Averaged kinetic models for flows on unstructured networks
Michael Herty Christian Ringhofer
Kinetic & Related Models 2011, 4(4): 1081-1096 doi: 10.3934/krm.2011.4.1081
We derive a kinetic equation for flows on general, unstructured networks with applications to production, social and transportation networks. This model allows for a homogenization procedure, yielding a macroscopic transport model for large networks on large time scales.
keywords: Network dynamics kinetic theory asymptotic analysis.
The coolest path problem
Martin Frank Armin Fügenschuh Michael Herty Lars Schewe
Networks & Heterogeneous Media 2010, 5(1): 143-162 doi: 10.3934/nhm.2010.5.143
We introduce the coolest path problem, which is a mixture of two well-known problems from distinct mathematical fields. One of them is the shortest path problem from combinatorial optimization. The other is the heat conduction problem from the field of partial differential equations. Together, they make up a control problem, where some geometrical object traverses a digraph in an optimal way, with constraints on intermediate or the final state. We discuss some properties of the problem and present numerical solution techniques. We demonstrate that the problem can be formulated as a linear mixed-integer program. Numerical solutions can thus be achieved within one hour for instances with up to 70 nodes in the graph.
keywords: Shortest Path. Integer Programming Heat Equation
Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws
Mapundi K. Banda Michael Herty
Mathematical Control & Related Fields 2013, 3(2): 121-142 doi: 10.3934/mcrf.2013.3.121
Suitable numerical discretizations for boundary control problems of systems of nonlinear hyperbolic partial differential equations are presented. Using a discrete Lyapunov function, exponential decay of the discrete solutions of a system of hyperbolic equations for a family of first--order finite volume schemes is proved. The decay rates are explicitly stated. The theoretical results are accompanied by computational results.
keywords: Conservation laws Lyapunov functions. controllability
Analytical and numerical investigations of refined macroscopic traffic flow models
Michael Herty Reinhard Illner
Kinetic & Related Models 2010, 3(2): 311-333 doi: 10.3934/krm.2010.3.311
We continue research on generalized macroscopic models of conservation type as started in [15]. In this paper we keep the characteristic (for traffic) non-locality removed in [15] by Taylor expansion and discuss the merits and problems of such an expansion. We observe that the models satisfy maximum principles and conclude that "triggers'' are needed in order to cause traffic jams (braking waves) in traffic guided by such models. Several such triggers are introduced and discussed. The models are refined further in order to properly address non-monotonic (in speed) traffic regimes, and the inclusion of an individual reaction time is discussed in the context of a braking wave. A number of numerical experiments are conducted to exhibit our findings.
keywords: Fokker-Planck Models Mathematical Modeling. Traffic Flow
Optimal control for continuous supply network models
Claus Kirchner Michael Herty Simone Göttlich Axel Klar
Networks & Heterogeneous Media 2006, 1(4): 675-688 doi: 10.3934/nhm.2006.1.675
We consider a supply network where the flow of parts can be controlled at the vertices of the network. Based on a coarse grid discretization provided in [6] we derive discrete adjoint equations which are subsequently validated by the continuous adjoint calculus. Moreover, we present numerical results concerning the quality of approximations and computing times of the presented approaches.
keywords: conservation laws networks Supply chains adjoint calculus.
A smooth model for fiber lay-down processes and its diffusion approximations
Michael Herty Axel Klar Sébastien Motsch Ferdinand Olawsky
Kinetic & Related Models 2009, 2(3): 489-502 doi: 10.3934/krm.2009.2.489
In this paper we improve and investigate a stochastic model and its associated Fokker-Planck equation for the lay-down of fibers on a conveyor belt in the production process of nonwoven materials which has been developed in [2]. The model is based on a stochastic differential equation taking into account the motion of the fiber under the influence of turbulence. In the present paper we remove an obvious drawback of the model, namely the non-differentiability of the paths of the process. We develop a model with smoother trajectories and investigate the relations between the different models looking at different scalings and diffusion approximations. Moreover, we compare the numerical results to simulations of the full physical process.
keywords: diffusion approximation. Fokker-Planck equations fiber dynamics
Performance bounds for the mean-field limit of constrained dynamics
Michael Herty Mattia Zanella
Discrete & Continuous Dynamical Systems - A 2017, 37(4): 2023-2043 doi: 10.3934/dcds.2017086

In this work we are interested in the mean-field formulation of kinetic models under control actions where the control is formulated through a model predictive control strategy (MPC) with varying horizon. The relation between the (usually hard to compute) optimal control and the MPC approach is investigated theoretically in the mean-field limit. We establish a computable and provable bound on the difference in the cost functional for MPC controlled and optimal controlled system dynamics in the mean-field limit. The result of the present work extends previous findings for systems of ordinary differential equations. Numerical results in the mean-field setting are given.

keywords: Model predictive control mean-field limits consensus models
On Stop-and-Go waves in dense traffic
Michael Herty Reinhard Illner
Kinetic & Related Models 2008, 1(3): 437-452 doi: 10.3934/krm.2008.1.437
From a Vlasov-type kinetic equation with nonlocal braking and acceleration forces, taken as a traffic model for higher densities, we derive macroscopic equations generalizing the second order model of conservation laws suggested by Aw and Rascle [1] and Zhang [19]. The nonlocality remains present in these equations, but more conventional, local equations are derived by using suitable Taylor expansion. A second order model of this type is discussed in some detail and is shown to possess traveling wave solutions that resemble stop-and-go waves in dense traffic. A phase space analysis suggests that inside the class of such traveling waves there are steady solutions that are stable.
keywords: nonlinear stability stop-and-go waves traffic flow nonlocal equations.
Fokker-Planck asymptotics for traffic flow models
Michael Herty Lorenzo Pareschi
Kinetic & Related Models 2010, 3(1): 165-179 doi: 10.3934/krm.2010.3.165
Starting from microscopic interaction rules we derive kinetic models of Fokker-Planck type for vehicular traffic flow. The derivation is based on taking a suitable asymptotic limit of the corresponding Boltzmann model. As particular cases, the derived models comprise existing models. New Fokker-Planck models are also given and their differences to existing models are highlighted. Finally, we report on numerical experiments.
keywords: Fokker-Planck Equations. Traffic Flow Modeling
Coupling of non-local driving behaviour with fundamental diagrams
Michael Herty Reinhard Illner
Kinetic & Related Models 2012, 5(4): 843-855 doi: 10.3934/krm.2012.5.843
We present an extended discussion of a macroscopic traffic flow model [18] which includes non-local and relaxation terms for vehicular traffic flow on unidirectional roads. The braking and acceleration forces are based on a behavioural model and on free flow dynamics. The latter are modelled by using different fundamental diagrams. Numerical investigations for different situations illustrate the properties of the mathematical model. In particular, the emergence of stop-and-go waves is observed for suitable parameter ranges.
keywords: Traffic flow mathematical modeling.

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