About Dynamical Systems Appearing in the Microscopic Traffic Modeling

Motivated by microscopic traffic modeling, we analyze dynamical systems which have a piecewise linear concave dynamics not necessarily monotonic. We introduce a deterministic Petri net extension where edges may have negative weights. The dynamics of these Petri nets are well-defined and may be described by a generalized matrix with a submatrix in the standard algebra with possibly negative entries, and another submatrix in the minplus algebra. When the dynamics is additively homogeneous, a generalized additive eigenvalue may be introduced, and the ergodic theory may be used to define a growth rate under additional technical assumptions. In the traffic example of two roads with one junction, we compute explicitly the eigenvalue and we show, by numerical simulations, that these two quantities (the additive eigenvalue and the growth rate) are not equal, but are close to each other. With this result, we are able to extend the well-studied notion of fundamental traffic diagram (the average flow as a function of the car density on a road) to the case of two roads with one junction and give a very simple analytic approximation of this diagram where four phases appear with clear traffic interpretations. Simulations show that the fundamental diagram shape obtained is also valid for systems with many junctions. To simulate these systems, we have to compute their dynamics, which are not quite simple. For building them in a modular way, we introduce generalized parallel, series and feedback compositions of piecewise linear concave dynamics.Comment: PDF 38 page

Duality and separation theorems in idempotent semimodules

We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to separate a point from a convex set. We also show that the projection minimizes the analogue of Hilbert's projective metric. We develop more generally a theory of dual pairs for idempotent semimodules. We obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings. We illustrate the results by showing polyhedra and half-spaces over the max-plus semiring.Comment: 24 pages, 5 Postscript figures, revised (v2

Optimisation de la gestion d'une maison chauffée par le soleil et une pompe à chaleur

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Résolution numérique de problèmes de commande optimale de chaînes de Markov observées imparfaitement

Nous résolvons numériquement le problème de commande optimale de chaînes de Markov en observations incomplètes. Grâce à une quantification des filtres optimaux possibles on approxime le support de la loi du filtre optimal. On résout l'équation de la programmation sur cet espace d'états quantifiés. Sur deux exemples on montre l'effectivité de ce point de vue : - un problème d'embauche d'une secrétaire, - une problème de renouvellement optima

The Dynamic Equations of the Tree Morphogenesis GreenLab Model

We explicit the dynamic equations followed by a tree during its growth according to the Greenlab model. In a first part we explicit the Organogenesis equations. In the second part, we recall the equations which, using a macroscopic photosynthesis point of view, describe with a great precision the evolution of the organ sizes. In a third part we establish the morphogenesis equations describing the geometry of a tree. The three parts are illustrated with an example

Simple modelling and control of plasma current profile

International audienceThe purpose of this paper is to present a simplified model and control law of the current and temperature profile in a tokamak plasma. Based on a description of the plasma as a magnetised uid, the model is expressed in the form of coupled one dimensional transport-diffusion equations. A simple feedback is used to obtain a given stationary profile. The numerical simulations are done in the Scilab/Scicos environment

A Minplus Derivation of the Fundamental Car-Traffic Law

We give deterministic models and a stochastic model of the traffic on a circular road without overtaking. The average speed is the eigenvalue of a minplus matrix describing the dynamics of the system in the first case and a Lyapounov exponentof a minplus stochastic matrix in the second one.The eigenvalue and the Lyapounov exponent are computed explicitly. From these formulae we derive the fundamental law that links the flow to the density of vehicle on the road. Numerical experiments using the maxplus toolbox of Scilab confirm the theoretical results obtained

Traffic Assignment and Gibbs-Maslov Semirings

The Traffic Assignment problem consists in determining the routes used by sets of network users taking into account the link congestions. In deterministic modelling, Wardrop Equilibriums are computed. They can be reduced to huge non-linear multiflow problems in the simplest cases. In stochastic modelling, Logit Assignments are used. They are obtained, mainly, by substituting the minplus semiring by the «Gibbs-Maslov semirings»[ ], in the deterministic assignment computations

Piecewise linear concave dynamical systems appearing in the microscopic traffic modeling

Motivated by microscopic traffic modeling, we analyze dynamical systems which have a piecewise linear concave dynamics not necessarily monotonic. We introduce a deterministic Petri net extension where edges may have negative weights. The dynamics of these Petri nets are uniquely defined and may be described by a generalized matrix with a submatrix in the standard algebra with possibly negative entries, and another submatrix in the minplus algebra. When the dynamics is additively homogeneous, a generalized additive eigenvalue is introduced, and the ergodic theory is used to define a growth rate. In the traffic example of two roads with one junction, we compute explicitly the eigenvalue and we show, by numerical simulations, that these two quantities (the additive eigenvalue and the average growth rate) are not equal, but are close to each other. With this result, we are able to extend the well-studied notion of fundamental traffic diagram (the average flow as a function of the car density on a road) to the case of roads with a junction and give a very simple analytic approximation of this diagram where four phases appear with clear traffic interpretations. Simulations show that the fundamental diagram shape obtained is also valid for systems with many junctions

On the Convergence of the Algorithm for Bilevel Programming Problems by Clegg and Smith

In [1,2], Smith and colleagues present an algorithm for solving the bilevel programming problem. We show that the points reached by the algorithm are not stationary points of bilevel programs in general. We further show that, with a minor modification, this method can be expressed as an inexact penalty method for gap function-constrained bilevel programs