Abstract
We study the pursuit-evasion problem for a model game with simple motions when pursuer controls are subject to both integral and geometric constraints while evader controls are only subject to geometric ones. Depending on initial conditions of the players and parametric values participating in control constraints, we prove the theorem of alternative. To solve the pursuit problem, we propose a parallel pursuit strategy (Π-strategy) that ensures optimal convergence of the players and study its structure depending on the parameters. To solve the evasion problem, we find lower bounds on the convergence that also depend on given parameters. This work develops and extends the works of Isaacs, Petrosyan, Pshenichnyi and other researchers, including the author.
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Pontryagin, L.S., Izbrannye nauchnye trudy, tom 2: Differentsial’nye uravneniya. Teoriya operatorov. Optimal’noe upravlenie. Differentsial’nye igry (Selected Works, vol. II: Differential Equations. Operator Theory. Optimal Control. Differential Games), Moscow: Nauka, 1988.
Krasovskii, N.N., Upravlenie dinamicheskoi sistemoi. Zadacha o minimume garantirovannogo rezul’tata (Control in a Dynamic System. The Minimal Guaranteed Result Problem), Moscow: Nauka, 1985.
Isaaks, R., Differential Games. A Mathematical Theory with Applications to Warfare and Pursuit, Control, and Optimization, New York: Wiley, 1952. Translated under the title Differentsial’nye igry, Moscow: Mir, 1967.
Petrosyan, L.A., On One Family of DIfferential Survival Games in the Space R n, Dokl. Akad. Nauk SSSR, 1965, vol. 161, no. 1, pp. 52–54.
Petrosyan, L.A., Differentsial’nye igry presledovaniya (Differential Pursuit Games), Leningrad: Leningr. Gos. Univ., 1977.
Azamov, A., On the Quality Problem for Simple Pursuit Games with Constraints, Serdika. B”lgarsko Mat. Spis., 1986, vol. 12, pp. 38–43.
Grigorenko, N.L., Matematicheskie metody upravleniya neskol’kimi dinamicheskimi protsessami (Mathematical Control Methods for Several Dynamic Processes), Moscow: Mosk. Gos. Univ., 1980.
Pshenichnyi, B.N., Simple Pursuit with Several Objects, Kibernetika, 1976, no. 3, pp. 145–146.
Pshenichnyi, B.N. and Ostapenko, V.V., Differentsial’nye igry (Differential Games), Kiev: Naukova Dumka, 1992.
Rikhsiev, B.B., Differentsial’nye igry s prostymi dvizheniyami (Differential Games with Simple Motions), Tashkent: Fan, 1989.
Satimov, N.Yu., Metody resheniya zadachi presledovaniya v teorii differentsial’nykh igr (Methods for Solving Pursuit Problems in the Theory of Differential Games), Tashkent: NUUz, 2003.
Azamov, A.A. and Samatov, B.T., Π-strategy. An Elementary Introduction to the Theory of Differential Games, Tashkent: National Univ. Uzb., 2000.
Chikrii, A.A., Conflict-Controlled Processes, Boston: Kluwer, 1997.
Ushakov, V.N., Extremal Strategies in Differential Games with Integral Constraints, Prikl. Mat. Mekh., 1972, vol. 36, no. 1, pp. 15–23.
Chernous’ko, F.L. and Melikyan, A.A., Igrovye zadachi upravleniya i poiska (Game Problems in Search and Control), Moscow: Nauka, 1978.
Subbotin, A.I. and Chentsov, A.G., Optimizatsiya garantii v zadachakh upravleniya (Optimizing Guarantees in Control Problems), Moscow: Nauka, 1981.
Kim, D.P., Metody poiska i presledovaniya podvizhnykh ob”ektov (Methods for Search and Pursuit of Moving Objects), Moscow: Nauka, 1989.
Samatov, B.T., The Differential Game with “A Survival Zone” with Different Classes of Admissible Control Functions, in Game Theory Appl., Hauppauge: Nova Sci. Publ., 2008, vol. 13, pp. 143–150.
Azamov, A.A., Kuchkarov, A.Sh., and Samatov, B.T., On the Relation between the Feasibility of Pursuit, Controllability, and Stability in Whole Problems in Linear Systems with Heterogeneous Constraints, Prikl. Mat. Mekh., 2007, vol. 71, no. 2, pp. 259–263.
Alekseev, V.M., Tikhomirov, V.M. and Fomin, S.V., Optimal’noe upravlenie (Optimal Control), Moscow: Nauka, 1979.
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Original Russian Text © B.T. Samatov, 2013, published in Avtomatika i Telemekhanika, 2013, No. 7, pp. 17–28.
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Samatov, B.T. The pursuit-evasion problem under integral-geometric constraints on pursuer controls. Autom Remote Control 74, 1072–1081 (2013). https://doi.org/10.1134/S0005117913070023
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DOI: https://doi.org/10.1134/S0005117913070023