Abstract: In this paper the general synthesis problem of optimal control systems with the criterion of transient responses as a positive integral functional(3)is discussed. In the first part it is assumed that the motion of controlled object is described by a system of ordinary differential equations and that the final states of the system form a bounded and closed convex region in n-dimentional euclidian phase space.A method is proposed for finding all optimal control functions which lead any starting state into the given final region of states.Some conclusions are obtained from the maximum principle by using transversal conditions of optimal trajectories in terminal points,and the particular properties of the stated problem are pointed out.The case of linear dif- ferential equations with integral quadratic functional criterion is investigated in detail. Further,in the second part the fundamental properties of isoloss regions,the rela- tions between the isoloss region and optimal control functions are indicated.As a direct result a partial differential equation determining the optimal loss-function J(x)is found and the connection between function J(x)and optimal vector control function u(x)is also stated.The methods proposed are practically the extension of the me- thods used by us for designing time optimal control systems as seen in[5,6,7]. Finally,an example is illustrated with optimal trajectories shown in phase plane. The necessary numerical data is calculated by an analog computer with high accuracy.