Abstract
The paper deals with the construction of universal closed-loop strategies for two-person nonzero-sum differential game of a special type. The dynamics of the first player is defined by its own position and control. The dynamics of the second player is defined by its own control and the position of both players. The strategies are constructed based on the solution of a system of Hamilton–Jacobi equations. The system of Hamilton–Jacobi equations has a hierarchical type. A generalized solution of the system of Hamilton–Jacobi equations belongs to the class of multivalued mappings. We show the relationship between the values of the players and the generalized solution of the system of Hamilton–Jacobi equations.
Similar content being viewed by others
REFERENCES
Warga, J., Optimal Control of Differential and Functional Equations, New York–London: Academic Press, 1972. Translated under the title: Optimal’noe upravlenie differentsial’nymi i funktsional’nymi uravneniyami, Moscow: Nauka, 1977.
Kolpakova, E.A., On the solution of a system of Hamilton–Jacobi equations of special form, Proc. Steklov Inst. Math. (Suppl.), 2018, vol. 301, no. 1, pp. 103–114.
Kolpakova, E.A., A construction of Nash equilibrium based on a system of Hamilton–Jacobi equations of special type, Mat. Teor. Igr Pril., 2017, vol. 9, no. 4, pp. 39–53.
Krasovskii, N.N. and Subbotin, A.I., Pozitsionnye differentsial’nye igry (Positional–Differential Games), Moscow: Nauka, 1974.
Lakhtin, A.S. and Subbotin, A.I., Minimax and viscosity solutions to discontinuous first-order partial differential equations, Dokl. Ross. Akad. Nauk, 1998, vol. 359, no. 4, pp. 452–455.
Subbotin, A.I., Generalized Solutions of First-Order PDEs. The Dynamical Optimization Perspective, Basel: Birkhäuser, 1994. Translated under the title: Obobshchennye resheniya uravnenii s chastnymi proizvodnymi pervogo poryadka: perspektivy dinamicheskoi optimizatsii, Moscow–Izhevsk: Inst. Komp’yut. Issled., 2003.
Subbotina, N.N., The method of characteristics for Hamilton–Jacobi equations and applications to dynamical optimization, J. Math. Sci. (New York), 2006, vol. 135, no. 3, pp. 2955–3090.
Aliprantis, C.D. and Border, K.C., Infinite Dimensional Analysis: A Hitchhiker’s Guide, Berlin–Heidelberg: Springer–Verlag, 2006.
Basar, T. and Olsder, G.J., Dynamic Noncooperative Game Theory, Philadelphia: SIAM, 1999.
Bressan, A. and Shen, W., Semi-cooperative strategies for differential games, Int. J. Game Theory, 2004, vol. 32, pp. 1–33.
Cardaliaguet, P. and Plaskacz, S., Existence and uniqueness of a Nash equilibrium feedback for a simple nonzero-sum differential game, Int. J. Game Theory, 2003, vol. 32, pp. 33–71.
Case, J.H., Toward a theory of many player differential games, SIAM J. Control, 1969, vol. 7, no. 2, pp. 179–197.
Friedman, A., Differential Games, New York: Wiley-Interscience, 1971.
Rockafellar, R.T. and Wets, R.J.-B., Variational Analysis, Berlin–Heidelberg: Springer–Verlag, 1998.
Starr, A.W. and Ho, Y.C., Non-zero sum differential games, J. Optim. Theory Appl., 1969, vol. 3, no. 3, pp. 184–206.
Funding
This work was supported by the Russian Science Foundation, project no. 17-11-01093.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by V. Potapchouck
Rights and permissions
About this article
Cite this article
Kolpakova, E.A. Feedback Strategies in a Nonzero-Sum Differential Game of Special Type. Autom Remote Control 82, 889–901 (2021). https://doi.org/10.1134/S000511792105012X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S000511792105012X