scholarly journals Optimality, reduction and collective motion

2015 ◽  
Vol 471 (2177) ◽  
pp. 20140606 ◽  
Author(s):  
Eric W. Justh ◽  
P. S. Krishnaprasad
Keyword(s):  
Collective Motion ◽  
Optimal Control Problems ◽  
Coupling Parameter ◽  
Rigid Motion ◽  
Particle Model ◽  
Explicit Integration ◽  
Hidden Symmetry ◽  
Necessary Conditions For Optimality ◽  
Special Solutions ◽  
Optimality Principles

The planar self-steering particle model of agents in a collective gives rise to dynamics on the N -fold direct product of SE (2), the rigid motion group in the plane. Assuming a connected, undirected graph of interaction between agents, we pose a family of symmetric optimal control problems with a coupling parameter capturing the strength of interactions. The Hamiltonian system associated with the necessary conditions for optimality is reducible to a Lie–Poisson dynamical system possessing interesting structure. In particular, the strong coupling limit reveals additional (hidden) symmetry, beyond the manifest one used in reduction: this enables explicit integration of the dynamics, and demonstrates the presence of a ‘master clock’ that governs all agents to steer identically. For finite coupling strength, we show that special solutions exist with steering controls proportional across the collective. These results suggest that optimality principles may provide a framework for understanding imitative behaviours observed in certain animal aggregations.

scholarly journals Symmetry and reduction in collectives: cyclic pursuit strategies

2013 ◽  
Vol 469 (2158) ◽  
pp. 20130264 ◽  
Author(s):  
Kevin S. Galloway ◽  
Eric W. Justh ◽  
P. S. Krishnaprasad
Keyword(s):  
Invariant Manifold ◽  
Collective Motion ◽  
Shape Space ◽  
Relative Equilibria ◽  
Loop Dynamics ◽  
Cyclic Pursuit ◽  
Special Solutions ◽  
Feedback Laws ◽  
Low Dimensional ◽  
Joint Shape

We specify and analyse models that capture the geometry of purposeful motion of a collective of mobile agents, with a focus on planar motion, dyadic strategies and attention graphs which are static, directed and cyclic. Strategies are formulated as constraints on joint shape space and are implemented through feedback laws for the actions of individual agents, here modelled as self-steering particles. By reduction to a labelled shape space (using a redundant parametrization to account for cycle closure constraints) and a further reduction through time rescaling, we characterize various special solutions (relative equilibria and pure shape equilibria) for cyclic pursuit with a constant bearing (CB) strategy. This is accomplished by first proving convergence of the (nonlinear) dynamics to an invariant manifold (the CB pursuit manifold), and then analysing the closed-loop dynamics restricted to the invariant manifold. For illustration, we sketch some low-dimensional examples. This formulation—involving strategies, attention graphs and sensor-driven steering laws—and the resulting templates of collective motion, are part of a broader programme to interpret the mechanisms underlying biological collective motion.

scholarly journals Optimal Control Problems for Nonlinear Variational Evolution Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Eun-Young Ju ◽  
Jin-Mun Jeong
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Necessary Conditions ◽  
Parabolic Type ◽  
Control Problems ◽  
Optimal Controls ◽  
Solution Mapping ◽  
Necessary Conditions For Optimality ◽  
Semilinear Parabolic ◽  
Variational Evolution

We deal with optimal control problems governed by semilinear parabolic type equations and in particular described by variational inequalities. We will also characterize the optimal controls by giving necessary conditions for optimality by proving the Gâteaux differentiability of solution mapping on control variables.

scholarly journals Body size affects the strength of social interactions and spatial organization of a schooling fish ( Pseudomugil signifer )

2017 ◽  
Vol 4 (4) ◽  
pp. 161056 ◽  
Author(s):  
Maksym Romenskyy ◽  
James E. Herbert-Read ◽  
Ashley J. W. Ward ◽  
David J. T. Sumpter
Keyword(s):  
Spatial Organization ◽  
Collective Motion ◽  
Developmental Stages ◽  
Statistical Properties ◽  
Inversion Method ◽  
Particle Model ◽  
Fish Schools ◽  
Schooling Fish ◽  
Particle Speed ◽  
Fish Interactions

While a rich variety of self-propelled particle models propose to explain the collective motion of fish and other animals, rigorous statistical comparison between models and data remains a challenge. Plausible models should be flexible enough to capture changes in the collective behaviour of animal groups at their different developmental stages and group sizes. Here, we analyse the statistical properties of schooling fish ( Pseudomugil signifer ) through a combination of experiments and simulations. We make novel use of a Boltzmann inversion method, usually applied in molecular dynamics, to identify the effective potential of the mean force of fish interactions. Specifically, we show that larger fish have a larger repulsion zone, but stronger attraction, resulting in greater alignment in their collective motion. We model the collective dynamics of schools using a self-propelled particle model, modified to include varying particle speed and a local repulsion rule. We demonstrate that the statistical properties of the fish schools are reproduced by our model, thereby capturing a number of features of the behaviour and development of schooling fish.

Collective motion in quantum mechanics

1957 ◽  
Vol 239 (1218) ◽  
pp. 399-412 ◽  
Keyword(s):  
Collective Motion ◽  
Electron Gas ◽  
Quantum Mechanical ◽  
Particle Model ◽  
Coupling Theory ◽  
Rotating System ◽  
Plasma Oscillations ◽  
Nucleon Recoil ◽  
The Moment ◽  
General Method

A general method of discussing quantum-mechanical problems involving collective motion is proposed, in which the emphasis is placed on consideration of sets of states rather than single states, and in which the additional collective co-ordinates are not redundant but used to describe the sets. The method is applied to a number of relatively simple examples: plasma oscillations of an electron gas; some problems of nuclear structure including the α-particle model and the collective model; and to two problems in meson field theory, concerning nucleon isobars in strong-coupling theory and concerning nucleon recoil. One of the main aims is to determine as well as possible the parameters of the collective motion; in particular, a formula is given for the moment of inertia of a rotating system.

scholarly journals A Turing test for collective motion

2015 ◽  
Vol 11 (12) ◽  
pp. 20150674 ◽  
Author(s):  
J. E. Herbert-Read ◽  
M. Romenskyy ◽  
D. J. T. Sumpter
Keyword(s):  
Large Scale ◽  
Collective Motion ◽  
Model Fitting ◽  
Turing Test ◽  
Statistical Properties ◽  
Biological Research ◽  
Particle Model ◽  
Fish Schools ◽  
The Public ◽  
Alan Turing

A widespread problem in biological research is assessing whether a model adequately describes some real-world data. But even if a model captures the large-scale statistical properties of the data, should we be satisfied with it? We developed a method, inspired by Alan Turing, to assess the effectiveness of model fitting. We first built a self-propelled particle model whose properties (order and cohesion) statistically matched those of real fish schools. We then asked members of the public to play an online game (a modified Turing test) in which they attempted to distinguish between the movements of real fish schools or those generated by the model. Even though the statistical properties of the real data and the model were consistent with each other, the public could still distinguish between the two, highlighting the need for model refinement. Our results demonstrate that we can use ‘citizen science’ to cross-validate and improve model fitting not only in the field of collective behaviour, but also across a broad range of biological systems.

Local interaction rules and collective motion in black neon tetra (Hyphessobrycon herbertaxelrodi) and zebrafish (Danio rerio).

2019 ◽  
Vol 133 (2) ◽  
pp. 143-155 ◽  
Author(s):  
Vicenç Quera ◽  
Elisabet Gimeno ◽  
Francesc S. Beltran ◽  
Ruth Dolado
Keyword(s):  
Danio Rerio ◽  
Collective Motion ◽  
Local Interaction

On closed-loop equilibrium strategies for mean-field stochastic linear quadratic problems

2020 ◽  
Vol 26 ◽  
pp. 41
Author(s):  
Tianxiao Wang
Keyword(s):  
Optimal Control ◽  
Closed Loop ◽  
Optimal Control Problems ◽  
Mean Field ◽  
Open Loop ◽  
Time Inconsistency ◽  
Control Problems ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Equilibrium Strategies

This article is concerned with linear quadratic optimal control problems of mean-field stochastic differential equations (MF-SDE) with deterministic coefficients. To treat the time inconsistency of the optimal control problems, linear closed-loop equilibrium strategies are introduced and characterized by variational approach. Our developed methodology drops the delicate convergence procedures in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. When the MF-SDE reduces to SDE, our Riccati system coincides with the analogue in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. However, these two systems are in general different from each other due to the conditional mean-field terms in the MF-SDE. Eventually, the comparisons with pre-committed optimal strategies, open-loop equilibrium strategies are given in details.

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