solution of long but finite time horizon optimal control problems with steady state optimal controls. A key ingredient of the analysis of the turnpike is the linear quadratic regulator problem and the convergence of the solution of the associated differential Riccati equation as the terminal time approaches infinity. This convergence has been investigated in linear systems theory in the 1980s. We extend classical system theoretic results for the investigation of turnpike properties of standard state space systems and descriptor systems. We present conditions for turnpike in the nondetectable case and for impulse controllable descriptor systems. For the latter, in line with the theory for standard linear systems, we establish existence and convergence of solutions to a generalized differential Riccati equation.Jan Heiland; Enrique Zuazuaarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/348Tue, 27 Oct 2020 13:11:57 +0100Large-time asymptotics in deep learning
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/347
It is by now well-known that practical deep supervised learning may roughly be cast as an optimal control problem for a specific discrete-time, nonlinear dynamical system called an artificial neural network. In this work, we consider the continuous-time formulation of the deep supervised learning problem, and study the latter’s behavior when the final time horizon increases, a fact that can be interpreted as increasing the number of layers in the neural network setting.
When considering the classical regularized empirical risk minimization problem, we show that, in long time, the optimal states converge to zero training error, namely approach the zero training error regime, whilst the optimal control parameters approach, on an appropriate scale, minimal norm parameters with corresponding states precisely in the zero training error regime. This result provides an alternative theoretical underpinning to the notion that neural networks learn best in the overparametrized regime, when seen from the large layer perspective.
We also propose a learning problem consisting of minimizing a cost with a state tracking term, and establish the well-known turnpike property, which indicates that the solutions of the learning problem in long time intervals consist of three pieces, the first and the last of which being transient short-time arcs, and the middle piece being a long-time arc staying exponentially close to the optimal solution of an associated static learning problem. This property in fact stipulates a quantitative estimate for the number of layers required to reach the zero training error regime.
Both of the aforementioned asymptotic regimes are addressed in the context of continuous-time and continuous space-time neural networks, the latter taking the form of nonlinear, integro-differential equations, hence covering residual neural networks with both fixed and possibly variable depths.Carlos Esteve; Borjan Geshkovski; Dario Pighin; Enrique Zuazuaarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/347Tue, 27 Oct 2020 13:11:56 +0100The Inverse Problem for Hamilton-Jacobi equations and Semiconcave Envelopes
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/346
We study the inverse problem, or inverse design problem, for a time-evolution Hamilton-Jacobi equation. More precisely, given a target function [katex]u_T[/katex] and a time horizon [katex]T > 0[/katex], we aim to construct all the initial conditions for which the viscosity solution coincides with [katex]u_T[/katex] at time [katex]T[/katex]. As it is common in this kind of nonlinear equations, the target might not be reachable. We first study the existence of at least one initial condition leading the system to the given target. The natural candidate, which indeed allows determining the reachability of [katex]u_T[/katex] , is the one obtained by reversing the direction of time in the equation, considering [katex]u_T[/katex] as terminal condition. In this case, we use the notion of backward viscosity solution, that provides existence and uniqueness for the terminal-value problem. We also give an equivalent reachability condition based on a differential inequality, that relates the reachability of the target with its semiconcavity properties. Then, for the case when [katex]u_T[/katex] is reachable, we construct the set of all initial conditions for which the solution coincides with [katex]u_T[/katex] at time [katex]T[/katex]. Note that in general, such initial conditions are not unique. Finally, for the case when the target [katex]u_T[/katex] is not necessarily reachable, we study the projection of [katex]u_T[/katex] on the set of reachable targets, obtained by solving the problem backward and then forward in time. This projection is then identified with the solution of a fully nonlinear obstacle problem, and can be interpreted as the semiconcave envelope of [katex]u_T[/katex] , i.e. the smallest reachable target bounded from below by [katex]u_T[/katex] .C Esteve; Enrique Zuazuaarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/346Tue, 27 Oct 2020 13:11:55 +0100Averaged dynamics and control for heat equations with random diffusion
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/345
Abstract. This paper deals with the averaged dynamics for heat equations in the degenerate case where the diffusivity coefficient, assumed to be constant, is allowed to take the null value. First we prove that the averaged dynamics is analytic. This allows to show that, most often, the averaged dynamics enjoys the property of unique continuation and is approximately controllable. We then determine if the averaged dynamics is actually null controllable or not depending on how the density of averaging behaves when the diffusivity vanishes. In the critical density threshold the dynamics of the average is similar to the $\frac{1}{2}$-fractional Laplacian, which is wellknown to be critical in the context of the controllability of fractional diffusion processes. Null controllability then fails (resp. holds) when the density weights more (resp. less) in the null diffusivity regime than in this critical regime.J.A. Bárcena; Enrique Zuazuaarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/345Tue, 27 Oct 2020 10:43:13 +0100On Linear Bilevel Optimization Problems with Complementarity-Constrained Lower Levels
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/344
We consider a novel class of linear bilevel optimization models with a lower level that is a linear optimization problem with complementarity constraints (LPCC). We present different single-level reformulations depending on whether the linear complementarity problem (LCP) as part of the lower-level constraint set depends on the upper-level decisions or not as well as on whether the LCP matrix is positive definite or positive semidefinite. Moreover, we illustrate the connection to linear trilevel models that can be seen as a special case of the considered class of bilevel problems under some additional assumptions. Finally, we provide two generic and illustrative bilevel models from the fields of transportation and energy to show the practical relevance of the newly introduced class of bilevel problems and show related theoretical results.Steven A. Gabriel; Marina Leal; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/344Wed, 21 Oct 2020 08:47:36 +0200Why there is no need to use a big-M in linear bilevel optimization: A computational study of two ready-to-use approaches
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/343
Linear bilevel optimization problems have gained increasing attention both in theory as well as in practical applications of Operations Research (OR) during the last years and decades. The latter is mainly due to the ability of this class of problems to model hierarchical decision processes. However, this ability makes bilevel problems also very hard to solve. Since no general-purpose solvers are available, a "best-practice" has developed in the applied OR community, in which not all people want to develop tailored algorithms but "just use" bilevel optimization as a modeling tool for practice. This best-practice is the big-M reformulation of the Karush-Kuhn-Tucker (KKT) conditions of the lower-level problem - an approach that has been shown to be highly problematic by Pineda and Morales (2019). Choosing invalid values for M yields solutions that may be arbitrarily bad. Checking the validity of the big-Ms is however shown to be as hard as solving the original bilevel problem in Kleinert et al. (2019). Nevertheless, due to its appealing simplicity, especially w.r.t. the required implementation effort, this ready-to-use approach still is the most popular method. Until now, there has been a lack of approaches that are competitive both in terms of implementation effort and computational cost.
In this note we demonstrate that there is indeed another competitive ready-to-use approach: If the SOS-1 technique is applied to the KKT complementarity conditions, adding the simple additional root-node inequality developed by Kleinert et al. (2020) leads to a competitive performance - without having all the possible theoretical disadvantages of the big-M approach.Thomas Kleinert; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/343Mon, 19 Oct 2020 11:30:53 +0200On the transport limit of singularly perturbed convection-diffusion problems on networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/342
We consider singularly perturbed convection-diffusion equations on one-dimensional
networks (metric graphs) as well as the transport problems arising in the
vanishing diffusion limit. Suitable coupling condition at inner vertices are derived that
guarantee conservation of mass as well as dissipation of a mathematical energy which
allows us to prove stability and well-posedness. For single intervals and appropriately
specified initial conditions, it is well-known that the solutions of the convection-diffusion
problem converge to that of the transport problem with order O(sqrt(eps)) in the L1(L2)-
norm with diffusion eps -> 0. In this paper, we prove a corresponding result for problems
on one-dimensional networks. The main difficulty in the analysis is that the number and
type of coupling conditions changes in the singular limit which gives rise to additional
boundary layers at the interior vertices of the network. Since the values of the solution
at these network junctions are not known a-priori, the asymptotic analysis requires a
delicate choice of boundary layer functions that allows to handle these interior layers.Herbert Egger; Nora Philippiarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/342Tue, 13 Oct 2020 13:25:12 +0200A hybrid discontinuous Galerkin method for transport equations on networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/341
We discuss the mathematical modeling and numerical discretization of
5 transport problems on one-dimensional networks. Suitable coupling conditions are
derived that guarantee conservation of mass across network junctions and dissipation
of a mathematical energy which allows us to prove existence of unique solutions.
We then consider the space discretization by a hybrid discontinuous Galerkin
method which provides a suitable upwind mechanism to handle the transport prob10
lem and allows to incorporate the coupling conditions in a natural manner. In addition,
the method inherits mass conservation and stability of the continuous problem.
Order optimal convergence rates are established and illustrated by numerical tests.Herbert Egger; Nora Philippiarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/341Tue, 13 Oct 2020 11:10:23 +0200Limits of stabilizabilizy for a semilinear model for gas pipeline flow
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/338
We present a positive and a negative stabilization result for a semilinear
model of gas flow in pipelines. For feedback boundary conditions we obtain an
unconditional stabilization result in the absence and conditional instability in
the presence of the source term. We also obtain unconditional instability for the
corresponding quasilinear model given by the isothermal Euler equationsMartin Gugat; Michael Hertypreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/338Thu, 08 Oct 2020 14:14:22 +0200Modeling, Control and Numerics of Gas Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/339
In this article we survey recent progress on mathematical results on gas flow in pipe
networks with a special focus on questions of control and stabilization. We briefly present
the modeling of gas flow and coupling conditions for flow through vertices of a network. Our
main focus is on gas models for spatially one-dimensional flow governed by hyperbolic balance
laws. We survey results on classical solutions as well as weak solutions. We present results
on well–posedness, controllability, feedback stabilization, the inclusion of uncertainty in the
models and numerical methods.Martin Gugat; Michael Hertypreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/339Thu, 08 Oct 2020 14:14:08 +0200Producer’s Best Response in Pay-as-clear Day-ahead Electricity Market with Uncertain Demand
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/336
We deal with several sources of uncertainty in electricity markets. The independent system operator (ISO) maximizes the social welfare using chance constraints to hedge against discrepancies between the estimated and real electricity
demand. We find an explicit solution of the ISO problem, and use it to tackle the
problem of a producer. In our model, production as well as income of a producer
are determined based on the estimated electricity demand predicted by the ISO, that
is unknown to producers. Thus, each producer is hedging against the uncertainty of
prediction of the demand using the value-at-risk approach. To illustrate our results, a
numerical study of a producer’s best response given a historical distribution of both
estimated and real electricity demand is provided.Martin Branda; René Henrion; Miroslav Pistekarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/336Tue, 06 Oct 2020 16:09:18 +0200Solving joint chance constrained problems using regularization and Benders' decomposition
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/335
In this paper we investigate stochastic programs with joint chance constraints. We consider discrete scenario set and reformulate the problem by adding auxiliary variables. Since the resulting problem has a difficult feasible set, we regularize it. To decrease the dependence on the scenario number, we propose a numerical method by iteratively solving a master problem while adding Benders cuts. We find the solution of the slave problem (generating the Benders cuts) in a closed form and propose a heuristic method to decrease the number of cuts. We perform a numerical study by increasing the number of scenarios and compare our solution with a solution obtained by solving the same problem with continuous distribution.Lukas Adam; Martin Branda; Holger Heitsch; René Henrionarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/335Tue, 06 Oct 2020 16:00:16 +0200Multi-period investment pathways - Modeling approaches to design distributed energy systems under uncertainty
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/334
Multi-modal distributed energy system planning is applied in the context of smart grids, industrial energy supply,and in the building energy sector. In real-world applications, these systems are commonly characterized by existing system structures of different age where monitoring and investment are conducted in a closed-loop, with the iterative possibility to invest. The literature contains two main approaches to approximate this computationally intensive multiperiod
investment problem. The first approach simplifies the temporal decision-making process collapsing the multistage decision to a two-stage decision, considering uncertainty in the second stage decision variables. The second approach considers multi-period investments under the assumption of perfect foresight. In this work, we propose a
multi-stage stochastic optimization problem that captures multi-period investment decisions under uncertainty and solves the problem to global optimality, serving as a first-best benchmark to the problem. To evaluate the performance of conventional approaches applied in a multi-year setup and to solve the multi-period problem at lower computational effort, we propose a rolling horizon heuristic that on the one hand reveals the performance of conventional approaches
applied in a multi-period set-up and on the other hand enables planners to identify approximate solutions to the original
multi-stage stochastic problem. Additionally, we consider an open-loop version of the rolling horizon algorithm to evaluate how single-period investments perform with respect to the entire scenario tree and compared to multi-period investments.
We conduct a real-world case study and investigate solution quality as well as the computational performance of the proposed approaches. Our findings indicate that the approximation of multi-period investments by two-stage stochastic approaches yield the best results regarding constraint satisfaction, while deterministic multi-period approximations yield
better economic and computational performance.Markus Bohlayer; Adrian Bürger; Markus Fleschutz; Marco Braun; Gregor Zöttlpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/334Mon, 05 Oct 2020 16:00:14 +0200Economic comparison of electric fuels produced at excellent locations for renewable energies: A Scenario for 2035
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/333
The use of electric fuels (e-fuels) enables CO2-neutral mobility and opens therefore an alternative to fossil-fuel-fired engines or battery-powered electric motors. This paper compares the cost-effectiveness of Fischer-Tropsch diesel, methanol, and hydrogen stored as cryogenic liquid (LH2) or in form of liquid organic hydrogen carriers (LOHCs). The production cost of those fuels are to a large extent driven by the energy-intensive electrolytic water splitting. The option of producing e-fuels in Germany competes with international locations with excellent conditions for renewable energy harvesting and thus very low levelized cost of electricity. We developed a mathematical model that covers the entire process chain. Starting with the production of the required resources such as fresh water, hydrogen, carbon dioxide, carbon monoxide, electrical and thermal energy, the subsequent chemical synthesis, the transport to filling stations in Germany and finally the energetic utilization of the fuels in the vehicle. We found that the choice of production site can have a major impact on the mobility cost using the respective fuels. Especially in case of diesel production, the levelized cost of electricity driven by the full load hours of the applied renewable energy source have a huge impact. An LOHC-based system is shown to be less dependent on the kind of electricity source compared to other technologies due to its comparatively low electricity consumption and the low cost for the hydrogenation units. The length of the transportation route and the price of the filling station infrastructure, on the other hand, clearly increase mobility cost for LOHC and LH2.Philipp Runge; Christian Sölch; Jakob Albert; Peter Wasserscheid; Gregor Zöttl; Veronika Grimmpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/333Mon, 05 Oct 2020 15:03:53 +0200Network structured kinetic models of social interactions
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/332
The aim of this paper is to study the derivation of appropriate meso- and macroscopic
models for interactions as appearing in social processes. There are two main characteristics
the models take into account, namely a network structure of interactions, which we treat
by an appropriate mesoscopic description, and a different role of interacting agents. The
latter differs from interactions treated in classical statistical mechanics in the sense that
the agents do not have symmetric roles, but there is rather an active and a passive agent.
We will demonstrate how a certain form of kinetic equations can be obtained to describe
such interactions at a mesoscopic level and moreover obtain macroscopic models from
monokinetics solutions of those.
The derivation naturally leads to systems of nonlocal reaction-diffusion equations (or
in a suitable limit local versions thereof), which can explain spatial phase separation
phenomena found to emerge from the microscopic interactions. We will highlight the
approach in three examples, namely the evolution and coarsening of dialects in human
language, the construction of social norms, and the spread of an epidemic.Martin Burgerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/332Sun, 04 Oct 2020 21:10:45 +0200Boundary feedback stabilization of a semilinear model for the flow in star-shaped gas networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/330
The flow of gas through a pipeline network
can be modelled by a
coupled system of 1-d quasilinear hyperbolic equations.
In this system, the influence of
certain source terms that model friction effects is essential.
Often for the solution of
control problems it is convenient to replace the quasilinear model
by a simpler semilinear model.
In this paper, we analyze the behavior of such
a semilinear model on a star-shaped network.
The model is derived from the diagonal form of the quasilinear model by replacing the eigenvalues by
the sound speed multiplied by
1 or -1 respectively.
Thus in the corresponding eigenvalues
the influence of the gas velocity is neglected,
which is justified in the applications
since it is much smaller than the sound speed in
the gas.
For a star-shaped network of horizontal pipes
for suitable coupling conditions
we present boundary feedback laws
that stabilize the system state
exponentially fast
to a position of rest
for sufficiently small initial data.
We show the exponential decay of
the $H^1$-norm
for arbitrarily long pipes.
This is remarkable since in general
even for linear systems, for certain source terms
the system can become exponentially unstable
if the space interval is too long.
Our proofs
are based upon
observability inequalities
for the $L^2$ and the $H^1$-norm.Martin Gugat; Jan Giesselmnannpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/330Sat, 03 Oct 2020 14:35:04 +0200The Finite-Time Turnpike Phenomenon for Optimal Control Problems: Stabilization by Non-Smooth Tracking Terms
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/329
In this paper, problems of optimal control are considered where in the objective function, in addition to the control cost there is a tracking term that measures the distance to a desired stationary state. The tracking term is given by some norm and therefore it is in general not differentiable. In the optimal control problem, the initial state is prescribed. We assume that the system is either exactly controllable in the classical sense or nodal profile controllable.
We show that both for systems that are governed by ordinary differential equations and for infinite-dimensional systems, for example for boundary control systems governed by the wave equation, under certain assumptions the optimal system state is steered exactly to the desired state after finite time.Martin Gugat; Michael Schuster; Enrique Zuazuapreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/329Fri, 02 Oct 2020 13:30:42 +0200Data-Driven Snapshot Calibration via Monotonic Feature Matching
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/327
Snapshot matrices of hyperbolic equations have a slow singular value decay, resulting in inefficient reduced-order models. We develop on the idea of inducing a faster singular value decay by computing snapshots on a transformed spatial domain, or the so-called snapshot calibration/transformation. We are particularly interested in problems involving shock collision, shock rarefaction-fan collision, shock formation, etc. For such problems, we propose a realizable algorithm to compute the spatial transform using monotonic feature matching. We consider discontinuities and kinks as features, and by carefully partitioning the parameter domain, we ensure that the spatial transform has properties that are desirable both from a theoretical and an implementation standpoint. We use these properties to prove that our method results in a fast $m$-width decay of a so-called calibrated manifold. A crucial observation we make is that due to calibration, the $m$-width does not only depend on $m$ but also on the accuracy of the full order model, which is in contrast to elliptic and parabolic problems that do not need calibration. The method we propose only requires the solution snapshots and not the underlying partial differential equation (PDE) and is therefore, data-driven. We perform several numerical experiments to demonstrate the effectiveness of our method.Neeraj Sarna; Jan Giesselmnann; Peter Bennerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/327Tue, 22 Sep 2020 08:36:06 +0200