The Pontryagin derivative in optimal control

A basic feature of Pontryagin's maximum principle is its native Hamiltonian format, inherent in the principle regardless of any regularity conditions imposed on the optimal problem under consideration. It canonically assigns to the problem a family of Hamiltonian systems, indexed with the control parameter, and complements the family with the maximum condition, which makes it possible to solve the initial value problem for the system by “dynamically” eliminating the parameter as we proceed along the trajectory, thus providing extremals of the problem. Much has been said about the maximum condition since its discovery in 1956, and all achievements in the field were mainly credited to it, whereas the Hamiltonian format of the maximum principle has always been taken for granted and never been discussed seriously. Meanwhile, the very possibility of formulating the maximum principle is intimately connected with its native Hamiltonian format and with the parametrization of the problem with the control parameter. Both these starting steps were made by L. S. Pontryagin in 1955 from scratch, in fact, out of nothing, and eventually led to the discovery of the maximum principle. Since the present volume is dedicated to the centenary of the birth of Lev Semenovich Pontryagin, I decided to return to this now semi-historical topic and give a short exposition of the Hamiltonian format of the maximum principle.