A numerical treatment based on Bernoulli Tau method for computing the open-loop Nash equilibrium in nonlinear differential games

Document Type : Research Article


Department of Applied Mathematics, Shahed University, Tehran, Iran.


The Tau method based on the Bernoulli polynomials is implemented efficiently to approximate the Nash equilibrium of open-loop kind in non-linear differential games over a finite time horizon. By this treatment, the system of two-point boundary value problems of differential game ex-tracted from Pontryagin’s maximum principle is transferred to a system of algebraic equations that Newton’s iteration method can be used for solving it. Also, for the mentioned approximation by the Bernoulli polynomials, the convergence analysis and the error upper bound are discussed. To demonstrate the applicably and accuracy of the proposed approach, some illustrated examples are presented at the final.


Main Subjects

1.Abbasbandy, S., Shivanian, E., and Vajravelu, K. Mathematical prop-erties of -curve in the frame work of the homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul., 16(11) (2011), 4268–4275.
2. Bhrawy, A., Tharwat, M., and Yildirim, A. A new formula for fractional integrals of Chebyshev polynomials: Application for solving multi-term fractional differential equations, Appl. Math. Model., 37(6) (2013), 4245–4252.
3. Bhrawy, A. H. and Zaky, M. A. A method based on the Jacobi tau approx-imation for solving multi-term time-space fractional partial differential equations, J. Comput. Phys., 281 (2015), 876–895.
4. Bhrawy, A. H., Zaky, M. A., and Baleanu, D. New numerical approxima-tions for space-time fractional burgers’ equations via a Legendre spectral-collocation method, Rom. Rep. Phys., 67(2) (2015), 340–349.
5. Bressan, A. Bifurcation analysis of a non-cooperative differential game with one weak player, J. Differ. Equ., 248(6), (2010), 1297–1314.
6. Bressan, A. Noncooperative differential games, Milan J. Math., 79(2) (2011), 357–427.
7. Canuto, C., Hussaini, M. Y., Quarteroni, A., Thomas Jr, A., et al. Spectral methods in fluid dynamics, Springer Science & Business Media, Berlin/Heidelberg, Germany, (2012).
8. Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A. Spectral methods: fundamentals in single domains, Springer Science & Business Media, Berlin/Heidelberg, Germany, (2007).
9. Carlson, D. A. and Leitmann, G. An extension of the coordinate transfor-mation method for open-loop Nash equilibria, J. Optim. Theory Appl., 123(1) (2004), 27–47.
10. Cesari, L. Optimization-theory and applications: problems with ordinary differential equations, Springer Science & Business Media, Berlin/Heidel-berg, Germany, (2012).
11. Dockner, E. J., Jorgensen, S., Van Long, N., and Sorger, G. Differen-tial games in economics and management science, Cambridge University Press, London, (2000).
12. Ehtamo, H. and Raivio, T. On applied nonlinear and bilevel programming or pursuit-evasion games, J. Optim. Theory Appl., 108(1) (2001), 65–96.
13. Engwerda, J. LQ dynamic optimization and differential games, John Wi-ley & Sons, NJ, USA, (2005).
14. Engwerda, J. C. On the open-loop Nash equilibrium in lq-games, J. Econ. Dyn. Control, 22(5) (1998), 729–762.
15. Erickson, G. Dynamic models of advertising competition, Springer Science & Business Media, Berlin/Heidelberg, Germany, (2002).
16. Ghaneai, H. and Hosseini, M. Variational iteration method with an aux-iliary parameter for solving wave-like and heat-like equations in large domains, Comput. Math. Appl., 69(5) (2015), 363–373.
17. Grosset, L. A note on open loop Nash equilibrium in linear-state differ-ential games, Appl. Math. Sci., 8 (2014), 7239–7248.
18. Heydari, M., Loghmani, G. B., Rashidi, M. M. and Hosseini, S. M. A numerical study for off-centered stagnation flow towards a rotating disc, Propuls. Power Res., 4(3) (2015), 169–178.
19. Heydari M., Loghmani G. B. and Hosseini S. M. Exponential Bernstein functions: an effective tool for the solution of heat transfer of a micropolar fluid through a porous medium with radiation, Comput. Appl. Math., 36(1) (2017), 647–675.
20. Horie, K. and Conway, B. A. Genetic algorithm preprocessing for nu-merical solution of differential games problems, J. Guid. Control Dyn., 27(6), (2004), 1075–1078.
21. Hosseini, E., Barid Loghmani, G., Heydari, M., and Wazwaz, A.-M. A numerical study of electrohydrodynamic flow analysis in a circular cylin-drical conduit using orthonormal Bernstein polynomials, Comput. Meth-ods Differ. Equ., 5(4) (2017), 280–300.
22. Hosseini, E., Loghmani, G. B., Heydari, M. and Rashidi, M. M. Inves-tigation of magneto-hemodynamic flow in a semi-porous channel using orthonormal Bernstein polynomials, Eur. Phys. J. Plus, 132(7) (2017), 1–16.
23. Jafari, S. and Navidi, H. A game-theoretic approach for modeling com-petitive diffusion over social networks, Games, 9(1) (2018), 8.
24. Jiménez-Lizárraga, M., Basin, M., Rodríguez, V., and Rodríguez, P. Open-loop Nash equilibrium in polynomial differential games via state-dependent Riccati equation, Automatica J. IFAC, 53 (2015), 155–163.
25. Johnson, P. A. Numerical solution methods for differential game prob-lems, PhD thesis, Massachusetts Institute of Technology, (2009).
26. Keshavarz, E., Ordokhani, Y., and Razzaghi, M. A numerical solution for fractional optimal control problems via Bernoulli polynomials, J. Vib.Control, 22(18) (2016), 3889–3903.
27. Nikooeinejad, Z., Delavarkhalafi, A., and Heydari, M. A numerical so-lution of open-loop Nash equilibrium in nonlinear differential games based on Chebyshev pseudospectral method, J. Comput. Appl. Math., 300 (2016), 369–384.
28. Nikooeinejad, Z., Delavarkhalafi, A., and Heydari, M. Application of shifted Jacobi pseudospectral method for solving (in) finite-horizon min-max optimal control problems with uncertainty, Int. J. Control, 91(3) (2018), 725–739.
29. Nikooeinejad, Z., Heydari, M., Saffarzadeh, M., Loghmani, G. B. and Engwerda, J. Numerical Simulation of Non-cooperative and Cooperative Equilibrium Solutions for a Stochastic Government Debt Stabilization Game, Comput. Econ., (2021), 1–27.
30. Nikooeinejad, Z., Delavarkhalafi, A., Heydari, M. and Wazwaz, A. M. A computational method for solving the system of Hamilton-Jacobi-Bellman PDEs in nonzero-sum fixed-final-time differential games, Trans. A. Raz-madze Math. Inst., 175(1) (2021), 83–100.
31. Nikooeinejad, Z. and Heydari, M. Nash equilibrium approximation of some class of stochastic differential games: A combined Chebyshev spec-tral collocation method with policy iteration, J. Comput. Appl. Math., 362 (2019), 41–54.
32. Nikooeinejad, Z., Heydari M. and Loghmani G. B. Numerical solution of two-point BVPs in infinite-horizon optimal control theory: A combined quasilinearization method with exponential Bernstein functions, Int. J. Comput. Math., 98(11) (2021), 2156–2174.
33. Rabiei, K., Ordokhani, Y., and Babolian, E. Numerical solution of 1d and 2d fractional optimal control of system via Bernoulli polynomials, Int. J. Appl. Comput., 4(1) (2018), 1–17.
34. Shen, J., Tang, T., and Wang, L.-L. Spectral methods: algorithms, anal-ysis and applications, Springer Science & Business Media, Berlin/Heidel-berg, Germany, (2011).
35. Sorger, G. Competitive dynamic advertising: A modification of the case game, J. Econ. Dyn. Control, 13(1) (1989), 55–80.
36. Starr, A. W. and Ho, Y.-C. Further properties of nonzero-sum differential games, J. Optim. Theory Appl., 3(4) (1969a), 207–219.
37. Starr, A. W. and Ho, Y.-C. Nonzero-sum differential games, J. Optim. Theory Appl., 3(3) (1969b), 184–206.
38. Tameh, M. S. and Shivanian, E. Fractional shifted Legendre tau method to solve linear and nonlinear variable-order fractional partial differential equations, Math. Sci., 15(1) (2021), 11–19.
39. Tari, A., Rahimi, M., Shahmorad, S., and Talati, F. Development of the tau method for the numerical solution of two-dimensional linear Volterra integro-differential equations, Comput. Methods Appl. Math., 9(4) (2009), 421–435.
40. Tohidi, E., Bhrawy, A., and Erfani, K. A collocation method based on Bernoulli operational matrix for numerical solution of generalized pan-tograph equation, Appl. Math. Model., 37(6) (2013), 4283–4294.
41. Torabi, S. M., Tari, A., and Shahmorad, S. Two-step collocation methods for two-dimensional Volterra integral equations of the second kind, J. Appl. Anal., 25(1) (2019), 1–11.
42. Wu, C., Teo, K. L., and Wu, S. Min-max optimal control of linear systems with uncertainty and terminal state constraints, Automatica J. IFAC, 49(6) (2013), 1809–1815.
43. Yazdaniyan, Z., Shamsi, M., Foroozandeh, Z., and de Pinho, M. d. R. A numerical method based on the complementarity and optimal control formulations for solving a family of zero-sum pursuit-evasion differential games, J. Comput. Appl. Math., 368, (2020), 112535
44. Yeung, D. W. and Petrosjan, L. A. Cooperative stochastic differential games, Springer Science & Business Media, Berlin/Heidelberg, Germany, (2006).