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This paper deals with various applications of conservation laws on networks. In particular we consider the car traffic, described by the Lighthill-Whitham-Richards model and by the Aw-Rascle-Zhang model, the telecommunication case, by using the model introduced by D'Apice-Manzo-Piccoli and, finally, the case of a gas pipeline, modeled by the classical $p$-system. For each of these models we present a review of some results about Riemann and Cauchy problems in the case of a network, formed by a single vertex with $n$ incoming and $m$ outgoing arcs.
We consider $n$ tubes exiting a junction and filled with a non viscous isentropic or isothermal fluid. In each tube a copy of the $p$-system in Euler coordinates is considered. The aim of the presentation is to compare three different notions of solutions at the junctions: p-solutions, Q-solutions and P-solutions.
The aim of this paper is to address the following questions: which models, among fluido-dynamic ones, are more appropriate to describe urban traffic? While a rich debate was developed for the complicate dynamics of highway traffic, some basic problems of urban traffic are not always appropriately discussed. We analyze many recent, and less recent, models focusing on three basic properties. The latter are necessary to reproduce correctly queue formation at lights and junctions, and their backward propagation on an urban network.
We prove existence and uniqueness of solutions, continuous dependence from the initial datum and stability with respect to the boundary condition in a class of initial--boundary value problems for systems of balance laws. The particular choice of the boundary condition allows to comprehend models with very different structures. In particular, we consider a juvenile-adult model, the problem of the optimal mating ratio and a model for the optimal management of biological resources. The stability result obtained allows to tackle various optimal management/control problems, providing sufficient conditions for the existence of optimal choices/controls.
We consider the Lighthill-Whitham-Richards traffic flow model on a network composed by an arbitrary number of incoming and outgoing arcs connected together by a node with a buffer. Similar to , we define the solution to the Riemann problem at the node and we prove existence and well posedness of solutions to the Cauchy problem, by using the wave-front tracking technique and the generalized tangent vectors.
We consider a hyperbolic conservation law with discontinuous flux. Such a partial differential equation arises in different applications, in particular we are motivated by a model of traffic flow. We provide a new formulation in terms of Riemann Solvers. Moreover, we determine the class of Riemann Solvers which provide existence and uniqueness of the corresponding weak entropic solutions.
We consider the Lighthill-Whitham-Richards traffic flow model on a network composed by a single junction $J$ with $n$ incoming roads, $m$ outgoing roads and $m$ buffers, one for each outgoing road. We introduce a concept solution at $J$, which is compared with that proposed in . Finally we study the Cauchy problem and, in the special case of $n \le 2$ and $m \le 2$, we prove existence of solutions to the Cauchy problem, via the wave-front tracking method.
This paper deals with coupling conditions between the classical microscopic Follow The Leader model and a phase transition (PT) model. We propose a solution at the interface between the two models. We describe the solution to the Riemann problem.
This work is devoted to the solution to Riemann Problems for the $p$-system at a junction, the main goal being the extension to the case of an ideal junction of the classical results that hold in the standard case.
On conditions that prevent steady-state controllability of certain linear partial differential equations
In this paper, we investigate the connections between controllability properties of distributed systems and existence of non zero entire functions subject to restrictions on their growth and on their sets of zeros. Exploiting these connections, we first show that, for generic bounded open domains in dimension $n\geq 2$, the steady--state controllability for the heat equation with boundary controls dependent only on time, does not hold. In a second step, we study a model of a water tank whose dynamics is given by a wave equation on a two-dimensional bounded open domain. We provide a condition which prevents steady-state controllability of such a system, where the control acts on the boundary and is only dependent on time. Using that condition, we prove that the steady-state controllability does not hold for generic tank shapes.
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