Lyapunov-Krotov inequality and sufficient conditions in optimal control (Q1776282)

The paper is devoted to the analysis of the canonical theory based on the use of a family of solutions of the Lyapunov-Krotov inequality. Classical smooth optimal control with terminal constraints is studied and it is established that if the sufficient conditions with the family of solutions of the Lyapunov-Krotov inequality hold, then the gradients of ``essential'' functions of the family along the optimal trajectory coincide with the adjoint components of the Pontryagin extremals in the sense of Milyutin. There are presented some examples with singularities that are characteristic for the non-existence of the smooth Bellman and Krotov functions, namely, the presence of invariant sets and abnormal Pontryagin extremals and extremals with the set of normed Lagrange multipliers consisting of more than one point. The examples show that the application of the canonical theory can be efficient in these situations, while the smooth versions of the Bellman and Krotov functions cannot be applied here in principle.