Abstract
For a dynamical system whose motion is described by neutral-type differential equations in Hale’s form, we consider a minimax–maximin differential game with a quality index evaluating the motion history realized up to the terminal time. The control actions of the players are subject to geometric constraints. The game is formalized in classes of pure positional strategies with a memory of the motion history. It is proved that the game has a value and a saddle point. The proof is based on the choice of an appropriate Lyapunov–Krasovskii functional for the construction of control strategies by the method of an extremal shift to accompanying points.
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REFERENCES
N. N. Krasovskii and A. I. Subbotin, Positional Differential Games (Nauka, Moscow, 1974) [in Russian].
Yu. S. Osipov, “On the theory of differential games of systems with aftereffect,” J. Appl. Math. Mech. 35 (2), 262–272 (1971).
N. N. Krasovskii, Control of a Dynamical System (Nauka, Moscow, 1985) [in Russian].
J. Hale, Theory of Functional Differential Equations (Springer, New York, 1977).
N. Yu. Lukoyanov and A. R. Plaksin, “Differential games for neutral-type systems: An approximation model,” Proc. Steklov Inst. Math. 291, 190–202 (2015).
M. I. Gomoyunov, N. Yu. Lukoyanov, and A. R. Plaksin, “Existence of a value and a saddle point in positional differential games for neutral-type systems,” Proc. Steklov Inst. Math. 299 (Suppl. 1), S37–S48 (2017).
M. I. Gomoyunov and A. R. Plaksin, “On the basic equation of differential games for neutral-type systems,” J. Appl. Math. Mech. 82 (6), 675–689 (2018). doi 10.31857/S003282350002733-6
M. I. Gomoyunov and Yu. N. Lukoyanov, “On the numerical solution of differential games for neutral-type linear systems,” Proc. Steklov Inst. Math. 301 (Suppl. 1), S44–S56 (2018). doi 10.1134/S0081543818050048
A. V. Kryazhimskii, “On stable position control in differential games,” J. Appl. Math. Mech. 42 (6), 1055–1060 (1980).
V. I. Maksimov, “A differential game of guidance for systems with a deviating argument of neutral type,” in Problems of Dynamic Control (IMM UNTs AN SSSR, Sverdlovsk, 1981), pp. 33–45 [in Russian].
N. N. Krasovskii, “On the application of A. M. Lyapunov’s second method for time-delay equations,” Prikl. Mat. Mekh. 20 (3), 315–327 (1956).
N. N. Krasovskii, Some Problems in the Theory of Motion Stability (Fizmatgiz, Moscow, 1959) [in Russian].
A. F. Filippov, Differential Equations with Discontinuous Righthand Sides (Nauka, Moscow, 1985; Springer, Berlin, 1988).
N. Yu. Lukoyanov, Functional Hamilton–Jacobi Equations and Control Problems with Hereditary Information (Izd. Ural. Fed. Univ., Yekaterinburg, 2011) [in Russian].
Funding
This work was supported by the Russian President’s Grant for Young Russian Scientists no. MK-3566.2019.1.
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Translated by E. Vasil’eva
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Lukoyanov, N.Y., Plaksin, A.R. On the Theory of Positional Differential Games for Neutral-Type Systems. Proc. Steklov Inst. Math. 309 (Suppl 1), S83–S92 (2020). https://doi.org/10.1134/S0081543820040100
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DOI: https://doi.org/10.1134/S0081543820040100