Geodesic
Minimization of resource stock in an impulse differential game with a non-convex terminal set
Čelâbinskij fiziko-matematičeskij žurnal, Tome 6 (2021) no. 1, pp. 22-33.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider a linear differential game of a given duration. Reachable sets of players are $n$-dimensional balls. An impulse constraint is imposed on the choice of the first player's control. Abilities of the first player are determined by the stock of resources that can be used by the player at formation of his control. Control of the second player has geometrical constraints. The terminal set of a game is determined by a condition for assigning the norm of a phase vector to a segment with positive ends. The goal of the first player is to lead a phase vector to the terminal set at fixed time. The goal of the second player is opposite. This article solves the problem of finding the smallest initial stock of resources at which the first player can guarantee the achievement of his goal.
Keywords: control, impulse control, differential game, non-convex terminal set. 
@article{CHFMJ_2021_6_1_a2,
     author = {I. V. Izmestyev and V. I. Ukhobotov},
     title = {Minimization of resource stock in an impulse differential game with a non-convex terminal set},
     journal = {\v{C}el\^abinskij fiziko-matemati\v{c}eskij \v{z}urnal},
     pages = {22--33},
     publisher = {mathdoc},
     volume = {6},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHFMJ_2021_6_1_a2/}
}
TY  - JOUR
AU  - I. V. Izmestyev
AU  - V. I. Ukhobotov
TI  - Minimization of resource stock in an impulse differential game with a non-convex terminal set
JO  - Čelâbinskij fiziko-matematičeskij žurnal
PY  - 2021
SP  - 22
EP  - 33
VL  - 6
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHFMJ_2021_6_1_a2/
LA  - ru
ID  - CHFMJ_2021_6_1_a2
ER  - 
%0 Journal Article
%A I. V. Izmestyev
%A V. I. Ukhobotov
%T Minimization of resource stock in an impulse differential game with a non-convex terminal set
%J Čelâbinskij fiziko-matematičeskij žurnal
%D 2021
%P 22-33
%V 6
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHFMJ_2021_6_1_a2/
%G ru
%F CHFMJ_2021_6_1_a2
I. V. Izmestyev; V. I. Ukhobotov. Minimization of resource stock in an impulse differential game with a non-convex terminal set. Čelâbinskij fiziko-matematičeskij žurnal, Tome 6 (2021) no. 1, pp. 22-33. http://geodesic.mathdoc.fr/item/CHFMJ_2021_6_1_a2/

[1] Krasovskiy N.N., Theory of motion control, Nauka Publ., Moscow, 1968, 475 pp. (In Russ.)

[2] Krasovskiy N.N., “On a problem of tracking”, Journal of Applied Mathematics and Mechanics, 27:2 (1963), 363–377 (In Russ.) | Zbl

[3] Krasovskiy N.N., Repin Yu.M., Tret'yakov V.E., “Some game situations in the theory of control systems”, News of USSR Academy of Sciences. Technical cybernetics, 1965, no. 4, 3–23 (In Russ.)

[4] Krasovskiy N.N., Tret'yakov V.E., “To the problem of a pursuit in the case of constraints on the pulses of control forces”, Differential equations, 2:5 (1966), 587–599 (In Russ.)

[5] Subbotina N.N., Subbotin A.I., “Alternative for the encounter-evasion differential game with constraints on the momenta of the players' controls”, Journal of Applied Mathematics and Mechanics, 39:3 (1975), 376–385 | DOI | MR | Zbl

[6] Serov V.P., Chentsov A.G., “On a programmed linear game-theoretic guidance problem with constraints on the control force impulse”, Automation and Remote Control, 54, suppl. 5 (1993), 755–768 (In Russ.) | Zbl

[7] Kumkov S.I., Patsko V.S., “Information sets in the pulse control problem”, Automation and Remote Control, 58:7, part 2 (1997), 1224–1234 | MR | Zbl

[8] Chikrii A.A., Matichin I.I., “Linear differential games with impulse control of players”, Proceedings of the Steklov Institute of Mathematics, no. 1, 2005, S68–S81 | MR | Zbl

[9] Tukhtasinov M., “A linear differential game of a pursuit with impulse and integrally constrained controls of the players”, Proceedings of the Institute of Mathematics and Mechanics of UB RAS, 22, no. 3, 2016, 273–282 (In Russ.) | MR

[10] Ukhobotov V.I., “A linear differential game with constraints imposed on the control impulses”, Journal of Applied Mathematics and Mechanics, 52:3 (1988), 277–283 | DOI | MR | Zbl

[11] Ukhobotov V.I., Method of the one-dimensional design in linear differential games with integral constraints, Chelyabinsk State University, Chelyabinsk, 2005, 124 pp. (In Russ.)

[12] Ukhobotov V.I., Zaitseva O.V., “A linear problem of pulse encounter at a given time under interference”, Proceedings of the Steklov Institute of Mathematics, 272, no. 1, 2011, S215–S228 | DOI | MR

[13] Ukhobotov V.I., “On a class of linear differential games with impulse controls”, Journal of Applied Mathematics and Mechanics, 38:4 (1974), 550–557 | DOI | MR | Zbl

[14] Ukhobotov V.I., Izmest'ev I.V., “Single-type problem of the pulse meeting in fixed time with a terminal set in the form of the ring”, Bulletin of Udmurt University. Mathematics. Mechanics. Computer science, 25:2 (2015), 197–211 (In Russ.) | Zbl

[15] Ukhobotov V.I., Izmest'ev I.V., “Synthesis of controls in a single-type game problem of the pulse meeting in fixed time with a terminal set in the form of the ring”, Bulletin of Udmurt University. Mathematics. Mechanics. Computer science, 27:1 (2017), 69–85 (In Russ.) | MR | Zbl

[16] Kolmogorov A.N., Fomin S.V., Elements of the theory of functions and functional analysis, Nauka Publ., Moscow, 1972, 496 pp. (In Russ.) | MR

[17] Ukhobotov V. I., Izmest'ev I. V., “Impulse differential game with fixed time and one-dimensional aim”, Constructive Nonsmooth Analysis and Related Topics., International Conference, dedicated to the memory of V.F. Demyanov (CNSA), IEEE Xplore (St. Petersburg, May 22–27, 2017), 2017

[18] Pshenichnyi B.N., Convex analysis and extremal problems, Nauka Publ., Moscow, 1980, 320 pp. (In Russ.) | MR

[19] Kudryavtsev L.D., Course of mathematical analysis, v. 1, Vysshaya shkola Publ., Moscow, 1981, 687 pp. (In Russ.) | MR