An optimal control problem for a linear nonstationary system is under consideration. The behavior of this system is estimated by the values of convex objective function on its terminal state that belong to the bounded convex set. The method of construction of optimal open-loop solution is proposed. It consists from the linearization of the considered problem and special finishing procedure. On the base of this method the algorithm of optimal controller work is realized. It allows to synthesis an optimal feedback control. Results are illustrated on the example of fourth-order dynamical system.