This paper investigates a nonlocal control problem of a linear non-stationary dynamic system described by an ordinary differential equation subject to an integral constraint imposed on the components of the state vector over a certain subinterval of the system’s operating time. Conditions for complete controllability of the dynamic system under the imposed integral constraints are formulated. Conditions ensuring the existence of a solution to the control problem with such a nonlocal constraint are obtained. A constructive approach to solving the control problem is proposed, and the corresponding solution is constructed. The continuity and non-uniqueness of the control function are demonstrated. As an illustration, a control problem for the oscillatory motion of a material point with classical boundary conditions and an integral condition is solved.