The evasion problem in a nonlinear differential game with discrete control

A. Ya. Narmanov ; K. A. Shchelchkov

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 52 (2018), pp. 75-85.

Résumé

A two-agent differential game is considered. The game is described by the following system of differential equations: $\dot x = f(x, v) + g(x, u),$ where $x \in \mathbb R^k$, $u \in U$, $v \in V$. The evader's admissible control set is a finite subset of phase space. The pursuer's admissible control set is a compact subset of phase space. The pursuer's purpose is to avoid an encounter, that is, to ensure a system position no closer than some neighborhood of zero. Sufficient conditions for avoidance of an encounter in the class of piecewise open-loop strategies on infinite and any finite-time intervals are obtained. The conditions are superimposed on the velocity vectogram at the zero point of phase space. When the game is considered on an infinite time interval, the conditions provide the evader with some advantage. The properties of a positive basis play a major role in proving the theorems.

Détail

Keywords: differential game, nonlinear system, avoidance of an encounter, discrete control.

Bibliographie

[1] Krasovskii N. N., Subbotin A. I., Positional differetial games, Nauka, M., 1974, 456 pp. | MR

[2] Krasovskii N. N., Control of dynamic system, Nauka, M., 1985, 520 pp.

[3] Subbotin A. I., Chentsov A. G., Optimization of guarantee in control problems, Nauka, M., 1981, 288 pp. | MR

[4] Pontryagin L. S., Selected scientific works, v. 2, Nauka, M., 1988, 575 pp.

[5] Pontryagin L. S. Mishchenko E. F., “A problem on the escape of one controlled object from another”, Sov. Math. Dokl., 10 (1969), 1488–1490 | MR | Zbl

[6] Pontryagin L. S., Mishchenko E. F., “Evasion problem in a linear differential games”, Differ. Uravn., 7:3 (1971), 436–445 (in Russian) | MR | Zbl

[7] Satimov N. Yu., Rikhsiev B. B., Methods of solving the problem of avoiding encounter in mathematical control theory, Fan, Tashkent, 2000, 176 pp. | MR

[8] Pshenichnyi B. N., Ostapenko V. V., Differential games, Naukova Dumka, Kiev, 1992, 261 pp. | MR

[9] Gusyatnikov P. B., Differential game theory, Moscow Institute of Physics and Technology, M., 1982, 99 pp.

[10] Chikrii A. A., “Evasion problem in nonlinear differential games”, Kibernetika, 1975, no. 3, 65–68 (in Russian)

[11] Rzymowski W., Method of construction of the evasion strategy for differential game with many pursuers, PWN, Warszawa, 1986, 43 pp. | MR

[12] Mishchenko E. F., Nikol'skii M. S., Satimov N. Yu., “The problem of avoiding encounter in $n$-person differential games”, Proceedings of the Steklov Institute of Mathematics, 143 (1980), 111–136

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

[14] Ibragimov G. I., Hasim R. M., “Pursuit and evasion differential games in Hilbert space”, International Game Theory Review, 12:3 (2010), 239–251 | DOI | MR | Zbl

[15] Kumkov S. S., Ménec S. L., Patsko V. S., “Zero-sum pursuit-evasion differential games with many objects: survey of publications”, Dynamic Games and Applications, 7:4 (2017), 609–633 DOI: {10.1007/s13235-016-0209-z} | DOI | MR | Zbl

[16] Brooks R. R., Pang Jing-En, Griffin C., “Game and information theory analysis of electronic countermeasures in pursuit-evasion games”, IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans, 38:6 (2008), 1281–1294 | DOI | MR

[17] Petrov N. N., Shchelchkov K. A., “On the interrelationship of two problems on evasion with many evaders”, J. Appl. Math. Mech., 80:4 (2016), 333–338 | DOI | MR

[18] Petrov N. N., “On the controllability of autonomous systems”, Differ. Uravn., 4:4 (1968), 606–617 (in Russian) | Zbl

[19] Petrov N. N., “Local controllability of autonomous systems”, Differ. Uravn., 4:7 (1968), 1218–1232 (in Russian) | Zbl

[20] Narmanov A. Ya., Petrov N. N., “Nonlocal problems in the theory of optimal processes. I”, Differential Equations, 21:4 (1985), 398–406 | MR | Zbl | Zbl

[21] Narmanov A. Ya., “Stability of completely controllable systems”, Differential Equations, 36:10 (2000), 1475–1483 | DOI | MR | Zbl

[22] Narmanov A. Ya., “On stability of totally controlled systems”, Siberian Adv. Math., 11:4 (2001), 110–125 | MR | MR | Zbl | Zbl

[23] Bannikov A. S., Petrov N. N., “On a nonstationary problem of group pursuit,”, Proceedings of the Steklov Institute of Mathematics, 271, suppl. 1 (2010), 41–52 | DOI | MR | Zbl

[24] Petrov N. N., “A certain simple pursuit problem with phase constraints”, Automation and Remote Control, 53:5 (1992), 639–642 | MR | Zbl

[25] Petrov N. N., “One problem of group pursuit with fractional derivatives and phase constraints”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 27:1 (2017), 54–59 (in Russian) | DOI | MR | Zbl

[26] Petrov N. N., Solov'eva N. A., “Multiple capture in Pontryagin's recurrent example with phase constraints”, Proceedings of the Steklov Institute of Mathematics, 293, suppl. 1 (2016), 174–182 | DOI

[27] Vinogradova M. N., Petrov N. N., Solov'eva N. A., “Capture of two coordinated evaders in a linear recurrent differential games”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 19, no. 1, 2013, 41–48 (in Russian) | MR

[28] Shchelchkov K. A., “A nonlinear pursuit problem with discrete control and incomplete information”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 28:1 (2018), 111–118 (in Russian) | DOI | MR