## Suboptimal universal strategies in an optimum-time game problem.(English. Russian original)Zbl 0885.90132

J. Appl. Math. Mech. 59, No. 5, 677-683 (1995); translation from Prikl. Mat. Mekh. 59, No. 5, 707-713 (1995).
Summary: A differential game in which the payoff functional is the time required for the phase point to reach the target set is considered. A construction of $$\varepsilon$$-optimal strategies, similar to the standard construction when the value function is everywhere differentiable, is proposed. The difference is that the gradient of a non-smooth and discontinuous value function is replaced by a certain quasigradient.

### MSC:

 91A23 Differential games (aspects of game theory)
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### References:

 [1] Krasovskii, N. N., Control of a Dynamical System (1985), Nauka: Nauka Moscow · Zbl 0090.06101 [2] Krasovskii, N. N., Differential games. Approximative and formal models, Mat. Sbornik, 107, 4, 541-571 (1978) [3] Kononenko, A. R., On equilibrium positional strategies in non-antagonistic differential games, Dokl. Akad. Nauk SSSR, 231, 2, 285-288 (1976) [4] Subbotina, N. N., Universal optimal strategies in positional differential games, Diferents. Uravn., 19, 11, 1890-1896 (1983) · Zbl 0543.90104 [5] Subbotin, A. I., Continuous and discontinuous solutions of boundary-value problems for first-order partial differential equations, Dokl. Ross. Akad. Nauk, 323, 1, 30-34 (1992) · Zbl 0787.35022 [6] Crandall, M. G.; Ishii, H.; Lions, P.-L., Uniqueness of viscosity solutions of Hamilton-Jacobi equations revisited, J. Math. Soc. Japan, 39, 4, 581-596 (1987) · Zbl 0644.35016 [7] Subbotin, A. I., Discontinuous solutions of a Dirichlet type boundary-value problem for the first-order partial differential equation, Russian. J. Numer. Anal. Math. Modelling, 8, 2, 145-164 (1993) · Zbl 0821.35027 [8] Clarke, F. H., Optimization and Nonsmooth Analysis (1983), John Wiley: John Wiley New York · Zbl 0727.90045 [9] Garnysheva, G. G.; Subbotin, A. I., A strategy of minimax shooting in the direction of the quasi-gradient, Prikl. Mat. Mekh., 58, 4, 5-11 (1994) · Zbl 0885.90131 [10] Krasovskii, N. N.; Subbotin, A. I., Positional Differential Games (1974), Nauka: Nauka Moscow · Zbl 0298.90067 [11] Isaacs, R., Differential Games (1965), John Wiley: John Wiley New York · Zbl 0152.38407 [12] Kruzhkov, S. N., Generalized solutions of Hamilton-Jacobi equations of the eikonal type, I. Mat. Sbornik, 98, 3, 450-493 (1975)
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