I'm finishing my PhD thesis about applications of optimal control theory in the field of energy harvesting. In the course of my PhD I dealt with different ways to compute optimal controls, and I found myself trapped between two-points boundary value problems of Pontryagin Maximum Principle, algebraic optimization techniques of Model Predictive Control (Direct Method) and Dynamic Programming of Hamilton-JAcobi-Bellman equation. I felt very unconfortable with these methods and began researching for different techniques. After a couple of months of intense googling and searching in Genesis Library I found the book by Krotov, Global Method in Optimal Control Theory. This book contains a set of sufficient conditions for optimality of a control law $u$ different from Hamilton-Jacobi-Bellman equations, together with a particular numeric technique that is incredibly easier to implement than the PMP and has the advantage of converging the global minimizer of the functional.
Personally, I have several reasons to prefer Krotov's method among the other methods. But I'm very concerned by the fact that this method is
almost absent in english scientific literature with the exception
of optimal control of quantum processes. Meanwhile in russian literature
there is rich collection of articles and different perspective of this
method, I feel very uncomfortable regarding the fact that this method
are being ignored in most Optimal Control Literature in the "western" countries.
Because of that I'm writing this post to invite the members of this community that have had some kind of experience or just heard about this method to share their opinions about the reasons why Krotov's method is almost absent in engineering and Optimal Control literature.
It's worth someone else looking, but it appears the Krotov method considers only problems with fixed initial and final points. It's worth noting that sometimes the Pontryagin's maximum principle can yield a necessary and sufficient condition for the optimal trajectory, depending on the uniqueness and smoothness of the costate trajectory and the optimal control. So there may be nothing special about the theory.