The problem is investigated of optimal control of a system, described with the model of fractional-order double integrator, in which the initial and final conditions depend on the appropriate time-point selection. Several cases are considered, physically similar to the system transfer from a state of the rest to a state of the uniform linear motion, to a uniformly accelerated and to a periodic motion, and from a uniform to a uniformly accelerated motion. The dependencies are analyzed of the control norm from the control time and from the fractional differentiation index value.