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.
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