Variational Analysis of Infinite Order

The report will present an approach to the (numerical) solution of problems of optimal control over ordinary and distributed systems based on exact (without residual terms) representations of the increment of the target functional.
The approach is based on (explicit or implicit) duality and includes elements of the theory of local extremum, independent of the Pontryagin principle.
The main result is a series of necessary conditions for the optimality of program control, involving "positional" control comparisons. These conditions can be applied to "discard" suboptimal extremals of the maximum principle and provide an indirect method of monotonic descent by the functional, free from the parameters of "depth" or "step".
The main time of the report will be devoted to the classical smooth (nonlinear) problem. In the final part, we will discuss the transfer of the ideas presented to some control tasks of distributed systems in an associated Banach space.