This paper contains a nonstandard formulation of the well-known lemma on substitution homomorphisms stated as the canonical duality between the family of all smooth mappings of one smooth manifold into another and the family of all homomorphisms of algebras of smooth scalar functions on these manifolds. This formulation gives the lemma the maximum possible generality and emphasizes the fundamental symmetry of the problem: the duality between “conjugation” (transition from mappings of manifolds to homomorphisms of algebras of smooth functions on them) and “co-conjugation” (transition from homomorphisms to mappings).

An invariant dual formulation of the Pontryagin maximum principle is given for the time-optimal case.

An invariant formulation of the Pontryagin Maximum Principle (PMP) is given. It is proved that the Pontryagin derivative P-X coincides on vector fields X is an element of Vect M, (M - the configuration space of the problem), with the Lie bracket ad(X), and the flow generated on the cotangent bundle T*M by the vector field P-X is bundle-preserving.

In the late 1950's, the group of Soviet mathematicians consisting of L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko made fundamental contributions to optimal control theory. Much of their work was collected in their monograph, The Mathematical Theory of Optimal Processes. Subsequently, Professor Gamkrelidze made further important contributions to the theory of necessary conditions for problems of optimal control and general optimization problems. In the present monograph, Professor Gamkrelidze presents his current view of the fundamentals of optimal control theory. It is intended for use in a one-semester graduate course or advanced undergraduate course. We are now making these ideas available in English to all those interested in optimal control theory. West Lafayette, Indiana, USA Leonard D. Berkovitz Translation Editor Vll Preface This book is based on lectures I gave at the Tbilisi State University during the fall of 1974. It contains, in essence, the principles of general control theory and proofs of the maximum principle and basic existence theorems of optimal control theory. Although the proofs of the basic theorems presented here are far from being the shortest, I think they are fully justified from the conceptual view point. In any case, the notions we introduce and the methods developed have one unquestionable advantage -they are constantly used throughout control theory, and not only for the proofs of the theorems presented in this book.

The 9th of March 2012 was the 90th anniversary of the birth of the prominent Russian mathematician Academician Evgenii Frolovich Mishchenko, one of the creators of modern mathematical control theory and the theory of oscillations.

март апрель т. 67, вып. 2 (404) УСПЕХИ МАТЕМАТИЧЕСКИХ НАУК МАТЕМАТИЧЕСКАЯ ЖИЗНЬ Евгений Фролович Мищенко (к девяностолетию со дня рождения) 9 марта 2012 г. исполнилось 90 лет со дня рождения выдающегося русского математика, одного из создателей современной математической теории управления и теории колебаний, академика Евгения Фроловича Мищенко.Е. Ф. Мищенко родился в деревне Хотиловка Владимирской области.Среднее образование он получил в железнодорожной школе поселка Ново-Вязники той же области.Когда Женя учился в 9-м классе, он случайно наткнулся в библиотеке местного клуба на книгу П. С. Александрова и А. Н. Колмогорова "Введение в теорию функций действительного переменного".Он ее прочитал и понял, и это привело юного школьника в восторг, -у него возникло желание стать математиком, хоть учителем, но стать.Как раз в это время на заработки из Нижнего Новгорода приехал аспирант знаменитого физика А. А. Андронова Александр Иванович Егоров.Он поступил в школу учителем математики, да так и остался в Ново-Вязниках.А. И. Егоров начал заниматься со способным школьником и посоветовал ему добыть задачи всесоюзной математической олимпиады, а для этого написать письмо в Москву профессору Павлу Сергеевичу Александрову, который в то время был президентом Московского математического общества и курировал олимпиады.В 1939 г.Женя написал письмо этому известному ученому, где сообщил, что прочел его книжку.Каково же было его удивление, когда через пару недель он получил конверт и там три письма от П. С. Александрова.Одно письмо было адресовано самому Евгению с разъяснением, как добраться до Москвы, а оттуда до деревни Комаровка, где находилась дача Александрова.Второе письмо -маме, с просьбой отпустить сына и обещанием оплатить все расходы, в частности дорогу.Третьедиректору школы, с просьбой, чтобы Женю отпустили из школы на три дня.Мама собрала сына, и он отправился в свое первое самостоятельное путешествие

On 17 February 2012 Lev Dmitrievich Kudryavtsev passed away. He was a well-known expert in the theory of functions and differential equations, a corresponding member of the Russian Academy of Sciences, a professor and doctor of the physical and mathematical sciences, a laureate of the USSR State Prize and the Prize of the Government of the Russian Federation, and a member of the European Academy of Sciences.

An invariant formulation of the maximum principle in optimal control is presented, and some second-order invariants are discussed.

The major achievements of mathematical analysis from Newton and Euler to modern applications of mathematics in physical sciences, engineering and other areas are presented in this volume. Its three parts cover the methods of analysis: representation methods, asymptotic methods and transform methods. The authors - the well-known analysts M.A. Evgrafov and M.V. Fedoryuk - have not simply presented a compendium of techniques but have stressed throughout the underlying unity of the various methods. The fundamental ideas are clearly presented and illustrated with interesting and non-trivial examples. References, together with guides to the literature, are provided for those readers who wish to go further.

Abstract: According to the goal of the Conference in Bedlewo, dedicated to the 50-th anniversary of Optimal Control theory, and considering that the 2008 year marked the centennial birthday of Lev Semenovich Pontryagin, I decided to devote my talk to a brief account on the discovery of the maximum principle and to an analysis of its basic feature, the Hamiltonian format.

According to the goal of the Conference in Bedlewo, dedicated to the 50-th anniversary of Optimal Control theory, and considering that the 2008 year marked the centennial birthday of Lev Semenovich Pontryagin, I decided to devote my talk to a brief account on the discovery of the maximum principle and to an analysis of its basic feature, the Hamiltonian format.

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.

In this paper, we study basic differential invariants of the pair (vector field, foliation). As a result, we establish a dynamic interpretation and a generalization of the Levi-Civita connection and Riemannian curvature treated as invariants of the geodesic flow on the tangent bundle.