Geodesic
Generalized Hopf formula for the value function in the positional differential game “Boy and Crocodile”
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 3, pp. 229-240 Cet article a éte moissonné depuis la source Math-Net.Ru

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The paper proposes a new formula for the minimax solution to the Cauchy boundary value problem for the Hamilton–Jacobi equation in the case when the Hamiltonian depends on time and the gradient in the phase variable of the minimax solution. This formula is a generalization of the Hopf formula. It is shown using a specific example that this formula is true for the minimax solution of the Hamilton–Jacobi equation in the Cauchy problem, which arises in the positional differential game “Boy and Crocodile.” The proposed formula describes the value function in this differential game.
Keywords: positional differential game, value function, Hamilton–Jacobi equation, Hopf formula, directional derivative, minimax solution. 
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N. N. Subbotina; A. S. Rodin. Generalized Hopf formula for the value function in the positional differential game “Boy and Crocodile”. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 3, pp. 229-240. http://geodesic.mathdoc.fr/item/TIMM_2024_30_3_a17/

[1] Krasovskii N.N., Teoriya upravleniya dvizheniem, Nauka, M, 1968, 475 pp.

[2] Krasovskii N.N., Subbotin A.I., Pozitsionnye differentsialnye igry, Nauka, M., 1974, 456 pp. | MR

[3] Subbotin A.I., Obobschennye resheniya uravnenii v chastnykh proizvodnykh pervogo poryadka. Perspektivy dinamicheskoi optimizatsii, In-t kompyuter. issled., M.; Izhevsk, 2003, 336 pp.

[4] Subbotin A.I., “Generalization of the main equation of differential game theory”, J. Optim. Theory Appl., 43:1 (1984), 103–133 | DOI | MR | Zbl

[5] Subbotin A. I., Minimaksnye neravenstva i uravneniya Gamiltona — Yakobi, Nauka, M., 1991, 216 pp.