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

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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

Mattia Bongini ^1, , Massimo Fornasier ^1, and Dante Kalise ^2,

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

Abstract
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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.

Keywords: Cucker-Smale model, Multi-agent systems, feedback stabilization, decentralized control., consensus emergence.

Mathematics Subject Classification: Primary: 34D45; Secondary: 35B36, 93A14, 93B52, 93D15, 58F1.

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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