On the maximization of the horizontal range and the brachistochrone with an accelerating force and viscous friction

Abstract

The problem of maximizing the horizontal coordinate of a point moving in the vertical plane driven by gravity, viscous friction, which is proportional to the nth degree of the velocity, and an accelerating force is considered, along with the related brachistochrone problem. The optimal control problem is reduced to a boundary value problem for a system of two nonlinear differential equations. The qualitative analysis of the trajectories of this system is performed, and their characteristic features that allow us to elaborate the results obtained in other studies are revealed. The optimality of the found extremals is discussed.