Geodesic
The functional differential inclusions with impulses and with the right-hand side not necessarily convex-valued with respect to switching
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, no. 1 (2014), pp. 3-48 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The Cauchy problem for the functional differential inclusion with Volterra's multivalued map not necessarily convex-valued with respect to switching and with impulses is considered. For this problem, the definition of a generalized solution is introduced, and the questions of existence and extendibility of generalized solutions are studied. Notions of a near-realization and realization of the distance to an arbitrary summable function by the set of generalized solutions are formulated. For a set of generalized solutions of functional differential inclusions with impulses and with multivalued map not necessarily convex-valued with respect to switching, estimations are found similar to A. F. Filippov's estimations. The generalized principle of density is proved.
Keywords: functional differential inclusion, convex-valued with respect to switching, generalized solution, the near-realization and realization of the distance to a given summable function by the set of generalized solutions, a-priori boundedness. 
@article{IIMI_2014_1_a0,
     author = {A. I. Bulgakov and O. V. Filippova},
     title = {The functional differential inclusions with impulses and with the right-hand side not necessarily convex-valued with respect to switching},
     journal = {Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta},
     pages = {3--48},
     year = {2014},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IIMI_2014_1_a0/}
}
TY  - JOUR
AU  - A. I. Bulgakov
AU  - O. V. Filippova
TI  - The functional differential inclusions with impulses and with the right-hand side not necessarily convex-valued with respect to switching
JO  - Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
PY  - 2014
SP  - 3
EP  - 48
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/IIMI_2014_1_a0/
LA  - ru
ID  - IIMI_2014_1_a0
ER  - 
%0 Journal Article
%A A. I. Bulgakov
%A O. V. Filippova
%T The functional differential inclusions with impulses and with the right-hand side not necessarily convex-valued with respect to switching
%J Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
%D 2014
%P 3-48
%N 1
%U http://geodesic.mathdoc.fr/item/IIMI_2014_1_a0/
%G ru
%F IIMI_2014_1_a0
A. I. Bulgakov; O. V. Filippova. The functional differential inclusions with impulses and with the right-hand side not necessarily convex-valued with respect to switching. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, no. 1 (2014), pp. 3-48. http://geodesic.mathdoc.fr/item/IIMI_2014_1_a0/

[1] Wazewski A., “Sur une generalisation de la notion des solutions d'une equation au contingent”, Bull. Acad. Polon. Sci., Ser. Math. Astr. Phys., 10:1 (1962), 11–15 | MR | Zbl

[2] Bulgakov A. I., “Functional-differential inclusions with nonconvex right side”, Differ. Uravn., 26:11 (1990), 1872–1878 (in Russian) | MR | Zbl

[3] Bressan A., “On a bang-bang principle for nonlinear systems”, Boll. Unione Math. Italiana, 1, suppl. (1980), 53–59 | MR | Zbl

[4] Zheleztsov N. A., The method of point transformation and the problem of the forced vibrations of an oscillator with combined friction, American Mathematical Society, 1951, 49 pp. | MR

[5] Zabreiko P. P., Krasnosel'skii M. A., Lifshits E. L., “Elastic-plastic oscillator”, Doklady Akademii Nauk SSSR, 190:2 (1970), 266–268 (in Russian) | MR

[6] Kantorovich L. V., Akilov G. N., Function analysis, Nauka, M., 1984, 752 pp. | MR | Zbl

[7] Tikhonov A. N., “Functional equations of Volterra's type and their applications to certain problems of mathematical physics”, Byulleten' Moskovskogo universiteta. Sektsiya A, 1:8 (1938), 1–25 (in Russian) | MR

[8] Ioffe A. D., Tikhomirov V. M., Theory of extremal problems, Nauka, M., 1974, 480 pp. | MR | Zbl

[9] Borisovich B. G., Gel'man B. D., Myshkis A. D., Obukhovskii V. V., Introduction to the theory of multivalued mappings and differential inclusions, KomKniga, M., 2005, 216 pp.

[10] Fillipov A. F., Differential equations with discontinuous righthand side, Nauka, M., 1985, 224 pp. | MR

[11] Bulgakov A. I., Belyaeva O. P., Machina A. N., “Functional-differential inclusion with multivalued map not necessarily convex-valued with respect to switching”, Vestn. Udmurt. Univ. Mat., 2005, no. 1, 3–20 (in Russian) | MR

[12] Bulgakov A. I., “Continuous branches of multivalued mappings and functional-differential inclusions”, Differ. Uravn., 22:10 (1986), 1659–1670 (in Russian) | MR | Zbl

[13] Plis A., “Trajectories and quasitrajectories of an orientor field”, Bull. Acad. Polon. Sci., Ser. Math. Astr. Phys., 11:6 (1963), 369–370 | MR | Zbl

[14] Turowicz A., “Remarque sur la definition des quasitrajectoires d'une system de commande nonlineaire”, Bull. Acad. Polon. Sci., Ser. Math. Astr. Phys., 11:6 (1963), 367–368 | MR | Zbl

[15] Puchkov N. P., Bulgakov A. I., Grigorenko A. A., Korobko A. I., Korchagina E. V., Machina A. N., Filippova O. V., Shlykova I. V., “About some problems of functional differential inclusions”, Vestn. Tambov. Gos. Tekhn. Univ., 14:4 (2009), 947–974 (in Russian)

[16] Finogenko I. A., “On sliding mode controlled discontinuous systems with aftereffect”, Izv. Ross. Akad. Nauk Teor. Sist. Upr., 2004, no. 4, 19–26 (in Russian) | MR

[17] Bulgakov A. I., Vasil'ev V. V., Efremov A. A., “On the principle of density for functional-differential inclusions of neutral type”, Vestn. Tambov. Univ., Ser. Estestv. Tekh. Nauki, 6:3 (2001), 308–315 (in Russian)