Optimal Controlled Descent in the Atmosphere and the Modified Brachistochrone Problem

Abstract

Abstract The problem of maximization of the horizontal coordinate and minimization of the fuel expenditures of mass-point moving in the vertical plane driven by gravity, linear and quadratic 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 nonlinear differential equations. Thrust control depending on the velocity and slope angle is designed. It is established that the extreme thrust control program consists either of single arc with intermediate thrust control, or two arcs, starting with maximum thrust and ending with the intermediate thrust or three arcs: ”intermediate-maximum-intermediate”.