On the unimprovability of full memory strategies in the risk minimization problem

Methods from the theory of guaranteeing positional control are used to study the risk minimization problem, i.e., the problem of optimal control under dynamic disturbances in a formalization based on the Savage criterion. A control system described by an ordinary differential equation is considered. The values of control actions and disturbance at each moment lie in known compact sets. Realizations of the disturbance are also subject to an unknown functional constraint from a given set of functional constraints. Realizations of the control are formed by full memory positional strategies. The quality functional, which is defined on motions of the control system, is assumed to be continuous on the corresponding space of continuous functions. New conditions that provide the unimprovability of the class of full memory positional strategies under program constraints and $L_2$-compact constraints on the disturbance are presented.