On an optimal control strategy in a kinetic social dynamics model

Damián A. Knopoff 1 , 2  and Germán A. Torres 3 , 4
  • 1 Facultad de Matemática, Astronomía, Física y Computación, Universidad Nacional de Córdoba,, Córdoba, Argentina
  • 2 Centro de Investigación y Estudios de Matemática, CONICET, Godoy Cruz 2290,, Buenos Aires, Argentina
  • 3 Facultad de Ciencias Exactas y Naturales y Agrimensura, Universidad Nacional del Nordeste,, Corrientes, Argentina
  • 4 Instituto de Modelado e Innovacion Tecnologica, CONICET, Godoy Cruz 2290,, Buenos Aires, Argentina

Abstract

Kinetic models have so far been used to model wealth distribution in a society. In particular, within the framework of the kinetic theory for active particles, some important models have been developed and proposed. They involve nonlinear interactions among individuals that are modeled according to game theoretical tools by introducing parameters governing the temporal dynamics of the system. In this present paper we propose an approach based on optimal control tools that aims to optimize this evolving dynamics from a social point of view. Namely, we look for time dependent control variables concerning the distribution of wealth that can be managed, for instance, by the government, in order to obtain a given social profile.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • 1. K. D. Bailey, Sociology and the new systems theory: Toward a theoretical synthesis. Suny Press, 1994.

  • 2. N. Bellomo, F. Colasuonno, D. Knopoff, and J. Soler, From a systems theory of sociology to modeling the onset and evolution of criminality, Networks & Heterogeneous Media, vol. 10, no. 3, 2015.

  • 3. N. Bellomo, D. Knopoff, and J. Soler, On the diffcult interplay between life, “complexity", and mathematical sciences, Mathematical Models and Methods in Applied Sciences, vol. 23, no. 10, pp. 1861-1913, 2013.

  • 4. M. Dolfin and M. Lachowicz, Modeling opinion dynamics: how the network enhances consensus, Networks & Heterogeneous Media, vol. 10, no. 4, pp. 421-441, 2015.

  • 5. D. Knopoff, On a mathematical theory of complex systems on networks with application to opinion formation, Mathematical Models and Methods in Applied Sciences, vol. 24, no. 2, pp. 405-426, 2014.

  • 6. A. Bellouquid, E. De Angelis, and D. Knopoff, From the modeling of the immune hallmarks of cancer to a black swan in Biology, Mathematical Models and Methods in Applied Sciences, vol. 23, no. 5, pp. 949-978, 2013.

  • 7. D. A. Knopoff and J. M. Sánchez Sansó, A kinetic model for horizontal transfer and bacterial antibiotic resistance, International Journal of Biomathematics, vol. 10, no. 04, p. 1750051, 2017.

  • 8. M. Delitala, P. Pucci, and M. Salvatori, From methods of the mathematical kinetic theory for active particles to modeling virus mutations, Mathematical Models and Methods in Applied Sciences, vol. 21, no. supp01, pp. 843-870, 2011.

  • 9. M. Dolfin, L. Leonida, and N. Outada, Modeling human behavior in economics and social science, Physics of Life Reviews, 2017.

  • 10. M. L. Bertotti and M. Delitala, From discrete kinetic and stochastic game theory to modelling complex systems in applied sciences, Mathematical Models and Methods in Applied Sciences, vol. 14, no. 7, pp. 1061-1084, 2004.

  • 11. M. L. Bertotti and M. Delitala, Conservation laws and asymptotic behavior of a model of social dynamics, Nonlinear Analysis: Real World Applications, vol. 9, no. 1, pp. 183-196, 2008.

  • 12. D. Knopoff, On the modeling of migration phenomena on small networks, Mathematical Models and Methods in Applied Sciences, vol. 23, no. 3, pp. 541-563, 2013.

  • 13. N. Bellomo, M. A. Herrero, and A. Tosin, On the dynamics of social conflicts: looking for the black swan, Kinetic and related models, vol. 6, no. 3, pp. 459-479, 2013.

  • 14. N. N. Taleb, The black swan: The impact of the highly improbable. Random house, 2007.

  • 15. M. Dolfin, D. Knopoff, L. Leonida, and D. M. A. Patti, Escaping the trap of "blocking": a kinetic model linking economic development and political competition, Kinetic and Related Models, vol. in press, 2016.

  • 16. B. Düring, D. Matthes, and G. Toscani, Kinetic equations modelling wealth redistribution: a comparison of approaches, Physical Review E, vol. 78, no. 5, p. 056103, 2008.

  • 17. G. Ajmone Marsan, N. Bellomo, and L. Gibelli, Stochastic evolutionary differential games toward a systems theory of behavioral social dynamics, Mathematical Models and Methods in Applied Sciences, vol. 26, no. 6, pp. 1051-1093, 2016.

  • 18. L. Pareschi and G. Toscani, Interacting multiagent systems: kinetic equations and Monte Carlo meth- ods. OUP Oxford, 2013.

  • 19. D. Cass and K. Shell, The Hamiltonian approach to dynamic economics. Academic Press, 2014.

  • 20. W. Stadler, Multicriteria Optimization in Engineering and in the Sciences, vol. 37. Springer Science & Business Media, 2013.

  • 21. G. Albi, M. Herty, and L. Pareschi, Kinetic description of optimal control problems and applications to opinion consensus, Communications in Mathematical Sciences, vol. 13, no. 6, pp. 1407-1429, 2015.

  • 22. A. Barrea and M. E. Hernández, Optimal control of a delayed breast cancer stem cells nonlinear model, Optimal Control Applications and Methods, vol. 37, no. 2, pp. 248-258, 2016.

  • 23. M. Gerdts, Optimal control of ODEs and DAEs. Walter de Gruyter, 2012.

  • 24. S. Lenhart and J. T. Workman, Optimal control applied to biological models. Crc Press, 2007.

  • 25. UNU-IHDP., Inclusive wealth report 2012: measuring progress toward sustainability. Cambridge University Press, 2012.

  • 26. M. Jarvis, G.-M. Lange, K. Hamilton, D. Desai, B. Fraumeni, B. Edens, S. Ferreira, H. Li, L. Chakraborti, W. Kingsmill, et al., The changing wealth of nations: measuring sustainable de- velopment in the new millennium. 2011.

OPEN ACCESS

Journal + Issues

Search