Geodesic
Damping of a system of linear oscillators using the generalized dry friction
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 2, pp. 168-177 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The problem of damping a system of linear oscillators is considered. The problem is solved by using a control in the form of dry friction. The motion of the system under the control is governed by a system of differential equations with a discontinuous right-hand side. A uniqueness and continuity theorem is proved for the phase flow of this system. Thus, the control in the form of generalized dry friction defines the motion of the system of oscillators uniquely.
Keywords: optimal control, DiPerna-Lions theory, singular ODE. 
@article{TIMM_2015_21_2_a13,
     author = {A. I. Ovseevich and A. K. Fedorov},
     title = {Damping of a system of linear oscillators using the generalized dry friction},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {168--177},
     year = {2015},
     volume = {21},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2015_21_2_a13/}
}
TY  - JOUR
AU  - A. I. Ovseevich
AU  - A. K. Fedorov
TI  - Damping of a system of linear oscillators using the generalized dry friction
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2015
SP  - 168
EP  - 177
VL  - 21
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TIMM_2015_21_2_a13/
LA  - ru
ID  - TIMM_2015_21_2_a13
ER  - 
%0 Journal Article
%A A. I. Ovseevich
%A A. K. Fedorov
%T Damping of a system of linear oscillators using the generalized dry friction
%J Trudy Instituta matematiki i mehaniki
%D 2015
%P 168-177
%V 21
%N 2
%U http://geodesic.mathdoc.fr/item/TIMM_2015_21_2_a13/
%G ru
%F TIMM_2015_21_2_a13
A. I. Ovseevich; A. K. Fedorov. Damping of a system of linear oscillators using the generalized dry friction. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 2, pp. 168-177. http://geodesic.mathdoc.fr/item/TIMM_2015_21_2_a13/

[1] Ovseevich A.I., Fedorov A.K., “Asimptoticheskoe optimalnoe upravlenie v forme sinteza dlya sistemy lineinykh ostsillyatorov”, Dokl. akad. nauk, 2013, no. 3, 266–270 | DOI | MR | Zbl

[2] Fedorov A.K., Ovseevich A.I., Asymptotic control theory for a system of linear oscillators, Preprint arXiv:1308.6090, arXiv: arXiv:1308.6090

[3] Ovseevich A.I., Fedorov A.K., “Dvizhenie sistemy ostsillyatorov pod deistviem obobschennogo sukhogo treniya”, Avtomatika i telemekhanika, 2015, no. 5, 121–129

[4] DiPerna R.J., Lions P.L., “Ordinary differential equations, transport theory and Sobolev spaces”, Invent. Math., 98:3 (1989), 511–547 | DOI | MR | Zbl

[5] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze [i dr.], Matematicheskaya teoriya optimalnykh protsessov, Nauka, M., 1983, 393 pp.

[6] Kalman R.E., “Ob obschei teorii sistem upravleniya”, Tr. 1-go Kongressa Mezhdunar. federatsii po avtomaticheskomu upravleniyu, Moskva, 1960, 481–492

[7] Arnold V.I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974, 432 pp. | MR

[8] Chernousko F.L., “O postroenii ogranichennogo upravleniya v kolebatelnykh sistemakh”, Prikl. matematika i mekhanika, 1988, no. 4, 549–558 | MR | Zbl

[9] Ovseevich A.I., “O polnoi upravlyaemosti lineinykh dinamicheskikh sistem”, Prikl. matematika i mekhanika, 1989, no. 5, 845–848 | MR | Zbl

[10] Ovseevich A.I., “Limit behaviour of attainable and superattainable sets”, Proc. Conf. Modeling, Estimation and Filtering of Systems with Uncertainty, Hungary, 1990, 324–333 | MR

[11] Goncharova E.V., Ovseevich A.I., “Sravnitelnyi analiz asimptoticheskoi dinamiki mnozhestv dostizhimosti lineinykh sistem”, Izv. RAN. Teoriya i sistemy upravleniya, 2007, no. 4, 5–13 | MR | Zbl

[12] Ovseevich A.I., “Singularities of attainable sets”, Russian J. Math. Physics, 5:3 (1998), 389–398 | MR

[13] Schneider R., Convex bodies: The Brunn-Minkowski theory, Cambridge University Press, Cambridge, 1993, 490 pp. | MR | Zbl

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

[15] Ovseevich A.I., “Irregular dynamic systems according to R.J. DiPerna and P.L. Lions”, Funct. Anal. Other Math., 4:1 (2012), 57–70 | MR

[16] Bogaevskii I.A., “Razryvnye gradientnye differentsialnye uravneniya i traektorii v variatsionnom ischislenii”, Mat. sb., 197:12 (2006), 11–42 | DOI | MR