The maximum principle in problems of optimal control of processes described by hybrid functional differential equations

In view of the fact that in the investigation and solution of smooth optimal control problems the linearization of the equations of state of a controlled process is usually used, the idea naturally appeared to apply here the available modern results on the theory of linear functional differential equations. The following problem was posed: to obtain such necessary optimality conditions in the form of the maximum principle, which should be valid for all classes of equations with deviating argument known to us. This allowed us to construct our own scheme for obtaining the necessary conditions for optimality in the theory of extremal problems. With its help, it was possible to briefly and in a rather general form state the theory of the maximum principle. With the help of the mentioned scheme of the maximum principle and the theory of linear functional differential equations, necessary optimality conditions are obtained and proved in smooth optimal control problems.