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September  2015, 35(9): 4071-4094. doi: 10.3934/dcds.2015.35.4071

(Un)conditional consensus emergence under perturbed and decentralized feedback controls

1. 

Technische Universität München, Fakultät Mathematik, Boltzmannstraße 3, D-85748 Garching, Germany

2. 

Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstraße 69, 4040 Linz

Received  April 2014 Revised  September 2014 Published  April 2015

We study the problem of consensus emergence in multi-agent systems via external feedback controllers. We consider a set of agents interacting with dynamics given by a Cucker-Smale type of model, and study its consensus stabilization by means of centralized and decentralized control configurations. We present a characterization of consensus emergence for systems with different feedback structures, such as leader-based configurations, perturbed information feedback, and feedback computed upon spatially confined information. We characterize consensus emergence for this latter design as a parameter-dependent transition regime between self-regulation and centralized feedback stabilization. Numerical experiments illustrate the different features of the proposed designs.
Citation: Mattia Bongini, Massimo Fornasier, Dante Kalise. (Un)conditional consensus emergence under perturbed and decentralized feedback controls. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4071-4094. doi: 10.3934/dcds.2015.35.4071
References:
[1]

L. Bakule, Decentralized control: An overview, Annu. Rev. Control, 32 (2008), 87-98. doi: 10.1016/j.arcontrol.2008.03.004.

[2]

M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. U.S.A., 105 (2008), 1232-1237. doi: 10.1073/pnas.0711437105.

[3]

M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat, Sparse Stabilization and Control of Alignment Models, Math. Mod. Meth. Appl. Sci. (M3AS), 25 (2015), 521-564. doi: 10.1142/S0218202515400059.

[4]

J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences (eds. G. Naldi, L. Pareschi, G. Toscani and N. Bellomo), Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, (2010), 297-336. doi: 10.1007/978-0-8176-4946-3_12.

[5]

F. Cucker and C. Huepe, Flocking with informed agents, MathS In Action, 1 (2008), 1-25. doi: 10.5802/msia.1.

[6]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.

[7]

F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x.

[8]

A. Filipov, Differential Equations with Discontinuous Righthand Sides, Volume 18 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.

[9]

M. Fornasier and F. Solombrino, Mean-field optimal control, ESAIM, Control Optim. Calc. Var., 20 (2014), 1123-1152. doi: 10.1051/cocv/2014009.

[10]

S.-Y. Ha, T. Ha and J.-H. Kim, Asymptotic dynamics for the cucker-smale-type model with the rayleigh friction, J. Phys. A: Math. Theor.l, 43 (2010), 315201, 19 pp. doi: 10.1088/1751-8113/43/31/315201.

[11]

S.-Y. Ha, T. Ha and J.-H. Kim, Emergent behavior of a Cucker-Smale type particle model with nonlinear velocity couplings, IEEE Trans. Automat. Control, 55 (2010), 1679-1683. doi: 10.1109/TAC.2010.2046113.

[12]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence models, analysis, and simulation, J. Artif. Soc. Soc. Simulat., 5 (2002), 1-24.

[13]

P. Ignaciuk and A. Bartoszewicz, Congestion Control in Data Transmission Networks, Springer, 2013. doi: 10.1007/978-1-4471-4147-1.

[14]

R. M. Murray, Recent research in cooperative control of multivehicle systems, J. Dyn. Syst. Meas. Control , 129 (2007), 571-583. doi: 10.1115/1.2766721.

[15]

R. Olfati-Saber and R. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control, 49 (2004), 1520-1533. doi: 10.1109/TAC.2004.834113.

[16]

A. A. Peters, R. H. Middleton and O. Mason, Leader tracking in homogeneous vehicle platoons with broadcast delays, Automatica, 50 (2014), 64-74. doi: 10.1016/j.automatica.2013.09.034.

[17]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, SIGGRAPH Comput. Graph., 21 (1987), 25-34. doi: 10.1145/37401.37406.

[18]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.

show all references

References:
[1]

L. Bakule, Decentralized control: An overview, Annu. Rev. Control, 32 (2008), 87-98. doi: 10.1016/j.arcontrol.2008.03.004.

[2]

M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. U.S.A., 105 (2008), 1232-1237. doi: 10.1073/pnas.0711437105.

[3]

M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat, Sparse Stabilization and Control of Alignment Models, Math. Mod. Meth. Appl. Sci. (M3AS), 25 (2015), 521-564. doi: 10.1142/S0218202515400059.

[4]

J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences (eds. G. Naldi, L. Pareschi, G. Toscani and N. Bellomo), Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, (2010), 297-336. doi: 10.1007/978-0-8176-4946-3_12.

[5]

F. Cucker and C. Huepe, Flocking with informed agents, MathS In Action, 1 (2008), 1-25. doi: 10.5802/msia.1.

[6]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.

[7]

F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x.

[8]

A. Filipov, Differential Equations with Discontinuous Righthand Sides, Volume 18 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.

[9]

M. Fornasier and F. Solombrino, Mean-field optimal control, ESAIM, Control Optim. Calc. Var., 20 (2014), 1123-1152. doi: 10.1051/cocv/2014009.

[10]

S.-Y. Ha, T. Ha and J.-H. Kim, Asymptotic dynamics for the cucker-smale-type model with the rayleigh friction, J. Phys. A: Math. Theor.l, 43 (2010), 315201, 19 pp. doi: 10.1088/1751-8113/43/31/315201.

[11]

S.-Y. Ha, T. Ha and J.-H. Kim, Emergent behavior of a Cucker-Smale type particle model with nonlinear velocity couplings, IEEE Trans. Automat. Control, 55 (2010), 1679-1683. doi: 10.1109/TAC.2010.2046113.

[12]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence models, analysis, and simulation, J. Artif. Soc. Soc. Simulat., 5 (2002), 1-24.

[13]

P. Ignaciuk and A. Bartoszewicz, Congestion Control in Data Transmission Networks, Springer, 2013. doi: 10.1007/978-1-4471-4147-1.

[14]

R. M. Murray, Recent research in cooperative control of multivehicle systems, J. Dyn. Syst. Meas. Control , 129 (2007), 571-583. doi: 10.1115/1.2766721.

[15]

R. Olfati-Saber and R. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control, 49 (2004), 1520-1533. doi: 10.1109/TAC.2004.834113.

[16]

A. A. Peters, R. H. Middleton and O. Mason, Leader tracking in homogeneous vehicle platoons with broadcast delays, Automatica, 50 (2014), 64-74. doi: 10.1016/j.automatica.2013.09.034.

[17]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, SIGGRAPH Comput. Graph., 21 (1987), 25-34. doi: 10.1145/37401.37406.

[18]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.

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