 | | Journal of Applied
Mathematics and Mechanics
Russian Academy of Sciences | | Founded
in January 1936
(Translated from 1958)
Issued 6 times a year
ISSN 0021-8928
(print version) |
Archive of Issues
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| In Russian (ΟΜΜ): | | 9713
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| In English (J. Appl. Math. Mech.): | | 799 |
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| "Nikolai Nikolayevich Krasovskii (7 September 1924-4 April 2012)," J. Appl. Math. Mech. 76 (4), 493-495 (2012) |
| Year |
2012 |
Volume |
76 |
Issue |
4 |
Pages |
493-495 |
| Title |
Nikolai Nikolayevich Krasovskii (7 September 1924-4 April 2012) |
| Author(s) |
|
| Abstract |
Academician Nikolai Nikolayevich Krasovskii, an eminent Russian scientist who enriched science with fundamental achievements in a number of areas of mathematics and mechanics, was a member of the Editorial Board of the journal "Appled Mathematics and Mechanics" for more than 40 years.
He was born on the 7 September 1924 in Ekaterinburg into the family of a physician who was well known in the cities. Having finished at school, he worked from 1941 to 1943 as an electrician in the S. Ordzhonikidze factory. Entering the S. M. Kirov Ural Polytechnic Institute (UPI) in 1943, he completed his studies there in January 1949 with the title of engineer in the plastic and heat treatment of metals. In second year at this institute, he started to engage in scientific research under the guidance of Professor-algebraist S. N. Chernikov. After graduating from the UPI, he worked for ten years in the Department of Higher Mathematics as assistant, senior lecturer, professor and head of department. In the years from 1959-1970, he worked in the A. M. Gorkii Ural State University (UrGU), initially as head of the Department of Theoretical Mechanics and, later, as head of computational mathematics and (from 1965) as head of applied mathematics, a department set up under his initiative. From 1970 to 1977, he headed the Institute of Mathematics and Mechanics (IMM) of the Academy of Sciences of the USSR that had been set up on the basis of the Sverdlovsk branch of the V. A. Steklov Mathematical Institute under the initiative of Academicians I. M. Vinogradov and M. V. Keldysh. This institute later became part of the Ural Scientific Centre (and, subsequently, the Ural Branch of the Russian Academy of Sciences). Up to the last days, his life and scientific activity were associated with the IMM.
The priorities in his scientific interests fell within the sphere of the qualitative theory of differential equations, the theory of stability, the theory of the stabilization of motion, mathematical control theory and differential game theory. Together with this, he also showed a constant interest in other areas of mathematics and mechanics and the latest achievements in these areas, supporting their development in the Ural region. The breadth of his scientific interests had already been revealed as a young man under the influence of his teachers Ye. A. Barbashin, N. P. Yerugin, I. G. Malkin and N. G. Chetayev. He successfully continued the scientific traditions and scientific studies in the field of the qualitative theory of differential equations and stability theory going back to A. M. Lyapunov.
Briefly listing the fundamental scientific results he obtained in these areas, we first note that he proved the existence of Lyapunov functions, satisfying Lyapunov's theorems on asymptotic stability and instability. He further suggested an approach to the solution of optimal preset control problems based on the ideas and methods of functional analysis. Linear problems with functionals having the sense of a norm in suitable functional spaces were reduced by him to a problem of moments, the solution of which is associated with one of the most important theorems in functional analysis, the Hahn-Banach theorem on the extension of a linear functional. This approach enables missing information to be obtained on the boundary conditions for the adjoint system of Pontryagin's maximum principle.
He constructed a theory of stability for systems with an aftereffect. He showed for the first time that the motion of such systems is naturally described by ordinary differential equations in the functional space of segments of the trajectories. In the case of linear systems with an aftereffect, the evolution of the segments of the trajectories is determined by a linear transformation possessing a semi-group property that enables one to invoke results from the theory of semi-groups of linear transformations. Asymptotic stability criteria were obtained in terms of the spectrum of the generating operator of a semi-group of transformations. Using this approach, he extended the second Lyapunov method to systems with an aftereffect. In so doing, the use of Lyapunov-Krasovskii functionals in the role of Lyapunov functions was found to be natural and this enabled him to prove a theorem on asymptotic stability by assuming the converse. The theory of optimal stabilization was carried over to systems with an aftereffect for the first time by him. In the case of a linear stabilization problem with a quadratic characteristic functional, he pointed out the explicit form of the linear functional giving the solution of the problem.
He developed the basic concept of the stability of stochastic systems perturbed by Markov processes. Fundamental definitions of stochastic stability, that became generally accepted, were introduced by him as well as the concept of stochastic Lyapunov functions. A number of conditions guaranteeing stochastic stability, were obtained in terms of these functions.
He developed a theory of positional differential games. The central features of the theory are the formulization of the strategies and motions and also the establishment of alternative conditions for the solvability of game control problems and the proof of general theorems concerning the existence of the value and the saddle point of a game. A unified form of describing the dynamics of an auxiliary model system he proposed became a convenient instrument for the study of a game. The problem of the unification of positional differential games was solved and optimal strategies that were stable to perturbations were constructed.
He was the author of about three hundred scientific papers, including six monographs. He headed the Ural Scientific School in the theory of stability of motion and was also the founder and head of the Ural Scientific School for mathematical control theory. Among his pupils are engineers and teachers, doctors and masters of science, and corresponding members and academicians of the Russian Academy of Sciences.
His activity as director of the IMM (1970-1977) and his authority and energy ensured a qualitatively new level of development of the institute. He initiated and supported many areas of applied studies in mechanics and new technologies. He paid a great deal of attention to providing the institute with first-class computational facilities and to the development of computational work in the Ural region.
He made a weighty contribution to mathematical education in the Ural region both as an outstanding organizer and a brilliant lecturer. Under new specialities and specializations, he created two new departments in the UrGU: a department of computational mathematics and a department of applied mathematics. A large number of special courses were developed and presented that were original in their content and included new scientific achievements, among which were those accomplished by himself.
He devoted a considerable amount of time and energy to publicizing the achievements of fundamental science among applied researchers, engineers and physicians. For them, he developed and presented a number of courses and published textbooks and pamphlets describing ideas and new methods in mathematics, mechanics and informatics in an easily understood manner. It is symbolic that the American Institute of Electrical and Electronics Engineers (IEEE), when awarding him a prize in 2003, noted his "innovative ideas that were taken up both by theoreticians as well as by engineers and skilled workers".
He took problems in school mathematical education in the country and in the Ural region particularly seriously. He was closely associated with a middle school from the very beginning of his creative activity. From year to year, he constantly met pupils and teachers, presented them with lectures, carried out studies and participated in the work of the school olympiads. He repeatedly visited mathematics lessons. In the 1980s, he formed part of the Commission on School Education at the Department of Mathematics of Academy of Sciences of the USSR and carried out a large amount of work on evaluating the programmes operating in schools for general education and those for students of mathematics. The reforms in school education and, in particular, school mathematical education put into practice in the last decade caused him great anxiety. He considered that the majority of the arguments put forward in favour of these reforms were far from justified and that the actions carried out by the reformers could have very negative consequences for Russian school education. It is only to be hoped that teachers, who are considerable experts in the subject and brilliant popularizers, are not further changed in their vital vocation and profound understanding of their mission.
His authority among scholars was extremely high. They listened to his opinion and checked with him when solving urgent problems in science and education. In 1964, he was elected as a Corresponding Member of the Academy of Sciences of the USSR and as an Acting Member of the Academy of Sciences of the USSR in 1968. He formed part of the Presidium of the Russian Academy of Sciences and the Bureau of the Department of Mechanics and Control Processes of the Academy of Sciences of the USSR, was a member of the Presidium of the National Committee for Theoretical and Applied Mechanics and joined the editorial boards of leading scientific publications.
His scientific achievements and teaching activity were highly valued by the state (he was a Hero of Socialist Labour, winner of the Lenin and State prizes and a Cavalier of the Orders of the Soviet Union and Russia) and by the scientific community he was awarded the Lomonosov Great Gold Medal of the Russian Academy of Sciences, the Lyapunov Gold Medal, the Demidov Prize in Physics and Mathematic's, the "Triumph" Prize, Doctor honoris causa of the Hungarian Academy of Sciences, the award of the International Association of Electrical and Electronic Engineers (IEEE) and was prize-winner of the "Eminent Scholars" program of the Foundation for the Promotion of Russian Science).
Efficiency, great will-power and benevolence in personal contacts were his inalienable human qualities. He was a genuine citizen who cared deeply about the destiny of Russia. |
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