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

Abstract

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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.

How to cite

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Giambò, 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

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  1. H. Brezis, Analyse fonctionnelle. Masson, Paris (1983).  
  2. 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.  
  3. F. Giannoni and P. Piccione, An existence theory for relativistic brachistochrones in stationary space-times. J. Math. Phys. 39 (1998) 6137-6152.  
  4. J. Nash, The embedding problem for Riemannian manifolds. Ann. Math.63 (1956) 20-63.  
  5. L.A. Pars, An Introduction to the Calculus of Variations. Heinemann, London (1962).  
  6. V. Perlick, The brachistochrone problem in a stationary space-time. J. Math. Phys.32 (1991) 3148-3157.  

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