# American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

All Issues

## Networks & Heterogeneous Media

June 2017 , Volume 12 , Issue 2

Special issue on analysis and control on networks: Trends and perspectives

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2017, 12(2): i-ii doi: 10.3934/nhm.201702i +[Abstract](402) +[HTML](17) +[PDF](130.1KB)
Abstract:
2017, 12(2): 173-189 doi: 10.3934/nhm.2017007 +[Abstract](451) +[HTML](13) +[PDF](448.8KB)
Abstract:

The paper examines the model of traffic flow at an intersection introduced in [2], containing a buffer with limited size. As the size of the buffer approaches zero, it is proved that the solution of the Riemann problem with buffer converges to a self-similar solution described by a specific Limit Riemann Solver (LRS). Remarkably, this new Riemann Solver depends Lipschitz continuously on all parameters.

2017, 12(2): 191-215 doi: 10.3934/nhm.2017008 +[Abstract](487) +[HTML](15) +[PDF](533.7KB)
Abstract:

In this paper we formulate a theory of measure-valued linear transport equations on networks. The building block of our approach is the initial and boundary-value problem for the measure-valued linear transport equation on a bounded interval, which is the prototype of an arc of the network. For this problem we give an explicit representation formula of the solution, which also considers the total mass flowing out of the interval. Then we construct the global solution on the network by gluing all the measure-valued solutions on the arcs by means of appropriate distribution rules at the vertexes. The measure-valued approach makes our framework suitable to deal with multiscale flows on networks, with the microscopic and macroscopic phases represented by Lebesgue-singular and Lebesgue-absolutely continuous measures, respectively, in time and space.

2017, 12(2): 217-243 doi: 10.3934/nhm.2017009 +[Abstract](620) +[HTML](249) +[PDF](819.2KB)
Abstract:

We study numerically a coagulation-fragmentation model derived by Niwa [17] and further elaborated by Degond et al. [5]. In [5] a unique equilibrium distribution of group sizes is shown to exist in both cases of continuous and discrete group size distributions. We provide a numerical investigation of these equilibria using three different methods to approximate the equilibrium: a recursive algorithm based on the work of Ma et. al. [12], a Newton method and the resolution of the time-dependent problem. All three schemes are validated by showing that they approximate the predicted small and large size asymptotic behaviour of the equilibrium accurately. The recursive algorithm is used to investigate the transition from discrete to continuous size distributions and the time evolution scheme is exploited to show uniform convergence to equilibrium in time and to determine convergence rates.

2017, 12(2): 245-258 doi: 10.3934/nhm.2017010 +[Abstract](567) +[HTML](11) +[PDF](391.7KB)
Abstract:

We study well-posedness of scalar conservation laws with moving flux constraints. In particular, we show the Lipschitz continuous dependence of BV solutions with respect to the initial data and the constraint trajectory. Applications to traffic flow theory are detailed.

2017, 12(2): 259-275 doi: 10.3934/nhm.2017011 +[Abstract](413) +[HTML](12) +[PDF](433.9KB)
Abstract:

We consider the initial boundary value problem for the phase transition traffic model introduced in [9], which is a macroscopic model based on a 2×2 system of conservation laws. We prove existence of solutions by means of the wave-front tracking technique, provided the initial data and the boundary conditions have finite total variation.

2017, 12(2): 277-295 doi: 10.3934/nhm.2017012 +[Abstract](584) +[HTML](11) +[PDF](473.2KB)
Abstract:

In this paper we investigate a time-optimal control problem in the space of positive and finite Borel measures on \begin{document}$\mathbb R^d$\end{document}, motivated by applications in multi-agent systems. We provide a definition of admissible trajectory in the space of Borel measures in a particular non-isolated context, inspired by the so called optimal logistic problem, where the aim is to assign an initial amount of resources to a mass of agents, depending only on their initial position, in such a way that they can reach the given target with this minimum amount of supplies. We provide some approximation results connecting the microscopical description with the macroscopical one in the mass-preserving setting, we construct an optimal trajectory in the non isolated case and finally we are able to provide a Dynamic Programming Principle.

2017, 12(2): 297-317 doi: 10.3934/nhm.2017013 +[Abstract](531) +[HTML](19) +[PDF](454.6KB)
Abstract:

In this paper we present a macroscopic phase transition model with a local point constraint on the flow. Its motivation is, for instance, the modelling of the evolution of vehicular traffic along a road with pointlike inhomogeneities characterized by limited capacity, such as speed bumps, traffic lights, construction sites, toll booths, etc. The model accounts for two different phases, according to whether the traffic is low or heavy. Away from the inhomogeneities of the road the traffic is described by a first order model in the free-flow phase and by a second order model in the congested phase. To model the effects of the inhomogeneities we propose two Riemann solvers satisfying the point constraints on the flow.

2017, 12(2): 319-337 doi: 10.3934/nhm.2017014 +[Abstract](400) +[HTML](12) +[PDF](487.4KB)
Abstract:

We characterize the space of all exactly reachable states of an abstract boundary control system using a semigroup approach. Moreover, we study the case when the controls of the system are constrained to be positive. The abstract results are then applied to study flows in networks with static as well as dynamic boundary conditions.

2017  Impact Factor: 1.187