Optimal Trajectories in the Brachistochrone Problem with an Accelerating Forceстатья
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Дата последнего поиска статьи во внешних источниках: 26 сентября 2016 г.
Аннотация:The woparametric family of optimal curves is studied within the brachistochrone problem under the action of an accelerating tractive force in a dynamic statement. The problem is reduced to a standard timeoptimal control problem. The normal component of the support reaction is used as control. It is noted that the optimal control formula, which does not include adjoint variables, has a singularity at the initial point at zero velocity. A system of differential equations is derived, for which solving the Cauchy problem with initial conditions makes it possible to obtain the optimal trajectories for the case of a quasiconstant accelerating force and viscous friction. In the case of a constant accelerating force and the absence of both dry and viscous friction, the system used for obtaining optimal trajectories is reduced to a simpler form without the singularity at the initial point. The selfsimilarity property of trajectories with no friction is proved, based on which all optimal trajectories may be obtained from a multitude of optimal trajectories with fixed initial conditions and different finite tangent slopes by scaling. It is shown that trajectories with viscous friction do not have the selfsimilarity property.
DOI: 10.1134/S1064230715040139