Geodesic
On linear differential games with convex integral constraints
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 4, pp. 308-319 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We consider the problem of the approach of a trajectory of a linear conflict-controlled process to a linear subspace in the case of general integral constraints on the player's controls. Using the technique of set-valued mappings and convex analysis (epigraph of a function, recession cone), we obtain sufficient conditions for problem solvability in the class of measurable controls. The relation to a dual description based on conjugate functions is specified. Termination conditions for a game with impulse controls are established. The results are exemplified by means of model problems with various types of constraints on the players' controls.
Keywords: differential game, integral constrains, set-valued mapping, recession cone, impulse control. 
@article{TIMM_2013_19_4_a29,
     author = {A. A. Chikrii and A. A. Belousov},
     title = {On linear differential games with convex integral constraints},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {308--319},
     year = {2013},
     volume = {19},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2013_19_4_a29/}
}
TY  - JOUR
AU  - A. A. Chikrii
AU  - A. A. Belousov
TI  - On linear differential games with convex integral constraints
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2013
SP  - 308
EP  - 319
VL  - 19
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TIMM_2013_19_4_a29/
LA  - ru
ID  - TIMM_2013_19_4_a29
ER  - 
%0 Journal Article
%A A. A. Chikrii
%A A. A. Belousov
%T On linear differential games with convex integral constraints
%J Trudy Instituta matematiki i mehaniki
%D 2013
%P 308-319
%V 19
%N 4
%U http://geodesic.mathdoc.fr/item/TIMM_2013_19_4_a29/
%G ru
%F TIMM_2013_19_4_a29
A. A. Chikrii; A. A. Belousov. On linear differential games with convex integral constraints. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 4, pp. 308-319. http://geodesic.mathdoc.fr/item/TIMM_2013_19_4_a29/

[1] Pontryagin L. S., Izbrannye nauchnye trudy, v. 2, Nauka, M., 1988, 576 pp.

[2] Krasovskii N. N., Igrovye zadachi o vstreche dvizhenii, Nauka, M., 1970, 420 pp.

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

[4] Osipov Yu. S., Kryazhimskii A. V., Inverse problems for ordinary differential equations: dynamical solutions, Gordon and Breach Science Publishers, Basel, 1995, 625 pp. | MR | Zbl

[5] Osipov Yu. S., Kryazhimskii A. V., Maksimov V. I., Metody dinamicheskogo vosstanovleniya vkhodov upravlyaemykh sistem, Izd-vo UrO RAN, Ekaterinburg, 2011, 292 pp.

[6] Subbotina N. N., “The method of characteristics for Hamilton-Jacobi equations and applications to dynamical optimization”, J. Math. Sci., 135:3 (2006), 2955–3091 | DOI | MR

[7] Kurzhanskii A. B., Melnikov N. B., “O zadache sinteza upravlenii: alternirovannyi integral Pontryagina i uravnenie Gamiltona–Yakobi”, Mat. sb., 191:6 (2000), 69–100 | DOI | MR | Zbl

[8] Subbotin A. I., Chentsov A. G., Optimizatsiya garantii v zadachakh upravleniya, Nauka, M., 1981, 288 pp.

[9] Ushakov V. N., Guseinov Kh. T., Latushkin Ya. A., Lebedev P. D., “O sovpadenii maksimalnykh stabilnykh mostov v dvukh zadachakh o sblizhenii dlya statsionarnykh upravlyaemykh sistem”, Tr. In-ta matematiki i mekhaniki UrO RAN, 15, no. 3, 2009, 219–240

[10] Nikolskii M. S., “Pryamoi metod v lineinykh differentsialnykh igrakh s integralnymi ogranicheniyami”, Upravlyaemye sistemy, 2, Izd-vo SO AN SSSR, Novosibirsk, 1969, 49–58 | MR

[11] Nikolskii M. S., “Pryamoi metod v lineinykh differentsialnykh igrakh s obschimi integralnymi ogranicheniyami”, Differents. uravneniya, 8:6 (1972), 964–971 | MR | Zbl

[12] Nikolskii M. S., “Lineinye differentsialnye igry presledovaniya s integralnymi ogranicheniyami”, Differents. uravneniya, 28:2 (1992), 219–223 | MR

[13] Ushakov V. N., “Ekstremalnye strategii v differentsialnykh igrakh s integralnymi ogranicheniyami”, Prikl. matematika i mekhanika, 36:1 (1972), 15–23 | MR | Zbl

[14] Pshenichnyi B. N., Onopchuk Yu. N., “Lineinye differentsialnye igry s integralnymi ogranicheniyami”, Izv. AN SSSR. Tekhn. kibernetika, 1968, no. 1, 13–22 | MR

[15] Chikrii A. A., Belousov A. A., “O lineinykh differentsialnykh igrakh s integralnymi ogranicheniyami”, Tr. In-ta matematiki i mekhaniki UrO RAN, 15, no. 4, 2009, 290–301

[16] Chikrii A. A., Konfliktno upravlyaemye protsessy, Nauk. dumka, Kiev, 1992, 384 pp.

[17] Belousov A. A., “Differentsialnye igry s integralnymi ogranicheniyami na upravleniya v norme $L_1$”, Teoriya optimalnykh reshenii, sb. nauch. tr., In-t kibernetiki im. V. M. Glushkova NAN Ukrainy, Kiev, 2011, 3–9

[18] Belousov A. A., “Differentsialnye igry s integralnymi ogranicheniyami obschego vida”, Teoriya optimalnykh reshenii, sb. nauch. tr., In-t kibernetiki im. V. M. Glushkova NAN Ukrainy, Kiev, 2012, 30–35

[19] Rokafellar R., Vypuklyi analiz, Mir, M., 1973, 470 pp.

[20] Pshenichnyi B. N., Vypuklyi analiz i ekstremalnye zadachi, Nauka, M., 1980, 330 pp.

[21] Kuratovskii K., Topologiya, v. 2, Mir, M., 1969, 624 pp.

[22] Kisielewicz M., Differential inclusions and optimal control, Math. and Its Applications, 44, Kluwer Academic Publishers, Boston–London–Dordrecht, 1991, 260 pp. | MR

[23] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1981, 544 pp.

[24] Miller B. M., Rubinovich E. Ya., Optimizatsiya dinamicheskikh sistem s impulsnymi upravleniyami, Nauka, M., 2005, 429 pp.

[25] Filippov A. F., Differentsialnye uravneniya s razryvnoi pravoi chastyu, Nauka, M., 1985, 224 pp.