Skip to main content
SpringerLink
Log in

Sparse Control of Multiagent Systems

  • Chapter
  • First Online:
Active Particles, Volume 1

Abstract

In recent years, numerous studies have focused on the mathematical modeling of social dynamics, with self-organization, i.e., the autonomous pattern formation, as the main driving concept. Usually, first- or second-order models are employed to reproduce, at least qualitatively, certain global patterns (such as bird flocking, milling schools of fish, or queue formations in pedestrian flows, just to mention a few). It is, however, common experience that self-organization does not always spontaneously occur in a society. In this review chapter, we aim to describe the limitations of decentralized controls in restoring certain desired configurations and to address the question of whether it is possible to externally and parsimoniously influence the dynamics to reach a given outcome. More specifically, we address the issue of finding the sparsest control strategy for finite agent-based models in order to lead the dynamics optimally toward a desired pattern.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Given a real \(N\times N\) matrix \(A = (a_{ij})_{i,j = 1}^N\) and \(v\in {\mathbb R}^{dN}\), we denote by Av the action of A on \({\mathbb R}^{dN}\) by mapping v to \((a_{i1}v_{1} + \cdots + a_{iN}v_{N})_{i=1}^N\). Given a nonnegative symmetric \(N \times N\) matrix \(A = (a_{ij})_{i,j = 1}^N\), the Laplacian L of A is defined by \(L = D - A\), with \(D = \mathrm {diag} (d_{1}, \ldots , d_{N})\) and \(d_{k} = \sum _{j=1}^{N} a_{kj}\).

References

  1. S. M. Ahn and S.-Y. Ha. Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises. J. Math. Phys., 51(10):103301, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  2. G. Albi, M. Bongini, E. Cristiani, and D. Kalise. Invisible sparse control of self-organizing agents leaving unknown environments. To appear in SIAM J. Appl. Math., 2015.

    MATH  Google Scholar 

  3. F. Arvin, J. C. Murray, L. Shi, C. Zhang, and S. Yue. Development of an autonomous micro robot for swarm robotics. In Proceedings of the IEEE International Conference on Mechatronics and Automation (ICMA), pages 635–640. IEEE, 2014.

    Google Scholar 

  4. P. Bak. How nature works: the science of self-organized criticality. Springer Science & Business Media, 2013.

    Google Scholar 

  5. 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. P. Natl. Acad. Sci. USA, 105(4):1232–1237, 2008.

    Article  Google Scholar 

  6. S. Battiston, D. Delli Gatti, M. Gallegati, B. Greenwald, and J. Stiglitz. Liaisons dangereuses: Increasing connectivity, risk sharing, and systemic risk. J. Econ. Dyn. Control, 36(8):1121–1141, 2012.

    Article  MATH  Google Scholar 

  7. M. Bongini. Sparse Optimal Control of Multiagent Systems. PhD thesis, Technische Universität München, 2016.

    Google Scholar 

  8. M. Bongini and M. Fornasier. Sparse stabilization of dynamical systems driven by attraction and avoidance forces. Netw. Heterog. Media, 9(1):1–31, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Bongini, M. Fornasier, F. Frölich, and L. Hagverdi. Sparse control of force field dynamics. In International Conference on NETwork Games, COntrol and OPtimization, October 2014.

    Google Scholar 

  10. M. Bongini, M. Fornasier, O. Junge, and B. Scharf. Sparse control of alignment models in high dimension. Netw. Heterog. Media, 10(3):647–697, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Bongini, M. Fornasier, and D. Kalise. (Un)conditional consensus emergence under perturbed and decentralized feedback controls. Discrete Contin. Dyn. Syst., 35(9):4071–4094, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Borzì and S. Wongkaew. Modeling and control through leadership of a refined flocking system. Math. Models Methods Appl. Sci., 25(02):255–282, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Camazine, J.-L. Deneubourg, N. Franks, J. Sneyd, G. Theraulaz, and E. Bonabeau. Self-organization in biological systems. Princeton University Press, 2002.

    Google Scholar 

  14. E. J. Candès, J. K. Romberg, and T. Tao. Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math., 59(8):1207–1223, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Caponigro, M. Fornasier, B. Piccoli, and E. Trélat. Sparse stabilization and optimal control of the Cucker-Smale model. Math. Control Relat. Fields, 3(4):447–466, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Caponigro, M. Fornasier, B. Piccoli, and E. Trélat. Sparse stabilization and control of alignment models. Math. Models Methods Appl. Sci., 25(03):521–564, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. A. Carrillo, M. R. D’Orsogna, and V. Panferov. Double milling in self-propelled swarms from kinetic theory. Kinet. Relat. Models, 2(2):363–378, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. A. Carrillo, M. Fornasier, J. Rosado, and G. Toscani. Asymptotic flocking dynamics for the kinetic Cucker-Smale model. SIAM J. Math. Anal., 42(1):218–236, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  19. 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, Modeling and Simulation in Science, Engineering and Technology, pages 297–336. Birkhäuser Boston, 2010.

    Google Scholar 

  20. J. A. Carrillo, Y.-P. Choi, and M. Hauray. The derivation of swarming models: mean-field limit and Wasserstein distances. In Collective Dynamics from Bacteria to Crowds, pages 1–46. Springer, 2014.

    Google Scholar 

  21. E. Casas, C. Clason, and K. Kunisch. Approximation of elliptic control problems in measure spaces with sparse solutions. SIAM J. Control Optim., 50(4):1735–1752, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  22. Y.-L. Chuang, M. R. D’Orsogna, D. Marthaler, A. L. Bertozzi, and L. S. Chayes. State transitions and the continuum limit for a 2D interacting, self-propelled particle system. Phys. D, 232(1):33–47, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  23. F. R. K. Chung. Spectral graph theory, volume 92. American Mathematical Society, 1997.

    Google Scholar 

  24. C. Clason and K. Kunisch. A measure space approach to optimal source placement. Comput. Optim. Appl., 53(1):155–171, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  25. M. A. Cohen and S. Grossberg. Absolute stability of global pattern formation and parallel memory storage by competitive neural networks. IEEE Trans. Syst., Man, Cybern., Syst., 13(5):815–826, 1983.

    Google Scholar 

  26. J. Cortés and F. Bullo. Coordination and geometric optimization via distributed dynamical systems. SIAM J. Control Optim., 44(5):1543–1574, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  27. I. D. Couzin and N. R. Franks. Self-organized lane formation and optimized traffic flow in army ants. P. Roy. Soc. Lond. B Bio., 270(1511):139–146, 2003.

    Article  Google Scholar 

  28. I. D. Couzin, J. Krause, N. R. Franks, and S. A. Levin. Effective leadership and decision-making in animal groups on the move. Nature, 433:513–516, 2005.

    Article  Google Scholar 

  29. A. J. Craig and I. Flügge-Lotz. Investigation of optimal control with a minimum-fuel consumption criterion for a fourth-order plant with two control inputs; synthesis of an efficient suboptimal control. J. Fluids Eng., 87(1):39–58, 1965.

    Google Scholar 

  30. E. Cristiani, B. Piccoli, and A. Tosin. Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints. In Mathematical modeling of collective behavior in socio-economic and life sciences, pages 337–364. Springer, 2010.

    Google Scholar 

  31. E. Cristiani, B. Piccoli, and A. Tosin. Multiscale modeling of granular flows with application to crowd dynamics. Multiscale Model. Simul., 9(1):155–182, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  32. F. Cucker and J.-G. Dong. A general collision-avoiding flocking framework. IEEE Trans. Automat. Control, 56(5):1124–1129, 2011.

    Article  MathSciNet  Google Scholar 

  33. F. Cucker and J.-G. Dong. A conditional, collision-avoiding, model for swarming. Discrete Contin. Dynam. Systems, 34(3):1009–1020, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  34. F. Cucker and S. Smale. Emergent behavior in flocks. IEEE Trans. Automat. Control, 52(5):852–862, 2007.

    Article  MathSciNet  Google Scholar 

  35. F. Cucker and S. Smale. On the mathematics of emergence. Jpn. J. Math., 2(1):197–227, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  36. F. Cucker, S. Smale, and D. Zhou. Modeling language evolution. Found. Comput. Math., 4(5):315–343, 2004.

    MathSciNet  MATH  Google Scholar 

  37. S. Currarini, M. O. Jackson, and P. Pin. An economic model of friendship: Homophily, minorities, and segregation. Econometrica, 77(4):1003–1045, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  38. F. Dalmao and E. Mordecki. Cucker-Smale flocking under hierarchical leadership and random interactions. SIAM J. Appl. Math., 71(4):1307–1316, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  39. J. Dickinson. Animal social behavior. In Encyclopaedia Britannica Online. Encyclopaedia Britannica Inc., 2016.

    Google Scholar 

  40. D. L. Donoho. Compressed sensing. IEEE Trans. Inform. Theory, 52(4):1289–1306, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  41. M. R. D’Orsogna, Y.-L. Chuang, A. L. Bertozzi, and L. S. Chayes. Self-propelled particles with soft-core interactions: patterns, stability, and collapse. Phys. Rev. Lett., 96(10):104302, 2006.

    Article  Google Scholar 

  42. Y. Eldar and H. Rauhut. Average case analysis of multichannel sparse recovery using convex relaxation. IEEE Trans. Inform. Theory, 56(1):505–519, 2010.

    Article  MathSciNet  Google Scholar 

  43. J. A. Fax and R. M. Murray. Information flow and cooperative control of vehicle formations. IEEE Trans. Automat. Control, 49(9):1465–1476, 2004.

    Article  MathSciNet  Google Scholar 

  44. A. F. Filippov. Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers, 1988.

    Google Scholar 

  45. M. Fornasier and H. Rauhut. Recovery algorithms for vector-valued data with joint sparsity constraints. SIAM J. Numer. Anal., 46(2):577–613, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  46. M. Fornasier and H. Rauhut. Handbook of Mathematical Methods in Imaging, chapter Compressive Sensing, pages 187–228. Springer-Verlag, 2010.

    Google Scholar 

  47. S.-Y. Ha, J.-G. Liu, et al. A simple proof of the Cucker-Smale flocking dynamics and mean-field limit. Commun. Math. Sci., 7(2):297–325, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  48. 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(7):1679–1683, 2010.

    Article  MathSciNet  Google Scholar 

  49. G. Hardin. The tragedy of the commons. Science, 162(3859):1243–1248, 1968.

    Article  Google Scholar 

  50. J. Haskovec. A note on the consensus finding problem in communication networks with switching topologies. Appl. Anal., 94(5):991–998, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  51. R. Hegselmann and U. Krause. Opinion dynamics and bounded confidence models, analysis, and simulation. J. Artif. Soc. Soc. Simulat., 5(3), 2002.

    Google Scholar 

  52. R. Herzog, G. Stadler, and G. Wachsmuth. Directional sparsity in optimal control of partial differential equations. SIAM J. Control Optim., 50(2):943–963, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  53. E. F. Keller and L. A. Segel. Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol., 26(3):399–415, 1970.

    Article  MATH  Google Scholar 

  54. A. Kirman, S. Markose, S. Giansante, and P. Pin. Marginal contribution, reciprocity and equity in segregated groups: Bounded rationality and self-organization in social networks. J. Econ. Dyn. Control, 31(6):2085–2107, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  55. A. Koch and D. White. The social lifestyle of myxobacteria. Bioessays 20, pages 1030–1038, 1998.

    Article  Google Scholar 

  56. S. Mallat. A wavelet tour of signal processing: the sparse way. Academic press, 2008.

    Google Scholar 

  57. M. McPherson, L. Smith-Lovin, and J. M. Cook. Birds of a feather: Homophily in social networks. Annu. Rev. Sociol., pages 415–444, 2001.

    Google Scholar 

  58. B. Mohar. The Laplacian spectrum of graphs. In Y. Alavi, G. Chartrand, O. R. Oellermann, and A. J. Schwenk, editors, Graph theory, Combinatorics, and Applications, volume 2, pages 871–898. Wiley, 1991.

    Google Scholar 

  59. L. Moreau. Stability of multiagent systems with time-dependent communication links. IEEE Trans. Automat. Control, 50(2):169–182, 2005.

    Article  MathSciNet  Google Scholar 

  60. S. Motsch and E. Tadmor. Heterophilious dynamics enhances consensus. SIAM Rev., 56(4):577–621, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  61. J. F. Nash. Equilibrium points in \(N\)-person games. Proc. Natl. Acad. Sci. USA, 36(1):48–49, 1950.

    Article  MathSciNet  MATH  Google Scholar 

  62. H. Niwa. Self-organizing dynamic model of fish schooling. J. Theor. Biol., 171:123–136, 1994.

    Article  Google Scholar 

  63. F. Paganini, J. Doyle, and S. Low. Scalable laws for stable network congestion control. In Proceedings of the 40th IEEE Conference on Decision and Control, volume 1, pages 185–190. IEEE, 2001.

    Google Scholar 

  64. J. Parrish and L. Edelstein-Keshet. Complexity, pattern, and evolutionary trade-offs in animal aggregation. Science, 294:99–101, 1999.

    Article  Google Scholar 

  65. J. Parrish, S. Viscido, and D. Gruenbaum. Self-organized fish schools: An examination of emergent properties. Biol. Bull., 202:296–305, 2002.

    Article  Google Scholar 

  66. L. Perea, P. Elosegui, and G. Gómez. Extension of the Cucker-Smale control law to space flight formations. J. Guid. Control Dynam., 32(2):527–537, 2009.

    Article  Google Scholar 

  67. B. Perthame. Transport Equations in Biology. Basel: Birkhäuser, 2007.

    MATH  Google Scholar 

  68. L. Petrovic, M. Henne, and J. Anderson. Volumetric Methods for Simulation and Rendering of Hair. Technical report, Pixar Animation Studios, 2005.

    Google Scholar 

  69. C. W. Reynolds. Flocks, herds and schools: A distributed behavioral model. ACM SIGGRAPH Computer Graphics, 21(4):25–34, 1987.

    Article  Google Scholar 

  70. W. Romey. Individual differences make a difference in the trajectories of simulated schools of fish. Ecol. Model., 92:65–77, 1996.

    Article  Google Scholar 

  71. J. Shen. Cucker-Smale flocking under hierarchical leadership. SIAM J. Appl. Math., 68(3):694–719, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  72. M. B. Short, M. R. D’Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi, and L. B. Chayes. A statistical model of criminal behavior. Math. Models Methods Appl. Sci., 18(suppl.):1249–1267, 2008.

    Google Scholar 

  73. G. Stadler. Elliptic optimal control problems with \(L^1\)-control cost and applications for the placement of control devices. Comput. Optim. Appl., 44(2):159–181, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  74. H. G. Tanner, A. Jadbabaie, and G. J. Pappas. Flocking in fixed and switching networks. IEEE Trans. Automat. Control, 52(5):863–868, 2007.

    Article  MathSciNet  Google Scholar 

  75. J. Toner and Y. Tu. Long-range order in a two-dimensional dynamical xy model: How birds fly together. Phys. Rev. Lett., 75:4326–4329, 1995.

    Article  Google Scholar 

  76. T. Vicsek and A. Zafeiris. Collective motion. Phys. Rep., 517(3):71–140, 2012.

    Article  Google Scholar 

  77. 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(6):1226, 1995.

    Article  MathSciNet  Google Scholar 

  78. J. von Neumann and O. Morgenstern. Theory of Games and Economic Behavior. Princeton University Press, 1944.

    Google Scholar 

  79. G. Wachsmuth and D. Wachsmuth. Convergence and regularization results for optimal control problems with sparsity functional. ESAIM Control Optim. Calc. Var., 17(3):858–886, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  80. G. Weisbuch, G. Deffuant, F. Amblard, and J.-P. Nadal. Meet, discuss, and segregate! Complexity, 7(3):55–63, 2002.

    Article  Google Scholar 

  81. S. Wongkaew, M. Caponigro, and A. Borzì. On the control through leadership of the Hegselmann–Krause opinion formation model. Math. Models Methods Appl. Sci., 25(03):565–585, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  82. C. Yates, R. Erban, C. Escudero, L. Couzin, J. Buhl, L. Kevrekidis, P. Maini, and D. Sumpter. Inherent noise can facilitate coherence in collective swarm motion. Proceedings of the National Academy of Sciences, 106:5464–5469, 2009.

    Article  Google Scholar 

Download references

Acknowledgements

The authors acknowledge the support of the ERC-Starting Grant “High-Dimensional Sparse Optimal Control” (HDSPCONTR - 306274).

Author information

Authors and Affiliations

  1. Technische Universität München, Fakultät Mathematik, Boltzmannstraße 3, 85748, Garching, Germany

    Mattia Bongini & Massimo Fornasier

Authors
  1. Mattia Bongini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Massimo Fornasier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Massimo Fornasier .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, King Abdulaziz University, Jeddah, Saudi Arabia

    Nicola Bellomo

  2. Department of Mathematics, Imperial College London, London, United Kingdom

    Pierre Degond

  3. CSCAMM, University of Maryland, College Park, USA

    Eitan Tadmor

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Bongini, M., Fornasier, M. (2017). Sparse Control of Multiagent Systems. In: Bellomo, N., Degond, P., Tadmor, E. (eds) Active Particles, Volume 1 . Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-49996-3_5

Download citation

Publish with us

Policies and ethics

Search

Navigation