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The problem of analytical synthesis of the control acceleration for an unmanned aerial vehicle (UAV) during its flight along a complex trajectory, consisting of sequentially located horizontal flight sections, located at different heights relative to the earth's surface has been solved in the paper. The problem has been solved as an analytical definition of the optimal control of a linear non-stationary system for a specified minimized quality functional. The mathematical model of the system is presented in the form of differential equations of UAV motion in the vertical plane of a fixed coordinate system related to the earth's surface. A feature of the proposed methodology for solving the problem is the substantiation of the original form of the minimized functional and parameters included in the law of variation of the control acceleration obtained by known methods. As the components of the quality functional, the values of coordinates and velocity of the UAV are considered and they are specified at the corresponding points in space through which the UAV path must pass, in order to obtain the optimal curvature of the trajectory. The derived mathematical dependences make it possible to implement them on board of an aircraft and, ultimately, solve the problem of ensuring the minimum energy consumption when controlling an object (UAV). Computer simulation of the analytically obtained results in the form of the UAV flight trajectory and the processes of changing its acceleration and speed have shown the efficiency of the proposed technique and the prospects of its use at the initial stage of the synthesis of the UAV control system.