A weakly coupled model of differential equations for thief tracking
Simone Göttlich Camill Harter
Networks & Heterogeneous Media 2016, 11(3): 447-469 doi: 10.3934/nhm.2016004
In this work we introduce a novel model for the tracking of a thief moving through a road network. The modeling equations are given by a strongly coupled system of scalar conservation laws for the road traffic and ordinary differential equations for the thief evolution. A crucial point is the characterization at intersections, where the thief has to take a routing decision depending on the available local information. We develop a numerical approach to solve the thief tracking problem by combining a time-dependent shortest path algorithm with the numerical solution of the traffic flow equations. Various computational experiments are presented to describe different behavior patterns.
keywords: numerical simulations. Traffic flow networks thief trajectories
Optimal inflow control of production systems with finite buffers
Simone Göttlich Patrick Schindler
Discrete & Continuous Dynamical Systems - B 2015, 20(1): 107-127 doi: 10.3934/dcdsb.2015.20.107
We introduce the optimal inflow control problem for buffer restricted production systems involving a conservation law with discontinuous flux. Based on an appropriate numerical method inspired by the wave front tracking algorithm, we present two techniques to solve the optimal control problem efficiently. A numerical study compares the different optimization procedures and comments on their benefits and drawbacks.
keywords: numerical schemes conservation law with discontinuous flux optimal control. Production systems
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.
Evacuation dynamics influenced by spreading hazardous material
Simone Göttlich Sebastian Kühn Jan Peter Ohst Stefan Ruzika Markus Thiemann
Networks & Heterogeneous Media 2011, 6(3): 443-464 doi: 10.3934/nhm.2011.6.443
In this article, an evacuation model describing the egress in case of danger is considered. The underlying evacuation model is based on continuous network flows, while the spread of some gaseous hazardous material relies on an advection-diffusion equation. The contribution of this work is twofold. First, we introduce a continuous model coupled to the propagation of hazardous material where special cost functions allow for incorporating the predicted spread into an optimal planning of the egress. Optimality can thereby be understood with respect to two different measures: fastest egress and safest evacuation. Since this modeling approach leads to a pde/ode-restricted optimization problem, the continuous model is transferred into a discrete network flow model under some linearity assumptions. Second, it is demonstrated that this reformulation results in an efficient algorithm always leading to the global optimum. A computational case study shows benefits and drawbacks of the models for different evacuation scenarios.
keywords: Evacuation dynamic network flows optimization.
Semi-Markovian capacities in production network models
Simone Göttlich Stephan Knapp
Discrete & Continuous Dynamical Systems - B 2017, 22(9): 3235-3258 doi: 10.3934/dcdsb.2017090

In this paper, we focus on production network models based on ordinary and partial differential equations that are coupled to semi-Markovian failure rates for the processor capacities. This modeling approach allows for intermediate capacity states in the range of total breakdown to full capacity, where operating and down times might be arbitrarily distributed. The mathematical challenge is to combine the theory of semi-Markovian processes within the framework of conservation laws. We show the existence and uniqueness of such stochastic network solutions, present a suitable simulation method and explain the link to the common queueing theory. A variety of numerical examples emphasizes the characteristics of the proposed approach.

keywords: Production networks conservation laws semi-Markovian processes
Traffic light control: A case study
Simone Göttlich Ute Ziegler
Discrete & Continuous Dynamical Systems - S 2014, 7(3): 483-501 doi: 10.3934/dcdss.2014.7.483
This article is devoted to traffic flow networks including traffic lights at intersections. Mathematically, we consider a nonlinear dynamical traffic model where traffic lights are modeled as piecewise constant functions for red and green signals. The involved control problem is to find stop and go configurations depending on the current traffic volume. We propose a numerical solution strategy and present computational results.
keywords: conservation laws networks optimal control. Traffic problems
Time-continuous production networks with random breakdowns
Simone Göttlich Stephan Martin Thorsten Sickenberger
Networks & Heterogeneous Media 2011, 6(4): 695-714 doi: 10.3934/nhm.2011.6.695
Our main objective is the modelling and simulation of complex production networks originally introduced in [15, 16] with random breakdowns of individual processors. Similar to [10], the breakdowns of processors are exponentially distributed. The resulting network model consists of coupled system of partial and ordinary differential equations with Markovian switching and its solution is a stochastic process. We show our model to fit into the framework of piecewise deterministic processes, which allows for a deterministic interpretation of dynamics between a multivariate two-state process. We develop an efficient algorithm with an emphasis on accurately tracing stochastic events. Numerical results are presented for three exemplary networks, including a comparison with the long-chain model proposed in [10].
keywords: piecewise deterministic processes (PDPs). coupled PDE-ODE systems random breakdowns Markovian switching Production networks conservation laws
Optimization for a special class of traffic flow models: Combinatorial and continuous approaches
Simone Göttlich Oliver Kolb Sebastian Kühn
Networks & Heterogeneous Media 2014, 9(2): 315-334 doi: 10.3934/nhm.2014.9.315
In this article, we discuss the optimization of a linearized traffic flow network model based on conservation laws. We present two solution approaches. One relies on the classical Lagrangian formalism (or adjoint calculus), whereas another one uses a discrete mixed-integer framework. We show how both approaches are related to each other. Numerical experiments are accompanied to show the quality of solutions.
keywords: Traffic networks conservation laws combinatorial optimization. control of discretized PDEs adjoint calculus
Numerical discretization of Hamilton--Jacobi equations on networks
Simone Göttlich Ute Ziegler Michael Herty
Networks & Heterogeneous Media 2013, 8(3): 685-705 doi: 10.3934/nhm.2013.8.685
We discuss a numerical discretization of Hamilton--Jacobi equations on networks. The latter arise for example as reformulation of the Lighthill--Whitham--Richards traffic flow model. We present coupling conditions for the Hamilton--Jacobi equations and derive a suitable numerical algorithm. Numerical computations of travel times in a round-about are given.
keywords: Networks traffic flow Hamilton-Jacobi Equations.
Capacity drop and traffic control for a second order traffic model
Oliver Kolb Simone Göttlich Paola Goatin
Networks & Heterogeneous Media 2017, 12(4): 663-681 doi: 10.3934/nhm.2017027

In this paper, we illustrate how second order traffic flow models, in our case the Aw-Rascle equations, can be used to reproduce empirical observations such as the capacity drop at merges and solve related optimal control problems. To this aim, we propose a model for on-ramp junctions and derive suitable coupling conditions. These are associated to the first order Godunov scheme to numerically study the well-known capacity drop effect, where the outflow of the system is significantly below the expected maximum. Control issues such as speed and ramp meter control are also addressed in a first-discretize-then-optimize framework.

keywords: Traffic flow second order model on-ramp coupling numerical simulations optimal control

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