Range maximization and Brachistochrone problem with friction and thrust

Abstract

The range maximization problem and corresponding the brachistochrone problem with Coulomb, viscous friction and accelerating force are considered. Normal component of the reaction force and the thrust are considered as a control variables. A Maximum Principle procedure is applied to reduce optimal problem to the boundary-value problem for the system of two nonlinear differential equations. Solutions of this system are corresponding to the motion under singular control for normal component of the reaction force and ``bang-bang'' control for the thrust. The qualitative analysis of the solutions is carried out. It allows to install the characteristic properties of the trajectories, unnoticed in a number of publications as well as to confirm the hypothesis and the results of numerical calculations presented in other sources.