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.
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.
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.
We continue research on generalized macroscopic models of
conservation type as started in . In this paper we keep the
characteristic (for traffic) non-locality removed in  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.
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
 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
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 . 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.
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.
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  and Zhang
. 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
steady solutions that are stable.
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.
We present an extended discussion of a macroscopic traffic flow
model  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.