The problem of the optimization of a guaranteed result for the control system, described by an ordinary differential equation, and a continuous payoff functional, is considered. At every moment the values of the control and of the disturbance are in the given compact sets. The disturbances as functions of time are subject to functional constraints belonging to a given family of constraints. The actions of control are formed by the strategies with full memory. It is demonstrated, that optimal guaranteed result in this problem is equal to the value of the lower game. For the effectiveness of implemented control algorithm additional conditions on the system and appropriate ways of constructing an optimal strategy are specified.