One problem of the optimal control of a system with aftereffect in conditions of conflict

In the paper a time-optimal control problem is considered. Sufficient conditions for local optimality are obtained which are linked with necessary conditions of Pontryagin's maximum principle under assumption of total controllability of a system in variations. The problem is studied for a system described by a vector differential equation either ordinary or with aftereffect. In the case of conflict control, the optimal control problem is discussed for a criterion of the minmax-maxmin time when the system attains a given state. The model example is given and the corresponding numerical experiment is discussed.