# American Institute of Mathematical Sciences

• Previous Article
Integrability methods in the time minimal coherence transfer for Ising chains of three spins
• DCDS Home
• This Issue
• Next Article
Value iteration convergence of $\epsilon$-monotone schemes for stationary Hamilton-Jacobi equations
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.
 [1] Marco Caponigro, Massimo Fornasier, Benedetto Piccoli, Emmanuel Trélat. Sparse stabilization and optimal control of the Cucker-Smale model. Mathematical Control and Related Fields, 2013, 3 (4) : 447-466. doi: 10.3934/mcrf.2013.3.447 [2] Rui Li, Yingjing Shi. Finite-time optimal consensus control for second-order multi-agent systems. Journal of Industrial and Management Optimization, 2014, 10 (3) : 929-943. doi: 10.3934/jimo.2014.10.929 [3] Seung-Yeal Ha, Jinwook Jung, Peter Kuchling. Emergence of anomalous flocking in the fractional Cucker-Smale model. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5465-5489. doi: 10.3934/dcds.2019223 [4] Mei Luo, Jinrong Wang, Yumei Liao. Bounded consensus of double-integrator stochastic multi-agent systems. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022088 [5] Richard Carney, Monique Chyba, Chris Gray, George Wilkens, Corey Shanbrom. Multi-agent systems for quadcopters. Journal of Geometric Mechanics, 2022, 14 (1) : 1-28. doi: 10.3934/jgm.2021005 [6] Xi Zhu, Meixia Li, Chunfa Li. Consensus in discrete-time multi-agent systems with uncertain topologies and random delays governed by a Markov chain. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4535-4551. doi: 10.3934/dcdsb.2020111 [7] Zhongkui Li, Zhisheng Duan, Guanrong Chen. Consensus of discrete-time linear multi-agent systems with observer-type protocols. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 489-505. doi: 10.3934/dcdsb.2011.16.489 [8] Yibo Zhang, Jinfeng Gao, Jia Ren, Huijiao Wang. A type of new consensus protocol for two-dimension multi-agent systems. Numerical Algebra, Control and Optimization, 2017, 7 (3) : 345-357. doi: 10.3934/naco.2017022 [9] Ke Yang, Wencheng Zou, Zhengrong Xiang, Ronghao Wang. Fully distributed consensus for higher-order nonlinear multi-agent systems with unmatched disturbances. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1535-1551. doi: 10.3934/dcdss.2020396 [10] Xiaojin Huang, Hongfu Yang, Jianhua Huang. Consensus stability analysis for stochastic multi-agent systems with multiplicative measurement noises and Markovian switching topologies. Numerical Algebra, Control and Optimization, 2022, 12 (3) : 601-610. doi: 10.3934/naco.2021024 [11] Lining Ru, Xiaoping Xue. Flocking of Cucker-Smale model with intrinsic dynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6817-6835. doi: 10.3934/dcdsb.2019168 [12] Mauro Rodriguez Cartabia. Cucker-Smale model with time delay. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2409-2432. doi: 10.3934/dcds.2021195 [13] Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023 [14] Zhiyong Sun, Toshiharu Sugie. Identification of Hessian matrix in distributed gradient-based multi-agent coordination control systems. Numerical Algebra, Control and Optimization, 2019, 9 (3) : 297-318. doi: 10.3934/naco.2019020 [15] Hongru Ren, Shubo Li, Changxin Lu. Event-triggered adaptive fault-tolerant control for multi-agent systems with unknown disturbances. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1395-1414. doi: 10.3934/dcdss.2020379 [16] Ewa Girejko, Luís Machado, Agnieszka B. Malinowska, Natália Martins. On consensus in the Cucker–Smale type model on isolated time scales. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 77-89. doi: 10.3934/dcdss.2018005 [17] Daniel Lear, David N. Reynolds, Roman Shvydkoy. Grassmannian reduction of cucker-smale systems and dynamical opinion games. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5765-5787. doi: 10.3934/dcds.2021095 [18] Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic and Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034 [19] Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic and Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1 [20] Young-Pil Choi, Jan Haskovec. Cucker-Smale model with normalized communication weights and time delay. Kinetic and Related Models, 2017, 10 (4) : 1011-1033. doi: 10.3934/krm.2017040

2021 Impact Factor: 1.588