Solution for a System of Hamilton–Jacobi Equations of Special Type and a Link with Nash Equilibrium

  • Ekaterina A. Kolpakova
Part of the Static & Dynamic Game Theory: Foundations & Applications book series (SDGTFA)


The paper is concerned with systems of Hamilton–Jacobi PDEs of the special type. This type of systems of Hamilton–Jacobi PDEs is closely related with a bilevel optimal control problem. The paper aims to construct equilibria in this bilevel optimal control problem using the generalized solution for the system of the Hamilton–Jacobi PDEs. We introduce the definition of the solution for the system of the Hamilton–Jacobi PDEs in a class of multivalued functions. The notion of the generalized solution is based on the notions of minimax solution and M-solution to Hamilton–Jacobi equations proposed by Subbotin. We prove the existence theorem for the solution of the system of the Hamilton–Jacobi PDEs.



This work was supported by the Russian Fond of Fundamental Researches under grant No. 17-01-00074.


  1. 1.
    Averboukh, Y., Baklanov A.: Stackelberg solutions of differential games in the class of nonanticipative strategies. Dyn. Games Appl. 4(1), 1–9 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Basar, T., Olsder, G.J.: Dynamic Noncooperative Game Theory. SIAM, Philadelphia (1999)zbMATHGoogle Scholar
  3. 3.
    Bressan, A., Shen, W.: Semi-cooperative strategies for differential games. Int. J. Game Theory 32, 1–33 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cardaliaguet, P., Plaskacz, S.: Existence and uniqueness of a Nash equilibrium feedback for a simple nonzero-sum differential game. Int. J. Game Theory 32, 33–71 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, C.I., Cruz, J.B., Jr.: Stackelberg solution for two-person games with biased information patterns. IEEE Trans. Autom. Control 6, 791–798 (1972)CrossRefGoogle Scholar
  6. 6.
    Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27, 1–67 (1992)Google Scholar
  7. 7.
    Friedman, A.: Differential Games. Wiley, Hoboken (1971)zbMATHGoogle Scholar
  8. 8.
    Glimm, J.: Solutions in the large for nonlinear systems of equations. Commun. Pure Appl. Math. 18, 697–715 (1965)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lakhtin, A.S., Subbotin, A.I.: Multivalued solutions of first-order partial differential equations. Sb. Math. 189(6), 849–873 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lakhtin, A.S., Subbotin, A.I.: The minimax and viscosity solutions in discontinuous partial differential equations of the first order. Dokl. Akad. Nauk 359, 452–455 (1998)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Pergamon Press, New York (1964)Google Scholar
  12. 12.
    Starr, A.W., Ho, Y.C.: Non-zero sum differential games. J. Optim. Theory Appl. 3(3), 184–206 (1969)CrossRefGoogle Scholar
  13. 13.
    Subbotin, A.I.: Generalized Solutions of First-Order PDEs: The Dynamical Optimization Perspectives. Birkhauser, Boston (1995)CrossRefGoogle Scholar
  14. 14.
    Subbotina, N.N.: The method of characteristics for Hamilton–Jacobi equations and applications to dynamical optimization. J. Math. Sci. 135(3), 2955–3091 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Warga, J.: Optimal Control of Differential and Functional Equations. Academic Press, New York (1972)zbMATHGoogle Scholar
  16. 16.
    Ye, J.J.: Optimal strategies for bilevel dynamic problems. SIAM J. Control Optim. 35(2), 512–531 (1997)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Ekaterina A. Kolpakova
    • 1
  1. 1.Krasovskii Institute of Mathematics and Mechanics UrB RASYekaterinburgRussia

Personalised recommendations