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The problem of maximization of the horizontal coordinate of a mass-point moving in the vertical plane driven by gravity, non-linear viscous drag, and thrust is considered. The slope angle and the thrust are considered as a control variables. The problem is related to the modified brachistochrone problem. Principle maximum procedure allows to reduce the optimal control problem to the boundary value problem for a set of systems of two non-linear differential equations. The qualitative analysis of the trajectories of these systems is performed, and the characteristic features of the optimal solutions are determined. Thrust control depending on the velocity and slope angle is designed. Results obtained allow to construct quasi-optimal solutions for the more complex systems, where phase plane method is not applicable.

Optimal Thrust Programming Along the Brachistochronic Trajectory with Non-linear Drag