Abstract:
The paper deals with a zero-sum differential game, in which the dynamic of a conflict-controlled system is described by linear functional differential equations of neutral type and the quality index is the sum of two terms: the first term estimates the history of motion of the system realized by the terminal time, and the second term is an integral-quadratic estimation of the corresponding realizations of the players' controls. To calculate the value and construct the optimal control laws in this differential game, we propose an approach based on solving a suitable auxiliary differential game, in which the motion of a conflict-controlled system is described by ordinary differential equations and the quality index contains an estimation of the motion at the terminal time only. To find the value and the saddle point in the auxiliary differential game, we apply the so-called upper convex hull method, which leads to an effective solution in the case under consideration due to the specific structure of the quality index and the geometric constraints on the control actions of the players. The efficiency of the approach is illustrated by an example, and the results of numerical simulations are presented. The constructed optimal control laws are compared with the optimal control procedures with finite-dimensional approximating guides, which were developed by the authors earlier.
Keywords:
differential games, neutral-type systems, optimal control strategies, numerical methods.
Citation:
M. I. Gomoyunov, N. Yu. Lukoyanov, “On the numerical solution of differential games for neutral-type linear systems”, Trudy Inst. Mat. i Mekh. UrO RAN, 23, no. 1, 2017, 75–87; Proc. Steklov Inst. Math. (Suppl.), 301, suppl. 1 (2018), 44–56
\Bibitem{GomLuk17}
\by M.~I.~Gomoyunov, N.~Yu.~Lukoyanov
\paper On the numerical solution of differential games for neutral-type linear systems
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2017
\vol 23
\issue 1
\pages 75--87
\mathnet{http://mi.mathnet.ru/timm1385}
\crossref{https://doi.org/10.21538/0134-4889-2017-23-1-75-87}
\elib{https://elibrary.ru/item.asp?id=28409369}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2018
\vol 301
\issue , suppl. 1
\pages 44--56
\crossref{https://doi.org/10.1134/S0081543818050048}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000453520500008}
Linking options:
https://www.mathnet.ru/eng/timm1385
https://www.mathnet.ru/eng/timm/v23/i1/p75
This publication is cited in the following 15 articles:
V. N. Ushakov, A. A. Ershov, “K konstruirovaniyu reshenii igrovoi zadachi s fiksirovannym momentom okonchaniya”, Tr. IMM UrO RAN, 30, no. 3, 2024, 255–273
Vladimir N. Ushakov, Aleksandr M. Tarasev, Andrei V. Ushakov, “Minimaksnaya differentsialnaya igra s fiksirovannym momentom okonchaniya”, MTIP, 16:3 (2024), 77–112
V. N. Ushakov, A. M. Tarasyev, A. A. Ershov, “A Supplement to Krasovskii's Unification Method in Differential Game Theory”, Dokl. Math., 2024
V. N. Ushakov, A. V. Ushakov, O. A. Kuvshinov, “Sblizhenie konfliktno upravlyaemykh sistem na konechnom promezhutke vremeni”, Izv. IMI UdGU, 64 (2024), 70–96
V. N. Ushakov, A. M. Tarasyev, A. A. Ershov, “Concerning one supplement to unification method of N.N. Krasovskii in differential games theory”, Dokl. RAN. Math. Inf. Proc. Upr., 519 (2024), 65–71
V. N. Ushakov, A. A. Ershov, “On the Construction of Solutions to a Game Problem with a Fixed End Time”, Proc. Steklov Inst. Math., 327:S1 (2024), S239
V. N. Ushakov, A. M. Tarasyev, A. V. Ushakov, “Minimax Differential Game with a Fixed End Moment”, Dokl. Math., 110:S2 (2024), S495
MUHAMMAD AKBAR, RASHID NAWAZ, SUMBAL AHSAN, KOTTAKKARAN SOOPPY NISAR, KAMAL SHAH, EMAD E. MAHMOUD, M. M. ALQARNI, “FRACTIONAL POWER SERIES APPROACH FOR THE SOLUTION OF FRACTIONAL-ORDER INTEGRO-DIFFERENTIAL EQUATIONS”, Fractals, 30:01 (2022)
M. Gomoyunov, “Solution to a zero-sum differential game with fractional dynamics via approximations”, Dyn. Games Appl., 10:2 (2020), 417–443
Mikhail I. Gomoyunov, “On Representation Formulas for Solutions of Linear Differential Equations with Caputo Fractional Derivatives”, Fract Calc Appl Anal, 23:4 (2020), 1141
N. Yu. Lukoyanov, A. R. Plaksin, “On the Theory of Positional Differential Games for Neutral-Type Systems”, Proc. Steklov Inst. Math. (Suppl.), 309, suppl. 1 (2020), S83–S92
Nikolai Lukoyanov, Mikhail Gomoyunov, “Differential Games on Minmax of the Positional Quality Index”, Dyn Games Appl, 9:3 (2019), 780
M. I. Gomoyunov, N. Yu. Lukoyanov, A. R. Plaksin, “Uravneniya Gamiltona-Yakobi v zadachakh dinamicheskoi optimizatsii sistem neitralnogo tipa”, Vestnik Tambovskogo universiteta. Seriya: estestvennye i tekhnicheskie nauki, 23:122 (2018), 268–277
Mikhail Gomoyunov, Anton Plaksin, “On Hamilton-Jacobi equations for neutral-type differential games”, IFAC-PapersOnLine, 51:14 (2018), 171
A. R. Plaksin, “Ob uravnenii Gamiltona–Yakobi–Aizeksa–Bellmana dlya sistem neitralnogo tipa”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:2 (2017), 222–237
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