Geodesic
On the numerical solution of differential games for neutral-type linear systems
Trudy Instituta matematiki i mehaniki, Tome 23 (2017) no. 1, pp. 75-87 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The paper deals with a zero-sum differential game, in which the dynamic of a conflict-controlled system is described by linear functional differential equations of neutral type and the quality index is the sum of two terms: the first term estimates the history of motion of the system realized by the terminal time, and the second term is an integral-quadratic estimation of the corresponding realizations of the players' controls. To calculate the value and construct the optimal control laws in this differential game, we propose an approach based on solving a suitable auxiliary differential game, in which the motion of a conflict-controlled system is described by ordinary differential equations and the quality index contains an estimation of the motion at the terminal time only. To find the value and the saddle point in the auxiliary differential game, we apply the so-called upper convex hull method, which leads to an effective solution in the case under consideration due to the specific structure of the quality index and the geometric constraints on the control actions of the players. The efficiency of the approach is illustrated by an example, and the results of numerical simulations are presented. The constructed optimal control laws are compared with the optimal control procedures with finite-dimensional approximating guides, which were developed by the authors earlier.
Keywords: differential games, neutral-type systems, optimal control strategies, numerical methods. 
@article{TIMM_2017_23_1_a9,
     author = {M. I. Gomoyunov and N. Yu. Lukoyanov},
     title = {On the numerical solution of differential games for neutral-type linear systems},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {75--87},
     year = {2017},
     volume = {23},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2017_23_1_a9/}
}
TY  - JOUR
AU  - M. I. Gomoyunov
AU  - N. Yu. Lukoyanov
TI  - On the numerical solution of differential games for neutral-type linear systems
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2017
SP  - 75
EP  - 87
VL  - 23
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TIMM_2017_23_1_a9/
LA  - ru
ID  - TIMM_2017_23_1_a9
ER  - 
%0 Journal Article
%A M. I. Gomoyunov
%A N. Yu. Lukoyanov
%T On the numerical solution of differential games for neutral-type linear systems
%J Trudy Instituta matematiki i mehaniki
%D 2017
%P 75-87
%V 23
%N 1
%U http://geodesic.mathdoc.fr/item/TIMM_2017_23_1_a9/
%G ru
%F TIMM_2017_23_1_a9
M. I. Gomoyunov; N. Yu. Lukoyanov. On the numerical solution of differential games for neutral-type linear systems. Trudy Instituta matematiki i mehaniki, Tome 23 (2017) no. 1, pp. 75-87. http://geodesic.mathdoc.fr/item/TIMM_2017_23_1_a9/

[1] Krasovskii N.N., Subbotin A.I., Pozitsionnye differentsialnye igry, Nauka, M., 1974, 456 pp. | MR

[2] Krasovskii N.N., Upravlenie dinamicheskoi sistemoi, Nauka, M., 1985, 516 pp.

[3] Krasovskii A.N., Krasovskii N.N., Control under lack of information, Birkhauser, Berlin etc., 1995, 322 pp. | MR

[4] Subbotina N.N., “Universalnye optimalnye strategii v pozitsionnykh differentsialnykh igrakh”, Differents. uravneniya, 19:11 (1983), 1890–1896 | MR | Zbl

[5] Ushakov V.N., “K svoistvu stabilnosti v igrovoi zadache o sblizhenii s fiksirovannym momentom okonchaniya”, Differents. uravneniya, 47:7 (2011), 1034–1046 | MR | Zbl

[6] Gomoyunov M.I., Lukoyanov N.Yu., Plaksin A.R., “Suschestvovanie tseny i sedlovoi tochki v pozitsionnykh differentsialnykh igrakh dlya sistem neitralnogo tipa”, Tr. In-ta matematiki i mekhaniki UrO RAN, 22:2 (2016), 101–112 | DOI | MR

[7] Krasovskii N.N., Nekotorye zadachi teorii ustoichivosti dvizheniya, Fizmatgiz, M., 1959, 211 pp.

[8] Lukoyanov N.Yu., “K voprosu vychisleniya tseny differentsialnoi igry dlya pozitsionnogo funktsionala”, Prikl. matematika i mekhanika, 62:2 (1998), 188–198 | MR | Zbl

[9] Lukoyanov N.Yu., “Ob odnoi differentsialnoi igre s integralnym kriteriem kachestva”, Differents. uravneniya, 30:11 (1994), 1905–1913 | MR | Zbl

[10] Lukoyanov N.Yu., Reshetova T.N., “Zadachi konfliktnogo upravleniya funktsionalnymi sistemami vysokoi razmernosti”, Prikl. matematika i mekhanika, 62:4 (1998), 586–597 | MR

[11] Gomoyunov M.I., Lukoyanov N.Yu., “Optimizatsiya garantii v funktsionalno-differentsialnykh sistemakh s posledeistviem po upravleniyu”, Prikl. matematika i mekhanika, 76:4 (2012), 515–525 | MR

[12] Gomoyunov M.I., “Lineino-vypuklye zadachi optimizatsii garantii pri zapazdyvanii v upravlenii”, Izv. IMI UdGU, 2015, no. 1(45), 37–105 | MR | Zbl

[13] Gomoyunov M., Plaksin A., “On a problem of guarantee optimization in time-delay systems”, IFAC PapersOnLine, 48:25 (2015), 172–177 | DOI

[14] Lukoyanov N.Yu., “Konechnomernye povodyri sistem neitralnogo tipa”, Mezhdunar. konf. “Dinamicheskie sistemy: obratnye zadachi, ustoichivost i protsessy upravleniya”, posvyaschen. 80-letiyu akad. Yu.S. Osipova, tez. dokl. (Moskva, 22–23 sentyabrya 2016 g.), 67–70

[15] Hale J.K., Lunel S.M.V., Introduction to functional differential equations, Appl. Math. Sci., 99, Springer-Verlag, New York, 1993, 447 pp. | DOI | MR | Zbl

[16] Hale J.K., Meyer K.R., A class of functional equations of neutral type, Mem. Amer. Math. Soc., 76, 1967, 65 pp. | MR | Zbl

[17] Kent G.A., Optimal control of functional differential equations of neutral type, Thesis (Ph.D.)-Brown University, ProQuest LLC, Ann Arbor, 1971, 132 pp. | MR