Geodesic
Features of the support reaction in the range maximization problem in a resistant medium
Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2018) no. 2, pp. 147-158 
A. V. Zarodnyuk; D. I. Bugrov; O. Yu. Cherkasov. Features of the support reaction in the range maximization problem in a resistant medium. Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2018) no. 2, pp. 147-158. http://geodesic.mathdoc.fr/item/FPM_2018_22_2_a8/
@article{FPM_2018_22_2_a8,
     author = {A. V. Zarodnyuk and D. I. Bugrov and O. Yu. Cherkasov},
     title = {Features of the support reaction in the range maximization problem in a resistant medium},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {147--158},
     year = {2018},
     volume = {22},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2018_22_2_a8/}
}
TY  - JOUR
AU  - A. V. Zarodnyuk
AU  - D. I. Bugrov
AU  - O. Yu. Cherkasov
TI  - Features of the support reaction in the range maximization problem in a resistant medium
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2018
SP  - 147
EP  - 158
VL  - 22
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/FPM_2018_22_2_a8/
LA  - ru
ID  - FPM_2018_22_2_a8
ER  - 
%0 Journal Article
%A A. V. Zarodnyuk
%A D. I. Bugrov
%A O. Yu. Cherkasov
%T Features of the support reaction in the range maximization problem in a resistant medium
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2018
%P 147-158
%V 22
%N 2
%U http://geodesic.mathdoc.fr/item/FPM_2018_22_2_a8/
%G ru
%F FPM_2018_22_2_a8

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

The horizontal coordinate's maximization problem of a mass-point as well as the corresponding brachistochrone problem is considered. The mass-point is supposed to be moving in a vertical plane under the influence of gravity and viscous drag that is proportional to the $n$th degree of the velocity. The analysis of the reaction force, which is considered as control along the extremal curve, is provided. It is established that the reaction of the basement can change its sign no more than one time, moreover, it changes only from negative values to positive values.

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

[2] Braison A., Kho Yu-Shi, Prikladnaya teoriya optimalnogo upravleniya, Mir, M., 1972

[3] Golubev Yu. F., “Brakhistokhrona s treniem”, Izv. RAN. Teor. i sist. upravl., 2010, no. 5, 41–52 | Zbl

[4] Gurman V. I., Ni Min Kan, “Vyrozhdennye zadachi optimalnogo upravleniya. I”, Avtomat. i telemekh., 2011, no. 3, 36–50 | Zbl

[5] Zarodnyuk A. V., Cherkasov O. Yu., “Kachestvennyi analiz optimalnykh traektorii dvizheniya materialnoi tochki v soprotivlyayuscheisya srede i zadacha o brakhistokhrone”, Izv. RAN. Teor. i sist. upravl., 2015, no. 1, 41–49 | DOI | MR | Zbl

[6] Lokshin B. Ya., Cherkasov O. Yu., “O strukture optimalnykh traektorii dvizheniya vraschayuschegosya tela v soprotivlyayuscheisya srede”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 1990, no. 2, 63–67

[7] Panasyuk A. I., Panasyuk V. I., Asimptoticheskaya magistralnaya optimizatsiya upravlyaemykh sistem, Nauka i tekhnika, Minsk, 1986 | MR

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

[9] Cherkasov O. Yu., Yakushev A. G., “Singular arcs in the optimal evasion against a proportional navigation vehicle”, JOTA, 113:2 (2002), 211–226 | DOI | MR | Zbl

[10] Cherkasov O. Yu., Zarodnyuk A. V., “Brachistochrone problem with Coulomb friction and viscouse drag: qualitative analysis”, Preprints 1st Conf. on Modelling, Identification and Control of Nonlinear System, MICNON 2015 (St. Petersburg, Russia, 24–26 June, 2015), IFAC, 1018–1023

[11] Jeremić O., Šalinić S., Obradović A., Mitrović Z., “On the brachistochrone of a variable mass particle in general force fields”, Math. Comput. Modelling, 54 (2011), 2900–2912 | DOI | MR | Zbl

[12] Kelley H. J., “A transformation approach to singular subarcs in optimal trajectory and control problems”, SIAM J. Control, 2 (1964), 234–240 | MR | Zbl

[13] Šalinić S., “Contribution to the brachistochrone problem with Coulomb friction”, Acta Mech., 208 (2009), 97–115 | DOI

[14] Vratanar B., Saje M., “On the analytical solution of the brachistochrone problem in a non-conservative field”, Int. J. Non-Linear Mech., 33 (1998), 489–505 | DOI | MR | Zbl