Optimal guarantee under the disturbances of Caratheodory type

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 actions of the disturbance are assumed to be generated by an unknown function of the Caratheodory type, i.e. by the function continuous with respect to the spatial variable for every value of time variable and measurable with respect to the time variable for every value of spatial one. The actions of control are formed by the strategies with full memory.
It is demonstrated, that for a class of control systems the optimal guaranteed result in this problem is equal to the value of the lower game, i.e. to the value of the optimal guaranteed result in the class of quasi-strategies. The optimal strategy with full memory, that allows numerical implementation, is provided. An illustrative nonlinear example is given.