Optimal trajectories in the brachistochrone problem with an accelerating force

Abstract

The wo-parametric 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 time-optimal 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 quasi-constant 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 self-similarity 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 self-similarity property.