Geodesic
A construction of Nash equilibrium based on system of Hamilton--Jacobi equations of special type
Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 9 (2017) no. 4, pp. 39-53.

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The paper is devoted to the Nash program strategies' construction in nonzero-sum differential games with two players applying the solution of a strong coupled system of Hamilton–Jacobi equations. The system of Hamilton–Jacobi equations is of the special type where the first equation of the system doesn't depend on the second one, and the second equation depends on the derivative of the first equation's solution. We show that the solution of the system of Hamilton–Jacobi equations should be considered in the class of multivalued maps. We propose a generalized solution for the system of Hamilton–Jacobi equations and prove the existence theorem for such generalized solution. To conclude we consider an example illustrating the Nash program strategies' construction.
Keywords: hierarchical differential games, Nash equilibrium, generalized solution, system of Hamilton–Jacobi equations. 
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Ekaterina A. Kolpakova. A construction of Nash equilibrium based on system of Hamilton--Jacobi equations of special type. Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 9 (2017) no. 4, pp. 39-53. http://geodesic.mathdoc.fr/item/MGTA_2017_9_4_a2/

[1] Varga Dzh., Optimalnoe upravlenie differentsialnymi t funktsionalnymi uravneniyami, Nauka, M., 1977

[2] Lakhtin A. S., Subbotin A. I., “Minimaksnye i vyazkostnye resheniya razryvnykh uravnenii s chastnymi proizvodnymi pervogo poryadka”, Doklady Akademii Nauk, 359:4 (1998), 452–455

[3] Lakhtin A. S., Subbotin A. I., “Mnogoznachnye resheniya uravnenii s chastnymi proizvodnymi pervogo poryadka”, Matematicheskii sbornik, 189:6 (1998), 33–58 | DOI

[4] Petrosyan L. A., Zenkevich N. A., Shevkoplyas E. V., Teoriya igr, BKhV-Peterburg, Sankt-Peterburg, 2012

[5] Subbotin A. I., Obobschennye resheniya uravnenii s chastnymi proizvodnymi pervogo poryadka: perspektivy dinamicheskoi optimizatsii, Institut kompyuternykh issledovanii, M.–Izhevsk, 2003

[6] Subbotina N. N., “Metod kharakteristik dlya uravnenii Gamiltona–Yakobi i ego prilozheniya v dinamicheskoi optimizatsii”, Sovremennaya matematika i ee prilozheniya, 20, 2004, 1–129

[7] Bressan A., Shen W., “Semi-cooperative strategies for differential games”, International Journal of Game Theory, 32 (2004), 1–33 | DOI | MR

[8] Cardaliaguet P., Plaskacz S., “Existence and uniqueness of a Nash equilibrium feedback for a simple nonzero-sum differential game”, International Journal of Game Theory, 32 (2003), 33–71 | DOI | MR

[9] Case J. H., “Toward a theory of many player differential games”, SIAM J. Control, 7:2 (1969), 179–197 | DOI | MR

[10] Friedman A., Differential Games, Wiley-Interscience, 1971 | MR

[11] Starr A. W., Ho Y. C., “Non-zero Sum Differential Games”, Journal of Optimization Theory and Applications, 3:3 (1969), 184–206 | DOI | MR