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
Received May 27, 2024
Revised June 6, 2024
Accepted June 24, 2024
Nina Nikolaevna Subbotina, Dr. Phys.-Math. Sci., Prof., Corresponding Member of RAS, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620000 Russia,
e-mail: subb@uran.ru
Aleksei Semenovich Rodin, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: alexey.rodin.ekb@gmail.com
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Cite this article as: 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 Mekhaniki UrO RAN, 2024, vol. 30, no. 3, pp. 229–240.
