Control improvement and approximately optimal synthesis in the neighborhood of a support trajectory

We consider problems of optimal control improvement and synthesis for continuous and discrete processes with a unified scheme that includes searching for an, at least, locally optimal process and implementing it via an approximately optimal positional control with Krotov's estimate. For regular problems, this scheme is based on local approximation of the well-known Krotov-Bellman relations; for degenerate problems, on the corresponding generalized relations. We show how this scheme relates to the A. M. Letov's ACOC theory and give specific examples.