Geodesic
Qualitative analysis of the brachistochrone problem with a dry friction and maximization of horizontal distance
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 4 (2016), pp. 54-59 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The range maximization problem of a particle moving in a vertical plane under the action of gravity and dry friction and the corresponding brachistochrone problem are considered. The optimal control problem is reduced to a boundary value problem for a system of two nonlinear differential equations. A qualitative anaslysis of the trajectories of this system is carried out, their typical features are found and illustrated by numerical solving of the boundary value problem. It is shown that the normal component of the support reaction should be positive when moving along the optimal curve. The optimality of the found extremal trajectories is discussed. 
@article{VMUMM_2016_4_a8,
     author = {A. V. Zarodnyuk and O. Yu. Cherkasov},
     title = {Qualitative analysis of the brachistochrone problem with a dry friction and maximization of horizontal distance},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {54--59},
     year = {2016},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2016_4_a8/}
}
TY  - JOUR
AU  - A. V. Zarodnyuk
AU  - O. Yu. Cherkasov
TI  - Qualitative analysis of the brachistochrone problem with a dry friction and maximization of horizontal distance
JO  - Vestnik Moskovskogo universiteta. Matematika, mehanika
PY  - 2016
SP  - 54
EP  - 59
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/VMUMM_2016_4_a8/
LA  - ru
ID  - VMUMM_2016_4_a8
ER  - 
%0 Journal Article
%A A. V. Zarodnyuk
%A O. Yu. Cherkasov
%T Qualitative analysis of the brachistochrone problem with a dry friction and maximization of horizontal distance
%J Vestnik Moskovskogo universiteta. Matematika, mehanika
%D 2016
%P 54-59
%N 4
%U http://geodesic.mathdoc.fr/item/VMUMM_2016_4_a8/
%G ru
%F VMUMM_2016_4_a8
A. V. Zarodnyuk; O. Yu. Cherkasov. Qualitative analysis of the brachistochrone problem with a dry friction and maximization of horizontal distance. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 4 (2016), pp. 54-59. http://geodesic.mathdoc.fr/item/VMUMM_2016_4_a8/

[1] Sussmann H.J., Willems J.C., “300 years of optimal control: from the brachystochrone to the maximum principle”, IEEE Control Systems Magazine, 17:3 (1997), 32–44 | DOI

[2] Mednikov V.N., Dinamika poleta i pilotirovanie samoletov, Izd-vo VVA, Monino, 1976

[3] Aleksandrov V.V., Voronin L.I., Glazkov Yu.N., Ishlinskii A.Yu., Sadovnichii V.A., Matematicheskie zadachi dinamicheskoi imitatsii aerokosmicheskikh poletov, Izd-vo MGU, M., 1995

[4] Golubev Yu. F., “Brakhistokhrona s treniem”, Izv. RAN. Teoriya i sistemy upravleniya, 2010, no. 5, 41–52 | Zbl

[5] Wensrich C.M., “Evolutionary solutions to the brachistochrone problem with Coulomb friction”, Mech. Res. Communs., 31:2 (2004), 151–159 | DOI | Zbl

[6] Ashby N., Brittin W.E., Love W.F., Wyss W., “Brachistochrone with Coulomb friction”, Amer. J. Phys., 43:10 (1975), 902–905 | DOI | MR

[7] Lipp S.C., “Brachistochrone with Coulomb friction”, SIAM J. Control Optim., 35:2 (1997), 562–584 | DOI | MR | Zbl

[8] Salinic S., Obradovich A., Mitrovic Z., Rusov S., “Brachistochrone with limited reaction of constraint in an arbitrary force field”, Nonlinear Dyn., 69:1 (2012), 211–222 | DOI | MR | Zbl

[9] Cherkasov O.Yu., Zarodnyuk A.V., “Brachistochrone problem with Coulomb friction and viscous drag: qualitative analysis”, Proc. 1st IFAC Conf. on Modelling, Identification and Control of Nonlinear Systems, MICNON 2015 (June 24–26, St. Petersburg), St. Petersburg, 2015, 1028–1033

[10] Zarodnyuk A,V., Cherkasov O.Yu., “K zadache o brakhistokhrone s lineinym vyazkim treniem”, Vestn. Mosk. un-ta. Matem. Mekhan., 2015, no. 3, 65–69 | Zbl

[11] Pontryagin L. S., Boltyanskii V.G., Gamkrelidze R.V., Mischenko E.F., Matematicheskaya teoriya optimalnykh protsessov, Nauka, M., 1983 | MR

[12] Kelley H.J., Kopp R.E., Moyer H.G., “Singular extremal”, Topics in Optimization, ed. G. Leitmann, Academic Press, N.Y.–L., 1967, 63–101 | DOI | MR