Dynamic control problem under interference with a given set of correction momenta

We consider a dynamic control system under interference. A set of correction momenta of the controls is given. The problem of phase point retention in a given collection of sets at correction momenta is considered. Instantaneous change of a position is admissible. Necessary and sufficient conditions for the possibility of retention are found. As an example, we consider a discrete linear control problem under interference and with the one-dimensional aim. The condition of one-dimensionality of the aim means that the modulus of the value of a given linear function of the phase variables at a fixed moment of the control process end should not be more than a given number. For this problem, necessary and sufficient conditions are found in an explicit form, the fulfillment of which guarantees the existence of an admissible control that ensures the achievement of the aim for any admissible realization of the interference. This control is constructed in an explicit form, and information about the realized value of the interference is not used. We constructed the interference which guarantees that the aim will not be reached at any admissible control from the initial state that does not satisfy the obtained conditions.