Аннотация:Abstract: The motion of a point mass in a vertical plane under the action of gravity forces,viscous friction, support reaction of the curve and the thrust is considered. The slope angleand the thrust are treated as control variables. The amount of the propellant is given. The aimis to maximize the horizontal coordinate of the particle. Time of the process is given. The interrelatedBrachistochrone problem is also considered. For the case of frictionless motion, it isshown that optimal thrust control is bang-bang-type, and trajectory consists of two arcs, startingwith maximum thrust, and ending with zero thrust. Optimal synthesis in the three dimensionalspace “mass-velocity-slope angle” is designed. For the case of linear viscous frictionthe arc with singular thrust includes in the extremal trajectory. It is shown that optimal thrustprogram consists of either two arcs, maximum thrust at the beginning and zero thrust at theend, or three arcs: maximum thrust at the beginning, then intermediate (singular) thrust andzero thrust at the end. The control logic of the thrust is similar to the Goddard problem. Theresults of numerical simulation for the case of linear viscous friction illustrating the theoreticalconclusions are presented.