A set of extremals was found for the brachistochrone problem with taking into account dry and random viscous friction. The problem is solved by searching the speed-optimal normal component of the reaction (control) of a plane curve, whose shape is to be determined. The investigation of a differential of the functional was done by Okhotsimsky?Pontryagin method. An analytical expression in terms of phase coordinates for the optimal reaction of the brachistochrone curve is given. This expression gives the solution of a classical brachistochrone problem in the absence of any friction and in the presence of friction ? corresponding optimal curves. Parametrical formulas are presented for a brachistochrone curve under influence of dry and viscous friction. Properties of brachistochrone curves are investigated.
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