On the Theory of Positional Differential Games for Neutral-Type Systems
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 con...
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| Published in | Proceedings of the Steklov Institute of Mathematics Vol. 309; no. Suppl 1; pp. S83 - S92 |
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| Main Authors | , |
| Format | Journal Article Conference Proceeding |
| Language | English |
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Moscow
Pleiades Publishing
01.08.2020
Springer Nature B.V |
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| 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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| AbstractList | 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. |
| Author | Lukoyanov, N. Yu Plaksin, A. R. |
| Author_xml | – sequence: 1 givenname: N. Yu surname: Lukoyanov fullname: Lukoyanov, N. Yu email: nyul@imm.uran.ru organization: Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ural Federal Universit – sequence: 2 givenname: A. R. surname: Plaksin fullname: Plaksin, A. R. email: a.r.plaksin@gmail.com organization: Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ural Federal Universit |
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| Cites_doi | 10.1134/S0081543815080155 10.31857/S003282350002733-6 10.1016/0021-8928(78)90054-0 10.1007/978-1-4612-9892-2 10.1016/0021-8928(71)90032-3 10.1007/978-94-015-7793-9 10.1134/S0081543817090061 10.1134/S0081543818050048 |
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| Keywords | control theory neutral-type systems differential games |
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| References | N. N. Krasovskii, Control of a Dynamical System (Nauka, Moscow, 1985) [in Russian]. 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, Some Problems in the Theory of Motion Stability (Fizmatgiz, Moscow, 1959) [in Russian]. 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 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 A. V. Kryazhimskii, “On stable position control in differential games,” J. Appl. Math. Mech. 42 (6), 1055–1060 (1980). 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, “On the application of A. M. Lyapunov’s second method for time-delay equations,” Prikl. Mat. Mekh. 20 (3), 315–327 (1956). N. Yu. Lukoyanov, Functional Hamilton–Jacobi Equations and Control Problems with Hereditary Information (Izd. Ural. Fed. Univ., Yekaterinburg, 2011) [in Russian]. A. F. Filippov, Differential Equations with Discontinuous Righthand Sides (Nauka, Moscow, 1985; Springer, Berlin, 1988). 8010_CR2 8010_CR1 8010_CR6 8010_CR5 8010_CR4 8010_CR3 8010_CR11 8010_CR10 8010_CR9 8010_CR8 8010_CR7 8010_CR13 8010_CR12 8010_CR14 |
| References_xml | – reference: 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]. – reference: N. N. Krasovskii and A. I. Subbotin, Positional Differential Games (Nauka, Moscow, 1974) [in Russian]. – reference: N. N. Krasovskii, Some Problems in the Theory of Motion Stability (Fizmatgiz, Moscow, 1959) [in Russian]. – reference: A. V. Kryazhimskii, “On stable position control in differential games,” J. Appl. Math. Mech. 42 (6), 1055–1060 (1980). – reference: N. Yu. Lukoyanov, Functional Hamilton–Jacobi Equations and Control Problems with Hereditary Information (Izd. Ural. Fed. Univ., Yekaterinburg, 2011) [in Russian]. – reference: Yu. S. Osipov, “On the theory of differential games of systems with aftereffect,” J. Appl. Math. Mech. 35 (2), 262–272 (1971). – reference: 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). – reference: 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). – reference: J. Hale, Theory of Functional Differential Equations (Springer, New York, 1977). – reference: 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 – reference: N. Yu. Lukoyanov and A. R. Plaksin, “Differential games for neutral-type systems: An approximation model,” Proc. Steklov Inst. Math. 291, 190–202 (2015). – reference: N. N. Krasovskii, Control of a Dynamical System (Nauka, Moscow, 1985) [in Russian]. – reference: A. F. Filippov, Differential Equations with Discontinuous Righthand Sides (Nauka, Moscow, 1985; Springer, Berlin, 1988). – reference: 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 – ident: 8010_CR5 doi: 10.1134/S0081543815080155 – ident: 8010_CR7 doi: 10.31857/S003282350002733-6 – ident: 8010_CR9 doi: 10.1016/0021-8928(78)90054-0 – ident: 8010_CR4 doi: 10.1007/978-1-4612-9892-2 – ident: 8010_CR2 doi: 10.1016/0021-8928(71)90032-3 – ident: 8010_CR11 – ident: 8010_CR10 – ident: 8010_CR12 – ident: 8010_CR13 doi: 10.1007/978-94-015-7793-9 – ident: 8010_CR1 – ident: 8010_CR14 – ident: 8010_CR3 – ident: 8010_CR6 doi: 10.1134/S0081543817090061 – ident: 8010_CR8 doi: 10.1134/S0081543818050048 |
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| SubjectTerms | Differential equations Differential games Game theory Geometric constraints Mathematics Mathematics and Statistics Minimax technique Saddle points |
| Title | On the Theory of Positional Differential Games for Neutral-Type Systems |
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