Abstract The Brachistochrone problems with various penalties for fuel expense for a mass moving in a vertical plane in a uniform field of gravity are considered. The resistance of the medium is considered viscous. The lift force or normal component of the reaction force of the curve and thrust are considered as control variables. The optimal control problems are reduced to a boundary value problems for a system of two nonlinear differential equations. An analytical analysis of the resulting system allows for obtaining the structure of extremal trajectories and studying their asymptotic behavior. Thrust control is designed as a function of the velocity and the angle of inclination. The structure of the extremal thrust control program is defined, and the sequence of the arcs is found analytically.

中文翻译：

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修正的极速问题中带有燃料消耗惩罚的范围最大化问题

摘要 考虑了在均匀重力场中在垂直平面上运动的质量对燃料费用的各种惩罚的极速问题。介质的阻力被认为是粘性的。升力或曲线和推力的反作用力的法向分量被视为控制变量。将最优控制问题简化为两个非线性微分方程组的边值问题。对所得系统的分析分析允许获得极值轨迹的结构并研究它们的渐近行为。推力控制设计为速度和倾斜角的函数。定义了极值推力控制程序的结构，并通过解析找到了弧的顺序。