# The brachistochrone problem with frictional forces

Roberto Giambò; Fabio Giannoni

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 5, page 187-206
- ISSN: 1292-8119

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topGiambò, Roberto, and Giannoni, Fabio. "The brachistochrone problem with frictional forces." ESAIM: Control, Optimisation and Calculus of Variations 5 (2010): 187-206. <http://eudml.org/doc/197275>.

@article{Giambò2010,

abstract = { In this paper we show the existence of the solution for the
classical brachistochrone problem under the action of a
conservative field in presence of frictional forces. Assuming that
the frictional forces and the potential grow at most
linearly, we prove the existence of a minimizer on the travel
time between any two given points, whenever the initial velocity
is great enough. We also prove the uniqueness of the minimizer
whenever the given points are sufficiently close.
},

author = {Giambò, Roberto, Giannoni, Fabio},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Minimal travel time; non linear constraints.; minimal travel time; non linear constraints; brachistochrone problem},

language = {eng},

month = {3},

pages = {187-206},

publisher = {EDP Sciences},

title = {The brachistochrone problem with frictional forces},

url = {http://eudml.org/doc/197275},

volume = {5},

year = {2010},

}

TY - JOUR

AU - Giambò, Roberto

AU - Giannoni, Fabio

TI - The brachistochrone problem with frictional forces

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 5

SP - 187

EP - 206

AB - In this paper we show the existence of the solution for the
classical brachistochrone problem under the action of a
conservative field in presence of frictional forces. Assuming that
the frictional forces and the potential grow at most
linearly, we prove the existence of a minimizer on the travel
time between any two given points, whenever the initial velocity
is great enough. We also prove the uniqueness of the minimizer
whenever the given points are sufficiently close.

LA - eng

KW - Minimal travel time; non linear constraints.; minimal travel time; non linear constraints; brachistochrone problem

UR - http://eudml.org/doc/197275

ER -

## References

top- H. Brezis, Analyse fonctionnelle. Masson, Paris (1983).
- F. Giannoni, P. Piccione and J.A. Verderesi, An approach to the relativistic brachistochrone problem by sub-Riemannian geometry. J. Math. Phys. 38 (1997) 6367-6381.
- F. Giannoni and P. Piccione, An existence theory for relativistic brachistochrones in stationary space-times. J. Math. Phys. 39 (1998) 6137-6152.
- J. Nash, The embedding problem for Riemannian manifolds. Ann. Math.63 (1956) 20-63.
- L.A. Pars, An Introduction to the Calculus of Variations. Heinemann, London (1962).
- V. Perlick, The brachistochrone problem in a stationary space-time. J. Math. Phys.32 (1991) 3148-3157.

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