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The paper deals with the problem of optimization of a guaranteed (worst case)result for a control system described by an ordinary differential equation. Thedisturbances as functions of time are subject to functional constraintsbelonging to a given family of constraints. The latter family is known to theside that forms the control actions. The controlling side uses positionalfull-memory strategies and does not observe the disturbance. When theconstraints family consists of $L_p$-compact sets the optimal guaranteed resultis non-improvable in the sense that it coincides with that obtained in theclass of quasi-strategies -- nonanticipatory transformations of disturbancesinto controls.  In this paper for the effectiveness of implemented control algorithm anadditional condition on the system and appropriate ways of constructing anoptimal strategy are specified.

Guaranteed control design under $L_p$-compact constraints on the disturbance