Geodesic
Two game-theoretic problems of approach
Sbornik. Mathematics, Tome 212 (2021) no. 9, pp. 1228-1260 Cet article a éte moissonné depuis la source Math-Net.Ru

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A nonlinear conflict control system in a finite-dimensional Euclidean space on a finite time interval is considered. Two interrelated game-theoretic problems of making a system approach a compact set at a fixed moment of time are studied. A method for constructing approximate solutions to game problems of approach is presented. Most attention is paid to problems related to constructing approximations of the solvability sets of game problems in the phase space. Bibliography: 35 titles.
Keywords: control system, target set, game-theoretic problem of approach, differential inclusion, solvability set. 
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A. A. Ershov; A. V. Ushakov; V. N. Ushakov. Two game-theoretic problems of approach. Sbornik. Mathematics, Tome 212 (2021) no. 9, pp. 1228-1260. http://geodesic.mathdoc.fr/item/SM_2021_212_9_a2/

[1] L. Pontryagin, “Linear differential games. I”, Soviet Math. Dokl., 8 (1967), 769–771 | MR | Zbl

[2] L. S. Pontryagin, “Linear differential games. II”, Soviet Math. Dokl., 8 (1967), 910–912 | MR | Zbl

[3] L. S. Pontryagin, “Matematicheskaya teoriya optimalnykh protsessov i differentsialnye igry”, Topologiya, obyknovennye differentsialnye uravneniya, dinamicheskie sistemy, Sbornik obzornykh statei. 2. K 50-letiyu instituta, Tr. MIAN SSSR, 169, Izd-vo AN SSSR, M., 1985, 119–158 | MR | Zbl

[4] M. S. Nikol'skii, “On the alternating integral of Pontryagin”, Math. USSR-Sb., 44:1 (1983), 125–132 | DOI | MR | Zbl

[5] M. S. Nikol'skii, “Approximate computation of the least guaranteed estimate in linear differential games with a fixed duration”, J. Appl. Math. Mech., 46:4 (1982), 550–552 | DOI | MR | Zbl

[6] E. S. Polovinkin, “Stability of terminal sets and optimization of pursuit time in differential games”, Differential Equations, 20:3 (1984), 330–341 | MR | Zbl

[7] 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 | DOI | MR | Zbl

[8] A. Azamov, “Semistability and duality in the theory of the Pontryagin alternating integral”, Dokl. Math., 37:2 (1988), 355–359 | MR | Zbl

[9] N. N. Krasovskii, Igrovye zadachi o vstreche dvizhenii, Nauka, M., 1970, 420 pp. | MR | Zbl

[10] N. N. Krasovskiĭ, A. I. Subbotin, “On the structure of differential games”, Soviet Math. Dokl., 11 (1970), 143–147 | MR | Zbl

[11] N. N. Krasovskii, A. I. Subbotin, Game-theoretical control problems, Springer Ser. Soviet Math., Springer-Verlag, New York, 1988, xii+517 pp. | MR | MR | Zbl | Zbl

[12] A. I. Subbotin, A. G. Chentsov, Optimizatsiya garantii v zadachakh upravleniya, Nauka, M., 1981, 288 pp. | MR | Zbl

[13] B. N. Pshenichnyi, “The structure of differential games”, Soviet Math. Dokl., 10 (1969), 70–72 | MR | Zbl

[14] B. N. Pshenichnyi, A. A. Chikrii, “The problem of evading contact in differential games”, U.S.S.R. Comput. Math. Math. Phys., 14:6 (1974), 46–56 | DOI | MR | Zbl

[15] Yu. S. Osipov, “Minimax absorption in difference-differential games”, Soviet Math. Dokl., 13 (1972), 337–341 | MR | Zbl

[16] A. V. Kryazhimskii, Yu. S. Osipov, “On an algorithmic criterion of the solvability of game problems for linear controlled systems”, Proc. Steklov Inst. Math. (Suppl.), 2000, suppl. 1, S154–S162 | MR | Zbl

[17] Yu. S. Osipov, Izbrannye trudy, Izd-vo Mosk. un-ta, M., 2009, 654 pp.

[18] M. I. Gusev, A. B. Kuržanskiĭ, “On equilibrium situations in multicriteria game problems”, Soviet Math. Dokl., 17:4 (1976), 1176–1180 | MR | Zbl

[19] A. B. Kurzhanskiĭ, “Pontryagin's alternated integral in the theory of control synthesis”, Proc. Steklov Inst. Math., 224 (1999), 212–225 | MR | Zbl

[20] 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 | DOI | DOI | MR | Zbl

[21] F. L. Chernousko, A. A. Melikyan, Igrovye zadachi upravleniya i poiska, Nauka, M., 1978, 270 pp. | MR | Zbl

[22] F. L. Chernousko, Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem. Metod ellipsoidov, Nauka, M., 1988, 319 pp. | MR | Zbl

[23] F. L. Chernous'ko, “Ellipsoidal approximation of attainability sets of a linear system with indeterminate matrix”, J. Appl. Math. Mech., 60:6 (1996), 921–931 | DOI | MR | Zbl

[24] L. A. Petrosyan, “Differential games with incomplete information”, Soviet Math. Dokl., 11 (1970), 1524–1527 | MR | Zbl

[25] L. A. Petrosyan, G. V. Tomskiĭ, “Dynamic games with complete information and their applications to games with incomplete information”, Differential Equations, 18:4 (1982), 429–434 | MR | Zbl

[26] L. A. Petrosjan, N. A. Zenkevich, “Conditions for sustainable cooperation”, Autom. Remote Control, 76:10 (2015), 1894–1904 | DOI | MR | Zbl

[27] P. Cardaliaguet, M. Quincampoix, P. Saint-Pierre, “Pursuit differential games with state constraints”, SIAM J. Control Optim., 39:5 (2000), 1615–1632 | DOI | MR | Zbl

[28] M. Bardi, M. Falcone, P. Soravia, “Numerical methods for pursuit-evasion games via viscosity solutions”, Stochastic and differential games, Ann. Internat. Soc. Dynam. Games, 4, Birkhäuser Boston, Boston, MA, 1999, 105–175 | DOI | MR | Zbl

[29] M. Falcone, “Numerical methods for differential games based on partial differential equations”, Int. Game Theory Rev., 8:2 (2006), 231–272 | DOI | MR | Zbl

[30] N. Lukoyanov, M. Gomoyunov, “Differential games on minmax of the positional quality index”, Dyn. Games Appl., 9:3 (2019), 780–799 | DOI | MR | Zbl

[31] V. Patsko, M. Quincampoix, “Preface: DGAA special issue on pursuit-evasion games and differential games with incomplete information”, Dyn. Games Appl., 9:3 (2019), 569–572 | DOI | MR | Zbl

[32] A. M. Taras'yev, V. N. Ushakov, A. P. Khripunov, “On a computational algorithm for solving game control problems”, J. Appl. Math. Mech., 51:2 (1987), 167–172 | DOI | MR | Zbl

[33] V. N. Ushakov, A. A. Uspenskii, “On a supplement to the stability property in differential games”, Proc. Steklov Inst. Math., 271 (2010), 286–305 | DOI | MR | Zbl

[34] V. N. Ushakov, V. I. Ukhobotov, A. E. Lipin, “An addition to the definition of a stable bridge and an approximating system of sets in differential games”, Proc. Steklov Inst. Math., 304 (2019), 268–280 | DOI | DOI | MR | Zbl

[35] A. Bressan, B. Piccoli, Introduction to the mathematical theory of control, AIMS Ser. Appl. Math., 2, Am. Inst. Math. Sci. (AIMS), Springfield, MO, 2007, xiv+312 pp. | MR | Zbl