Geodesic
$\alpha$-systems of differential inclusions and their unification
Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 7 (2015) no. 2, pp. 85-116.

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

In this article, $\alpha$-systems of differential inclusions are introduced on a bounded time segment $[t_0,\vartheta]$ and $\alpha$-weakly invariant sets in $[t_0,\vartheta] \times \mathbb R^n$ are defined, where $\mathbb R^n$ is a phase space of the differential inclusions. Problems are studied connected with bringing the motions (trajectories) of differential inclusions in an $\alpha$-system to a given compact set $M \subset \mathbb R^n$ at the time $\vartheta$. Questions are discussed of finding the solvability set $W \subset [t_0, \vartheta] \times \mathbb R^n$ of problem of bringing the motions of $\alpha$-system to $M$ and calculating the maximal $\alpha$-weakly invariant set $W^c \subset [t_0, \vartheta] \times \mathbb R^n$. The notion is introduced of quasi-Hamiltonian of $\alpha$-system ($\alpha$-Hamiltonian), which we see as important for studying problems of bringing motions of $\alpha$-system to $M$.
Keywords: differential inclusion, guidance problem, Hamiltonian, invariance, weak invariance. 
@article{MGTA_2015_7_2_a5,
     author = {Vladimir N. Ushakov and Sergey A. Brykalov and Grigory V. Parshikov},
     title = {$\alpha$-systems of differential inclusions and their unification},
     journal = {Matemati\v{c}eska\^a teori\^a igr i e\"e prilo\v{z}eni\^a},
     pages = {85--116},
     publisher = {mathdoc},
     volume = {7},
     number = {2},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MGTA_2015_7_2_a5/}
}
TY  - JOUR
AU  - Vladimir N. Ushakov
AU  - Sergey A. Brykalov
AU  - Grigory V. Parshikov
TI  - $\alpha$-systems of differential inclusions and their unification
JO  - Matematičeskaâ teoriâ igr i eë priloženiâ
PY  - 2015
SP  - 85
EP  - 116
VL  - 7
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MGTA_2015_7_2_a5/
LA  - ru
ID  - MGTA_2015_7_2_a5
ER  - 
%0 Journal Article
%A Vladimir N. Ushakov
%A Sergey A. Brykalov
%A Grigory V. Parshikov
%T $\alpha$-systems of differential inclusions and their unification
%J Matematičeskaâ teoriâ igr i eë priloženiâ
%D 2015
%P 85-116
%V 7
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MGTA_2015_7_2_a5/
%G ru
%F MGTA_2015_7_2_a5
Vladimir N. Ushakov; Sergey A. Brykalov; Grigory V. Parshikov. $\alpha$-systems of differential inclusions and their unification. Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 7 (2015) no. 2, pp. 85-116. http://geodesic.mathdoc.fr/item/MGTA_2015_7_2_a5/

[1] Guseinov Kh. G., Moiseev A. N., Ushakov V. N., “Ob approksimatsii oblastei dostizhimosti upravlyaemykh sistem”, Prikladnaya matematika i mekhanika, 62:2 (1998), 179–187 | MR

[2] Guseinov Kh. G., Ushakov V. N., “Silno i slabo invariantnye mnozhestva otnositelno differentsialnogo vklyucheniya”, Dokl. AN SSSR, 303:4 (1988), 794–797

[3] Krasovskii N. N., “K zadache unifikatsii differentsialnykh igr”, Dokl. AN SSSR, 226:6 (1976), 1260–1263 | MR

[4] Krasovskii N. N., “Unifikatsiya differentsialnykh igr”, Tr. In-ta matematiki i mekhaniki, 24, UNTs AN SSSR, Sverdlovsk, 1977, 32–45 | MR

[5] Kurzhanskii A. B., “Ob analiticheskom opisanii puchka vyzhivayuschikh traektorii differentsialnoi sistemy”, Doklady AN SSSR, 287:5 (1986), 1047–1050 | MR

[6] Kurzhanskii A. B., Izbrannye trudy, Izd. Moskovskogo Universiteta, 2009

[7] Subbotin A. I., Minimaksnye neravenstva i uravneniya Gamiltona–Yakobi, Nauka, M., 1991 | MR

[8] Tarasev A. M., Ushakov V. N., O postroenii stabilnykh mostov v minimaksnoi igre sblizheniya-ukloneniya, Dep. v VINITI, No 2454-83

[9] Ushakov V. N., Matviichuk A. R., Ushakov A. V., “Approksimatsiya mnozhestv dostizhimosti i integralnykh voronok differentsialnykh vklyuchenii”, Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternye nauki, 2011, no. 4, 23–39 | MR | Zbl

[10] Filippov A. F., Differentsialnye uravneniya s razryvnoi pravoi chastyu, Nauka, M., 1985 | MR

[11] Chernousko F. L., Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem. Metod ellipsoidov, Nauka, M., 1988 | MR

[12] Aubin J. P., Cellina A., Differential inclusions: set-valued maps and viability theory, Springer-Verlag, New York, 1984 | MR | Zbl

[13] Basar T., Olsder G. J., Clsder G., Dynamic noncooperative game theory, Classics in Applied Mathematics, 200, SIAM, 1995 | MR

[14] Chernousko F. L., Ovseevich A. I., “Properties of the optimal ellipsoids approximating the reachable sets of uncertain systems”, Journal of optimization theory and applications, 120:2 (2004), 223–246 | DOI | MR | Zbl

[15] Crandall M. G., Evans L. C., Lions P.-L., “Some properties of viscosity solutions of Hamilton–Jacobi equations”, Transactions of the American Mathematical Society, 282:2 (1984), 487–502 | DOI | MR | Zbl

[16] Guseinov K. G., Moiseyev A. N., Ushakov V. N., “On approximation of reachable sets of differential inclusions”, 8th International Symposium on Dynamic Games and Applications (Maastricht, 1998), 235–238

[17] Krasovskii N. N., Subbotin A. I., Game-theoretical control problems, Springer-Verlag, 1988 | MR

[18] Kurzhanski A. B., Valyi I., Ellipsoidal calculus for estimation and control, Birkhauser, 1997 | MR | Zbl

[19] Lee E. B., Markus L., Foundations of optimal control theory, Tech. rep., DTIC Document, 1967 | MR

[20] Quincampoix M., Veliov V. M., “Optimal control of uncertain systems with incomplete information for the disturbances”, SIAM Journal on Control and Optimization, 43:4 (2005), 1373–1399 | DOI | MR | Zbl