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Networks & Heterogeneous Media

2007 , Volume 2 , Issue 4

Special Issue on
Modelling and Control of Physical Networks

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Preface
Alexandre M. Bayen , Naomi Ehrich Leonard and  Benedetto Piccoli
2007, 2(4): i-ii doi: 10.3934/nhm.2007.2.4i +[Abstract](20) +[PDF](47.3KB)
Abstract:
This special issue was linked to a workshop, having the same title, held at the Centro De Giorgi of the Scuola Normale Superiore di Pisa on April 6th 2007 and centered around the broad theme: modelling and control of physical networks. The workshop was a one-day satellite event of Hybrid Systems: Computation and Control (HSCC 2007). We want to acknowledge sponsored projects for this workshop, in particular: Italian project PRIN2005 “Metodi di viscosit, metrici e di teoria del controllo in equazioni alle derivate parziali nonlineari”, coordinated by I. Capuzzo Dolcetta; INDAM Project 2005 “Traffic flows and optimization on complex networks”, coordinated by B. Piccoli (www.altamatematica.it); Istituto per le Applicazioni del Calcolo “Mauro Picone” (www.iac.cnr.it). More information can be found at www.crm.sns.it.

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Comparison of the performance of four Eulerian network flow models for strategic air traffic management
Dengfeng Sun , Issam S. Strub and  Alexandre M. Bayen
2007, 2(4): 569-595 doi: 10.3934/nhm.2007.2.569 +[Abstract](58) +[PDF](1308.3KB)
Abstract:
Four Eulerian network models are implemented to model high altitude air traffic flow. Three of the models use the framework of discrete time dynamical systems, while the fourth consists of a network of partial differential equations. The construction of these models is done using one year of air traffic data. The four models are applied to high altitude traffic for six Air Route Traffic Control Centers in the National Airspace System and surrounding airspace. Simulations are carried out for a full day of data for each of the models, to assess their predictive capabilities. The models’ predictions are compared to the recorded flight data. Several error metrics are used to characterize the relative accuracy of the models. The efficiency of the respective models is also compared in terms of computational time and memory requirements for the scenarios of interest. Control strategies are designed and implemented on similar benchmark scenarios for two of the models. They use techniques such as adjoint-based optimization, as well as mixed integer linear programming. A discussion of the four models’ structural differences explains why one model may outperform another.
Stable synchronization of rigid body networks
Sujit Nair and  Naomi Ehrich Leonard
2007, 2(4): 597-626 doi: 10.3934/nhm.2007.2.597 +[Abstract](35) +[PDF](422.5KB)
Abstract:
We address stable synchronization of a network of rotating and translating rigid bodies in three-dimensional space. Motivated by applications that require coordinated spinning spacecraft or diving underwater vehicles, we prove control laws that stably couple and coordinate the dynamics of multiple rigid bodies. We design decentralized, energy shaping control laws for each individual rigid body that depend on the relative orientation and relative position of its neighbors. Energy methods are used to prove stability of the coordinated multi-body dynamical system. To prove exponential stability, we break symmetry and consider a controlled dissipation term that requires each individual to measure its own velocity. The control laws are illustrated in simulation for a network of spinning rigid bodies.
On rendezvous control with randomly switching communication graphs
Luca Schenato and  Sandro Zampieri
2007, 2(4): 627-646 doi: 10.3934/nhm.2007.2.627 +[Abstract](39) +[PDF](305.5KB)
Abstract:
In this paper we analyze randomized coordination control strategies for the rendezvous problem of multiple agents with unknown initial positions. The performance of these control strategies is measured in terms of three metrics: average relative agents’ distance, total input energy consumption, and number of packets per unit time that each agent can receive from the other agents. By considering an LQ-like performance index, we show that a-priori knowledge about the first and second order statistics of agents’ initial position can greatly improve performance as compared to rendezvous control strategies based only on relative distance feedback. Moreover, we show that randomly switching communication topologies, as compared to static communication topologies, require very little information exchange to achieve high performance even when the number of agents grows very large.
Spatial instabilities and size limitations of flocks
J. J. P. Veerman , B. D. Stošić and  A. Olvera
2007, 2(4): 647-660 doi: 10.3934/nhm.2007.2.647 +[Abstract](33) +[PDF](274.0KB)
Abstract:
The movement of flocks with a single leader (and a directed path from it to every agent) can be stabilized over time as has been shown before (for details see [3] and prior references therein, shorter descriptions are given in [1, 4]). But for large flocks perturbations in the movement of the leader may nonetheless grow to a considerable size as they propagate throughout the flock and before they die out over time. We calculate the effect of this “finite size resonance” in two simple cases, and indicate two applications of these ideas. The first is that if perturbations grow as the size of the flock gets larger, then the size of the flock will have a natural limitation. Our examples suggest that for flocks with a symmetric communication graph perturbations tend to grow much slower than in the asymmetric case. The second application concerns a simple traffic-like problem. Suppose the leader accelerates from standstill to a given velocity and a large flock is supposed to follow it. The acceleration of the leader is the ‘perturbation’.
A continuum-discrete model for supply chains dynamics
Gabriella Bretti , Ciro D’Apice , Rosanna Manzo and  Benedetto Piccoli
2007, 2(4): 661-694 doi: 10.3934/nhm.2007.2.661 +[Abstract](118) +[PDF](2670.2KB)
Abstract:
This paper is focused on continuum-discrete models for supply chains. In particular, we consider the model introduced in [10], where a system of conservation laws describe the evolution of the supply chain status on sub-chains, while at some nodes solutions are determined by Riemann solvers. Fixing the rule of flux maximization, two new Riemann Solvers are defined. We study the equilibria of the resulting dynamics, moreover some numerical experiments on sample supply chains are reported. We provide also a comparison, both of equilibria and experiments, with the model of [15].
Null controllability of degenerate parabolic operators with drift
Piermarco Cannarsa , Genni Fragnelli and  Dario Rocchetti
2007, 2(4): 695-715 doi: 10.3934/nhm.2007.2.695 +[Abstract](35) +[PDF](267.6KB)
Abstract:
We give null controllability results for some degenerate parabolic equations in non divergence form with a drift term in one space dimension. In particular, the coefficient of the second order term may degenerate at the extreme points of the space domain. For this purpose, we obtain an observability inequality for the adjoint problem using suitable Carleman estimates.
Robust feedforward boundary control of hyperbolic conservation laws
Xavier Litrico , Vincent Fromion and  Gérard Scorletti
2007, 2(4): 717-731 doi: 10.3934/nhm.2007.2.717 +[Abstract](46) +[PDF](367.8KB)
Abstract:
The paper proposes a feedforward boundary control to reject measured disturbances for systems modelled by hyperbolic partial differential equations obtained from conservation laws. The controller design is based on fre- quency domain methods. Perfect rejection of measured perturbations at one boundary is obtained by controlling the other boundary. This result is then extended to design robust open-loop controller when the model of the system is not perfectly known, e.g. in high frequencies. Frequency domain comparisons and time-domain simulations illustrates the good performance of the feedforward boundary controller.
Adjoint calculus for optimization of gas networks
Michael Herty and  Veronika Sachers
2007, 2(4): 733-750 doi: 10.3934/nhm.2007.2.733 +[Abstract](125) +[PDF](537.9KB)
Abstract:
We consider an optimization problem arising in the context of gas transport in pipe networks. To compensate the pressure loss due to friction and to guarantee a desired (time dependent) outflow profile, compressor stations are included in the network. These compressor stations are relatively cost-intensive, so that a cost effective control is required. In the presented model the compressors are special vertices of the network. We derive an adjoint calculus for gas networks to solve the optimization problem and prove well–posedness of forward and adjoint coupling conditions. Furthermore, numerical examples illustrate the obtained results.
Lyapunov stability analysis of networks of scalar conservation laws
Georges Bastin , B. Haut , Jean-Michel Coron and  Brigitte d'Andréa-Novel
2007, 2(4): 751-759 doi: 10.3934/nhm.2007.2.751 +[Abstract](64) +[PDF](180.6KB)
Abstract:
It is shown how an entropy-based Lyapunov function can be used for the stability analysis of equilibria in networks of scalar conservation laws. The analysis gives a sufficient stability condition which is weaker than the condition which was previously known in the literature. Various extensions and generalisations are briefly discussed. The approach is illustrated with an application to ramp-metering control of road traffic networks.
Asymptotical compliance optimization for connected networks
Giuseppe Buttazzo and  Filippo Santambrogio
2007, 2(4): 761-777 doi: 10.3934/nhm.2007.2.761 +[Abstract](42) +[PDF](222.2KB)
Abstract:
We consider the problem of the optimal location of a Dirichlet region in a two-dimensional domain $\Omega$ subject to a force $f$ in order to minimize the compliance of the configuration. The class of admissible Dirichlet regions among which we look for the optimum consists of all one-dimensional connected sets (networks) of a given length $L$. Then we let $L$ tend to infinity and look for the $\Gamma$-limit of suitably rescaled functionals, in order to identify the asymptotical distribution of the optimal networks. The asymptotically optimal shapes are discussed as well and links with average distance problems are provided.

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