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CRITICAL THRESHOLDS IN FLOCKING HYDRODYNAMICS WITH NONLOCAL ALIGNMENT
"... We study the largetime behavior of Eulerian systems augmented with nonlocal alignment. Such systems arise as hydrodynamic descriptions of agentbased models for selforganized dynamics, e.g., CuckerSmale and MotschTadmor models [4, 17]. We prove that in analogy with the agentbased models, the ..."
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We study the largetime behavior of Eulerian systems augmented with nonlocal alignment. Such systems arise as hydrodynamic descriptions of agentbased models for selforganized dynamics, e.g., CuckerSmale and MotschTadmor models [4, 17]. We prove that in analogy with the agentbased models, the presence of nonlocal alignment enforces strong solutions to selforganize into a macroscopic flock. This then raises the question of existence of such strong solutions. We address this question in one and twodimensional setups, proving global regularity for subcritical initial data. Indeed, we show that there exist critical thresholds in the phase space of initial configuration which dictate the global regularity vs. a finite time blowup. In particular, we explore the regularity of nonlocal alignment in the presence of vacuum.
Sparse Stabilization of Dynamical Systems Driven by Attraction and Avoidance Forces ⋆
"... We address dynamical systems of agents driven by attraction and repulsion forces, modelling cohesion and collision avoidance. When the total energy, which is composed of a kinetic part and a geometrical part describing the balance between attraction and repulsion forces, is below a certain threshold ..."
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We address dynamical systems of agents driven by attraction and repulsion forces, modelling cohesion and collision avoidance. When the total energy, which is composed of a kinetic part and a geometrical part describing the balance between attraction and repulsion forces, is below a certain threshold, then it is known that the agents will converge to a dynamics where mutual space confinement is guaranteed. In this paper we question the construction of a stabilization strategy, which requires the minimal amount of external intervention for nevertheless inducing space confinement, also when the initial energy threshold is violated. Our main result establishes that if the initial energy exceeds the threshold mainly because of its kinetic component, then a sparse control instantaneously applied with enough strength on the most rowdy agent, i.e., the one with maximal speed, will be able to steer in finite time the system to an energy level under the threshold.
Towards a unified theory of consensus
, 2014
"... ISR develops, applies and teaches advanced methodologies of design and analysis to solve complex, hierarchical, heterogeneous and dynamic problems of engineering technology and systems for industry and government. ISR is a permanent institute of the University of Maryland, within the ..."
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ISR develops, applies and teaches advanced methodologies of design and analysis to solve complex, hierarchical, heterogeneous and dynamic problems of engineering technology and systems for industry and government. ISR is a permanent institute of the University of Maryland, within the
154 PUBLICATIONS 1,483 CITATIONS SEE PROFILE
, 2014
"... Kinetic description of optimal control problems and applications to opinion consensus ..."
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Kinetic description of optimal control problems and applications to opinion consensus
MULTISCALE PROBLEMS ON COLLECTIVE DYNAMICS AND IMAGE PROCESSING: THEORY, ANALYSIS AND NUMERICS
, 2014
"... Multiscale problems appear in many contexts. In this thesis, we study two different subjects involving multiscale problems: (i) collective dynamics, and (ii) image processing. For collective dynamics, we concentrate on flocking models, in particular, CuckerSmale and MotschTadmor systems. These ..."
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Multiscale problems appear in many contexts. In this thesis, we study two different subjects involving multiscale problems: (i) collective dynamics, and (ii) image processing. For collective dynamics, we concentrate on flocking models, in particular, CuckerSmale and MotschTadmor systems. These models characterize the emergent behaviors of selforganized dynamics. We study flocking systems in three different scales, from microscopic agentbased models, through mesoscopic kineitc discriptions, to macroscopic fluid systems. Global existence theories are developed for all three scales, with the proof of asymptotic flocking behaviors. In the macroscopic level, a critical threhold phenomenon is addressed to obtain global regularity. Similar idea is implemented to other fluid systems as well, like EulerPoisson equations. In the kinetic level, a discontinuous Galerkin method is introduced to overcome the numerical difficulty due to the precence of δsingularity. For image processing, we apply the idea of multiscale image representation to construct uniformly bounded solutions for div U = F. Despite the fact that the equation
PROBLEMS IN DISTRIBUTED CONTROL SYSTEMS, CONSENSUS AND FLOCKING NETWORKS
, 2015
"... The present thesis discusses the consensus problem from a unification perspective. A general stability theory is developed discussing the majority of linear and nonlinear consensus networks with emphasis on the rate of convergence as an explicit estimate of the systems ’ parameters. The discussion b ..."
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The present thesis discusses the consensus problem from a unification perspective. A general stability theory is developed discussing the majority of linear and nonlinear consensus networks with emphasis on the rate of convergence as an explicit estimate of the systems ’ parameters. The discussion begins from the classical deterministic linear consensus problem in discrete and continuoustime setting. Vital assumptions are dropped and new types of nonuniform convergence are proven. All the related past results turn out to be only special cases of the developed framework, the central contribution of which is the derivation of explicit estimates on the rate of convergence. We proceed with the study of communication regimes that are governed by stochastic measures and we show that this setup is general enough to include many proposed stochastic settings as special cases. We highlight the strong interdependence between stochastic and deterministic signals and comment on how the imposed probabilistic regularity simply recaptures the deterministic sufficient conditions for consensus. An important variant of the linear model is the delayed one where it is dis
MeanField Pontryagin Maximum Principle
, 2015
"... We derive a Maximum Principle for optimal control problems with constraints given by the coupling of a system of ODEs and a PDE of Vlasovtype. Such problems arise naturally as Γlimits of optimal control problems subject to ODE constraints, modeling, for instance, external interventions on crowd d ..."
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We derive a Maximum Principle for optimal control problems with constraints given by the coupling of a system of ODEs and a PDE of Vlasovtype. Such problems arise naturally as Γlimits of optimal control problems subject to ODE constraints, modeling, for instance, external interventions on crowd dynamics. We obtain these firstorder optimality conditions in the form of Hamiltonian flows in the Wasserstein space of probability measures with forwardbackward boundary conditions with respect to the first and second marginals, respectively. In particular, we recover the equations and their solutions by means of a constructive procedure, which can be seen as the meanfield limit of the Pontryagin Maximum Principle applied to the discrete optimal control problems, under a suitable scaling of the adjoint variables.
Modeling opinion dynamics: how the network enhances consensus
"... In this paper we analyze emergent collective phenomena in the evolution of opinions in a society structured into few interacting nodes of a network. The presented mathematical structure combines two dynamics: a first one on each single node and a second one among the nodes, i.e. in the network. The ..."
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In this paper we analyze emergent collective phenomena in the evolution of opinions in a society structured into few interacting nodes of a network. The presented mathematical structure combines two dynamics: a first one on each single node and a second one among the nodes, i.e. in the network. The aim of the model is to analyze the effect of a network structure on a society with respect to opinion dynamics and we show some numerical simulations addressed in this direction, i.e. comparing the emergent behaviors of a consensusdissent dynamic on a single node when the effect of the network is not considered, with respect to the emergent behaviors when the effect of a network structure linking few interacting nodes is considered. We adopt the framework of the Kinetic Theory for Active Particles (KTAP), deriving a general mathematical structure which allows to deal with nonlinear features of the interactions and representing the conceptual framework toward the derivation of specific models. A specific model is derived from the general mathematical structure by introducing a consensusdissent dynamics of interactions and a qualitative analysis is given. 1Keywords and phrases: Active particles; opinion dynamics; nonlinear interactions; network.