Guaranteed escaping strategies

Igor I. Shevchenko

Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 7 (2015) no. 4, pp. 71-97.

Résumé

To generate evasion strategies and evaluate corresponding guaranteed miss distances from $E$ to $\mathcal P_{j_1,\ldots,j_n} = \{P_{j_1},\ldots, P_{j_n}\}$, $ n \geq 3$, we set up two basic problems for the players with simple motions. In the first one, $E$ maximizes the miss distance to $P_a\in \mathcal P_{j_1,\ldots,j_n}$ when she moves along a given straight-line. In the second one, $E$ seeks to cross the intercept $P_b P_c$ just once and to maximize the miss distance to either of $P_b$ and $P_c$ during the infinite period of manoeuvring. In the game with a group of three or more pursuers, for a given history, we evaluate the minimum of the guaranteed miss distances when $E$ passing between $P_b$ and $ P_c$, $\forall b,c \in \{j_1,\ldots,j_n\}, b\not = c,$ and the guaranteed miss distance to $P_a$, $\forall a \in \{j_1,\ldots,j_n\}\backslash\{b,c\}$. After that, we are able to choose the best alternative for assigning $b$ and $c$.

Détail

Keywords: maximizing miss distances, passing between two slower pursuers, alternative games, memory strategies.

Bibliographie

[1] Abramyants T. G., Maslov E. P., Rubinovich E. Ya., “Upravlenie podvizhnymi ob'ektami v usloviyakh iskusstvenno organizovannoi nepolnoty informatsii”, Problemy upravleniya, 2005, no. 4, 75–81 | MR

[2] Aizeks R., Differentsialnye igry, Mir, M., 1967 | MR

[3] Vaisbord E. M., Zhukovskii V. I., Vvedenie v differentsialnye igry neskolkikh lits i ikh prilozheniya, Sovetskoe radio, M., 1980 | MR

[4] Grigorenko N. L., Differentsialnye igry presledovaniya neskolkimi ob'ektami, Izd-vo Mosk. un-ta, M., 1983

[5] Grigorenko N. L., Matematicheskie metody upravleniya neskolkimi dinamicheskimi protsessami, Izd-vo Moskovskogo un-ta, M., 1990

[6] Zak V. L., “Zadacha ukloneniya ot mnogikh presledovatelei pri nalichii fazovogo ogranicheniya”, DAN SSSR, 265:5 (1982), 1051–1053 | MR | Zbl

[7] Ibragimov G. I., “Ob odnoi zadache optimalnogo presledovaniya neskolkimi ob'ektami odnogo”, Prikladnaya matematika i mekhanika, 62:2 (1998), 187–192 | MR

[8] Krasovskii N. N., Subbotin A. I., Pozitsionnye differentsialnye igry, Nauka, M., 1974 | MR

[9] Krasovskii N. N., Kotelnikova A. N., “Unifikatsiya differentsialnykh igr, obobschennye resheniya uravnenii tipa Gamiltona–Yakobi, stokhasticheskii povodyr”, Differentsialnye uravneniya, 45:11 (2009), 1618–1633 | MR | Zbl

[10] Kumkov S. S., Patsko V. S., Le Menek S., “Dva slabykh presledovatelya v igre protiv odnogo ubegayuschego”, Avtomatika i telemekhanika, 2014, no. 10, 73–96

[11] Petrov N. N., Teoriya igr, Izd-vo Udm. un-ta, Izhevsk, 1997 | MR

[12] Petrosyan L. A., Tomskii G. V., Geometriya prostogo presledovaniya, Nauka, Novosibirsk, 1983 | MR

[13] Pontryagin L. S., “Matematicheskaya teoriya optimalnykh protsessov i differentsialnye igry”, Tr. Mat. in-ta AN SSSR, 169, 1985, 119–158 | Zbl

[14] Pshenichnyi B. N., “Prostoe presledovanie neskolkimi ob'ektami”, Kibernetika, 1976, no. 3, 145–146

[15] Subbotin A. I., Obobschennye resheniya uravnenii v chastnykh proizvodnykh pervogo poryadka. Perspektivy dinamicheskoi optimizatsii, Institut kompyuternykh issledovanii, M.–Izhevsk, 2003

[16] Chernousko F. L., “Odna zadacha ukloneniya ot mnogikh presledovatelei”, Prikladnaya matematika i mekhanika, 40:1 (1976), 14–24 | MR | Zbl

[17] Chikrii A. A., Konfliktno-upravlyaemye protsessy, Naukova dumka, Kiev, 1992

[18] Shevchenko I. I., “Garantirovannoe sblizhenie s dalnim iz ubegayuschikh”, Avtomatika i telemekhanika, 2008, no. 5, 101–119

[19] Shevchenko I. I., “Formirovanie garantiruyuschikh strategii alternativnogo presledovaniya s pamyatyu”, Matematicheskaya teoriya igr i ee prilozheniya, 4:4 (2012), 114–128 | Zbl

[20] Basar T., Olsder G. J., Dynamic Noncooperative Game Theory, SIAM, 1999 | MR | Zbl

[21] Chernousko F. L., Zak V. L., “On differential games of evasion from many pursuers”, JOTA, 46:4 (1985), 461–470 | DOI | MR | Zbl

[22] Crandall M. G., Lions P. L., “Viscosity solution of Hamilton–Jacobi equations”, Trans. Am. Math. Soc., 277:1 (1983), 1–42 | DOI | MR | Zbl

[23] Hagedorn P., Breakwell J. V., “A differential game with two pursuers and one evader”, JOTA, 18:1 (1976), 15–29 | DOI | MR | Zbl

[24] Mastellone S., Stipanović D. M., Graunke C. R., Intlekofer K. A., Spong M. W., “Formation control and collision avoidance for multi-agent nonholonomic systems: theory and experiments”, International Journal of Robotics Research, 13 (2008), 107–126 | DOI

[25] Petrosjan L. A., Differential games of pursuit, Series on Optimization, 2, World Scientific, Singapore, 1993 | MR

[26] Yavin Y., Pachter M. (eds.), Pursuit-Evasion Differential Games, International Series in Modern Applied Mathematics and Computer Science, Elsevier, 1987 | MR

[27] Shevchenko I., “Locally optimizing strategies for approaching the furthest evader”, Contributions to game theory and management, 5 (2012), 293–303 | MR

[28] Shevchenko I., “On reduction of alternative pursuit games”, Game Theory and Applications, 11 (2007), 125–137 | MR

[29] Stipanović D., Melikyan A., Hovakimyan N., “Some sufficient conditions for multi-player pursuit-evasion games with continuous and discrete observations”, Annals of the International Society of Dynamic Games, 10 (2009), 133–145 | MR | Zbl

[30] Stipanović D. M., Tomlin C. J., Leitmann G., “Monotone approximations of minimum and maximum functions and multi-objective problems”, Applied Mathematics Optimization, 66:3 (2012), 455–473 | DOI | MR | Zbl