Qualitative analysis of optimal trajectories of the point mass motion in a resisting medium and the brachistochrone problem

Abstract

The problem of maximization of the horizontal range and minimization of time for a point mass moving in a resisting medium is analyzed. Various models of medium resistance are considered. Using Pontryagin’s maximum principle, the optimal control problem is reduced to the boundary value problem for a system of two nonlinear differential equations. The qualitative analysis of the resulting system makes it possible to reveal previously unknown characteristics of optimal trajectories for a general model of the resistance force. This analysis is illustrated by a numerical simulation.