Fluid dynamic description of flocking via the Povzner–Boltzmann equation

Fornasier, Massimo; Haškovec, Jan; Toscani, Giuseppe

doi:10.1016/j.physd.2010.08.003

Cited by 95 publications

(56 citation statements)

References 22 publications

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“…Typical examples in socio-economical sciences, biology and engineering are represented by the problems of persuading voters to vote for a specific candidate, influencing buyers towards a given good or asset, forcing human crowds or a group of animals to follow a specific path or to reach a desired zone, or optimizing the traffic flow in road networks and supply chains [2,[6][7][8][9][10][11][12][13][14]. In this paper, we focus on control problems where the collective behaviour corresponds to the formation process of opinion consensus [15][16][17][18][19][20][21][22][23]. In particular, we consider models where the control strategy is based on hierarchical leadership.…”

Section: Introductionmentioning

confidence: 99%

Boltzmann-type control of opinion consensus through leaders

Albi

Pareschi

Zanella

2014

Phil. Trans. R. Soc. A.

114

159

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The study of formations and dynamics of opinions leading to the so-called opinion consensus is one of the most important areas in mathematical modelling of social sciences. Following the Boltzmann-type control approach recently introduced by the first two authors, we consider a group of opinion leaders who modify their strategy accordingly to an objective functional with the aim of achieving opinion consensus. The main feature of the Boltzmann-type control is that, owing to an instantaneous binary control formulation, it permits the minimization of the cost functional to be embedded into the microscopic leaders’ interactions of the corresponding Boltzmann equation. The related Fokker–Planck asymptotic limits are also derived, which allow one to give explicit expressions of stationary solutions. The results demonstrate the validity of the Boltzmann-type control approach and the capability of the leaders’ control to strategically lead the followers’ opinion.

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Section: Introductionmentioning

confidence: 99%

Boltzmann-type control of opinion consensus through leaders

Albi

Pareschi

Zanella

2014

Phil. Trans. R. Soc. A.

114

159

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“…The CS model has received lots of attention in literature from different perspectives, i.e., global and local flocking [8,18,19,32,34,36], collision avoiding [1,16], time-delay effect [9,10,21,24,27,44], hierarchical and rooted leadership, general digraph [20,38,39,47], application to flight navigation [43], noisy effects [2,17,23,33], mean-field limit [5,32,34], kinetic and hydrodynamic description [6,7,25,36,45], and variants of C-S model [41,42], etc. (see recent survey papers [14,40] for details).…”

Section: Introductionmentioning

confidence: 99%

Emergent behaviors of continuous and discrete thermomechanical Cucker–Smale models on general digraphs

Dong

Kim

2019

Math. Models Methods Appl. Sci.

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We present emergent dynamics of continuous and discrete thermomechanical Cucker-Smale(TCS) models equipped with temperature as an extra observable on general digraph. In previous literature, the emergent behaviors of the TCS models were mainly studied on a complete graph, or symmetric connected graphs. Under this symmetric setting, the total momentum is a conserved quantity. This determines the asymptotic velocity and temperature a priori using the initial data only. Moreover, this conservation law plays a crucial role in the flocking analysis based on the elementary ℓ 2 energy estimates. In this paper, we consider a more general connection topology which is registered by a general digraph, and the weights between particles are given to be inversely proportional to the metric distance between them. Due to this possible symmetry breaking in communication, the total momentum is not a conserved quantity, and this lack of conservation law makes the asymptotic velocity and temperature depend on the whole history of solutions. To circumvent this lack of conservation laws, we instead employ some tools from matrix theory on the scrambling matrices and some detailed analysis on the state-transition matrices. We present two sufficient frameworks for the emergence of mono-cluster flockings on a digraph for the continuous and discrete models. Our sufficient frameworks are given in terms of system parameters and initial data.Definition 1.1. [31, 35] Let C := (X, V, Θ) be a time-dependent configuration on the extended state space R N d × R N d × R d . Then the configuration C exhibits asymptotic mono-cluster flocking, if and only if the following conditions hold: sup 0≤t<∞ D(X(t)) < ∞, lim t→∞ D(V (t)) = 0, lim t→∞ D(Θ(t)) = 0. December 10, 2018 1:46 WSPC/INSTRUCTION FILE

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“…Equilibria and phase transitions in kinetic flocking models have been studied in [7,21,22]. Passage from kinetic to hydrodynamic descriptions of flocking has been investigated in [8,21,22,23,25,28]. Numerical simulation methods have been put forward in [2,29].…”

Section: Introductionmentioning

confidence: 99%

Kinetic Models for Topological Nearest-Neighbor Interactions

Blanchet

Degond

2017

J Stat Phys

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We consider systems of agents interacting through topological interactions. These have been shown to play an important part in animal and human behavior. Precisely, the system consists of a finite number of particles characterized by their positions and velocities. At random times a randomly chosen particle, the follower, adopts the velocity of its closest neighbor, the leader. We study the limit of a system size going to infinity and, under the assumption of propagation of chaos, show that the limit kinetic equation is a non-standard spatial diffusion equation for the particle distribution function. We also study the case wherein the particles interact with their K closest neighbors and show that the corresponding kinetic equation is the same. Finally, we prove that these models can be seen as a singular limit of the smooth rank-based model previously studied in Blanchet and Degond (J Stat Phys 163:41-60, 2016). The proofs are based on a combinatorial interpretation of the rank as well as some concentration of measure arguments.

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Fluid dynamic description of flocking via the Povzner–Boltzmann equation

Cited by 95 publications

References 22 publications

Boltzmann-type control of opinion consensus through leaders

Boltzmann-type control of opinion consensus through leaders

Emergent behaviors of continuous and discrete thermomechanical Cucker–Smale models on general digraphs

Kinetic Models for Topological Nearest-Neighbor Interactions

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