About the conditions of optimality in the minimax problem for controlling differential inclusion with delay

In the paper we considered one class controlling differential inclusions with delay. For the control system the minimax problem is researched. In the no smooth control problem the necessary and sufficient conditions of optimality are obtained. One of the approaches used in making decisions in conditions of incomplete information is the principle of guaranteed management [18]. This principle leads to minimax control problems for the ensemble of trajectories of a dynamical system. Such an approach for controlled differential inclusions was used in [16]. However, taking into account the specific form of the terminal functional, in this paper we use a different method based on the use of the minimax theorem from convex analysis. Control problems for such systems are more complex than systems without delay. Moreover, the specific of the constructions method of optimal control depends on the degree to which the delay factor is taken into account in the dynamics of the control system. In the case of linearity of system (3), the controllability conditions of the ensemble of trajectories are studied. In [23], for such systems, the control problem with terminal constraints was considered.

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