A Parabolic Approach to the Control of Opinion Spreading

  • Domènec Ruiz-BaletEmail author
  • Enrique Zuazua
Part of the Mathematics of Planet Earth book series (MPE, volume 6)


We analyse the problem of controlling to consensus a nonlinear system modelling opinion spreading. We derive explicit exponential estimates on the cost of approximately controlling these systems to consensus, as a function of the number of agents N and the control time horizon T. Our strategy makes use of known results on the controllability of spatially discretised semilinear parabolic equations. Both systems can be linked through time rescaling.


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This work has been partially funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 694126-DyCon), grant MTM2017-92996 of MINECO (Spain), the Marie Curie Training Network “Conflex”, the ELKARTEK project KK-2018/00083 ROAD2DC of the Basque Government, ICON of the French ANR and “Nonlocal PDEs: Analysis, Control and Beyond”, AFOSR Grant FA9550-18-1-0242.


  1. Biccari, U., Ko, D., Zuazua, E.: Dynamics and control for multi-agent networked systems: a finite difference approach. arXiv Math, to appear in M3AS (2019). MathSciNetCrossRefGoogle Scholar
  2. Boyer, F., Le Rousseau, J.: Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations. Annales de l’IHP Analyse non linéaire 31(5), 1035–1078 (2014). MathSciNetCrossRefGoogle Scholar
  3. Cañizo, J.A,, Carrillo, J.A,, Rosado, J.: A well-posedness theory in measures for some kinetic models of collective motion. Math. Models Methods Appl. Sci. 21(3), 515–539 (2011). MathSciNetCrossRefGoogle Scholar
  4. Coron, J.M.: Control and Nonlinearity. American Mathematical Society, Boston, MA, USA (2007)zbMATHGoogle Scholar
  5. Cucker, F., Smale, S.: Emergent behavior in flocks. IEEE Trans. Automatic Control 52(5), 852–862 (2007). MathSciNetCrossRefGoogle Scholar
  6. DyCon Toolbox. Universidad de Deusto & Universidad Autónoma de Madrid, Spain (2019) Accessed on 21.06.2019
  7. Fattorini, H., Russell, D.: Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations. Quarterly Appl. Math. 32, 45–69 (1974). MathSciNetCrossRefGoogle Scholar
  8. Fattorini, H.O., Russell, D.L.: Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Rat. Mech. Anal. 43(4), 272–292 (1971). MathSciNetCrossRefGoogle Scholar
  9. Fernández-Cara, E., Zuazua, E.: The cost of approximate controllability for heat equations: the linear case. Adv. Differential Equations 5(4-6), 465–514 (2000a)MathSciNetzbMATHGoogle Scholar
  10. Fernández-Cara, E., Zuazua, E.: Null and approximate controllability for weakly blowing up semilinear heat equations. Annales de l’Institut Henri Poincare (C) Non Linear Analysis 17(5), 583–616 (2000b). MathSciNetCrossRefGoogle Scholar
  11. Ha, S.Y., Tadmor, E.: From particle to kinetic and hydrodynamic descriptions of flocking. Kinetic & Related Models 1(3), 415–435 (2008). MathSciNetCrossRefGoogle Scholar
  12. Kearns, D.: A field guide to bacterial swarming motility. Nature Reviews Microbiology 8(9), 634–644 (2010). CrossRefGoogle Scholar
  13. Kuramoto, Y.: Self-entrainment of a population of coupled non-linear oscillators. In: Araki, H. (ed.) Mathematical Problems in Theoretical Physics. Lecture Notes in Physics, vol. 39, pp. 420–422, Springer, Berlin (1975). Google Scholar
  14. Liu, Y., Slotine, J.J., Barabási, A.: Controllability of complex networks. Nature 473, 167–173 (2011). CrossRefGoogle Scholar
  15. Liu, Y.Y., Barabási, A.L.: Control principles of complex systems. Rev. Mod. Phys. 88, 035006 (2016). CrossRefGoogle Scholar
  16. Macy, M.W., Willer, R.: From factors to actors: Computational sociology and agent-based modeling. Annu. Rev. Sociology 28, 143–166 (2002). CrossRefGoogle Scholar
  17. Motsch, S., Tadmor, E.: Heterophilious dynamics enhances consensus. SIAM Review 56(4), 577–621 (2014). MathSciNetCrossRefGoogle Scholar
  18. Piccoli, B., Rossi, F., Trélat, E.: Control to flocking of the kinetic Cucker–Smale model. SIAM J. Math. Anal. 47(6), 4685–4719 (2015). MathSciNetCrossRefGoogle Scholar
  19. Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd edn. Springer, Berlin (1998)CrossRefGoogle Scholar
  20. Sorrentino, F., Di Bernardo, M., Garofalo, F., Chen, G.: Controllability of complex networks via pinning. Phys. Rev. E 75(4), 046103 (2007). CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain
  2. 2.Fundación DeustoBilbaoBasque CountrySpain
  3. 3.Department of MathematicsFriedrich-Alexander-Universität Erlangen-NürnbergErlangenGermany

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