Geodesic
Vibrosolutions to differential equations in distributions with discontinuous regular functions on the right-hand side
Matematičeskie zametki, Tome 58 (1995) no. 1, pp. 12-21 
M. V. Basin. Vibrosolutions to differential equations in distributions with discontinuous regular functions on the right-hand side. Matematičeskie zametki, Tome 58 (1995) no. 1, pp. 12-21. http://geodesic.mathdoc.fr/item/MZM_1995_58_1_a1/
@article{MZM_1995_58_1_a1,
     author = {M. V. Basin},
     title = {Vibrosolutions to differential equations in distributions with discontinuous regular functions on the right-hand side},
     journal = {Matemati\v{c}eskie zametki},
     pages = {12--21},
     year = {1995},
     volume = {58},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1995_58_1_a1/}
}
TY  - JOUR
AU  - M. V. Basin
TI  - Vibrosolutions to differential equations in distributions with discontinuous regular functions on the right-hand side
JO  - Matematičeskie zametki
PY  - 1995
SP  - 12
EP  - 21
VL  - 58
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MZM_1995_58_1_a1/
LA  - ru
ID  - MZM_1995_58_1_a1
ER  - 
%0 Journal Article
%A M. V. Basin
%T Vibrosolutions to differential equations in distributions with discontinuous regular functions on the right-hand side
%J Matematičeskie zametki
%D 1995
%P 12-21
%V 58
%N 1
%U http://geodesic.mathdoc.fr/item/MZM_1995_58_1_a1/
%G ru
%F MZM_1995_58_1_a1

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

We study $n$-dimensional differential equations in distributions of the form $$ \dot x(t)=f(x,u,t)+ b(x,u,t)\dot u(t), $$ where $f(x,u,t)$ and $b(x,u,t)$ are piecewise continuous functions and $u(t)$ is an $m$-dimensional function of bounded variation with nondecreasing components. The notion of vibrosolution is introduced for equations of this type, and necessary and sufficient conditions for the existence of vibrosolutions are derived. The transition to an equivalent equation with measure is carried out, thus making it possible to explicitly calculate the jumps of the vibrosolutions at the points of discontinuity of $u(t)$.

[1] Krasnoselskii M. A., Pokrovskii A. V., Sistemy s gisterezisom, Nauka, M., 1983

[2] Orlov Yu. V., “Vibrokorrektnye differentsialnye uravneniya s merami”, Matem. zametki, 38:1 (1985), 110–119 | MR | Zbl

[3] Orlov Yu. V., Teoriya optimalnykh sistem s obobschennymi upravleniyami, Nauka, M., 1988

[4] Basin M. V., Orlov Yu. V., “Garantirovannoe otsenivanie sostoyaniya lineinykh dinamicheskikh sistem po diskretno-nepreryvnym nablyudeniyam”, Avtomatika i telemekhanika, 1992, no. 3, 36–45 | MR | Zbl

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

[6] Miller B. M., “Optimizatsiya dinamicheskikh sistem s obobschennym upravleniem”, Avtomatika i telemekhanika, 1989, no. 6, 23–34 | MR | Zbl

[7] Zavalischin S. T., Sesekin A. N., Impulsnye protsessy: modeli i prilozheniya, Nauka, M., 1991 | Zbl

[8] Kirillov A. A., Gvishiani A. D., Teoremy i zadachi funktsionalnogo analiza, Nauka, M., 1979

[9] Kurzweil J., “Generalized ordinary differential equations”, Chechosl. Math. Journ., 8:3 (1958), 360–388 | MR | Zbl

[10] Dykhta V. A., Kolokolnikova G. A., “Usloviya minimuma na mnozhestve posledovatelnostei v vyrozhdennoi variatsionnoi zadache”, Matem. zametki, 34:5 (1983), 735–744 | MR

[11] Shvarts L., Analiz, T. 2, Mir, M., 1972