Abstract:
In this article the concept of the lower alternating integral of Pontryagin, which differs from the analogous concept of A. Azamov (Mat. Sb. (N.S.), 1982, v. 118(160), № 3, p. 422–430), is introduced. The properties of this object are studied in comparison with those of the alternating integral of Pontryagin. A procedure is given for application of the lower alternating integral of Pontryagin to linear differential games of pursuit, with special consideration for the question of using the lower alternating integral of Pontryagin when there are countably many measurements of the phase vector. The results are illustrated in two known differential games: “the boy and the crocodile” and the “control example of Pontryagin”.
Bibliography: 18 titles.
Citation:
M. S. Nikol'skii, “On the lower alternating integral of Pontryagin in linear differential games of pursuit”, Math. USSR-Sb., 56:1 (1987), 33–47
\Bibitem{Nik85}
\by M.~S.~Nikol'skii
\paper On the lower alternating integral of Pontryagin in linear differential games of pursuit
\jour Math. USSR-Sb.
\yr 1987
\vol 56
\issue 1
\pages 33--47
\mathnet{http://mi.mathnet.ru/eng/sm2016}
\crossref{https://doi.org/10.1070/SM1987v056n01ABEH003022}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=805694}
\zmath{https://zbmath.org/?q=an:0621.90109|0607.90099}
Linking options:
https://www.mathnet.ru/eng/sm2016
https://doi.org/10.1070/SM1987v056n01ABEH003022
https://www.mathnet.ru/eng/sm/v170/i1/p35
This publication is cited in the following 15 articles:
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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. A. Ershov, “On the Construction of Solutions to a Game Problem with a Fixed End Time”, Proc. Steklov Inst. Math., 327:S1 (2024), S239
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V. N. Ushakov, A. V. Ushakov, “O sblizhenii upravlyaemoi sistemy na konechnom promezhutke vremeni”, Izv. IMI UdGU, 60 (2022), 111–154
A. A. Ershov, A. V. Ushakov, V. N. Ushakov, “Two game-theoretic problems of approach”, Sb. Math., 212:9 (2021), 1228–1260
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Iskanadjiev I., “Duality of the Alternating Integral for Quasi-Linear Differential Games”, Nonlinear Anal.-Model Control, 17:2 (2012), 169–181
A. Azamov, I. M. Iskanadjiev, “Pontryagin’s Alternating Integral for Differential Inclusions with Counteraction”, Contributions to Game Theory and Management, 5 (2012), 33–44
Dar'in, AN, “Control under indeterminacy and double constraints”, Differential Equations, 39:11 (2003), 1554
Daryin A., “Nonlinear synthesis for uncertain systems with diverse types of constraints”, Nonlinear Control Systems 2001, Ifac Symposia Series, 2002, 1511–1516
A. B. Kurzhanskii, N. B. Melnikov, “On the problem of control synthesis: the Pontryagin alternating integral and the Hamilton–Jacobi equation”, Sb. Math., 191:6 (2000), 849–881
A. B. Kurzhanskii, “Pontryagin's Alternated Integral in the Theory of Control Synthesis”, Proc. Steklov Inst. Math., 224 (1999), 212–225
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