3 REVAZ GAMKRELIDZE IS Not long ago the mathematical community warmly congratulated promi nent mathematician Revaz Gamkrelidze Corresp onding Member of the Russian Academy of Sciences and Member of the Georgian Academy of Sciences on his th birthday These years are marked by remarkable deeds and interesting events Revaz Gamkrelidze was b orn on February in the Georgian city of Kutaisi His father Valerian Gamkrelidze was a public gure who con tributed much to the publishing of scientic literature in Georgia When Revaz Gamkrelidze studied in a secondary school he read Courants excellent b o ok What is Mathematics and acquired a p ermanent passion for this science After school Revaz Gamkrelidze continued his education at Tbilisi State University As a gifted sophomore he was sent to Moscow to study at the mechanics and mathematics faculty of Moscow State University Here he met Lev Pontryagin an outstanding contemporary mathematician who lost his eyesight at the age of but whom Fate rewarded with an insight into the world of science Pontryagins maximum principle is a ne example of this insight The meeting of Lev Pontryagin and Revaz Gamkrelidze was that of a teacher and a pupil one of those rare lucky chances which leave an indelible mark in ones life Revaz Gamkrelidzes rst studies were dedicated to algebraic geometry and algebraic top ology He determined characteristic classes of complex algebraic manifolds In the mathematical literature this result is referred to as Gamkrelidzes formula In Revaz Gamkrelidze to ok up research into nonclassical varia tional problems related to the theory of automatic control systems for the ro cket industry He achieved remarkable results in this area Along with L Pontryagin V Boltyanski and E Mishchenko he laid the foundation of the mo dern mathematical theory of optimal pro cesses The rst pap er by the four authors was published in and evoked a great interest among sp ecialists In the pap er a general optimal control problem was formulated for the rst time and the maximum principle was prop osed as a metho d for its solution The maximum principle was at that time the most b eautiful and sophisticated theorem formulated as Pontryagins hypothesis Its pro of required four years of continuous strenuous research The ndings were presented in the monograph Mathematical Theory of Optimal Pro cesses which in won the authors the State Lenin Prize The monograph was translated into several foreign languages and in many countries it stimu lated research in this sp ecic mathematical area The achievements of Pontryagins school were rep orted at the interna tional conference on mathematical control theory held in at the Univer sity of Southern California USA The university situated in a picturesque 4 suburb of Los Angeles was regarded by mathematicians as a temple of dy namic programming with Richard Bellman as head priest Prior to Pontrya gins maximum principle Bellmans metho d of dynamic programming was the only one used in investigating optimal control systems R Bellmans school carried out research within the framework of the program sp ecially developed for the US Air Forces and nanced by the gigantic RAND Cor p oration Bellmans metho d of dynamic programming has certainly played a very imp ortant role but its main disadvantage is that it can not b e substanti ated in general terms and do es not hold for some cases In this resp ect Pontryagins maximum principle has no alternative the fact recognized by the participants of the ab ovementioned conference chaired by R Bellman himself However it to ok long years for these events to take place In the mean time the maximum principle remained unproven the Lenin prize was never thought of the gold star of honour did not decorate the lap el of Pontryagins suit In Revaz Gamkrelidze made the initial imp ortant step he proved the maximum principle for timeoptimal problem in the case of linear systems Pontryagins maximum principle was completely proved only three years later Even in a very concise form the pro of o ccupied typewritten pages To the same p erio d b elong Revaz Gamkrelidzes fundamental investiga tions into the theory of optimal pro cesses as well as into problems with b ounded phase co ordinates The results he obtained formed a part of his do ctoral thesis In the subsequent years Revaz Gamkrelidze discovered and worked on optimal sliding states He developed the notion of a quasiconvex set in a linear top ological space which underlies the general theory of extremal problems These results were rep orted at the International Congress of Mathematicians in Nice in Revaz Gamkrelidze studied quasilinear dierential games and worked out an evasion strategy for them Further he constructed an exp onential rep resentation of streams and invented a technique of chronological calculus In recent years Revaz Gamkrelidze has b een carrying out research and teaching activities in the USA Germany France and Great Britain Revaz Gamkrelidze pioneered and contributed much to the development of the optimal control theory in Georgia For a few decades he has b een heading the control theory chair at Tbilisi State University A course of lectures he read at the university was included in his monograph Funda mentals of Optimal Control which was translated into English and published in the United States In Georgia the monograph was awarded the Razmadze Prize Apart from his research and teaching work Revaz Gamkrelidze is also concerned with publishing matters He is the editorinchief of the Refer ativny Zhurnal Matematika Moscow the series Mo dern Problems of 5 Mathematics Russian Moscow Mo dern Mathematical Encyclopaedia Russian He is the scientic editor of the AmericanBritish series Mo dern Soviet Mathematics German series Soviet Mathematics and the series Russian Classics of Mathematics On his th birthday Revaz Gamkrelidze is as always full of intentions ideas and energy We wish him a long successful way in life strewn with further laurels Guram Kharatishvili March Tbilisi Selected Scientic Papers of R V Gamkrelidze Computation of Chern cycles of algebraic manifolds Russian Dokl Akad Nauk SSSR No Chern cycles of complex algebraic manifolds Russian Izv Akad Nauk SSSR Ser Mat No Theory of optimal pro cesses Russian Dokl Akad Nauk SSSR No with V G Boltyanski and L S Pontryagin Theory of optimal pro cesses in linear systems Russian Dokl Akad Nauk SSSR No Theory of timeoptimal pro cesses in linear systems Russian Izv Akad Nauk SSSR Ser Mat General Theory of optimal pro cesses Russian Dokl Akad Nauk SSSR No Timeoptimal pro cesses with b ounded phase co ordinates Russian Dokl Akad Nauk SSSR No Theory of optimal pro cesses I Russian Izv Akad Nauk SSSR Ser Mat with V G Boltyanski and L S Pontryagin Optimal control pro cesses with b ounded phase co ordinates Russian Izv Akad Nauk SSSR Ser Mat Mathematical Theory of Optimal Pro cesses John Wiley with L S Pontryagin V G Boltyanski and E F Mishchenko On sliding optimal regimes Russian Dokl Akad Nauk SSSR No On the theory of rst variation Russian Dokl Akad Nauk SSSR No On some extremal problems in the theory of dierential equations with applications to the theory of optimal control SIAM J Control Ser 6 A No Theory of rst variation in extremal problems Russian Soobshch Akad Nauk Gruzin SSR No with G L Kharatishvili Extremal problems in linear top ological spaces Russian Izv Akad Nauk SSSR Ser Mat No with G L Kharatishvili First order necessary conditions and axiomatics for extremal prob lems Russian Trudy Mat Inst Steklov Condition necessaires du premier ordre dans les problemes dextremum Proceedings of the International Congress of Mathematicians NiceSeptember GauthierVil lars with G L Kharatishvili A dierential game of evasion with nonlinear control SIAM J Control No An integral equation arising in the theory of games Russian Ba nach Center Publications A second order optimality principle for timeoptimal problems Rus sian Mat Sb No with A A Agrachev The exp onential representation of solutions of ordinary dierential equations In Proc Equadi V Conf Prague Springer Lecture Notes Principles of Optimal Control Theory Plenum Press The exp onential representation of ows and chronological calculus Russian Mat Sbornik No with A A Agrachev Chronological algebras and nonstationary vector elds Russian Itogi Nauki i Tekhniki Problemy Geometrii VINITI Moscow with A A Agrachev Dierentialgeometric and grouptheoretic metho ds in the theory of optimal control Russian Itogi Nauki i Tekhniki Problemy Geometrii VINITI Moscow with A A Agrachev and S A Vakhrameev Sliding regimes in optimal control theory Russian Trudy Mat Inst Steklov The index of extremality and quasiextremal controls Russian Dokl Akad Nauk SSSR No with A A Agrachev The Morse index and the Maslov index for extremals of controlled systems Russian Dokl Akad Nauk SSSR No with A A Agrachev Computing the Euler characteristic of intersections of real quadrics Russian Dokl Akad Nauk SSSR No with A A Agrachev Ordinary dierential equations on vector bundles and the chrono logical calculus Russian Itogi Nauki i Tekhniki Sovremennye Problemy Matematiki VINITI Moscow with A A Agrachev Quasiextremality for controlled systems Russian Itogi Nauki i Tekhniki Sovremennye Problemy Matematiki VINITI Moscow with A A Agrachev 7 Quadratic mappings and smo oth vectorfunctions Euler charac teristics of the level sets Russian Itogi Nauki i Tekhniki Sovremennye Problemy Matematiki VINITI