Necessary conditions for the optimality of the automaton part of a logical-dynamical system

A dynamical system controlled by an automaton with memory is considered. The continuous part of the system is described by differential equations, and the automaton part, by recurrence inclusions. The instants of time at which the state of the automaton part is changed are not known in advance and are determined during the optimization process. Moreover, modes with multiple switchings of the automaton part at a given instant of time are admitted. Necessary conditions for the optimality of a program control are obtained. The application of the optimality conditions is illustrated by examples.