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49
A macroscopic crowd motion model of gradient flow type
 MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
, 2010
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Pedestrian flows in bounded domains with obstacles
 Contin. Mech. Thermodyn
"... Abstract. In this paper we systematically apply the mathematical structures by timeevolving measures developed in a previous work to the macroscopic modeling of pedestrian flows. We propose a discretetime Eulerian model, in which the space occupancy by pedestrians is described via a sequence of Ra ..."
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Cited by 28 (9 self)
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Abstract. In this paper we systematically apply the mathematical structures by timeevolving measures developed in a previous work to the macroscopic modeling of pedestrian flows. We propose a discretetime Eulerian model, in which the space occupancy by pedestrians is described via a sequence of Radon positive measures generated by a pushforward recursive relation. We assume that two fundamental aspects of pedestrian behavior rule the dynamics of the system: On the one hand, the will to reach specific targets, which determines the main direction of motion of the walkers; on the other hand, the tendency to avoid crowding, which introduces interactions among the individuals. The resulting model is able to reproduce several experimental evidences of pedestrian flows pointed out in the specialized literature, being at the same time much easier to handle, from both the analytical and the numerical point of view, than other models relying on nonlinear hyperbolic conservation laws. This makes it suitable to address twodimensional applications of practical interest, chiefly the motion of pedestrians in complex domains scattered with obstacles. 1.
Handling congestion in crowd motion modeling
 519, Special issue on Crowd Dynamics: Results and Perspectives
"... Abstract. We address here the issue of congestion in the modeling of crowd motion, in the nonsmooth framework: contacts between people are not anticipated and avoided, they actually occur, and they are explicitly taken into account in the model. We limit our approach to very basic principles in te ..."
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Cited by 15 (1 self)
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Abstract. We address here the issue of congestion in the modeling of crowd motion, in the nonsmooth framework: contacts between people are not anticipated and avoided, they actually occur, and they are explicitly taken into account in the model. We limit our approach to very basic principles in terms of behavior, to focus on the particular problems raised by the nonsmooth character of the models. We consider that individuals tend to move according to a desired, or spontanous, velocity. We account for congestion by assuming that the evolution realizes at each time an instantaneous balance between individual tendencies and global constraints (overlapping is forbidden): the actual velocity is defined as the closest to the desired velocity among all admissible ones, in a least square sense. We develop those principles in the microscopic and macroscopic settings, and we present how the framework of Wasserstein distance between measures allows to recover the sweeping process nature of the problem on the macroscopic level, which makes it possible to obtain existence results in spite of the nonsmooth character of the evolution process. Micro and macro approaches are compared, and we investigate the similarities together with deep differences of those two levels of description. 1. Introduction. Congestion
Control of the continuity equation with a non local flow
 ESAIM Control Optim. Calc. Var
"... Abstract This paper focuses on the analytical properties of the solutions to the continuity equation with non local flow. Our driving examples are a supply chain model and an equation for the description of pedestrian flows. To this aim, we prove the well posedness of weak entropy solutions in a cl ..."
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Cited by 15 (6 self)
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Abstract This paper focuses on the analytical properties of the solutions to the continuity equation with non local flow. Our driving examples are a supply chain model and an equation for the description of pedestrian flows. To this aim, we prove the well posedness of weak entropy solutions in a class of equations comprising these models. Then, under further regularity conditions, we prove the differentiability of solutions with respect to the initial datum and characterize this derivative. A necessary condition for the optimality of suitable integral functionals then follows. Mathematics Subject Classification: 35L65, 49K20, 93C20
Heterophilious dynamics enhances consensus
, 2013
"... We review a general class of models for selforganized dynamics based on alignment. The dynamics of such systems is governed solely by interactions among individuals or “agents”, with the tendency to adjust to their ‘environmental averages’. This, in turn, leads to the formation of clusters, e.g., ..."
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Cited by 15 (2 self)
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We review a general class of models for selforganized dynamics based on alignment. The dynamics of such systems is governed solely by interactions among individuals or “agents”, with the tendency to adjust to their ‘environmental averages’. This, in turn, leads to the formation of clusters, e.g., colonies of ants, flocks of birds, parties of people, etc. A natural question which arises in this context is to understand when and how clusters emerge through the selfalignment of agents, and what type of “rules of engagement ” influence the formation of such clusters. Of particular interest to us are cases in which the selforganized behavior tends to concentrate into one cluster, reflecting a consensus of opinions, flocking or concentration of other positions intrinsic to the dynamics. Many standard models for selforganized dynamics in social, biological and physical science assume that the intensity of alignment increases as agents get closer, reflecting a common tendency to align with those who think or act alike. Moreover, “Similarity breeds connection,” reflects our intuition that increasing the intensity of alignment as the difference of positions decreases, is more likely to lead to a consensus. We argue here that the converse is true: when the dynamics is driven by local interactions, it is more likely to approach a consensus when the interactions among agents increase as a function of their difference in position. Heterophily — the tendency to bond more with those who are different rather than with those who are similar, plays a decisive rôle in the process of clustering. We point out that the number of clusters in heterophilious dynamics decreases as the heterophily dependence among agents increases. In particular, sufficiently strong heterophilious interactions enhance consensus.
Modeling MicroMacro Pedestrian Counterflow
 in Heterogeneous Domains, Nonlinear Phenomena in Complex Systems, Volume 14, Number 1
, 2011
"... Abstract We present a micromacro strategy able to describe the dynamics of crowds in heterogeneous media. Herein we focus on the example of pedestrian counterflow. The main working tools include the use of mass and porosity measures together with their transport as well as suitable application of ..."
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Abstract We present a micromacro strategy able to describe the dynamics of crowds in heterogeneous media. Herein we focus on the example of pedestrian counterflow. The main working tools include the use of mass and porosity measures together with their transport as well as suitable application of a version of RadonNikodym Theorem formulated for finite measures. Finally, we illustrate numerically our microscopic model and emphasize the effects produced by an implicitly defined social velocity.
Modeling selforganization in pedestrians and animal groups from macroscopic and microscopic viewpoints
 Mathematical Modeling of Collective Behavior in SocioEconomic and Life Sciences, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser
, 2010
"... Abstract. This paper is concerned with mathematical modeling of intelligent systems, such as human crowds and animal groups. In particular, the focus is on the emergence of different selforganized patterns from nonlocality and anisotropy of the interactions among individuals. A mathematical techni ..."
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Cited by 12 (6 self)
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Abstract. This paper is concerned with mathematical modeling of intelligent systems, such as human crowds and animal groups. In particular, the focus is on the emergence of different selforganized patterns from nonlocality and anisotropy of the interactions among individuals. A mathematical technique by timeevolving measures is introduced to deal with both macroscopic and microscopic scales within a unified modeling framework. Then selforganization issues are investigated and numerically reproduced at the proper scale, according to the kind of agents under consideration. 1. Selforganization in manyparticle systems One of the most outstanding expressions of intelligence in nonclassical physical systems, such as human crowds or animal groups, is their selforganization ability. Selforganization means that the individuals composing the system can give rise to complex patterns without using intercommunication as an essential mechanism. For instance, in normal conditions pedestrians are known to arrange in specific patterns, chiefly lanes (cf. Fig. 1ab), as demonstrated by many experimental investigations [19, 20, 23, 28, 29]. Lane formation may be fostered by a suitable setup of the space, as reported in [19, 23]: a test performed in a tunnel connecting two subway stations in Budapest showed that a series of columns, placed in the middle of the walkway, induce pedestrians to organize in two oppositely walking lanes, preventing each of them to expand up to the full width of the corridor. More in general, lanes form also spontaneously, i.e., without the need for being triggered by environmental factors, provided the density of pedestrians is sufficiently large [20]. This is particularly evident if one considers the case of two groups of people, walking in opposite directions, which meet and cross (see also [28]). Grouping and selforganization are well known and largely observed also in animals, see for example [34]. These phenomena are in fact ubiquitous, ranging from bird flocks in the sky to migrating lobsters on the sea floor. Many papers on this
An Eulerian Approach to the Analysis of Krause’s Consensus Model
 SIAM J. Control Optim
, 2012
"... Abstract. In this paper we analyze a class of multiagent consensus dynamical systems inspired by Krause’s original model. As in Krause’s, the basic assumption is the socalled bounded confidence: two agents can influence each other only when their state values are below a given distance threshold R ..."
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Cited by 11 (0 self)
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Abstract. In this paper we analyze a class of multiagent consensus dynamical systems inspired by Krause’s original model. As in Krause’s, the basic assumption is the socalled bounded confidence: two agents can influence each other only when their state values are below a given distance threshold R. We study the system under an Eulerian point of view considering (possibly continuous) probability distributions of agents and we present original convergence results. The limit distribution is always necessarily a convex combination of delta functions at least R far apart from each other: in other terms these models are locally aggregating. The Eulerian perspective provides the natural framework for designing a numerical algorithm, by which we obtain several simulations in 1 and 2 dimensions.